NUREG/CR-6823

 SAND2003-3348P

Handbook of Parameter

Estimation for Probabilistic 

Risk Assessment

Sandia National Laboratories

U.S. Nuclear Regulatory Commission

Office of Nuclear Regulatory Research

Washington, DC 20555-0001
 

 NUREG/CR-6823 

 SAND2003-3348P

Handbook of Parameter

Estimation for Probabilistic

Risk Assessment

Manuscript Completed: November 2002

Date Published: September 2003

Prepared by

C.L. Atwood1

, J.L. LaChance2, H.F. Martz3,

D.J. Anderson2

, M. Englehardt4, D. Whitehead2, 

T. Wheeler2

2

Sandia National Laboratories

Albuquerque, NM 87185-0748

1

Statwood Consulting

Silver Spring, MD 20910

3

Los Alamos National Laboratory

Los Alamos, NM 87545

4

Formerly with Idaho National Engineering and Environmental Laboratory

Idaho Falls, ID 83415

A.D. Salomon, NRC Technical Monitor 

Prepared for

Division of Risk Analysis and Applications

Office of Nuclear Regulatory Research

U.S. Nuclear Regulatory Commission

Washington, DC 20555-0001

NRC Job Code W6970
iii

ABSTRACT

Probabilistic risk assessment (PRA) is a mature technology that can provide a quantitative assessment of the risk from

accidents in nuclear power plants. It involves the development of models that delineate the response of systems and

operators to accident initiating events. Additional models are generated to identify the component failure modes required

to cause the accident mitigating systems to fail. Each component failure mode is represented as an individual “basic

event” in the systems models. Estimates of risk are obtained by propagating the uncertainty distributions for each of the

parameters through the PRA models.

The data analysis portion of a nuclear power plant PRA provides estimates of the parameters used to determine the

frequencies and probabilities of the various events modeled in a PRA. This handbook provides guidance on sources of

information and methods for estimating the parameters used in PRA models and for quantifying the uncertainties in the

estimates. This includes determination of both plant-specific and generic estimates for initiating event frequencies,

component failure rates and unavailabilities, and equipment non-recovery probabilities. 
iv
v

Table of Contents

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

LIST OF EXAMPLES ....................................................................... xix

FOREWORD .............................................................................. xxi

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii

ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

1.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

1.3 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

1.4 Contents of the Handbook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2

1.4.1 Identification of Probability Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2

1.4.2 Collection of Plant Specific Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2

1.4.3 Quantification of Probability Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3

1.4.3.1 Parameter Estimation from Plant-Specific Data . . . . . . . . . . . . . . . . . . . . . 1-3

1.4.3.2 Parameter Estimation from Existing Data Bases . . . . . . . . . . . . . . . . . . . . . 1-3

1.4.4 Advanced Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3

1.4.4.1 Analyzing Data for Trends and Aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3

1.4.4.2 Parameter Estimation Using Data from Different Sources . . . . . . . . . . . . . 1-3

1.5 How to Use This Handbook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3

2. BASIC EVENT PROBABILITY MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

2.2 Initiating Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2

2.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2

2.2.2 Probability Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4

2.2.3 Data Needed to Validate the Model and Estimate 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4

2.2.4 Case Studies: Validity of Model Assumptions in Examples . . . . . . . . . . . . . . . . . . . . . 2-5

2.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6

2.2.5.1 More General Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6

2.2.5.2 Non-randomness of t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6

2.3 Failure to Change State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7

2.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7

2.3.2 Failure on Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7

2.3.2.1 Probability Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7

2.3.2.2 Data Needed to Validate the Model and Estimate p . . . . . . . . . . . . . . . . . . 2-8

2.3.2.3 Case Studies: Validity of Model Assumptions in Examples . . . . . . . . . . . . 2-8

2.3.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-9

2.3.2.4.1 More General Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-9

2.3.2.4.2 Non-randomness of n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-9

2.3.3 Standby Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-10

2.3.3.1 Probability Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-10

2.3.3.1.1 Probability Model for the Data . . . . . . . . . . . . . . . . . . . . . . 2-10

2.3.3.1.2 Application of the Model to PRA . . . . . . . . . . . . . . . . . . . . 2-10

2.3.3.2 Data Needed to Validate the Model and Estimate 8 . . . . . . . . . . . . . . . . . 2-11

2.3.3.3 Case Studies: Validity of Model Assumptions in Examples . . . . . . . . . . . 2-11

2.3.4 Comparison of the Two Models for Failure to Change State . . . . . . . . . . . . . . . . . . . 2-12

2.3.4.1 Ease of Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12

2.3.4.2 Use in PRA Cut Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12
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vi

2.3.4.3 Estimates Obtained . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12

2.3.4.4 A Model Involving Both Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13

2.3.4.5 Choosing a Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13

2.4 Failure to Run during Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13

2.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14

2.4.2 Probability Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14

2.4.3 Data Needed to Validate the Model and Estimate 8 . . . . . . . . . . . . . . . . . . . . . . . . . . 2-15

2.4.4 Case Studies: Validity of Model Assumptions in Examples . . . . . . . . . . . . . . . . . . . . 2-15

2.4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-15

2.5 Recovery Times and Other Random Duration Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-15

2.5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16

2.5.2 Duration-Time Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16

2.5.3 Data Needed to Estimate Distribution of Durations and Validate Model . . . . . . . . . . 2-18

2.5.4 Case Studies: Validity of Model Assumptions in the Examples . . . . . . . . . . . . . . . . . 2-18

2.6 Unavailability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-18

2.6.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-19

2.6.2 Probability Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-19

2.6.3 Data Needed to Validate the Model and Estimate q . . . . . . . . . . . . . . . . . . . . . . . . . . 2-19

2.6.4 Case Study: Validity of Model Assumptions in Example . . . . . . . . . . . . . . . . . . . . . . 2-20

3. COMPONENT FAILURE AND BOUNDARY DEFINITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

3.1 Failure Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

3.2 Component Boundary Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

3.3 Failure Severity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

4. DATA SOURCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

4.1 Plant-Specific Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

4.1.1 Requirements on Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

4.1.1.1 Initiating Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

4.1.1.2 Component Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

4.1.1.3 Recovery Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2

4.1.2 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2

4.1.2.1 Regulatory Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2

4.1.2.2 Internal Plant Failure Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3

4.1.2.3 Maintenance Records . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3

4.1.2.4 Plant Logs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4

4.1.2.5 Component Exposure Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4

4.1.3 Plant-Specific Data Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4

4.1.3.1 Nuclear Plant Reliability Data System (NPRDS) . . . . . . . . . . . . . . . . . . . . 4-5

4.1.3.1.1 Limitations in the Data Available from the NPRDS . . . . . . . 4-5

4.1.3.2 Equipment Performance and Information Exchange (EPIX) System . . . . . 4-5

4.1.3.3 Reliability and Availability Data System (RADS) . . . . . . . . . . . . . . . . . . . . 4-6

4.2 Generic Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6

4.2.1 NRC-Sponsored Generic Data Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6

4.2.1.1 Current Data Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6

4.2.1.1.1 Severe Accident Risks Study Generic Data Base 

(NUREG-1150) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6

4.2.1.1.2 Evaluation of Loss of Offsite Power Events at Nuclear 

Power Plants: 1980 - 1996 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7

4.2.1.1.3 Rates of Initiating Events at U.S. Nuclear Power Plants: 

1987 - 1995 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-8

4.2.1.1.4 System Reliability Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-8
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4.2.1.1.5 Component Performance Studies . . . . . . . . . . . . . . . . . . . . . . 4-9

4.2.1.2 Historical Data Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9

4.2.2 DOE-Sponsored Generic Data Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9

4.2.2.1 Component External Leakage and Rupture Frequency Estimates . . . . . . . 4-10

4.2.2.2 Generic Component Failure Data Base for Light Water and Liquid 

Sodium Reactor PRAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-10

4.2.3 Industry Data Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-10

4.2.3.1 EQE, International . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-10

4.2.3.2 Science Applications International Corporation . . . . . . . . . . . . . . . . . . . . 4-11

4.2.3.3 Advanced Light Water Reactor Data Base . . . . . . . . . . . . . . . . . . . . . . . . 4-12

4.2.4 Foreign Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-12

4.2.4.1 Sweden’s T-Book for Nordic Nuclear Power Plants . . . . . . . . . . . . . . . . . 4-12

4.2.4.2 Sweden’s I-Book for Nordic Nuclear Power Plants . . . . . . . . . . . . . . . . . 4-13

4.2.5 Non-Nuclear Power Data Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13

4.2.5.1 Reliability Analysis Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13

4.2.5.2 Offshore Reliability Data Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13

4.2.5.3 IEEE-500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-14

4.2.6 Selection of Parameter Estimates from Existing Data Bases . . . . . . . . . . . . . . . . . . . . 4-14

5. PLANT-SPECIFIC DATA COLLECTION AND INTERPRETATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1

5.1 Initiating Event Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1

5.1.1 Initiating Event Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1

5.1.2 Data Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1

5.1.3 Initiating Event Data Allocation and Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2

5.1.4 Selection of Exposure Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2

5.2 Component Failure Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3

5.2.1 Component Data Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3

5.2.1.1 Data Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3

5.2.1.2 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4

5.2.2 Event Screening and Severity Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4

5.2.2.1 Event Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4

5.2.2.2 Event Severity Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4

5.2.3 Component Data Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-5

5.2.3.1 Component Failure Event Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-5

5.2.3.2 Allocation of Unavailability Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-7

5.2.4 Component Exposure Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-9

5.2.4.1 Time-Related Exposures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-9

5.2.4.2 Demand-Related Exposures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-10

5.2.4.2.1 Test Demands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-10

5.2.4.2.2 Automatic and Manual Initiation . . . . . . . . . . . . . . . . . . . . . 5-11

5.2.4.2.3 Corrective Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-11

5.2.5 Determination of Unavailable Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-11

5.3 Recovery Event Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-12

5.3.1 Recovery Data Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-12

5.3.2 Recovery Data Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-13

6. PARAMETER ESTIMATION AND MODEL VALIDATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

6.1.1 Chapter Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

6.1.2 Bayesian and Frequentist Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2

6.1.3 Uncertainties Other Than Parametric Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-3

6.1.3.1 Uncertainty from Nonrepresentativeness of the Data Sources . . . . . . . . . . . 6-3
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6.1.3.2 Uncertainty in the Data Counts Themselves . . . . . . . . . . . . . . . . . . . . . . . . 6-3

6.1.3.3 Uncertainty in the Correct Model to Use . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-4

6.2 Initiating Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-4

6.2.1 Frequentist or Classical Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5

6.2.1.1 Point Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5

6.2.1.2 Standard Deviation of the Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5

6.2.1.3 Confidence Interval for 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-6

6.2.2 Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-7

6.2.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-7

6.2.2.2 Choosing a Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-8

6.2.2.3 Estimation with a Discrete Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9

6.2.2.4 Estimation with a Conjugate Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-11

6.2.2.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-11

6.2.2.4.2 Update Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-11

6.2.2.5 Possible Conjugate Priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-12

6.2.2.5.1 Informative Priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-12

6.2.2.5.2 Noninformative Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-14

6.2.2.5.3 Constrained Noninformative Prior . . . . . . . . . . . . . . . . . . . . 6-15

6.2.2.5.4 Example Comparisons Using Above Priors . . . . . . . . . . . . . 6-15

6.2.2.6 Estimation with a Continuous Nonconjugate Prior . . . . . . . . . . . . . . . . . . 6-16

6.2.2.6.1 Direct Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . 6-16

6.2.2.6.2 Simple Random Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . 6-17

6.2.2.6.3 More Complicated Random Sampling . . . . . . . . . . . . . . . . . 6-17

6.2.2.7 Examples Involving Nonconjugate Priors . . . . . . . . . . . . . . . . . . . . . . . . . 6-18

6.2.2.7.1 Example with Lognormal Prior . . . . . . . . . . . . . . . . . . . . . . 6-18

6.2.2.7.2 Example with “Moment-Matching” Conjugate Prior . . . . . . 6-18

6.2.2.7.3 Comparison of Example Analyses . . . . . . . . . . . . . . . . . . . . 6-19

6.2.2.8 Analysis with Fixed Count and Random Time . . . . . . . . . . . . . . . . . . . . . 6-20

6.2.3 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-20

6.2.3.1 Poolability of Data Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-20

6.2.3.1.1 Graphical Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-21

6.2.3.1.2 Statistical Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-22

6.2.3.2 No Time Trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-25

6.2.3.2.1 Graphical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-25

6.2.3.2.2 Statistical Tests for a Trend in 8 . . . . . . . . . . . . . . . . . . . . . 6-26

6.2.3.3 No Multiple Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-28

6.2.3.4 Independence of Disjoint Time Periods . . . . . . . . . . . . . . . . . . . . . . . . . . 6-28

6.2.3.4.1 Graphical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-29

6.2.3.4.2 Statistical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-30

6.2.3.5 Consistency of Data and Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-30

6.3 Failures to Change State: 

Failure on Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-31

6.3.1 Frequentist or Classical Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-31

6.3.1.1 Point Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-31

6.3.1.2 Standard Deviation of Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-32

6.3.1.3 Confidence Interval for p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-32

6.3.2 Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-33

6.3.2.1 Estimation with a Discrete Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-33

6.3.2.2 Estimation with a Conjugate Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-35

6.3.2.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-35

6.3.2.2.2 Update Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-36

6.3.2.3 Possible Conjugate Priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-36
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6.3.2.3.1 Informative Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-36

6.3.2.3.2 Noninformative Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-37

6.3.2.3.3 Constrained Noninformative Prior . . . . . . . . . . . . . . . . . . . . 6-38

6.3.2.3.4 Example Comparison of Above Methods . . . . . . . . . . . . . . 6-38

6.3.2.4 Estimation with a Continuous Nonconjugate Prior . . . . . . . . . . . . . . . . . . 6-39

6.3.2.4.1 Direct Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . 6-39

6.3.2.4.2 Simple Random Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . 6-39

6.3.2.4.3 More Complicated Random Sampling . . . . . . . . . . . . . . . . . 6-39

6.3.2.5 Examples with Nonconjugate Priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-39

6.3.2.5.1 Lognormal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-40

6.3.2.5.2 Logistic-Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . 6-40

6.3.2.5.3 Exact Constrained Noninformative Distribution . . . . . . . . . 6-40

6.3.2.5.4 Maximum Entropy Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-41

6.3.2.5.5 Example Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-42

6.3.2.6 Estimation with Fixed Number of Failures and Random Number of 

Demands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-43

6.3.3 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-43

6.3.3.1 Poolability of Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-44

6.3.3.1.1 Graphical Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-44

6.3.3.1.2 Statistical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-45

6.3.3.2 No Time Trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-48

6.3.3.2.1 Graphical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-48

6.3.3.2.2 Statistical Tests for a Trend in p . . . . . . . . . . . . . . . . . . . . . 6-49

6.3.3.3 Independence of Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-49

6.3.3.4 Consistency of Data and Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-50

6.4 Failure to Change State: Standby Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-51

6.5 Failures to Run during Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-52

6.5.1 Estimates and Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-52

6.5.1.1 Likelihood-Based Methods: MLEs and Bayesian Methods . . . . . . . . . . . . 6-53

6.5.1.2 Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-53

6.5.1.3 Jeffreys Noninformative Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-53

6.5.1.4 Tests for Poolability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-54

6.5.2 Hazard Function Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-54

6.6 Recovery Times and Other Random Duration Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-55

6.6.1 Characterization of Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-56

6.6.1.1 Nonparametric Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-56

6.6.1.1.1 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-56

6.6.1.1.2 Percentiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-57

6.6.1.1.3 The Empirical Distribution Function . . . . . . . . . . . . . . . . . . 6-58

6.6.1.1.4 Histogram Estimate of the Density . . . . . . . . . . . . . . . . . . . . 6-58

6.6.1.2 Fitting a Parametric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-59

6.6.1.2.1 Lognormal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-59

6.6.1.2.2 Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-61

6.6.1.2.3 Gamma Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-62

6.6.1.2.4 Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-62

6.6.2 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-63

6.6.2.1 Poolability of Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-63

6.6.2.1.1 Graphical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-63

6.6.2.1.2 Statistical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-65

6.6.2.2 No Time Trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-66

6.6.2.2.1 Graphical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-66

6.6.2.2.2 Statistical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-67
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6.6.2.3 Goodness of Fit to Parametric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-68

6.6.2.3.1 Graphical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-68

6.6.2.3.2 Statistical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-70

6.6.2.4 Consistency of Data with Prior in Bayesian Parametric Estimation . . . . . 6-72

6.6.2.4.1 Exponential Durations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-72

6.6.2.4.2 Distributions Having Two or More Parameters . . . . . . . . . . 6-73

6.6.3 Nonparametric Density Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-73

6.6.3.1 Smoothing Techniques and Kernel Estimators . . . . . . . . . . . . . . . . . . . . . 6-73

6.6.3.1.1 The Rectangular Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-74

6.6.3.1.2 Boundary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-75

6.6.3.1.3 The Triangular Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-75

6.6.3.1.4 The Standard Normal Kernel . . . . . . . . . . . . . . . . . . . . . . . . 6-76

6.6.3.2 Choosing the Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-76

6.7 Unavailability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-77

6.7.1 Analysis of Detailed Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-77

6.7.1.1 Frequentist Point Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-78

6.7.1.2 Bayesian Estimation under Simple Assumptions . . . . . . . . . . . . . . . . . . . . 6-78

6.7.1.3 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-80

6.7.1.4 Bayesian Estimation under Other Assumptions . . . . . . . . . . . . . . . . . . . . . 6-80

6.7.2 Analysis of Summary Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-80

6.7.2.1 Data Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-80

6.7.2.2 Frequentist Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-82

6.7.2.3 Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-83

6.7.2.3.1 Noninformative Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-83

6.7.2.3.2 Informative Priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-83

6.7.2.4 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-84

6.7.3 Comparison of the Analyses with the Two Types of Data . . . . . . . . . . . . . . . . . . . . . . 6-84

7. TRENDS AND AGING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

7.2 Binned Poisson Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

7.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

7.2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2

7.2.2.1 General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2

7.2.2.2 Loglinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-3

7.2.2.3 Power-Law Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4

7.2.3 Bayesian Estimation with Loglinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4

7.2.4 Frequentist Estimation with Loglinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6

7.2.4.1 Point Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6

7.2.4.2 Confidence Intervals for a and b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7

7.2.4.3 Test for Presence of Trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7

7.2.4.4 Confidence Interval for 8(t) at Fixed t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8

7.2.4.5 Simultaneous Confidence Band at All t . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8

7.2.4.6 Alternative Using Least-Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-9

7.2.5 Comparison of Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-10

7.2.6 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-12

7.2.6.1 Graphical Check for Goodness of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-12

7.2.6.2 Statistical Test for Goodness of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-14

7.2.6.2.1 Test Based on Poisson Maximum Likelihood . . . . . . . . . . . 7-14

7.2.6.2.2 Test Based on Weighted Least-Squares Fit . . . . . . . . . . . . . 7-15

7.3 Unbinned Poisson Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-15

7.3.1 Bayesian Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-16
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7.3.2 Frequentist Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-16

7.4 Binomial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-16

7.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-16

7.4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17

7.4.2.1 General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17

7.4.2.2 Logit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17

7.4.2.3 Loglinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-18

7.4.3 Bayesian Estimation with Logit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-18

7.4.4 Frequentist Estimation with Logit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-19

7.4.4.1 Point Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-19

7.4.4.2 Confidence Intervals for a and b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-19

7.4.4.3 Test for Presence of Trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-19

7.4.4.4 Confidence Intervals and Confidence Bands . . . . . . . . . . . . . . . . . . . . . . . 7-20

7.4.4.5 Alternative Using Least-Squares Software . . . . . . . . . . . . . . . . . . . . . . . . 7-20

7.4.5 Comparison of Bayesian and Frequentist Estimation with Logit Model . . . . . . . . . . . 7-22

7.4.6 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-22

7.4.6.1 Graphical Check for Goodness of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-22

7.4.6.2 Statistical Test for Goodness of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-23

7.4.6.2.1 Test Based on Binomial Maximum Likelihood . . . . . . . . . . 7-23

7.4.6.2.2 Test Based on Weighted Least-Squares Fit . . . . . . . . . . . . . 7-23

7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-23

7.5.1 Generalized Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-23

7.5.2 The Many Appearances of the Chi-Squared Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-24

7.5.3 Nonparametric Estimators of 8(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-25

8. PARAMETER ESTIMATION USING DATA FROM DIFFERENT SOURCES . . . . . . . . . . . . . . . . . . . . . . . 8-1

8.1 The Hierarchical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1

8.2 The Parametric Empirical Bayes Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3

8.2.1 General Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3

8.2.2 MLE Equations for the Gamma-Poisson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3

8.2.3 MLE Equations for the Beta-Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-4

8.2.4 Adjustment for Uncertainty in the Estimate of g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6

8.2.4.1 Equations for the Gamma-Poisson Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6

8.2.4.2 Equations for the Beta-Binomial Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-7

8.2.5 Application to Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-7

8.2.5.1 Example 8.1, Initiating Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-8

8.2.5.2 Example 8.2, AFW Segment Failures to Start . . . . . . . . . . . . . . . . . . . . . . 8-10

8.3 The Hierarchical Bayes Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-12

8.3.1 General Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-12

8.3.2 Directed Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-13

8.3.3 Markov Chain Monte Carlo (MCMC) Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-13

8.3.3.1 Gibbs Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-14

8.3.3.2 Metropolis-Hastings Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-14

8.3.3.3 BUGS (Bayesian Inference Using Gibbs Sampling) . . . . . . . . . . . . . . . . . 8-15

8.3.4 Application to Example 8.1, Initiating Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-15

8.3.4.1 Development of Hyperprior Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 8-15

8.3.4.2 Example Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-16

8.3.5 Application to Example 8.2, AFW Segment Failures to Start . . . . . . . . . . . . . . . . . . . 8-16

8.3.5.1 Analysis with Beta Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-17

8.3.5.2 Analysis with Logistic-Normal Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-18

8.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-20

8.4.1 Hierarchical Bayes Is Still Bayes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-20
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8.4.2 The “Two-Stage” Bayesian Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-20

8.4.3 Lower Bounds on Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-20

8.4.4 Empirical Bayes as an Approximation to Hierarchical Bayes . . . . . . . . . . . . . . . . . . . 8-21

9. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1

APPENDICES

 A. BASICS OF PROBABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1

A.1 Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1

A.2 Basic Probability Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1

A.3 Basic Rules and Principles of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2

A.4 Random Variables and Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-5

A.4.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-5

A.4.2 Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-5

A.4.3 Cumulative Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-5

A.4.4 Reliability and Hazard Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-6

A.4.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-6

A.4.4.2 Relations among p.d.f., Reliability, and Hazard . . . . . . . . . . . . . . . . . . . . A-7

A.4.5 Joint, Marginal, and Conditional Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-7

A.4.6 Characterizing Random Variables and Their Distributions . . . . . . . . . . . . . . . . . . . . . A-8

A.4.6.1 Distribution Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-8

A.4.6.2 Mathematical Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-9

A.4.6.3 Moment-Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-10

A.4.6.4 Covariance and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-10

A.4.7 Distribution of a Transformed Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . A-11

A.5 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-11

A.6 Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-12

A.6.1 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-12

A.6.2 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-13

A.7 Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-14

A.7.1 The Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-14

A.7.2 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-15

A.7.3 The Lognormal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-16

A.7.4 The Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-17

A.7.5 The Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-18

A.7.6 The Gamma and Chi-Squared Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-18

A.7.7 The Inverted Gamma and Inverted Chi-Squared Distributions . . . . . . . . . . . . . . . . . A-20

A.7.8 The Beta Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-21

A.7.9 The Logistic-Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-22

A.7.10 Student’s t Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-23

A.7.11 F Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-23

A.7.12 Dirichlet Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-24

 B. BASICS OF STATISTICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1

B.1 Random Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1

B.2 Sample Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1

B.3 Statistical Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1

B.4 Frequentist Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2

B.4.1 Point Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2

B.4.2 Interval Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-5

B.4.3 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-6
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B.4.4 Goodness-of-Fit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-8

B.5 Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-11

B.5.1 Purpose and Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-11

B.5.2 Point and Interval Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-11

B.5.3 Prior Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-12

B.5.3.1 Noninformative Prior Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-12

B.5.3.2 Conjugate Prior Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-13

B.5.3.3 Other Prior Distribution Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-14

 C. STATISTICAL TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-1

 D. GLOSSARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-1

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-1
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LIST OF FIGURES

Figure Page

1.1 Relation of sections of this handbook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5

2.1 Probability density function (p.d.f.) and cumulative distribution function (c.d.f.). . . . . . . . . . . . . . . . . . 2-17

2.2 Uptime and downtime status for one system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-19

6.1 Schematic outline of Chapter 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

6.2 Two possible analysis paths for uncertain data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-4

6.3 Likelihood as a function of 8, for data of Example 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5

6.4 Confidence intervals from random data, all generated from the same process. . . . . . . . . . . . . . . . . . . . . . 6-7

6.5 Prior distribution and posterior distributions corresponding to three hypothetical data sets. . . . . . . . . . . . 6-8

6.6 Coarse discrete prior distribution for 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-10

6.7 Discrete posterior distribution of 8 based on 10 events in 6 years. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-10

6.8 Discrete prior and posterior distributions plotted together. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-10

6.9 Discrete prior and posterior distributions for 10 events in 6 years, with finely discretized prior. . . . . . . 6-10

6.10 Discrete prior and posterior distributions for 10 events in six years, with very finely discretized prior. . 6-10

6.11 Prior density for 8, gamma(1.53, 10.63). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13

6.12 Posterior density of 8, gamma(2.53, 15.52), for Example 6.1 with industry prior. 

The 5th and 95th percentiles are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13

6.13 Posterior cumulative distribution of 8 for Example 6.1 with industry prior. 

The 5th and 95th percentiles are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13

6.14 Jeffreys noninformative prior distribution for an event frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-14

6.15 Comparison of four point estimates and interval estimates for 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-16

6.16 Script for analyzing Example 6.3 using BUGS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-18

6.17 Two prior distributions having the same means and variances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-19

6.18 The same prior distributions as in the previous figure, with 8 plotted on a logarithmic scale. . . . . . . . . 6-19

6.19 Two posterior distributions, from priors in previous figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-19

6.20 The same posterior distributions as in the previous figure, with 8 plotted on logarithmic scale. . . . . . . . 6-20

6.21 MLEs and 90% confidence intervals for 8, based on each plant’s data and based on pooled data 

from all the plants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-21

6.22 MLEs and 90% confidence intervals for 8, each based on data from one calendar year. . . . . . . . . . . . . 6-25

6.23 Cumulative number of HPCI demands, by date. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-26

6.24 Scatter plot of rank(x) versus rank(y). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-29

6.25 Likelihood as a function of p, for the data of Example 6.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-32

6.26 Discrete informative prior distribution for p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-34

6.27 Discrete prior and posterior distribution for data in Example 6.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-34

6.28 Likelihood as a function of p, for ten times the data of Example 6.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-34

6.29 Discrete prior and posterior distributions for p, with 10 times as much data as in previous figure. . . . . . 6-35

6.30 Prior density for p, beta(4.2, 153.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-37

6.31 Posterior density for p, beta(5.2, 160.1). The 5th and 95th percentiles are shown. . . . . . . . . . . . . . . . . 6-37

6.32 Posterior cumulative distribution of p. The 5th and 95th percentiles are shown. . . . . . . . . . . . . . . . . . . 6-37

6.33 Jeffreys noninformative prior distribution for p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-37

6.34 Comparison of four point estimates and interval estimates for p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-38

6.35 Lognormal prior density and posterior density for p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-42

6.36 Script for analyzing Example 6.7 with BUGS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-43

6.37 MLEs and 90% confidence intervals for p, for three types of demand and for the pooled data. . . . . . . . 6-44

6.38 Point and interval estimates of p, each based on one year’s data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-48

6.39 Cumulative number of failures versus cumulative number of demands. . . . . . . . . . . . . . . . . . . . . . . . . . 6-48

6.40 Script for analyzing standby failure data exactly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-52
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6.41 Plot of cumulative failure count, a close approximation of plot of cumulative hazard function 

when only a small fraction of the systems fail. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-55

6.42 Empirical distribution function (EDF) for the data from group T in Example 6.12. . . . . . . . . . . . . . . . . 6-58

6.43 Histogram of the data from group T in Table 6.16, with bin width 50. . . . . . . . . . . . . . . . . . . . . . . . . . . 6-59

6.44 Histogram of same data, with bin width 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-59

6.45 One form of a box plot. The box shows the lower and upper quartiles, with the median marked. 

The whiskers show most of the range, from 4 to 118, and individual outlying points are plotted. . . . . . 6-64

6.46 A different style box plot of the same data. The box shows the upper and lower quartiles, with the 

median indicated by a stripe. The whiskers show much of the range, with dots marking outliers. . . . . 6-64

6.47 Side-by-side box plots of the three groups of data from Table 6.16, based on log10(recovery time). . . . 6-64

6.48 Log10(recovery time) plotted against event date, for data from groups S and T in Example 6.13. . . . . . . 6-66

6.49 Cumulative duration of LOSP events versus cumulative number of events. . . . . . . . . . . . . . . . . . . . . . . 6-67

6.50 Histogram of data from Table 6.19, with multiple of lognormal density overlaid. 

The skewness makes goodness of fit difficult to assess. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-68

6.51 Histogram of ln(time), with a multiple of a normal density overlaid. 

Fit appears as good as achievable without using a bimodal distribution. . . . . . . . . . . . . . . . . . . . . . . . . . 6-68

6.52 Empirical and theoretical reliability functions, where the reliability function is defined as 1 minus 

the c.d.f. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-69

6.53 Quantile-quantile plot of ln(recovery time) and fitted normal distribution. 

The points fall nearly on a straight line, indicating good fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-69

6.54 Quantile-quantile plot of raw recovery times against fitted normal distribution. 

The strong curvature indicates bad fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-69

6.55 Q-Q plot for checking exponential distribution in Example 6.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-70

6.56 Density estimate of the data from group T in Example 6.12, with rectangular kernel and 

bandwidth 25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-74

6.57 Density estimate of the data from group T in Example 6.12 with rectangular kernel and 

bandwidth 50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-74

6.58 Density estimate from group T of Example 6.13, with rectangular kernel and bandwidth 50, forced 

to be nonzero on positive axis only. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-75

6.59 Density estimate of the data from group T in Example 6.12, with standard normal kernel and 

bandwidth 25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-76

6.60 Density estimate of the data from group T in Example 6.12, with standard normal kernel and 

bandwidth 50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-76

6.61 Q-Q plot for checking whether durations have exponential distribution in Example 2.16. . . . . . . . . . . . 6-79

6.62 Q-Q plot for examining whether times between outages are exponential. . . . . . . . . . . . . . . . . . . . . . . . . 6-80

6.63 Q-Q plot for investigating normality of x when trains are pooled and months withing quarters 

are pooled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-82

7.1 Loglinear model, ln8(t) = a + bt, for a = 0.1 and three possible values of b. 

The vertical axis shows ln8(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-3

7.2 Same model as in previous figure, showing 8(t) instead of ln8(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-3

7.3 Power-law model, showing ln8(t) as a linear function of ln(t), with A = 1.0 and several values of b. . . . . 7-4

7.4 Same model as in previous figure, with 8(t) shown as function of t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4

7.5 Directed graph for analysis of Poisson trend in Example 6.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5

7.6 BUGS script for analyzing data of Example 6.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5

7.7 Posterior distribution of 8, assuming exponential trend in Example 6.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6

7.8 Frequency of unplanned HPCI demands, from Figure 6.22, with exponentially decreasing fitted 

trend line overlaid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7

7.9 Simultaneous 90% confidence band and band formed by individual 90% confidence intervals 

for Poisson event rate, 8(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-9
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7.10 Simultaneous 90% confidence band for 8(t), and 90% confidence interval for one frequency 

of special interest, 8(93). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-9

7.11 Fitted Poisson event occurrence rate and simultaneous 90% confidence band, based on three 

ways of fitting the HPCI unplanned-demands data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-11

7.12 Annual frequency of initiating events, with fitted exponentially decreasing 8(t) and 

simultaneous 90% confidence band on 8(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-13

7.13 Standardized residuals, also called the Pearson chi-squared residuals, for data of 7.12. . . . . . . . . . . . . . 7-13

7.14 Cumulative count of initiating events, for data of Table 7.3 (Example 2.1). . . . . . . . . . . . . . . . . . . . . . . 7-14

7.15 Plot of ln{ p(t)/[1 ! p(t)] } = a + bt, with a = !2 and three values of b. . . . . . . . . . . . . . . . . . . . . . . . . . 7-17

7.16 Plot of p(t) as a function of t, corresponding to Figure 7.15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17

7.17 BUGS script for analyzing data of Example 6.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-18

7.18 Posterior trend line for p with 90% credible band, for data of Table 7.5. In addition, annual 

estimates and 90% credible intervals are shown, based on constrained noninformative prior. . . . . . . . . 7-19

7.19 Annual estimates of p, fitted trend line and 90% confidence band for p(t), for data of Table 7.5. . . . . . 7-21

7.20 Estimate of 8(t) using standard normal kernel, with bandwidth h = 400. . . . . . . . . . . . . . . . . . . . . . . . . 7-26

7.21 Estimate of 8(t) using standard normal kernel, with bandwidth h = 200. . . . . . . . . . . . . . . . . . . . . . . . . 7-26

8.1 Hierarchical model for Poisson data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1

8.2 Plant-specific MLEs and 90% confidence intervals for 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-8

8.3 Plant-specific posterior means and 90% credible intervals for 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-8

8.4 Plant-specific MLEs and 90% confidence intervals for p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-10

8.5 Plant-specific posterior means and 90% credible intervals for p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-11

8.6 Hierarchical Bayes model for Poisson data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-12

8.7 Directed graph for the hierarchical Bayes analysis of Example 8.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-13

8.8 WinBUGS model specification for Example 8.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-16

8.9 Plant-specific posterior means and 90% credible intervals for 8, from hierarchical Bayes analysis. . . . . 8-17

8.10 WinBUGS model specification for a beta prior in Example 8.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-17

8.11 Plant-specific posterior means and 90% credible intervals for p, from hierarchical Bayes 

analysis with beta population-variability distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-18

8.12 WinBUGS model specification for a logistic-normal prior in Example 8.2. . . . . . . . . . . . . . . . . . . . . . . 8-19

8.13 Fitted population-variability distributions in Example 8.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-19

8.14 Plant-specific posterior means and 90% credible intervals for p, from hierarchical Bayes 

analysis with logistic-normal population-variability distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-19

A.1 Venn diagram, showing ten outcomes and three events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1

A.2 Probability density function (p.d.f.) and cumulative distribution function (c.d.f.). . . . . . . . . . . . . . . . . . A-6

A.3 The reliability function, hazard function and cumulative hazard function. . . . . . . . . . . . . . . . . . . . . . . . A-7

A.4 Density, showing quartiles, median, and mean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-9

A.5 Cumulative distribution function (c.d.f.) showing quartiles, median, and mean. . . . . . . . . . . . . . . . . . . . A-9

A.6 Three binomial probability distribution functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-13

A.7 Three Poisson probability distribution functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-14

A.8 Density of uniform(a, b) distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-15

A.9 Two normal densities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-15

A.10 Three lognormal densities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-17

A.11 Two exponential densities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-17

A.12 Four Weibull densities, all having 2 = 0 and all having the same ". . . . . . . . . . . . . . . . . . . . . . . . . . . . A-19

A.13 Gamma densities with four shape parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-20

A.14 Four inverted gamma densities, having the same scale parameter, $, and various shape parameters, ". A-20

A.15 Beta distributions with mean = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-21

A.16 Four beta distributions with mean 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-21

A.17 Three logistic-normal densities with median = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-22

A.18 Three logistic-normal densities with median = 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-23
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B.1 Probability of rejecting H0: : = 3.388, if in fact H0 is true (upper distribution), and if H0 is false with

: = 4.09 (lower distribution). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-8

B.2 Power curves when n = 5 and n = 10. The graph shows the probability of rejecting H0, as a function 

of the true :. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-8

B.3 The hypothesized distribution, the empirical distribution, and the Kolmogorov test statistic, D. . . . . . B-10

B.4 Schematic diagram of types of priors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-14
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LIST OF TABLES

Table Page

2.1 Kinds of models considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2

3.1 Examples of component boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2

6.1 Comparison of Bayesian and frequentist approaches in PRA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-3

6.2 Example 6.2, first sample (10 events in 6 years) with coarse prior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9

6.3 Comparison of Bayesian and frequentist estimates for the data in Example 6.2. . . . . . . . . . . . . . . . . . . 6-11

6.4 Comparison of estimates with 1 event in 4.89 reactor-critical-years. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-15

6.5 Posterior distributions from two analyses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-19

6.6 Quantities for calculation of chi-squared test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-22

6.7 Shutdown LOSP events at three sites, 1980-96. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-24

6.8 HPCI demands and reactor-critical-years. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-25

6.9 Calculations for analyzing LOSP dates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-29

6.10 Comparison of Bayesian distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-35

6.11 Comparison of estimates with one failure in eight demands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-39

6.12 Contingency table for Example 6.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-45

6.13 Counts, expected counts, and contributions to X2

, for Example 6.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-46

6.14 HPCI failures on demand, by year. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-48

6.15 Contingency table for successive outcomes in Example 6.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-50

6.16 Statistics based on the recovery times (minutes) of Example 6.12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-57

6.17 Detailed data for Example 2.16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-78

6.18 Summary data for Example 2.16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-78

6.19 Sample statistics for estimates of q, with different levels of aggregation. . . . . . . . . . . . . . . . . . . . . . . . . 6-81

6.20 Comparison of Bayesian results from two approaches, using noninformative priors. . . . . . . . . . . . . . . . 6-84

7.1 HPCI demands and reactor-critical-years (from Example 6.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

7.2 Posterior statistics for a and b, for loglinear model of Example 6.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6

7.3 Dates of initiating events at one plant, 1987-1995. (from Example 2.1) . . . . . . . . . . . . . . . . . . . . . . . . . 7-12

7.4 Initiating events and reactor-critical-years. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-13

7.5 HPCI failures on demand, by year (from Example 6.10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17

7.6 Posterior statistics for a and b, for loglinear model of Example 6.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-18

8.1 Portion of frequentist analysis results for Example 8.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-9

8.2 Portion of empirical Bayes analysis, without Kass-Steffey adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-9

8.3 Portion of empirical Bayes analysis, with Kass-Steffey adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-9

8.4 Portion of frequentist analysis results for Example 8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-11

8.5 Portion of empirical Bayes analysis, without Kass-Steffey adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . 8-11

8.6 Portion of empirical Bayes analysis, with Kass-Steffey adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-11

8.7 Portion of hierarchical Bayes analysis results for Example 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-17

8.8 Portion of hierarchical Bayes analysis results using beta prior for Example 8.2 . . . . . . . . . . . . . . . . . . . 8-18

8.9 Portion of hierarchical Bayes analysis results using logistic-normal prior for Example 8.2 . . . . . . . . . . 8-20

B.1 Features of Bayesian and Frequentist Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-3

B.2 Possible Hypothesis Test Outcomes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-6

C.1 Standard normal cumulative distribution function, M.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-1

C.2 Percentiles of the chi-squared distribution.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-4

C.3 Percentiles of the Sudent’s t distribution.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-6

C.4 90th and 10th percentiles of beta(", $) distribution.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-8

C.5 95th and 5th percentiles of beta(", $) distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-10

C.6 97.5th and 2.5th percentiles of beta(", $) distribution.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-12

C.7 Acceptance limits for the Kolmogorov test of goddness of fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-14

C.8 Parameters of constrained noninformative prior for binomial p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-15
Table of Contents

xix

LIST OF EXAMPLES

Example Page

2.1 Unplanned reactor trips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3

2.2 Shutdown loss of offsite power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3

2.3 Through-wall pipe leaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3

2.4 Temperature sensor/ transmitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3

2.5 HPCI failures to start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7

2.6 EDG failures to start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7

2.7 Steam binding in AFW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7

2.8 Failures of isolation valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7

2.9 EDG failures to run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14

2.10 AFW turbine train failures to run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14

2.11 Recovery times from loss of offsite power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16

2.12 Repair times for turbine-driven pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16

2.13 Time to failure of a component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16

2.14 Times to suppress fires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16

2.15 Gradual degradation until failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16

2.16 CVC unavailability for test and maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-19

6.1 Initiating events with loss of heat sink. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5

6.2 Confidence intervals from computer-generated data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-7

6.3 Small-break LOCAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-18

6.4 Shutdown LOSP events at five plants, 1980-96. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-21

6.5 Unplanned HPCI demands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-25

6.6 Dates of shutdown LOSP events and days between them. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-29

6.7 AFW turbine-train failure to start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-31

6.8 EDG failures to start on demand. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-44

6.9 Hypothetical unbalanced data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-47

6.10 Dates of HPCI failures and unplanned demands, 1987-1993. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-48

6.11 EDG failure-to-run times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-54

6.12 LOSP recovery times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-56

6.13 LOSP recovery times and event dates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-66

7.1 Thermal-fatigue leak events, by plant age. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

8.1 Initiating events from many plants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2

8.2 Failure to start of AFW motor-driven segments at many plants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2
xx
xxi

FOREWORD

During the last several years, both the U.S. Nuclear Regulatory Commission (NRC) and the nuclear industry have

recognized that probabilistic risk assessment (PRA) has evolved to the point where it can be used in a variety of

applications including as a tool in the regulatory decision-making process. The increased use of PRA has led to the

conclusion that the PRA scope and model must be commensurate with the applications. Several procedural guides and

standards have been and are being developed that identify requirements for the PRA models. For example, the “Standard

For Probabilistic Risk Assessment For Nuclear Power Plant Applications” published by The American Society of

Mechanical Engineers (ASME) in 2002 (ASME-RA-S-2002) defines requirements for PRA analysis used to develop riskinformed decisions for commercial nuclear power plants, and describes a process for applying these requirements in

specific applications. This handbook was generated to support these documents by providing a compendium of good

practices that a PRA analyst can use to generate the parameter distributions required for quantifying PRA models.

The increased use of risk assessment has also helped promote the idea that the collection and analysis of event data is

an important activity in and of itself. In particular, the monitoring of equipment performance and evaluation of

equipment trends can be used to enhance plant performance and reliability. The reference material provided in this

handbook can support those efforts.

This handbook provides references on sources of information and methods for estimating parameter distributions. This

includes determination of both plant-specific and generic estimates for initiating event frequencies, component failure

rates and unavailability, and equipment non-recovery probabilities, all of which directly supplement the ASME PRA

standard. 

This handbook provides the basic information needed to generate estimates of the parameters listed above. It begins by

describing the probability models and plant data used to evaluate each of the parameters. Possible sources for the plant

data are identified and guidance on the collection, screening, and interpretation is provided. The statistical techniques

(both Bayesian and classical methods) required to analyze the collected data and test the validity of statistical models

are described. Examples are provided to help the PRA analyst utilize the different techniques.

This handbook also provides advanced techniques that address modeling of time trends. Methods for combining data

from a number of similar, but not identical, sources are also provided. This includes empirical and hierarchical Bayesian

approaches. Again, examples are provided to guide the analyst.

This handbook does not provide guidance on parameter estimation for all types of events included in a PRA.

Specifically, common cause failure and human error probabilities are not addressed. In addition, guidance is not

provided with regard to the use of expert elicitation. For analysis of these events, the PRA analyst should consult other

sources, some of which are cited in Chapter 1.
xxii

ACKNOWLEDGMENTS

The authors wish to acknowledge Dennis C. Bley, Iyiin Chang, Mary T. Drouin, Leslie E. Lancaster, Gareth W. Parry,

Dale M. Rasmuson, David Robinson, Arthur D. Salomon, and Nathan O. Siu, who gave extensive and insightful written

comments. In particular, it was Mary Drouin’s initiative that started the project. She provided the early guidance and

support for the development of the document. Bley and Parry not only provided comments, but also contributed text that

was included in the final report. W. Jay Conover gave helpful advice on tests of serial independence, and David S.

Moore gave advice on the relation of the chi-squared statistic and the deviance. Robert F. Cavedo and Steven A. Eide

supplied information on the treatment of unavailability in current PRAs. Robert W. Youngblood contributed ideas and

references on the relation of the two models for failure to start. Steven M. Alferink, William J. Galyean, Cynthia D.

Gentillon, Dana L. Kelly, and Martin B. Sattison supplied examples and constructive comments. Ralph Nyman provided

information about the Swedish I-Book. Teresa Sype provided help in formatting the report. Christine E. White produced

many of the figures.
xxiii

ABBREVIATIONS

AC alternating current

AFW auxiliary feedwater

ALWR Advanced Light Water Reactor

ANO Arkansas Nuclear One

ANSI American National Standards Institute

ASEP Accident Sequence Evaluation Program

ASME American Society of Mechanical Engineers

ATWS anticipated transient without scram

BWR boiling water reactor

CDF core damage frequency

c.d.f. cumulative distribution function

ComEd Commonwealth Edison

ConEd Consolidated Edison

CPC Consumers Power Company

CREDO Centralized Reliability Data Organization

CVC chemical and volume control

DC direct current

DG diesel generator

DOE Department of Energy

ECCS emergency core cooling system

EDF empirical distribution function

EDG emergency diesel generator

EEI Edison Electric Institute

EF error factor

EPIX Equipment Performance and Information Exchange System

EPRI Electric Power Research Institute

FTR failure to run

FTS failure to start

GE General Electric

HCF Hardware Component Failure 

HEP human error probability

HHRAG Human and Hardware Reliability Analysis Group

HPCI high pressure coolant injection

HPP homogeneous Poisson process

HRA human reliability analysis

HVAC heating, ventilation and air conditioning 

IEEE Institute of Electrical and Electronics Engineers

IGSCC intergranular stress corrosion cracking

i.i.d. independent identically distributed

INEL Idaho National Engineering Laboratory

INEEL Idaho National Engineering and Environmental Laboratory

INPO Institute of Nuclear Power Operations

IPE Individual Plant Examination

IPEEE Individual Plant Examination of External Events

IPRDS In-Plant Reliability Data Base for Nuclear Power Plant Components

IREP Interim Reliability Evaluation Program 

LER Licensee Event Report

LCO limiting conditions of operation

LOCA loss of coolant accident

LOSP loss of offsite power

MCMC Markov Chain Monte Carlo
xxiv

MLE maximum likelihood estimate, or maximum likelihood estimator, depending on the context (see

estimate in glossary)

MOV motor-operated valve

MSE mean square error

MSIV main steam isolation valve

MTTF mean time to failure

MTTR mean time to repair

NA not applicable

NHPP nonhomogeneous Poisson process

NPE Nuclear Power Experience

NPP Nuclear Power Plant

NPRDS Nuclear Plant Reliability Data System

NRC U.S. Nuclear Regulatory Commission

NREP Nuclear Reliability Evaluation Program

NSAC Nuclear Safety Analysis Center

NUCLARR Nuclear Computerized Library for Assessing Reactor Reliability

OREDA Offshore Reliability Data

PCS Power Conversion System

PECO Philadelphia Electric Company

p.d.f. probability density or distribution function

PLG Pickard, Lowe and Garrick, Inc.

PORV power operated relief valve

PRA Probabilistic Risk Assessment

PSA Probabilistic Safety Assessment

PWR pressurized water reactor

RAC Reliability Analysis Center

RADS Reliability and Availability Data System

RCIC reactor core isolation cooling

RHR residual heat removal

RMIEP Risk Methods Integration and Evaluation Program

RPS reactor protection system

RSSMAP Reactor Safety Study Methodology Application Program

SAIC Scientific Applications International Corporation

SBLOCAs small-break loss-of-coolant accidents

SCSS Sequence Coding and Search System

SKi Swedish Nuclear Power Inspectorate

SRV safety relief valve

SSPI Safety System Performance Indicator

SwRI Southwest Research Institute

U.K. United Kingdom

U.S. United States
1-1

1. INTRODUCTION

1.1 Objective

The data analysis portion of a nuclear power plant

probabilistic risk assessment (PRA) provides estimates

of the parameters used to determine the frequencies and

probabilities of the various events modeled in a PRA.

The objective of this handbook is to provide methods

for estimating the parameters used in PRA models and

for quantifying the uncertainties in the estimates. 

1.2 Background

Probabilistic risk assessment is a mature technology

that can provide a quantitative assessment of the risk

from accidents in nuclear power plants. It involves the

development of models that delineate the response of

systems and operators to accident initiating events.

Additional models are generated to identify the

component failure modes required to cause the accident

mitigating systems to fail. Each component failure

mode is represented as an individual “basic event” in

the systems models. Estimates of risk are obtained by

propagating the uncertainty distributions for each of the

parameters through the PRA models.

During the last several years, both the U.S. Nuclear

Regulatory Commission (NRC) and the nuclear industry

have recognized that PRA has evolved to the point

where it can be used in a variety of applications,

including as a tool in the regulatory decision-making

process. The increased use of PRA has led to the

conclusion that the PRA scope and model must be

commensurate with the applications. Several

procedural guides and standards have been and are

being developed that identify requirements for the PRA

models. This handbook was generated to supplement

these documents. It provides a compendium of good

practices that a PRA analyst can use to generate the

parameter distributions required for quantifying PRA

models.

The increased use of risk assessment has also helped

promote the idea that the collection and analysis of

event data is an important activity in and of itself. In

particular, the monitoring of equipment performance

and evaluation of equipment trends can be used to

enhance plant performance and reliability. The

guidance provided in this handbook can support those

efforts.

1.3 Scope

This handbook provides guidance on sources of

information and methods for estimating parameter

distributions. This includes determination of both

plant-specific and generic estimates for initiating event

frequencies, component failure rates and

unavailabilities, and equipment non-recovery probabilities. 

This handbook provides the basic information needed

to generate estimates of the parameters listed above. It

begins by describing the probability models and plant

data used to evaluate each of the parameters. Possible

sources for the plant data are identified and guidance on

the collection, screening, and interpretation is provided.

The statistical techniques (both Bayesian and classical

methods) required to analyze the collected data and test

the validity of statistical models are described.

Examples are provided to help the PRA analyst utilize

the different techniques. 

This handbook also provides advanced techniques that

address modeling of time trends. Methods for

combining data from a number of similar, but not

identical, sources are also provided. These are the

empirical and hierarchical Bayesian approaches.

Again, examples are provided to guide the analyst.

This handbook does not provide guidance on parameter

estimation for all of the events included in a PRA.

Specifically, common cause failure and human error

probabilities are not addressed. In addition, guidance

is not provided with regard to the use of expert

elicitation. For these topics, the PRA analyst should

consult other sources, such as the following references:

Common cause failures

• NUREG/CR-5497 (Marshall et al. 1998),

• NUREG/CR-6268 (Kvarfordt et al. 1998),

• NUREG/CR-5485 (Mosleh et al. 1998),

• NUREG/CR-4780 (Mosleh et al. 1988), and

• EPRI NP-3967 (Fleming, 1985).

Human errors

• NUREG/CR-1278 (Swain and Guttman, 1983),

• NUREG/CR-4772 (Swain, 1987),
Introduction

1-2

• NUREG-1624 (NRC, 2000b), and

• EPRI TR-TR-100259 (Parry et al. 1992).

Expert Judgement

• NUREG/CR-6372 (Budnitz et al. 1997) and

• NUREG/CR-1563 (Kotra et al. 1996).

This list is not meant to be a comprehensive list of all of

the methodologies available for performing these types

of analyses. 

1.4 Contents of the Handbook

This section provides a road map of the contents of the

handbook and an overview discussion on how to use the

handbook to perform the elements of a data analysis.

The basics of probability and statistics described in

Appendices A and B, respectively, are provided as

reference material for the analyst. Appendix C provides

statistical tables for selected distribution types that can

be used in the data analysis.

1.4.1 Identification of Probability Models

The handbook provides guidance on the evaluation of

five types of parameters that are included in a PRA:

• initiating events,

• failures to start or change state,

• failures to run or maintain state,

• durations, and

• unavailability from being out of service.

A description of each of these parameters along with

examples, is provided in Chapter 2. Chapter 2 is

fundamental reading for all users of this handbook.

The first step in a data analysis is to determine the

appropriate probability models to represent the

parameter. Chapter 2 provides a detailed description

of the standard probability models for each event. This

includes a discussion of the assumptions on the physical

process inherent in the models and a description of the

kind of data that can be observed. The type of data

required to estimate the model parameter(s) are

described and example data sets are examined in the

light of the model assumptions. These examinations

illustrate the kind of thinking necessary for the data

analyst. Finally, a short discussion of related issues is

presented for the analyst to consider.

1.4.2 Collection of Plant Specific Data

Once probability models have been defined for the

basic events, plant-specific data should be evaluated for

the purpose of quantifying estimates of the probability

model parameters. Plant-specific data, if available in

sufficient quantity and quality, is the most desirable

basis for estimating parameter values. Chapter 5

discusses the process by which plant-specific data

should be identified, collected, screened, and

interpreted for applicability to the basic events defined

in the systems analysis and to their probability models.

To ensure that the collection and evaluation of plantspecific data is thorough, consistent, and accurate, the

steps laid out in Chapter 5 should be followed for

events defined in a PRA. The identification and

evaluation of appropriate sources of plant-specific data

for the basic events are discussed in Section 4.1. 

The process for collecting and evaluating data for

initiating events is discussed in Section 5.1. Guidance

is provided for screening the data, for grouping the data

into appropriate categories of initiating events, and for

evaluating the denominator associated with the data. 

The process for collecting and evaluating data for

component failures is discussed in Section 5.2. It is

critical that data be collected and processed accurately

according to the definition of the component boundary.

For example, it should be clearly noted whether or not

a pump’s control circuit is within or without the

physical boundaries of the component for purposes of

systems modeling. If failure of the control circuit has

been modeled separately from hardware failures of the

pump, then data involving failure of the pump should be

carefully evaluated to ensure that actuation failures and

other pump faults are not erroneously combined. This

process could result in some iteration between the

systems analysis task and the data collection task. It is

possible that system models may be simplified or

expanded based on insights derived during the data

collection. Chapter 3 describes the difference between

faults and failures, and discusses component boundary

definitions and failure severity as it relates to data

collection and analysis.

Other aspects of data collection for component failures

discussed in Section 5.2 include classification and

screening of the data, allocation of the data to

appropriate component failure modes, and exposure

evaluation (determining the denominator for parameter

estimates). 
Introduction

1-3

The collection of data for recovery events is described

in Section 5.3. Guidance is provided on where to find

recovery-related data and on how to interpret such data.

1.4.3 Quantification of Probability

Model Parameters

Once appropriate probability models have been selected

for each basic event, estimates for the model parameters

must be quantified. There are two basic approaches: 1)

statistical estimation based on available data; and 2)

utilization of generic parameter estimates based on

previous studies. Both approaches can incorporate

generic data. Several generic data sources currently

available and used throughout the nuclear PRA industry

are identified in Section 4.2.

1.4.3.1 Parameter Estimation from 

Plant-Specific Data

If the plant-specific data collection process yields data

of sufficient quantity and quality for the development of

parameter estimates, the statistical methods in Chapter

6 can be applied to the data to derive and validate

parameter estimates for the basic events. 

Chapter 6 discusses the statistical methods for

estimating the parameters of the probability models

defined in Chapter 2. Note that Appendix B discusses

basic concepts of statistics that will help the user to

understand the methods presented in Chapter 6.

For each type of event, two fundamental approaches are

presented for parameter estimation: classical

(frequentist) and Bayesian. An overview and

comparison of these two approaches are presented in

Section 6.1. The Bayesian approach is more commonly

used in PRA applications, but classical methods have

some use in PRA, as discussed in Section 6.1. 

The probability models discussed in Chapter 2 for each

type of event are applicable for most applications.

However, erroneous results can occur in some cases if

the assumptions of the model are not checked against

the data. In some applications (e.g., if the impact of

casual factors on component reliability is being

examined) it is imperative that the probability model

chosen for each basic event be validated given the

available data. It may seem sensible to first confirm the

appropriateness of the model and then estimate the

parameters of the model. However, validation of a

model is usually possible only after the model has been

assumed and the corresponding parameters have been

estimated. Thus, estimation methods are presented first

in Chapter 6 for each type of probability model; then

methods for validating the models against the available

data are presented. 

1.4.3.2 Parameter Estimation from Existing Data

Bases

If actual data are unavailable or of insufficient quality

or quantity then a generic data base will have to be

used. Several generic data sources currently available

and used throughout the nuclear PRA industry are

identified in Section 4.2. Section 4.2.6 provides

guidance on the selection of parameter estimates from

existing generic data bases. 

1.4.4 Advanced Methods

The last two chapters of the handbook describes some

methods for analyzing trends in data and Bayesian

approaches for combining data from a number of

similar sources. 

1.4.4.1 Analyzing Data for Trends and Aging

Data can be analyzed to assess the presence of time

trends in probability model failure rates and

probabilities (i.e., 8 and p). Such trends might be in

terms of calendar time or in terms of system age.

Ordinarily, the analysis of data to model time trends

involves complex mathematical techniques. However,

the discussion of Chapter 7 presents various approaches

that have been implemented in computer software. The

discussion in Chapter 7 focuses on the interpretation of

the computer output for application in PRA.

1.4.4.2 Parameter Estimation Using Data from

Different Sources

Two Bayesian approaches for combining data from a

number of similar, but not identical, sources are

discussed in Chapter 8. 

1.5 How to Use This Handbook

This handbook is intended for workers in probabilistic

risk assessment (PRA), especially those who are

concerned with estimating parameters used in PRA

modeling. Broadly speaking, three groups of readers

are anticipated: data collectors, who will be finding,

interpreting, and recording the data used for the

estimates; parameter estimators, who will be

constructing the parameter estimates from the data and

quantifying the uncertainties in the estimates; and (to a

lesser extent) PRA analysts, who will be using the
Introduction

1-4

estimated parameters. These three groups will find

their primary interests in different portions of the

handbook, as discussed below.

The major sections of the handbook can be grouped

into several areas:

• Foundation: Chapters 1 and 2;

• Data Collection: Chapters 3, 4, and 5;

• Parameter Estimation: Chapters 6, 7, and 8; and

• Supporting Material: Appendices, References,

Index.

These sections are shown in Figure 1.1, a schematic

representation of the contents of the handbook.

PRA analysts will be most interested in the foundational

material. Data collectors will need to read much of the

foundational material, and then read the chapters on

data collection. Parameter estimators will need to read

the foundational chapters, but may then wish to skip

directly to the relevant sections on parameter

estimation. The supporting material can be read by

anyone at any time.

The arrows in Figure 1.1 help the reader find the

quickest way to the sections of interest. For example,

the figure shows that Chapters 3-5 and Chapters 6-8 do

not refer to each other or assume material from the

other section, so it is possible to read from one section

and not the other. The only strong dependencies are

shown by the arrows: read Chapter 2 before starting

Chapter 3 or 6, read Chapter 3 before starting Chapter

4 or 5, and so forth. In practice, data collectors, data

analysts, and PRA analysts must work together, giving

feedback to each other. The handbook, on the other

hand, is formed of distinct segments, each of which can

be read in isolation from the others.

The material for PRA analysts and data collectors is

intended to be accessible by anyone with an engineering

background and some experience in PRA. The material

for data analysts, on the other hand, begins with

elementary techniques but eventually covers advanced

models and methods. These advanced topics will not

be needed in most cases, but are included as reference

material. 

To aid the reader, Appendices A and B summarize the

basics of probability and statistics, and Appendix C

provides useful statistical tables. A glossary of terms is

provided in Appendix D. Persons who have no

previous experience with probability or statistics will

need a more thorough introduction than is provided in

these sections of the handbook.
1-5

1. Introduction

App. A. Probability

2. Overview of PRA Events* Initiating events and basic events* Corresponding probability models App. B. Statis

6. Parameter Estimation and Model Validation* Initiating Events* Failures to Start* Failures to Run* Unavailability* Recovery Times, Durations

3. Definitions* Failure* Component boundary* Failure severity

5. Interpretation of Plant-SpecificData* Allocation* Exposure evaluation

4. Data Sources* Typical plant-specific* Current generic

7. Tr

8. Estimation with Data from Different Sourc

App. D Glossary 9. References Index

Figure 1.1 Relation of sections of this handbook. An arrow with a solid head indicates a strong dependence, with required information the arrow. An dashed arrow with a hollow head indicates that background or supplementary material flows in the direction of the arrowand index supply assistance for the entire handbook, so the arrows are not shown.

Foundation

 Data Collection Parameter Estimation
Introduction

1-6
2-1

2. BASIC EVENT PROBABILITY MODELS

2.1 Overview

This chapter introduces the models used for basic

events and for initiating events. This first section is an

overview, and the remaining sections of the chapter

give further detail.

Probabilistic risk assessment (PRA) considers various

possible accident sequences. An accident sequence

begins with an initiating event which challenges the

safety of the plant. Typically, one or more standby

safety systems are then demanded, and other, normally

operating, systems must continue operating to ensure

that no serious undesirable consequences occur. For

the systems to fail to bring the situation under control,

several components must either fail or be unavailable.

The logic events in the PRA model that represent these

failures or modes of unavailability are called basic

events.

It is not possible to predict precisely when an initiating

event or a component failure will occur, because the

processes that lead to their occurrences are complex.

Therefore, the initiating events and basic events are

modeled as resulting from random processes.

The first step in the data analysis task is, therefore, to

determine the appropriate probability model to represent the initiating event or basic event. (Probability is

reviewed in Appendix A, and the probability models

introduced here are presented more fully there.) These

probability models typically have one or more

parameters. The next major step is to estimate the

values of these parameters. This estimation is based on

the most applicable and available data. The process of

choosing data sources, extracting the data in an

appropriate form, and using it to estimate the

parameters is the main subject of this handbook.

Basic events are customarily divided into unavailability

(because the equipment is undergoing testing or

maintenance), failure to start or change state, and

failure to run (after successfully starting) or maintain

state to the end of the required mission time.

Unavailability and failure to run are each modeled in a

single way. On the other hand, two different probability

models have been used to represent a failure to start or

to change state. The first method is to model the

failures as having a constant probability of failure on

a demand. The second method is to model the failures

as occurring, in an unrevealed way, randomly in time.

The failed condition is then discovered at the time of

the demand. This is usually called the standby failurerate model. Both models are discussed here.

The above events are the typical ones considered in a

PRA. In addition, one must occasionally analyze

durations, such as the time to restore offsite power or

time to recover a failed component. Although such an

analysis is not needed for a typical accident sequence,

it is discussed in this handbook. Also, methods for

analyzing durations can be used when estimating

unavailability.

In summary, five topics are considered in the rest of this

chapter:

• initiating events,

• failures to start or change state (modeled in two

possible ways),

• failures to run or maintain state,

• durations, and

• unavailability from being out of service.

These topics are the subjects of Sections 2.2 through

2.6. Each section begins with examples of the data that

might be analyzed. This is followed by a brief

subsection presenting the assumptions of the usual

model for the random process (the result of underlying

physical mechanisms) and describing the kind of data

that can be observed. The next subsection summarizes

the data required to estimate the model parameter(s).

The example data sets are then examined in the light of

the model assumptions. These examinations illustrate

the kind of thinking necessary for the data analyst.

Finally, the section may conclude with a short

discussion of related issues.

As a preview, Table 2.1 indicates the models, the

parameters, and the data needed for each of the topics

in the above five bullets. The top line of the table also

indicates which section of Chapter 2 treats the topic. 

The term system is used to denote the set of hardware

for which data are collected; it may be an entire nuclear

power plant (NPP), or a system in the traditional sense,

such as the auxiliary feedwater (AFW) system, or a

train, component, or even piece part. This reduces the

need for phrases such as “system or component.”

The lengthiest part of each section below consists of the

examination of examples to see whether the
Basic Event Probability Models 

2-2

Table 2.1 Kinds of models considered.

2.2 Initiating

Events

2.3 Failures to Start or Change

State (2 models)

2.4 Failures to Run

or Maintain State

2.5 Durations 2.6 Unavailability

Typical Event

Event occurs

initiating an

accident

sequence

Standby system fails on demand System in operation

fails to run, or

component changes

state during mission

A condition

persists for a

random time

period

System is unavailable, intentionally

out of service,

when demanded

Parameter(s) to Estimate

8, event

frequency

For failure on

demand:

p, probability

of failure on

demand

For standby

failure:

8, rate of

occurrence of

standby

failures

8, rate of failure to

run

Parameters of

assumed

probability

distribution of

duration time

q, fraction of time

when system will

be out of service

Data Required to Estimate Parametersa

Number of

events, x, in

total time, t

Number of

failures, x, in

total number of

demands, n

Number of

failures, x, in

total standby

time, t

Number of failures,

x, in total running

time, t

Depends on

model, but

typically the

lengths of the

observed

durations

Onset times and

durations of

observed out-ofservice events; OR

observed fractions

of time when

system was out of

service

a.

 The data here are the minimal requirements to estimate the parameter. More detailed data are needed to check the

model assumptions.

assumptions of the probability model appear to be

satisfied. Verifying model assumptions is an important

part of good data analysis. Ways to investigate the

appropriateness of assumptions are considered in

Chapter 6, along with parameter estimation. The

present chapter, however, only introduces the

assumptions and illustrates their meanings through

examples. If the assumptions are clearly not satisfied,

some mention is given of ways to generalize the model,

although such generalizations are not presented until

Chapters 7 and 8 in this handbook.

Also, examples and extended discussion of examples

are printed in Arial font, to distinguish them from the

more general material.

2.2 Initiating Events

2.2.1 Examples

In the context of a nuclear-power-plant PRA, an

initiating event is any event that perturbs the steady

state operation of the plant, thereby initiating an

abnormal event such as a transient or a loss-of-coolant

accident within a plant. Initiating events begin

sequences of events that challenge plant control and

safety systems. Failure of these systems can lead to

core damage and a release of radioactivity to the

environment. However, the consideration of the

potential plant response to initiating events is irrelevant

when estimating their frequencies. 
Basic Event Probability Models

2-3

Here are several examples of data sets counting such

initiating events.

Example 2.1 Unplanned reactor trips

A U.S. commercial nuclear power plant had 34

unplanned reactor trips in 1987 through 1995. It

had its initial criticality on Jan. 3, 1987, and

experienced a total of 64651 critical hours, or

7.38 critical years (Poloski et al. 1999a).

Example 2.2 Shutdown loss of offsite power

In U.S. commercial nuclear power plants in 1980-

1996, there were 80 plant-centered loss-of-offsitepower (LOSP) events during shutdown. In that

period, the plants experienced 455.5 reactorshutdown years (Atwood et al. 1998).

Example 2.3 Through-wall pipe leaks

In world-wide experience of western-style

pressurized water reactors (PWR)s (3362

calendar years of operation), a single through-wall

leak event has been reported in large-diameter

piping ( Poloski et al. 1999a, Appendix J).

The final example of this section does not have

initiating events in the usual sense. However, the model

assumptions and the form of the data are exactly the

same as for initiating events. Therefore, such data can

be analyzed just like initiating-event data.

Example 2.4 Temperature sensor/transmitters

Eide et al. (1999a) report that temperature sensor/

transmitters in the reactor protection system

(RPS) of Westinghouse NPPs had 32 failures in

2264.1 component-years. These sensor/transmitters operate continuously, and when they fail they

are repaired or replaced in a relatively short time.

The number of failures is conservatively estimated

from sometimes incomplete Nuclear Plant

Reliability Data System (NPRDS) data, and the

number of component years is based on an

estimated number of components per loop.

These examples have several elements in common.

First, they involve a number of events that occurred,

and an exposure time, or time at risk, when the events

could have occurred. The next subsection will present

a simple probability model that gives rise to random

events in time. In addition, in each of the above

examples corrective action is taken after any event, so

that the system then resumes operation (the system is

repairable.) This means that the recorded operating

history consists of a sequence of random event

occurrences, which is summarized as a count of events

in some fixed time. This type of data will direct us to

a particular type of analysis, presented in Chapter 6.

The events may be the initiating events of an ordinary

PRA (Example 2.1), initiating events of a shutdown

PRA (Example 2.2), failures in a passive system

(Example 2.3), which incidentally happen to be

initiating events in a PRA. As mentioned above,

Example 2.4 does not describe initiating events in the

traditional PRA sense. However, the example may be

analyzed in the same way as the first three examples,

because the sensor/transmitter failures occur in a

continuously running system and they initiate quick

repair action. A PRA analyst would distinguish among

the examples based on their safety consequences. The

present discussion, however, adopts the viewpoint of

probability modeling, in which the important fact is not

the consequence of the events, but the way that they

occur randomly in time. Reactor trip initiators are the

prototypical examples of such events, but are not the

only examples.

The exposure time is the length of time during which

the events could possibly occur. In Example 2.1, the

exposure time is reactor-critical-years, because a

reactor trip can only occur when the reactor is at power.

Because only one plant is considered, “critical years”

can be used as shorthand for “reactor-critical-years.”

In Example 2.2, the event of interest is LOSP during

shutdown, so the exposure time must be the number of

reactor-shutdown-years in the study period. In Example

2.3, reactor-calendar-years are used, primarily because

more detailed worldwide data could not be easily

obtained. The model therefore assumes that a crack in

large-diameter piping could occur with equal

probability during operation and during shutdown. The

model also does not consider differences between

plants, such as differences in the total length of largediameter piping at a plant. In Example 2.4, the

exposure time is the number of component-years,

because the components operate constantly.

The possible examples are endless. The events could

be unplanned demands for a safety system, forced

outage events, or many other kinds of events that

resemble initiating events.
Basic Event Probability Models 

2-4

The data given in the above examples are expressed in

the crudest summary terms: a count of events in a total

exposure time. This is sufficient for the simple model

of this section. Section 2.5 will consider more sophisticated models using the exact event times.

The data could also be broken down into smaller

pieces. For example, the initiating event data could be

summarized for each calendar year, with an event count

and an exposure time reported separately for each year

from 1987 through 1995. This additional information

allows one to look for trends or other patterns, as

discussed in later chapters.

2.2.2 Probability Model

The assumptions concerning the physical process are

given here, along with a description of the kind of data

that can be observed.

It is standard to assume that the event count has a

Poisson distribution. As listed in Section A.6.2, the

usual assumptions (following Thompson 1981) for a

Poisson process are:

1. The probability that an event will occur in any

specified short exposure time period is

approximately proportional to the length of the

time period. In other words, there is a rate 8 > 0,

such that for any interval with short exposure time

)t the probability of an occurrence in the interval

is approximately 8 × )t.

2. Exactly simultaneous events do not occur.

3. Occurrences of events in disjoint exposure time

periods are statistically independent.

In addition, it is worthwhile to spell out the kind of data

that can be observed.

• A random number of events occur in some

prespecified, fixed time period. As a minimum, the

total number of events and the corresponding time

period are observed.

Under the above assumptions, the number of

occurrences X in some fixed exposure time t is a

Poisson distributed random variable with mean : = 8t,

Pr( ) / ! Xx e x . (2.1) x = = − μ μ

The probability distribution function (p.d.f.) is

sometimes used to abbreviate this: f(x) = Pr(X = x).

(Throughout this handbook, upper case letters are used

for random variables and lower case letters are used for

particular numbers.)

The parameter 8 is a rate or frequency. To make

things more clear, the kind of event is often stated, that

is, “initiating event rate” in Example 2.1, “throughwall-crack occurrence frequency” in Example 2.3, and

so forth. Because the count of events during a fixed

period is a unitless quantity, the mean number of

occurrences : is also unitless. However, the rate 8

depends on the units for measuring time. In other

words, the units of 8 are per unit of time, such as 1/year

or 1/reactor-critical-hour.

This model is called a Poisson process. It is extremely

simple, because it is completely specified by the

exposure time, t, and the one unknown parameter, 8.

Assumption 1 implies that the rate 8 does not change

over time, neither with a monotonic trend, nor

cyclically, nor in any other way. Assumption 2 says

that exactly simultaneous events do not occur. The only

way that they could occur (other than by incredible

coincidence) is if some synchronizing mechanism exists

– a common cause. Therefore, the operational

interpretation of Assumption 2 is that common-cause

events do not occur. Assumption 3 says that the past

history does not affect the present. In particular,

occurrence of an event yesterday does not make the

probability of another event tomorrow either more or

less likely. This says that the events do not tend to

occur in clusters, but nor do they tend to be

systematically spaced and evenly separated.

As stated above, a common cause that synchronizes

events violates Assumption 2. However, some

common-cause mechanisms do not exactly synchronize

the events. Instead, the second event may occur very

soon after the first, as a slightly delayed result of the

common cause. In this case, Assumption 3 is violated,

because the occurrence of one event increases the

probability of a second event soon after. One way or

the other, however, common-cause events violate the

assumptions of a Poisson process, by violating either

Assumption 2 or Assumption 3.

2.2.3 Data Needed to Validate the Model

and Estimate 8

Suppose that the Poisson model holds. Then any

reasonable estimator of 8 needs only two pieces of

information: the total exposure time, t, in the data

period, and the number of events, x, that occurred then.
Basic Event Probability Models

2-5

However, more information is needed to investigate

whether the Poisson model is valid. For example, the

data might cover a number of years or a number of

plants, and 8 might not be constant over time or the

same at all plants. These possibilities are not allowed

by the listed model assumptions. To study whether they

occur, the times and locations of the initiating events

should be recorded, or at least the data should be

partitioned into subsets, for example corresponding to

plants or years. Then the event count and exposure

time, xi

 and ti, should be given for each subset.

2.2.4 Case Studies: Validity of Model

Assumptions in Examples

Let us examine the reasonableness of the Poisson model

assumptions for Examples 2.1 through 2.4. Chapter 6

will address this issue by performing data analysis.

Here we will merely cite the results of published studies

and use critical thinking.

Example 2.1 Initiating Events

An initiating event is an event with the reactor critical,

causing an unplanned reactor trip. Assume that any

time interval starts on some date at some time and

ends on some date at some time, and that the length

of the interval, )t, is the number of critical years

contained between the start and stop of the time

interval. For example, if the time period is two 24-

hour days and the reactor was critical for half of that

time, then )t = 1/365 critical years.

Assumption 1 is violated in two ways. First, in the

industry as a whole, and presumably in individual

plants, the probability of an initiating event in an

interval of length )t (such as one critical day) has not

been constant. Instead, the probability dropped

substantially from 1987 to 1995. Equivalently, the

event rate, 8, dropped from 1987 to 1995. This

violation can be eliminated by considering only a

short time period for the study, such as one calendar

year instead of nine years. If, however, the whole

nine-year period is of interest, a more complicated

model must be used, such as one of the trend

models described in Chapter 7.

A second violation of Assumption 1 arises because

this particular plant was new at the start of the study

period, with initial criticality on January 3, 1987, and

commercial start on May 2, 1987. Many new plants

seem to experience a learning period for initiating

events, and this plant had 15 of its 34 initiating

events during the first six months of 1987. After that

initial period with a high event rate, the event rate

dropped sharply. This violation of Assumption 1 can

be resolved by eliminating data before the plant

reached a certain age. That is, not counting either

the operating time or the initiating events from the

plant until it has reached a certain age — excluding

that portion of the plant’s history from the universe

being studied.

Assumption 2 says that exactly simultaneous

initiating events do not occur. This is reasonable for

events at a single plant.

Assumption 3 says that the probability of an initiating

event in one time period does not depend on the

presence or absence of an initiating event in any

earlier time period. This assumption may be

challenged if the plant personnel learn from the first

event, thus reducing the probability of a second

event. This kind of dependence of one event on

another is not allowed by Assumption 3. Suppose,

however, that the learning is modeled as a general

kind of learning, so that the event rate decreases

over time but not as a clear result of any particular

event(s). This may justify using a Poisson model

with a trend in the event rate, as considered in detail

in Chapter 7.

There is a length of time when the reactor is down

after a reactor trip when an initiating event cannot

possibly occur. This does not violate Assumption 3

because during that time the plant has dropped out

of the study. Its shutdown hours are not counted in

the exposure time. Only when the reactor comes up

again does it begin contributing hours of exposure

time and possible initiating events.

Example 2.2 Shutdown LOSP

Just as with the previous example, consider the three

assumptions of the Poisson model. In this case,

because data come from the entire industry, 8 is

interpreted as the average rate for the entire industry.

First consider Assumption 1. The report that

analyzed this data (Atwood et al. 1998) found no

evidence of a trend in the time period 1980 through

1996. It did find evidence of differences between

plants, however. These differences can affect the

industry average, because plants enter the study

when they start up and leave the study when they

are decommissioned. When a plant with an

especially high or low event rate enters or leaves the

study, this will affect the industry average. However,

the event rate at the worst plant differed from the

industry average by only a factor of about 3.4, and

the best plant differed from the average by less than

that. Many plants (116) were considered. Therefore,

the effect of a single plant’s startup or

decommissioning should be small. Therefore, it

appears that the overall industry event rate was

approximately constant, as required by Assumption

1.
Basic Event Probability Models 

2-6

Assumption 2 rules out exactly simultaneous events.

In this example, however, events at sister units at a

single site are somewhat dependent, because a

common cause can result in LOSP events that are

simultaneous or nearly simultaneous at both units.

Of the 80 events in the data, two pairs of events

occurred together at sister units, each pair from a

common cause. Thus, simultaneous events do

occur, but they are not frequent. This departure from

Assumption 2 is probably not large enough to be

serious.

Assumption 3 requires statistical independence of

the number of events in disjoint time intervals. As

with Example 2.1, there may be some learning,

although the lack of trend indicates that any learning

is minimal.

In summary, the assumptions for the Poisson model

seem to be approximately satisfied.

Example 2.3 Through-Wall Leaks

This differs from the other examples in that the

number of events is very small. Any departures from

the Poisson assumptions cannot be seen in the data,

because so few events have occurred. With no

theoretical reason to postulate a trend or other

nonconstancy, or a high rate of multiple events, or

dependence between events, we accept the Poisson

assumptions. The assumptions may not be perfectly

true, and a different model may be more accurate,

but the Poisson model is simple, and good enough

for analyzing such a sparse data set.

Example 2.4 Temperature Sensor/Transmitters

A report by Eide et al. (1999a) divides the total study

time for instrumentation failures into two halves, and

finds a difference between 8 in 1984-1989 and 8 in

1990-1995. The example here is for 1990-1995 only.

Within this time period the report does not see strong

evidence of a trend. That is, a small trend may be

present, but the time period is too short, and the

failures too few, for any trend to be clear. Further,

because the components are regularly maintained, it

is reasonable to assume that the failure rate, 8, is

roughly constant, as required by Assumption 1. 

Assumption 2 requires that common-cause failures

be negligible. However, the report states that 14 of

the 32 component failures occurred during four

common-cause events. Thus, Assumption 2 is

seriously violated.

Finally, Assumption 3 requires independence of the

number of events in disjoint time intervals. The

report does not address this issue, but independence

appears plausible.

In summary, the example violates Assumption 2, but

probably satisfies the other two assumptions. One

way to deal with the violation of Assumption 2 would

be to model the independent failures and the

common-cause failures separately, although Eide et

al. do not do this.

2.2.5 Discussion

2.2.5.1 More General Models

The model considered thus far is a homogeneous

Poisson process (HPP), which has a constant event

occurrence rate, 8. The number of events in time t is a

Poisson random variable with parameter : = 8t. A

generalization is a nonhomogeneous Poisson process

(NHPP), in which 8 is a function of t. Such a model is

useful for analyzing trends. Chapter 6 includes ways to

test the assumptions of a homogeneous Poisson process,

and Chapter 7 includes ways to analyze data where a

trend is present.

When data come from the industry, one may consider

the differences between plants. Ways to model such

differences are discussed in Chapter 8 of this handbook.

The present chapter’s interest is restricted to 8 when no

such variation is present. Of course, if the data come

from only one plant, 8 refers to that plant and the issue

of differences typically does not arise.

Any mathematical model, such as the model for a

homogeneous Poisson process given here, is an

imperfect approximation of the true process that

generated the data. Data are used to validate or refute

the adequacy of the model. The data set may be sparse

— in the present context, this means that the data set

contains few events. In this case, two consequences

typically result: (a) it is difficult or impossible to see

evidence of departures from the model, and (b) the data

set contains too little information to allow realistic

estimation of the parameters of a more complicated

model. If, instead, the data set has many events,

departures from the model become visible, and typically

a more complicated model is appropriate. These

statements have been illustrated by the small and large

data sets given as examples.

2.2.5.2 Non-randomness of t

In the model considered here, the exposure time is

treated as fixed, and the number of events is treated as

random. This is a common type of data found in PRA

work. Sometimes, however, a fixed number of events

is specified by the data collector, and the corresponding

total time is random, as in the following two examples.
Basic Event Probability Models

2-7

One example occurs when equipment is tested until it

fails. That is, a predetermined number of items are

tested, say x items. Each item is run until it fails, and

the total running time of the items is random. The

second example occurs in a PRA context if the analyst

thinks that the event frequency has changed over time

and that only the recent history fully represents current

conditions. The analyst may then decide to consider

only the most recent events. If there are four recent

events, x is fixed at 4, and the corresponding time,

measured backwards from the present to the 4th event

in the past, is random.

These are examples of duration data with

exponentially distributed durations, discussed in

Section 2.5. The probability model is the Poisson

process presented above, but the data collection, and

resulting data analysis, are different. Because the time

t until the xth event can be called a waiting time, these

models are also sometimes called waiting time models.

2.3 Failure to Change State

This section considers two probability models, in

Subsections 2.3.2 and 2.3.3. First, however, example

data sets are given.

2.3.1 Examples

Here are four examples of failure to change state, three

with failure to start and one with failure to close.

Example 2.5 HPCI failures to start

At 23 BWRs in the 1987-1993 time period, the high

pressure coolant injection (HPCI) system had 59

unplanned attempts to start. The system failed to

start on 5 of these demands (Grant et al. 1999a).

The failures were typically erratic starts, which the

operator stabilized manually. These demands

occurred during 113.94 reactor-critical-years.

Example 2.6 EDG failures to start

Emergency diesel generators (EDGs) are

sometimes demanded because of unplanned loss of

power to a safety bus, and they are also tested

periodically, with one set of tests during each

operating cycle and another set of tests monthly. In

addition, a return-to-service test is normally

performed after maintenance of an EDG. At one

plant over an 18-month time period, the number of

such demands is counted, and the number of

failures to start is counted. 

Example 2.7 Steam binding in AFW

Between demands, steam binding can develop in

the AFW system, so that one or more pumps cannot

function when demanded. This is mentioned by

Wheeler et al. (1989), and by Nickolaus et al.

(1992).

Example 2.8 Failures of isolation valves

Nickolaus et al. (1992) review the causes of about

45 failures of air-operated and motor-operated

isolation valves. Some of the principal causes are

corrosion, instrument drift, and moisture in

instrument and control circuits. Other causes

include contamination and corrosion products in the

instrument air system, and debris in the system.

These are all conditions that can develop while the

valves are not being used.

2.3.2 Failure on Demand

All these examples involve a number of demands and a

number of failures, where the terms “demand” and

“failure” can be defined according to the purposes of

the study. Non-PRA contexts provide many other

examples of failures on demand. A simple example in

elementary probability or statistics courses is tossing a

coin n times, and counting the number of heads. Count

either a head or a tail as a “failure.” Just as in the PRA

examples, this example has a number of demands, with

a random number of the demands resulting in failures.

2.3.2.1 Probability Model

The standard model for such data assumes that the

number of failures has a binomial distribution. The

assumptions are listed in Appendix A.6.1. These

assumptions can be restated as two assumptions about

the physical process and one about the observable data:

1. On each demand, the outcome is a failure with

some probability p, and a success with probability

1 ! p. This probability p is the same for all

demands.

2. Occurrences of failures for different demands are

statistically independent; that is, the probability of

a failure on one demand is not affected by what

happens on other demands.

The following kind of data can be observed:
Basic Event Probability Models 

2-8

• A random number of failures occur during some

fixed, prespecified number of demands. As a

minimum, the total number of failures and number

of demands are observed.

Under these assumptions, the random number of

failures, X, in some fixed number of demands, n, has a

binomial (n, p) distribution.

(2.2)

Pr( ) ( )

,...,

X x , n

x

p p

x n

x nx == −

=

⎛

⎝

⎜ ⎞

⎠

⎟ − 1

0

where

 . n

x

n

xn x

⎛

⎝

⎜ ⎞

⎠

⎟ = −

!

!( )!

This distribution has two parameters, n and p, of which

only the second is unknown. (Although n may not

always be known exactly, it is treated as known in this

handbook. Lack of perfect knowledge of n, and other

uncertainties in the data, are discussed briefly in Section

6.1.3.2)

2.3.2.2 Data Needed to Validate the Model and

Estimate p

Suppose that the binomial model is appropriate. Then

any reasonable estimator of p needs only two pieces of

information: the number of demands, n, in the data

period, and the number of failures, x, that then

occurred.

However, more information is needed to investigate

whether the binomial model is valid. For example,

Assumption 1 assumes that p is the same on all

demands. If the data cover a number of years or a

number of systems or plants, p might not be constant

over time or the same at all systems or plants. To study

whether this is true, the times and locations of the

demands and failures should be recorded, or at least the

data should be partitioned into subsets, for example

corresponding to systems, plants, or years. Then the

failure and demand counts, xi

 and ni, should be given

for each subset.

2.3.2.3 Case Studies: Validity of Model

Assumptions in Examples

Let us examine Examples 2.5 through 2.8 to see if the

assumptions appear to be valid.

Example 2.5 HPCI Failures to Start

Assumption 1 says that the probability of failure on

demand is the same for every demand. If data are

collected over a long time period, this assumption

requires that the failure probability does not change.

Likewise, if the data are collected from various

plants, the assumption is that p is the same at all

plants.

In the HPCI example, the five failures do not reveal

any clear trend in time. However, one Licensee

Event Report (LER) mentions that a better-designed

switch had already been ordered before the HPCI

failure. This gives some evidence of a gradual

improvement in the HPCI system, which might be

visible with more data.

As for differences between plants, it happens that

three of the five failures occurred at a single plant.

Therefore, it might be wise to analyze that one plant

(three failures in nine demands) separately from the

rest of the industry (two failures in 50 demands). In

fact, Grant et al. (1995) did not analyze the data that

way, because they considered two types of failure to

start, and they also considered additional data from

full system tests performed once per operating cycle.

However, the high failure probability for the one plant

was recognized in the published analysis.

Assumption 2 says that the outcome of one demand

does not influence the outcomes of later demands.

Presumably, events at one plant have little effect on

events at a different plant. However, the experience

of one failure might cause a change in procedures or

design that reduces the failure probability on later

demands at the same plant. One of the five LERs

mentions a permanent corrective action as a result of

the HPCI failure, a change of piping to allow faster

throttling. This shows some evidence of dependence

of later outcomes on an earlier outcome at that plant.

Example 2.6 EDG Failures to Start

Assumption 1 says that every demand has the same

probability, p, of failure. This is certainly not true for

return-to-service tests, because such tests are

guaranteed to result in success. If the EDG does not

start on the test, maintenance is resumed and the

test is regarded as a part of the maintenance, not as

a return-to-service test. Therefore, any return-toservice tests should not be used with the rest of the

data.

As for the other demands, one must decide whether

the unplanned demands, operating-cycle tests, and

monthly tests are similar enough to have the same

value of p. Can plant personnel warm up or

otherwise prime the diesel before the test? Can an
Basic Event Probability Models

2-9

operator stop the test if the EDG is clearly having

trouble, and then not consider the event as a test?

If so, the different types of demands do not have the

same p, and they should not be analyzed as one

data set. For PRA purposes, one is normally most

interested in the failure probability on an actual unplanned demand. To estimate this, one should use

only data from unplanned demands and from tests

that closely mimic unplanned demands.

If the EDGs in the data set differ in some way, such

as having different manufacturers, this may also lead

to different values of p on different demands.

Analyzing the data while ignoring differences

between the individual EDGs will allow us to estimate

the average p, corresponding to failure to start for a

random EDG. However, this average p is not the

same as the p for a particular EDG.

Assumption 2 says that the outcome on one demand

does not affect the probability of failure on a different

demand. When the plant is very new there may be

some learning from individual failures, but when the

plant is mature, failure or success on one demand

should not change the chances of failure or success

on later demands. The only way for such

dependence to arise is if the first failure results from

a common cause. If the plant is mature and

common-cause failures are rare, then Assumption 2

is approximately satisfied.

Example 2.7 Steam binding in AFW

Assumption 1 says that every demand corresponds

to the same probability of failure. If the steam comes

from backflow through a check valve, it will build up,

and become more of a problem when the AFW

system has been unattended longer. Technically,

this is a violation of Assumption 1. However,

ignoring the differences between demands results in

estimating p for an average demand, and this may be

adequate for many purposes.

Assumption 2 says that the AFW pumps fail

independently of each other. However, steambinding of the AFW system was a recognized

common-cause mechanism in the 1970s and 1980s.

This means that Assumption 2 may be plausible if

interest is in the performance of a single AFW pump,

but not if interest is in an interconnected set of

pumps.

Section D-1 of Poloski et al. (1998) says that steam

binding has not been seen in 1987-1995 AFW

experience. Therefore, Example 2.7 is probably no

longer relevant, although it received great attention

at one time.

Example 2.8 Failures of isolation valves

The causes of valve failures postulated in this

example are degradations, so the probability of

failure increases over time, violating Assumption 1.

If failures from such causes are rare, then the

increase in failure probability may not be a problem.

In general, ignoring the differences results in

estimating an average p, averaged over components

that have been allowed to degrade for different

amounts of time. This may be acceptable.

As in Example 2.7, some of the mechanisms for

valve failure are common causes, violating the independence required by Assumption 2. The

seriousness of the violation depends on how many

multiple failures occur.

2.3.2.4 Discussion

2.3.2.4.1 More General Models

The model considered above has a constant failure

probability, p. A generalization would let p be a

function of time. Such a model is useful for analyzing

trends. Chapter 6 includes ways to test the assumptions

of the model assumed above, and Chapter 7 includes

ways to analyze data where a trend is present.

When data come from the industry, one might consider

the differences between plants, just as for initiating

events. Ways to model such differences are discussed

in Chapter 8. The present section’s interest is restricted

to p for the industry as a whole, the average of all the

plants. Of course, if the data come from only one plant,

p refers to that plant and the issue of differences

typically does not arise.

Any mathematical model is an imperfect approximation

of the true process that generated the data. When the

data set is sparse (few demands, or few or no failures,

or few or no successes), (a) it is difficult or impossible

to see evidence of departures from the model, and (b)

the data set is too small to allow realistic estimation of

the parameters of a more complicated model. When the

data set has many events, departures from the model

become visible, and a more complicated model may be

appropriate.

2.3.2.4.2 Non-randomness of n

One could argue that the numbers of demands in the

examples are not really fixed in advance. That is, no

one decided in advance to look at the outcomes of 59

unplanned HPCI demands. Instead, Grant et al.

decided to look at seven years of data from 23 plants,
Basic Event Probability Models 

2-10

and they observed that 59 demands had taken place.

The response to this argument is that we are actually

conditioning on the number of demands, that is, dealing

with conditional probabilities assuming that 59

demands take place. Conditioning on the number of

demands enables us to focus on the quantity of interest,

p. Treating both the number of failures and the number

of demands as random is needlessly complicated, and

yields essentially the same conclusions about p as do

the simpler methods in this handbook.

In the model considered here, the number of demands

is treated as fixed, and the number of failures is treated

as random. Sometimes, however, the number of

failures is specified in advance and the corresponding

number of demands is random. For example, the

analyst may believe that p has been changing, and that

only the most recent history is relevant. In this case, the

analyst might decide to consider only the most recent

failures and to treat the corresponding number of

demands as random. For example, if only the four most

recent failures are included, one would count

backwards from the present until x = 4 failures were

seen in the plant records, and record the corresponding

number of demands, n, regarded as an observation of a

random variable. This is a waiting time model, with n

equal to the waiting time until the 4th failure. Bayesian

analysis of such data is discussed briefly in Section

6.3.2.6.

2.3.3 Standby Failure

As stated in the introduction to this chapter, failure to

change state can be modeled in two ways. One way

was given in Section 2.3.2. The second way is given

here, in which the system (typically a component) is

assumed to transform to the failed state while the

system is in standby. This transition occurs at a random

time with a constant transition rate. The latent failed

condition ensures that the system will fail when it is

next demanded, but the condition is not discovered until

the next inspection, test, or actual demand.

2.3.3.1 Probability Model

The underlying assumption is that the transition to the

failed condition occurs randomly in time. Two settings

must be distinguished:

1. the data, the operational experiences in the past

that allow us to estimate 8, and

2. the application to PRA, in which the estimate of 8

is used to estimate the probability that a component

will fail when demanded.

These two settings are discussed in the next two

subsections.

2.3.3.1.1 Probability Model for the Data

It is customary to consider only the simplest model. 

1. Assuming that the system is operable at time t, the

probability that the system will fail during a short

time period from t to t + )t is approximately

proportional to the length of the exposure period,

)t. The probability does not depend on the

starting time of the period, t, or on anything else.

2. Failures of distinct systems, or of one system

during distinct standby periods, are independent of

each other.

The kind of observable data is spelled out here. It is

obvious, but is written down here for later comparison

with the data for similar models.

• At times unrelated to the state of the system, the

condition of each system (failed or not) can be

observed. As a minimum, the total number of

failures and the corresponding total standby time

are observed.

The times mentioned here can be scheduled tests or

unplanned demands.

Assumption 1 is essentially the same as for a Poisson

process in Section 2.2.2. It implies that there is a

proportionality constant, 8, satisfying

8)t . Pr(t < T # t + )t * T > t),

where T is the random time when the system becomes

failed. Then the probability that the system is failed

when observed at time t is

Pr(system is in failed state at time t) = 1 ! e!8t

 . (2.3)

This follows from Equation 2.6, given in Section 2.5 for

the exponential distribution. The parameter 8 is called

the standby failure rate. It is so named because the

failed condition develops while the system is in

standby, waiting for the next demand.

2.3.3.1.2 Application of the Model to PRA

The model is used to evaluate the probability of failure

on an unplanned demand. For this, one assumes that

there are periodic tests and the unplanned demand

occurs at a random time within the testing cycle. Then

the probability of failure on demand is approximated by
Basic Event Probability Models

2-11

p . 8ttest/2 , (2.4)

where 8 is the standby failure rate and ttest is the time

interval between tests.

A more accurate expression is the average of terms

from Equation 2.3, averaging over all the possible

demand times in the test interval:

( )

( )

p te ds

e t

test

s t

t

test

test

test

= −

=− −

−

−

∫ 1

1

1 1

0

λ

λ /( ) λ

This equation is approximated by Equation 2.4, as can

be verified by use of the second-order Taylor

expansion:

exp(!8t) . 1 + (!8t) + (!8t)

2

/2! . 

When more than one system is considered, the formulas

become more complicated. For example, suppose that

two systems (such as two pumps) are tested periodically

and at essentially the same time. Suppose that we are

interested in the event that both fail on an unplanned

demand. This is:

Pr(both fail)

(2.5)

= −

≈

−

∫ 1

1

3

2

0

2

t

e ds

t

test

s t

test

test ( )

( )/

λ

λ

When more systems are involved, or when testing is

staggered, the same ideas can be applied.

2.3.3.2 Data Needed to Validate the Model and

Estimate 8

Suppose that the standby failure rate model holds. If

the standby times are all similar, then an estimator of 8

needs only two pieces of information: the number of

failures, x, in the data period, and the corresponding

total standby time, t. If, instead, the standby times vary

substantially, then the total standby times should be

recorded separately for the failures and the successes,

as explained in Section 6.4.

To validate the model, the data could be partitioned.

As with initiating events, if the data come from various

years or plants, the data could be partitioned by year

and/or by plant, and the above information should be

given for each subset.

2.3.3.3 Case Studies: Validity of Model

Assumptions in Examples

Let us now examine the applicability of the model

assumptions in the examples given above. Much of the

discussion in Section 2.3.2.3 applies here as well. In

particular, when Section 2.3.2.3 sees a violation of an

assumption and suggests a remedy, an analogous

violation is probably present here, with an analogous

remedy.

Example 2.5 HPCI Failures to Start

Assumption 1 says that the probability of becoming

failed in a short time period is proportional to the

length of the time period, and on nothing else. As

discussed in Section 2.3.2.3, there is no clear

evidence of a trend in time. It may be, however, that

the probability of failure is higher at one plant than at

the other plants. If true, this would violate Assumption 1, and suggests that the outlying plant be

analyzed separately from the others.

Assumption 2 says that failures in distinct time

periods and locations are independent of each other.

As discussed in Section 2.3.2.3, there may be a very

small amount of learning, causing fewer failures later

in the history.

Example 2.6 EDG Failures to Start

Assumption 1 says that the probability of becoming

failed in a short time period is proportional to the

length of the time period, and on nothing else.

Section 2.3.2.3 discusses different types of tests of

EDGs. That discussion is applicable here as well. If

an EDG fails on one type of test more readily than on

another type of test, Assumption 1 is violated.

Another interpretation of this situation is that the

bulleted assumption on the data is false: it is not true

that a failed condition is always discovered on a test.

Some tests discover only major failed conditions

while other, more demanding tests discover less

obvious failed conditions. Just as mentioned in

Section 2.3.2.3, if the primary interest is the

probability of failure on an unplanned demand then

one should use only data from unplanned demands

and from tests that closely mimic unplanned

demands. 

Assumption 2 says that failures in distinct time

periods and locations are independent of each other.

As discussed in Section 2.3.2.3, this is probably true

if the plant is mature and if common-cause failures

are rare.
Basic Event Probability Models 

2-12

Example 2.7 Steam Binding in AFW

Assumption 1 says that the failed-condition event is

as likely to hit the system in one time interval as in

another of the same length. As discussed in Section

2.3.2.3, steam binding can result from a gradual

buildup, and become more of a problem when the

AFW system has been unattended longer. In this

case, Assumption 1 is violated. Ignoring this fact is

equivalent to treating the average of AFW conditions.

As discussed in Section 2.3.2.3, steam binding is a

common-cause mechanism. Therefore Assumption

2, independence of distinct AFW pumps, is violated.

Example 2.8 Failures of Isolation Valves

Just as discussed in Section 2.3.2.3, the causes

listed for Example 2.3 are degradations, violating

Assumption 1. However, it may be acceptable to

ignore the changes over time, and estimation of an

average parameter 8. Also, as discussed in Section

2.3.2.3, some of the mechanisms for valve failure are

common causes, violating the independence

required by Assumption 2. The seriousness of the

violation depends on how many multiple failures

occur. 

2.3.4 Comparison of the Two Models for

Failure to Change State

Two models have been presented for failure to change

state, the failure-on-demand model and the standbyfailure model. Several aspects of the models are

compared here.

2.3.4.1 Ease of Estimation

One great appeal of the standby-failure model is that the

analyst does not need knowledge of the number of

demands. Standby time is normally much easier to

obtain than a count of demands.

2.3.4.2 Use in PRA Cut Sets

The two models differ in their application to cut sets in

a PRA model. Consider failure of two redundant

components, each having the same probability of

failure. When the failure-on-demand model is used, we

have

Pr(both fail) = p2

 = [Pr(one fails)]2.

On the other hand, when the standby-failure model is

used and the two components are tested periodically at

the same time, with time t between tests, Equations 2.4

and 2.5 show that

Pr(one fails) . 8ttest/2

Pr(both fail) . (8ttest)

2

/3

so that

Pr(both fail) ... [Pr(one fails)]2

.

This fact is often ignored.

2.3.4.3 Estimates Obtained

The two models can produce different estimates of

basic event probabilities. For example, suppose that an

EDG is tested monthly by starting it. In 100 monthly

tests, 2 failures have been seen. A simple estimate of p,

the probability of failure on demand, is 2/100 = 0.02.

A simple estimate of 8, the standby failure rate, is

0.02/month. Now suppose that a basic event in a PRA

is that the EDG fails to start, when demanded at a

random time. Based on the estimate of p, the estimated

probability of the basic event is

Pr(EDG fails to start) = p . 0.02 .

Based on the estimate of 8 and Equation 2.4, the

estimated probability of the basic event is

Pr(EDG fails to start) . 8t/2

. (0.02/month)×(1 month)/2 = 0.01 .

The two models give estimates that differ by a factor

of two, with the failure-on-demand model being more

pessimistic than the standby-failure model. The reason

is simple: All, or virtually all, of the failures and

demands in the data occur at the end of test intervals.

However, unplanned demands might occur at any time

between tests. The standby-failure model says that demands soon after a successful test have smaller

probability of failure. The failure-on-demand model

says that all demands have the same probability of

failure.

The differences can be more extreme. For example,

suppose that two EDGs are tested monthly, and tested

at essentially the same time rather than in a staggered

way. According to the failure-on-demand model, the

probability that both EDGs fail to start is p2

, which is

estimated by (0.02)2

. On the other hand, according to

the standby-failure model, Equation (2.5) shows that the

same probability is approximately (8ttest)

2

/3, which is

estimated by (0.02)2

/3. The two models give estimates

that differ by a factor of three. More extreme examples

can be constructed.
Basic Event Probability Models

2-13

It might be mentioned that these numerical differences

between estimates disappear if only unplanned demands

are used in the data. However, unplanned demands are

rare, and so most analysts prefer to use test data if

possible.

2.3.4.4 A Model Involving Both Terms

The model described next postulates two reasons for the

observed randomness of failures.

One reason for the randomness of failures is that

demands are not all equally stressful. When a demand

occurs that is unusually harsh, the system will fail.

From the viewpoint of an outside observer, it appears

that failures just occur randomly with some probability

p, but the underlying cause is the variability in the

severity of the demands.

The other reason for randomness of the failures is that

the unattended system degrades, and becomes

inoperable at unpredictable times. This is simplified in

the standby-failure model by supposing that the system

changes suddenly from perfectly operable to completely

failed, with these transitions occurring at random times.

This leads to the standby-failure model, with failuretransition rate 8, and with probability of failure 8t at

time t after the last system restoration.

If just one of the two mechanisms described above is

considered, we are led to either the failure-on-demand

model or the standby-failure model. It is possible,

however, to construct a model that involves both terms,

corresponding to the two kinds of variation. In this

two-parameter model, the probability of failure is p + 8t

at time t after the last system restoration. (For example,

see Section 5.2.10 of Samanta et al. 1994.)

Lofgren and Thaggard (1992) state “it is virtually

impossible to directly determine from work

maintenance record descriptions whether the

component has failed from standby or demand stress

causes.” However, they look for patterns in data from

EDGs and motor-operated valves (MOVs) at a small

number of plants that use different test intervals. Their

data suggest that the standby-failure-rate model is most

appropriate for MOV failures, and the two-parameter

model is best for EDGs. 

In a similar spirit, the T-Book (TUD Office and Pörn

Consulting, 2000) uses the two-parameter model for

many components. The T-Book does not attempt to

identify which mechanism applies to which failures, but

instead estimates the two parameters from overall

patterns in the data. Some of the resulting estimates

have large uncertainties; for example, at a typical plant

the estimate of p for EDG failure to start has an error

factor of about 13. For components that cannot be

analyzed in this way, the T-Book uses the standbyfailure model. For details, see Pörn (1990).

2.3.4.5 Choosing a Model

No consensus exists among PRA workers as to which

model is most advantageous. In particular, the typical

mechanisms of failure are not understood well enough

to justify a theoretical basis for a model. Most current

work uses one of the two simple models given here:

failure on demand or standby failure. Therefore, this

handbook presents only these two models. The user

may choose between them.

2.4 Failure to Run during Mission

Aspects of this type of failure closely resemble the

initiating events of Section 2.2. One important

difference is in the kind of data normally present. The

difference is summarized here.

Example 2.4 of Section 2.2 is an example of continuously running components (temperature sensor/transmitters) that occasionally failed to run. When a

component failed, it was repaired or replaced in a

relatively short time, and resumed operation. That is,

the component was repairable. The present section

considers components or systems that do not run

continuously. Instead, they are occasionally demanded

to start, and then to run for some mission time. If they

fail during the mission, they are nonrepairable, that is,

they cannot be repaired or replaced quickly. Two

points deserve clarification:

• Some failures may be recoverable. They would

not be modeled as failures in the sense of causing

mission failure. Unrecoverable failures cause

mission failure, however.

• Given enough time, almost any system can be

repaired. During a mission, however, time is not

available. Because the component or system

cannot be repaired within the time constraints, it is

called “nonrepairable.”

As stated earlier, the word system is used in this

handbook for any piece of hardware for which data are

taken. In particular, components and trains are kinds of

systems.
Basic Event Probability Models 

2-14

2.4.1 Examples

Here are two examples of failures to run during

missions.

Example 2.9 EDG failures to run

Grant et al. (1999b) report that in 844 demands of

30 minutes or more for EDGs to run, there were

approximately 11 unrecovered failures to run in

the first 30 minutes. The count is approximate

because a few failure times were not given and

had to be inferred.

Example 2.10 AFW turbine train failures to run

Poloski et al. (1998) report that in 583 unplanned

demands of AFW system turbine trains, the train

failed to run 2 times, and the total running time

was 371 train-hours. The information is taken

from LERs, only 17% of which report running

times for the train. The total running time of 371

hours is an extrapolation from the LERs with

reported run times.

These examples are typical, in that hardly any of the

demands to run resulted in a failure. Therefore, for

most demands the time when failure would eventually

have occurred is unknown.

2.4.2 Probability Model

In principle, the times to failure are durations.

Section 2.5 deals with duration data, in the context of

recovery times. That section mentions various possible

distributions of time to failure, of which the simplest is

the exponential distribution.

Data for this section differ from data of Section 2.5,

however, because nearly all of the observed times in

this section are truncated before failure. This is

illustrated by the above examples. Therefore, the full

distribution of the time to failure cannot be observed.

In Example 2.9, no information is given about the

distribution of failures times after the first 30 minutes.

In Example 2.10, the average run time was only 38

minutes, and most AFW missions lasted for less than

one hour. In such cases the exponential distribution,

restricted to the observed time period, is a simple,

reasonable approximation of the observable portion of

the distribution.

Two assumptions are made concerning the physical

process:

1. Assuming that no failure has occurred by time t,

the probability that a failure will occur in a short

time period t to t + )t is approximately proportional to the length of the exposure period, )t. The

probability does not depend on the starting time of

the period, t, or on anything else.

2. Failures of distinct systems, or of one system

during distinct missions, are independent of each

other.

The kind of observable data is as follows:

 

• For each observed mission, the run time is

observable. Also, it is known whether the run

terminated in failure or in successful completion of

the mission. As a minimum, the total run time and

the number of failures to run are observed.

Assumption 1 implies that the time to failure is

exponentially distributed with parameter 8. The interpretation of 8 is that if the system is running, the

probability of failure in the next short interval of length

)t is approximately 8)t. That is

8)t . Pr(t < T # t + )t * T > t),

where T is the random time until failure. When defined

this way, 8 is sometimes called the failure rate, or rate

of failure to run. Many authors use the term hazard

rate, denoted by h, and discussed in Appendix A.4.4.

Note, the definition of 8 is different for repairable

systems (Section 2.2) and nonrepairable systems (the

present section), even though it is represented by the

same Greek letter and is called “failure rate” in both

cases. See Thompson (1981) for a reasonably clear

discussion of the subtle differences, and the glossary of

this handbook for a summary of the definitions. The

topic is discussed further in Appendix A.4.4.

It is instructive to compare the models for failure to run

and standby failure. The physical process is essentially

identical, but the observable data differs in the two

models. That is, Assumptions 1 and 2 in the two

sections agree except for small differences of wording.

However, the time of failure to run is observable,

whereas the time of transition to a standby failure is

never known.

It may also be somewhat instructive to compare the

Assumptions 1 and 2 here with the Assumptions 1-3 of

the Poisson process in Section 2.2.2. For the standby-
Basic Event Probability Models

2-15

failure model and the failure-to-run model, Assumptions 1 and 2 do not explicitly include an assumption

ruling out simultaneous failures. The reason is that

simultaneous failures are ruled out by the other two

assumptions: it is not meaningful for a system to fail

twice simultaneously; and distinct systems are assumed

to fail independently of each other, and therefore not

exactly simultaneously.

2.4.3 Data Needed to Validate the Model

and Estimate 8

Suppose that the time to failure has an exponential

distribution. Then, any reasonable estimator of 8 needs

only two pieces of information: the total running time,

t, in the data period, and the number of failures to run,

x, that occurred then.

However, more information is needed to investigate

whether the exponential distribution is valid. Assumption 1 says that 8 is constant during the mission. To

investigate this, the analyst should know the failure

times, that is, how long the failed pumps ran before

failing. The analyst should also know the mission

times, that is, how long the system ran when it did not

fail; often, however, this information is not recorded

and can only be estimated or approximated.

Implicit in Assumption 1 is that 8 is the same over all

the years of data, at all the plants where the data were

collected. To investigate this, the data should be

divided into subsets, corresponding to the different

plants and years. Then the failure count and running

time, xi

 and ti, should be given for each subset. This is

the exact analogue of what was said in Section 2.2.3 for

initiating events.

2.4.4 Case Studies: Validity of Model

Assumptions in Examples

Consider now whether the assumptions of the model are

plausible for the two examples.

Example 2.9 EDG Failures to Run

Assumption 1 says that a running EDG is as likely to

fail in one short time interval as in any other time

interval of the same length. That is, the EDG does

not experience burn-in or wear-out failures. The

reference report (Grant et al. 1999b) says that this is

not true over a 24-hr mission. Indeed, that report

divides the EDG mission into three time periods (first

half hour, from one-half hour to 14 hours, and from

14 to 24 hours) to account for different failure rates

during different time periods. Within the first half

hour, however, the data do not give reason for

believing that any short time interval is more likely to

have a failure than any other time interval.

Therefore, Assumption 1 can be accepted.

Assumption 2 is violated by common-cause failures.

It is also violated if a failure’s root cause is incorrectly

diagnosed, and persists on the next demand. If

these two conditions are rare the assumption may be

an adequate approximation. More subtle dependencies are difficult to detect from data.

Example 2.10 AFW Turbine Train Failures to Run

Assumption 1 says that a running turbine train is as

likely to fail in one short time interval as in any other

time interval of the same length. The data are too

sparse – only 2 observed failures – to confirm or

refute this assumption. The data are also too sparse

to confirm or refute Assumption 2, although failures

in separate plants are virtually certain to be

independent. In such a situation, it is common to

accept the simple model as adequate. A more

complicated model is justified only when a larger

data set is available.

2.4.5 Discussion

The exponential time to failure can also be derived as

the time to first failure in a Poisson process of

Section 2.2. This is possible because the time to first

failure and the times between subsequent failures are all

exponentially distributed when the failures follow a

Poisson process. The present context is simpler,

however, because the process ends after the first event,

failure to run. The Poisson-process assumptions about

hypothetical additional failures are irrelevant.

2.5 Recovery Times and Other

Random Duration Times

This section is about modeling of time data. Often, a

measurement of interest is a random duration time, such

as the time required to return a failed system to service

or the lifetime of a piece of hardware. The distinction

between random duration times here and events in time

in Sections 2.2 and 2.4 is that here the individual times

are measured on a continuous scale with units such as

minutes or hours, while the earlier data sets involve

discrete counts of the number of events occurring in a

total length of time.
Basic Event Probability Models 

2-16

2.5.1 Examples

Here are some examples involving random duration

times. They are only summarized here. Actual examples, with lists of durations times, will be analyzed in

Chapter 6.

Example 2.11 Recovery times from loss of

offsite power

A plant occasionally loses offsite power. When

this happens, the plant reports the time until

power is restored. Atwood et al. (1998) present

such durations for LOSP events in 1980-1996.

Example 2.12 Repair times for turbine-driven

pumps

A turbine-driven pump must occasionally be taken

out of service for unplanned maintenance. The

duration of time out of service for maintenance

may be extractable from maintenance records.

Example 2.13 Time to failure of a component

A typical power plant will have many individual

components such as compressors. When a

component is put into service, it operates

intermittently until it fails to perform its required

function for some reason. Høyland and Rausand

(1994) give an example of such data.

Example 2.14 Times to suppress fires

When a fire occurs in a nuclear power plant, the

time until the fire is suppressed is of interest.

Nowlen et al. (2002) report on analysis of such

suppression times. One difficulty is that the time

of fire onset often is not exactly known.

Example 2.15 Gradual degradation until failure

Examples 2.7 (steam binding) and 2.8 (failure of

isolation valves) involve gradual degradation,

which builds up until the system is inoperable.

The time until the system is inoperable can be

modeled as a duration time.

The common element in these examples is a duration

time that varies in an unpredictable way. In Examples

2.11 and 2.12, the recovery time is composed of several

factors such as the time to diagnose, perform and test

repairs, and the time to complete documentation

required before returning the plant to normal operating

conditions. Example 2.13 is a failure-to-run example,

similar to those of Section 2.4. This example differs

from that of Section 2.4, however, because here it is

assumed that virtually all of the times to failure are

recorded. In Section 2.4, on the other hand, most of the

systems did not fail during the test period or operational

mission. The severe truncation of the data in Section

2.4 meant that only a simple model could be

considered. The more complete data here allows

analysis of a more complex model. Example 2.14 is

complicated by the lack of exact knowledge of the

duration time. Finally, Example 2.15 gives a realistic

conceptual way to model the gradual degradations

encountered in Section 2.3.1, although good data are

unobtainable.

All five examples involve a duration time that is

uncertain due to random factors. Consequently, the

duration times are modeled as continuous random

variables.

2.5.2 Duration-Time Models

The duration, T, is random, following some probability

distribution. Two assumptions are made about the

process:

1. Each duration is statistically independent of the

others, and

2. All the random durations come from the same

probability distribution.

The data description is simple:

• The individual durations are observable. As a bare

minimum, the number of durations and the total

duration time are observed.

Assumptions 1 and 2 can be summarized by saying that

the durations are independent and identically

distributed. Independence means that one duration

does not influence the probability of any other duration.

The assumption of identical distributions means that

each random duration is as likely as any other to be

long or short. If the durations are from distinct

systems, the systems are assumed to be identical and to

act independently. If the durations are in sequence, as

for a system that alternates being up and down, the

assumption implies that no learning or long-term aging

takes place, and that each repair restores the system to

a condition as good as new. Such a process is called a

renewal process.
Basic Event Probability Models

2-17

Figure 2.1 Probability density function (p.d.f.) and

cumulative distribution function (c.d.f.).

The assumptions do not require a particular distribution

for the time between events. The most important such

distributions in PRA applications are: 

• lognormal,

• exponential,

• Weibull, and

• gamma.

These distributions are summarized in Appendix A.7.

An important part of the data analysis consists of

deciding on the form (or several plausible forms) of the

distribution. This will be discussed in Chapter 6. For

now, we simply note that these and other distributions

are possible.

There are different ways to specify a probability

distribution, and the next material summarizes some of

the concepts: their definitions, how to interpret them,

and how they are related to each other. The dataanalysis techniques of Chapter 6 will use these ways of

characterizing distributions. The usual convention is to

denote the random variables using capital letters, T, and

observed times as lower case, t. The letter T is used,

rather than some other letter such as X, because the

random quantities are times. As seen from the

examples, the durations may be times to repair, times to

failure, or other times. However, the concepts and

formulas are valid for any application.

The cumulative distribution function (c.d.f.) of a realvalued random variable T is defined as

F(t) = Pr(T # t)

for all real numbers t. The name is sometimes

abbreviated to distribution function. The c.d.f. is the

probability that the random variable T will assume a

value that is less than or equal to t. The c.d.f. is a

monotonically increasing function of t, with the limiting

properties F(0) = 0 and F(+4) = 1. [For random

variables that, unlike durations, can take negative

values, the limiting properties are F(!4) = 0 and F(+4)

= 1. That general case has few applications in this

handbook.]

The distribution is commonly used to characterize the

lifetimes, or recovery times, or some other kind of

durations, of a whole population of systems. The

population might be a large set of identical systems that

are operating in similar applications and with durations

that vary due to random influences. F(t) is the fraction

of items that have durations t or less, in a hypothetical

infinite population. 

A related function, denoted by f(t), is called a

probability density function (p.d.f.) for a continuously

distributed positive-valued random variable T. It is

related to the c.d.f. by

f t and

d

dt

() () = F t

F t f u du . t () ( ) = ∫

0

The variable u is a dummy variable of integration, and

t is the upper limit of the integral. An example of a

p.d.f. and the associated c.d.f. are shown in Figure 2.1.

It follows that probabilities corresponding to

occurrences in a small interval of time are

approximately proportional to the p.d.f.,

Pr(t < T # t + )t) . f(t))t.

Therefore, the ordinate of a p.d.f. has units of

“probability density” and not probability (as for a

c.d.f.). Thus, a p.d.f. determines how to assign

probability over small intervals of time. Now consider

an arbitrary interval from a to b. In this case we have

Pr( ) ( ) a T b f t dt . a

b <≤ = ∫

The simplest distribution is the exponential distribution. It arises when Assumption 1 of Section 2.4.2 is

satisfied. (That assumption is phrased as if T is a time

until failure.) In that case, the probability distribution

is exponential, and determined by a single parameter, 8.

The p.d.f. and c.d.f. are given by 

f(t) = 8e!8t

F(t) = 1 ! e!8t . (2.6)
Basic Event Probability Models 

2-18

When deriving the distribution mathematically from

Assumption 1, it is necessary to assume that F(0) = 0,

that is, failures at time 0 have zero probability.

Although not stated explicitly, this assumption is

implicit in the context of failure to run, because any

failures at time 0 would be counted as failures to start,

not failures to run.

2.5.3 Data Needed to Estimate

Distribution of Durations and

Validate Model

In general, a sample of observed durations is needed to

estimate the distribution of duration times. These

durations must independent and identically distributed,

that is, they must be generated by a process satisfying

the two assumptions given at the beginning of Section

2.5.2.

The special case when the times are assumed to have an

exponential (8) distribution is simpler. Only the

number of durations and the total duration time are

needed to estimate 8,. However, the individual

durations are still needed to investigate whether the

distribution is exponential or of some other form.

Incidentally, when the distribution is assumed to be

exponential, the model given here differs from the

standby-failure model (Section 2.3.3.1.1) and from the

failure-to-run model (Section 2.4.2) only by the kind of

data that can be observed.

To validate whether the distribution is the same for all

the data, extra information should be recorded for each

duration, the relevant circumstances of each duration.

The circumstances of interest are those that might affect

the durations, such as time of the event, system

location, and system condition just before the event.

2.5.4 Case Studies: Validity of Model

Assumptions in the Examples

Examples 2.11 through 2.13 all appear to satisfy the

assumptions of Section 2.5.2. Example 2.14 also

does, except that the durations are not observed

exactly.

In each case, all the distributions come from some

distribution. Discovering the form of that distribution

is a task for the data analyst.

One might ask whether the durations are statistically

independent. For example, does a long repair time

for a turbine-driven pump add an extra benefit to the

pump, so that the next few repair times will be short?

One might also ask, for each example, whether the

durations all come from the same probability distribution. For example, if the data cover a period of

years, has there been any long-term learning, so that

recovery times or repair times tend to be shorter than

at the start of the data period? Are different

durations associated with different systems for the

turbine-driven pumps, with different causes of loss of

offsite power, or with different kinds of fires?

The above are questions that could be investigated

during the data analysis, if enough durations have

been observed.

Example 2.14 is complicated by lack of exact

measurements of the durations. Bounds can be

given, and the analysis must be based on these

upper and lower bounds rather than on exact times.

Example 2.15 is different because the durations are

not observable at all. It might be theoretically

interesting to model the time until the system is in a

failed condition as a duration, but there is no monitor

on the pump or valve that says, “At this time the

system just became inoperable.” Therefore, the

durations are not directly observable, not even in

principle. Therefore, the methods of this handbook

are not applicable to this example.

Fortunately, degradation mechanisms have become

minor contributors to risk. When a degradation

mechanism is recognized as important, the natural

response is not to collect data to better estimate the

rate of degradation. Instead, the natural response is

(a) to shorten the interval between preventive

maintenance activities, and so to identify and correct

incipient degradation, or (b) to modify the plant to

mitigate or eliminate the problem. Examples are the

apparent elimination of steam-binding in AFW

pumps, mentioned above, and of intergranular stress

corrosion cracking (IGSCC) in BWR piping (Poloski

et al. 1999a, Appendix J).

2.6 Unavailability

This section considers test-and-maintenance

unavailability, corresponding to intentional removal of

the equipment from service for testing and/or

maintenance. This section does not consider

unavailability resulting from the hardware being in an

unrecognized failed condition; that topic was treated in

Section 2.3.3.

The discussion here is presented in terms of trains,

although other hardware configurations, such as

individual components, could be considered equally

well. A standby train, such as the single train of the

HPCI system or a motor-driven train of the AFW

system, is normally available if it should be demanded,
Basic Event Probability Models

2-19

Figure 2.2 Uptime and downtime status for one system.

but sometimes it is out of service for planned or

unplanned maintenance. The event of a train being

unavailable is called an outage, and the length of time

when it is unavailable is called an outage time or outof-service time. In a data set, the exposure time is the

time (e.g. number of hours) when the train should have

been available. The unavailability is the long-term

ratio of outage time to exposure time – the fraction of

time that the system is out of service when it should be

available. More precisely, the planned-maintenance

unavailability is the fraction of time that the system is

out of service for planned testing and maintenance, and

the unplanned-maintenance unavailability is defined

similarly. In summary, outage times are random but the

unavailability is a parameter, an unknown constant,

denoted here by q. Subscripts such as “planned” and

“unplanned” can be attached to q for clarity if needed.

2.6.1 Example

Example 2.16 CVC unavailability for test and

maintenance

Train outages of various durations occurred

during 15 calendar months at a plant with two

trains in the chemical and volume control (CVC)

system. For each month, the outage durations

are given by Atwood and Engelhardt (2003).

A way to picture the status of a standby train or other

repairable system uses a state variable, defined as S(t)

= 1 if the system is up at time t, and S(t) = 0 if it is

down at time t. A particular system history is illustrated

in Figure 2.2, from Engelhardt (1996). This figure

shows when a particular system was operating (S = 1)

or shut down (S = 0). A nominally identical system

would have a somewhat different history for the same

period, or the same system would have a different

history over a different time period of the same length.

As stated above, the long-term fraction of time when

the system is down is called the system unavailability.

2.6.2 Probability Model

The assumed underlying model is an alternating

renewal process. At any point in time a system is in

one of two states: “up” or “down,” corresponding in our

application to being available or out of service.

Initially, the system is up, and it remains up for a

random time Y1; it then goes down, and stays down for

a random time Z1. Then it then goes up for a time Y2,

and then down for a time Z2, and so forth. The

assumptions needed for the data analysis methods in

Chapter 6 are the following:

1. The random variables Yi

 have one distribution that

is continuous with a finite mean, and so do the

random variables Zi

.

2. All the random variables are independent of each

other

The sum of the down times, EZi

, is the total outage time

in the data. The sum of all the times, GYi

 + EZi, is the

exposure time – the time when the system should be

available. Time when the system is not required to be

available is not counted in either the up time or the

down time.

Two kinds of data can be considered:

• Detailed data: the onset time and duration of each

individual outage are recorded, as well as the total

time when the train should have been available;

and

• Summary data. Data totals are given for

“reporting periods,” such as calendar months. For

each reporting period, the total outage time and

exposure time are recorded.

Section 6.7 describes how to analyze both types of data.

2.6.3 Data Needed to Validate the Model

and Estimate q

The unavailability, q, can be estimated from either kind

of data. Enough data should be collected so that any

periodic, lengthy, planned outages are appropriately

represented – neither over-represented nor underrepresented.

In addition, if summary data are used, the methods

given in Chapter 6 combine reporting periods into

larger subsets of the data, at the very least so that the

aggregated subsets do not contain outage times of zero.

Therefore, a large enough set of summary data is
Basic Event Probability Models 

2-20

needed so that it consists of at least two (as a bare

minimum) subsets of approximately equal exposure

time, with each subset containing nonzero outage time.

To validate the model, any information that might be

related to unavailability should be recorded. For

example, if a motor-driven pump has most of its

scheduled maintenance during the plant’s refueling

outages, and the pump’s availability during shutdown is

of interest, then the data should indicate which outages

and exposure times correspond to reactor shutdown.

Separate analyses will probably need to be performed

for the time when the reactor is up and when the reactor

is down, to keep Assumption 1 from being violated.

2.6.4 Case Study: Validity of Model

Assumptions in Example

The ideas here are applicable to virtually any system,

with Example 2.16 being just one example.

The trains may undergo periodic, infrequent, lengthy

testing and maintenance, and less lengthy testing and

maintenance at more frequent intervals. This

periodicity of planned maintenance means that

Assumption 2 cannot be exactly true. The lengthiest

outages tend to be evenly spaced, not random as

assumed. However, more realistic assumptions would

be very difficult to work with.

It seems plausible that this deterministic periodicity

should lead to conservative estimates. That is, analysis

methods that assume pure randomness will tend to

overestimate the variance, so that the resulting

uncertainty in q is overestimated. However, this

conjecture has not been carefully investigated, and the

15 months of data in Example 2.16, analyzed in Section

6.7, do not support the conjecture.

Assumption 1, on the other hand, is surely correct. The

distributions are continuous, and it is inconceivable that

the durations for an operating power plant would have

infinite means.
3-1

3. COMPONENT FAILURE AND BOUNDARY DEFINITIONS

3.1 Failure Definitions

While the terms “faults” and “failures” are casually

used interchangeably, in the context of fault tree

analysis these terms have more distinct meanings.

Thus, for data analysis, it is necessary for one to

understand the distinctions. Generally speaking, all

failures are faults, but not all faults are failures. To put

it another way, failures comprise a subset of the larger

set of faults. For probabilistic risk assessment (PRA)

purposes, failures are regarded as basic (and undesired)

events which:

• render a component, subsystem, or system

incapable of performing its intended function, 

• represents a basic fault tree input that is not

analyzed further, and 

• require numerical estimates if quantification is to

be performed. 

Faults, on the other hand, are higher order events

(representing the occurrence or existence of an

undesired state of a component or set of components)

which are analyzed further, and ultimately resolved into

their constituent failures (Breeding, Leahy, and Young

1985; ANS and IEEE 1983; and Vesely et al. 1981).

The failures modeled in PRA can have many causes or

mechanisms. For example, failure of a motor-operated

valve (MOV) to open on demand can occur due to

physical problems with the valve (stem failure, disc

separation, etc.), problems with the motor operator

(motor failure, control circuit failure, breaker failure,

etc.), or due to loss of motive or control power. In

addition, the MOV may be unavailable due to test or

maintenance on its constituent parts. As such, each

failure (i.e., basic event) is the sum of the contributions

from each piece-part included in the component

boundary. Thus, it is critical to define what the

component boundary is in order to get the right data.

3.2 Component Boundary

Definitions

In order to collect failure data for components, it is

necessary to define component boundaries by

specifying the scope of each item to be considered as a

single entity. The PRA model and the data collection

should be coordinated so that the boundaries of the

components are defined identically. For example, all

pieces of an MOV are typically considered to be part of

a single “component” when collecting reliability data

even though the valve consists of various piece parts

(e.g., electric motor, gearbox, limit switches, torque

switches, reversing contacts and coils, stem, disc, valve

body, etc.) that may be separately identified in the plant

maintenance records. PRAs typically do not model

failures of every switch, relay, or contact in a control

circuit of a pump because that type of detail is difficult

to obtain from the plant data. Instead, failures of these

components are typically included with actual failures

of the pump to establish a pump failure rate. 

If generic data sources are used, it becomes the

responsibility of the analyst to ensure that the

component boundary definitions used in the generic

data source are compatible with the boundary

definitions used by the PRA being performed. 

Some typical examples of component boundaries are

shown in Table 3.l. The boundaries of a component

should include all components specific to the

component. However, the component boundary should

not include piece-parts that are shared with other

components modeled in the PRA. For example, the

component boundary for emergency-actuated valves

commonly includes the valve control circuit. However,

the components needed to generate an actuation signal

that initiates multiple components modeled in the PRA

should not be included as part of that specific valve

boundary. Similarly, a diesel generator boundary will

typically include the fuel day tank but the fuel oil

transfer pumps are not included since they are required

for operation of all the plant’s diesel generators.

3.3 Failure Severity

The raw data for a specific component will contain

some events not relevant to the component failure

modes being analyzed. These events can be screened

from further analysis. Some of the events will be

component failures that should be included in the data

assessment. The type of component failures will

determine how they are classified and subsequently

used to generate the required component failure data.

Component malfunction events are commonly classified

into one of the following three failure severity

categories:
Component Failure and Boundary Definitions

3-2

Table 3.1 Examples of component boundaries.

Component Component Boundary

Diesel

Generators

The diesel generator boundary includes the generator body, generator actuator, lubrication

system (local), fuel system (local), cooling components (local), startup air system, exhaust

and combustion air system, individual diesel generator control system, circuit breaker for

supply to safeguard buses and their associated local control circuit (coil, auxiliary contacts,

wiring, and control circuit contacts) with the exception of all the contacts and relays which

interact with other electrical or control systems.

Motor Pumps The pump boundary includes the pump body, motor/actuator, lubrication system cooling

components of the pump seals, the voltage supply breaker, and its associated local control

circuit (coil, auxiliary contacts, wiring, and control circuit contacts).

Turbine-Driven

Pumps

The turbine-driven pump boundary includes the pump body, turbine/actuator, lubrication

system (including pump), extractions, turbopump seal, cooling components, and local turbine

control system (speed). 

Motor-Operated

Valves

The valve boundary inc1udes the valve body, motor/actuator, the voltage supply breaker and 

its associated local open/close circuit (open/close switches, auxiliary and switch contacts, and

wiring and switch energization contacts). 

Air-Operated

Valves

The valve boundary includes the valve body, the air operator, associated solenoid-operated

valve, the power supply breaker or fuse for the solenoid valve, and its associated control

circuit (open/close switches and local auxiliary and switch contacts).

Fans The fan boundary includes the fan, the voltage supply breaker, and its associated control

circuit (open/close switches and local auxiliary and switch contacts).

Batteries The battery component boundary typically includes just the battery. Battery chargers are

modeled as separate components.

Bus Circuit

Breakers

A bus circuit breaker boundary includes the breaker and its associated control circuit

(open/close switches and local auxiliary and switch contacts).

• catastrophic failures,

• degraded failures, and

• incipient failures.

A catastrophic (complete) failure is one that prevents

the component from performing its mission as defined

in the PRA (Whitehead 1993). Catastrophic failures

require some kind of repair or replacement action on

the component in order to restore the component to

operability. For example, a valve that fails to open due

to a valve operator mechanical failure is a catastrophic

failure. 

A degraded failure is such that a component can

perform its mission, but at less than the optimum

performance level (Whitehead et al. 1993). An

incipient failure is such that there is no significant

degradation in performance but there are indications of

a developing fault (Whitehead et al. 1993). The

difference between the two is generally a matter of

severity. For example, an event involving pump shaft

vibration indicates possible damage to the pump

bearings. Severe vibration may be considered as

degraded failure if the pump produces less than

maximum flow. Shaft seizure or other failures could

occur within a few hours if the pump remains running

and thus would likely be removed from operation for

corrective maintenance. In contrast, minor vibration

may not result in degraded flow. This would thus be an

incipient failure. The significance of this event is that

it also could result in removal of the pump from

operation for inspection, lubrication, or some other

corrective action. Information about the types of

repairs made, the parts replaced, and the urgency of the

repairs often provides important insight about the

severity of these two types of component failures.
Component Failure and Boundary Definitions

3-3

Although both degraded and incipient failures will

typically lead to a corrective action, the corrective

action may or may not make the component unavailable

to perform its function. For example, maintenance on

the operator of a valve that is normally open will not

lead to the unavailability of the valve if it is required to

be open for system operation. This illustrates the

importance of ascertaining from event records the

modes of a component operation that a corrective action

would prevent. 

Sometimes the event information is so unclear and

incomplete that a definite classification of the severity

of a component malfunction event is not possible. For

example, Mosleh and Apostolakis (1985) cites one

maintenance work request issued at a nuclear power

plant that described the problem as follows: “Check

valve RHR-V-1A is leaking badly.” The maintenance

foreman’s description of the corrective action read:

“Fixed it, not leaking anymore!” No further

information was available. From the description given,

one cannot say for sure whether the leak was internal or

external, or whether it was large enough to result in

functional failure of the check valve. 

Unfortunately, the above example is not uncommon.

Descriptions of the malfunctions and repairs are often

very brief. The data analyst, then, is faced with the

difficult task of deciding whether to call a malfunction

a failure or not. The inability to distinguish between

severity levels of failures is particularly important as the

difference between the frequencies of catastrophic and

degraded modes of failures can be significant.

Therefore, in the absence of sufficient information, the

conservative assumption could be made that all such

events be recorded as catastrophic failures.

Unfortunately, conservative categorization of uncertain

events can lead to significantly higher failure rates.

Ultimately, the definition of failure from the system

analysis decides the classification of the data. Thus, the

failure of a component must match the definition of the

failure as described in the PRA model. A component

must fail to perform its function as defined in the

model. For example, a relief valve that opens at 1,115

psig instead of the required 1,110 psig is not failed,

although it may be described as failed by the governing

technical specifications, and a pump that delivers 645

gpm instead of the required 700 gpm is not failed if 645

gpm is sufficient for the function that it is required to

perform.
Component Failure and Boundary Definitions

3-4
4-1

4. DATA SOURCES

Two types of data sources can be utilized to produce

the various parameter estimates that are needed in a

probabilistic risk assessment (PRA). This chapter

identifies and discusses these two data sources.

Section 4.1 identifies and discusses plant-specific data

sources. Section 4.2 does the same for generic data

sources.

4.1 Plant-Specific Data Sources

Use of plant-specific data in a PRA produces risk

estimates that reflect the actual plant experience.

The scope of a plant-specific data analysis is

determined by the events that are included in the PRA

models. In general, plant-specific data are generally

reviewed for the following types of events:

1. The accident initiating events analyzed in the PRA.

2. The components included in system models

(generally fault trees). For components the

definition includes the component boundary and

failure mode. For unavailabilities due to

maintenance or testing it is necessary to know

whether the unavailabilities are to be specified at

the component, segment, train, or system level.

3. Some recovery events included in the PRA models.

Although most recovery events are analyzed using

human reliability analysis, the probabilities of

some events can be based upon a review of

operating experience.

Once the data needs are identified, the sources of raw

data at the plant are identified. In most cases, the

information needed may have to come from multiple

sources. For example, identification of maintenance

events and their duration may come from a control

room log, but other sources such as maintenance work

requests may be required to determine other

information such as whether a component had

experienced a catastrophic or degraded failure. 

There are many sources of raw data at a nuclear power

plant. Different plants have different means of

recording information on initiating events and

component failure and maintenance events. Since no

one source exists at a nuclear power plant that contains

all the necessary data, different sources must be

reviewed. The ease in which the plant-specific data can

be interpreted and the subsequent quality of the

resulting parameter estimates are a function of how well

the plant personnel recorded the necessary information.

Basic requirements associated with raw data sources

and some typical sources of raw data available at

nuclear power plants are identified in the following

sections.

4.1.1 Requirements on Data Sources

There are a variety of data sources that exist at a plant

and can be used in a data analysis. However, there are

some basic requirements that these raw data sources

should meet in order to be useful. Some typical

requirements, some of which were suggested in EPRI

TR-100381 (EPRI 1992), are delineated below.

4.1.1.1 Initiating Events

For reports on initiating events it is essential to include

the status of those systems that would be impacted as a

result of the event. This is typically not a problem since

the Licensee Event Report (LER) that is required to be

filed with the Nuclear Regulatory Commission (NRC)

following a plant trip usually contains this type of

information. It is also common for utilities to generate

additional detailed trip reports that delineate the cause

and effects of the event. Such reports need to specify

critical information needed for data analysis such as the

power level at the time of the plant trip and the

sequence of events, including the timing of individual

events.

4.1.1.2 Component Failures

For each event at a plant resulting in the unavailability

of a component, it is necessary that the raw data sources

identify the particular component or set of components

associated with the event. In order to determine if a

specific event contributes to a particular component

failure mode or to an unavailability due to the

component being in maintenance (either preventive or

corrective), it is necessary to be able to distinguish

between different degrees of degradation or failure.

The event reports should therefore specify whether

maintenance was required and if the maintenance was

corrective or preventive. If the component maintenance

is preventive there is generally no failure that initiates

the maintenance.
Data Sources

4-2

If an event involves corrective maintenance,

information is required to allow determination of the

severity of the failure (see Section 3.3 for definitions of

event severity). The ability to distinguish between

severity levels of failures is particularly important since

the difference between the frequencies of catastrophic

and degraded modes of failures can be significant. In

addition, information is required to determine the

component in which the failure actually occurred and

the mode of failure. Finally, it should be possible to

determine the time the component is unavailable during

each maintenance event. 

The data analysis may use plant data on component

unavailability that is being collected for other than PRA

purposes. The requirements for recording the data for

these other purposes may use definitions of severity and

failure modes that are different from the PRA

definitions. The definitions used for the data collection

programs should be determined and an appropriate

translation to the PRA basic events made.

4.1.1.3 Recovery Events

The information needed to estimate the probabilities

associated with recovering specific components or

systems from a failed state is similar to that needed for

component failures. Specific information pertaining to

the type of failure experienced by the component or

system (e.g., fail to operate, fail to start, fail to run), the

number of repair occurrences, and the time required to

perform the repair is needed to produce component

repair probabilities.

4.1.2 Data Sources 

Data sources that can provide information for

determining the number of initiating events include:

• internal plant failure records (e.g., scram reports or

unusual event reports),

• operator logs,

• LERs, and

• monthly operating reports/Gray Book.

Some data sources that typically provide information on

the occurrence of component failures include:

• LERs,

• internal plant failure records (e.g., failure reports,

trouble reports, or unusual event reports),

• maintenance records (e.g., maintenance work

orders, work request records),

• plant logs (e.g., control room log, component

history logs), and

• data bases (e.g., Equipment Performance and

Information Exchange System/Nuclear Plant

Reliability Data System).

The evaluation of component failure rates also requires

the number of demands and operating time for the

components. Sources of data for these parameters

include:

• monthly operating reports/Gray Book,

• component history logs,

• plant population lists,

• test procedures,

• plant operating procedures, and

• component demand or operating time counters

Repair information can be obtained from sources such

as:

• plant logs and

• maintenance work orders.

The type of information available in these sources and

their limitations are discussed in the following sections.

4.1.2.1 Regulatory Reports

All plants are required to submit LERs to the NRC for

all events meeting the 10 CFR 50.73 reporting criteria

presented in NUREG-1022 (NRC 2000a). LERs deal

with significant events related to the plant, including

plant shutdowns required by the technical

specifications, multiple train failures, engineered safety

feature actuations, and conditions outside the design

basis or not covered by plant procedures. An LER

includes an abstract that describes the major

occurrences during the event; the components, systems,

or human failures that contributed to the event; the

failure mode, mechanism, and effect of each failed

component; and an estimate of the elapsed time from

the discovery of the failure until the safety system train

was returned to service. A computerized search of LER

information is possible using the Sequence Coding and

Search System (SCSS) (Gallaher et al. 1984).

LERs generally provide a good description of the

causes of a reactor trip and subsequent events.

However, their value for obtaining component failure

data is very limited. The reporting criteria are limited

to safety-related trains or system failures, and therefore

LERs are not generally submitted for all failures.

Furthermore, LERs may not be submitted for every
Data Sources

4-3

safety-related component failure since individual

component failures do not have to be reported if

redundant equipment in the same system was operable

and available to perform the safety function. The

reporting criteria for LERs are also subject to

interpretation by the persons generating the reports and

thus can lead to inconsistencies in the LER data base.

Furthermore, there are other perceived deficiencies in

the LERs (Whitehead et al. 1993) that limit the

usefulness of the LER system for use in obtaining

estimates of component failure rates. The NRC staff

prepared NUREG-1022, Revision 1 (NRC 1998), to

address general issues in reporting that have not been

consistently applied. It covers some of the issues

identified above. 

The LER rule published in 1983 has recently been

amended and the reporting guidance in NUREG-1022,

Revision 2 (NRC 2000a) has been revised to eliminate

the burden of reporting events of little or no safety

significance, to better align the rules with the NRC’s

current needs and to clarify the reporting guidance

where needed. However, the rule still only requires the

reporting of failures leading to the unavailability of

safety-related system trains. Thus, LERs will not

provide failure data for all risk significant components.

In summary, LERs are a good source for identifying

and grouping initiating events. However, they have

very limited value for obtaining component failure data.

A plant’s Technical Specifications requires that a

monthly operating report be provided by the plant

licensee to the NRC. The scope of the information

requested of the licensees was originally identified in

Draft Regulatory Guide 1.16 (NRC 1975a) and includes

operating statistics and shutdown experience

information. The information requested to be included

in the monthly operating report contents was revised by

Generic Letter 97-02 (NRC 1997) and eliminated some

reporting requirements. Information that still must be

reported includes identification of all plant shutdowns,

whether they were forced or scheduled shutdowns, their

duration, the reason for the shutdown, the method of

shutting down the reactor, and corrective actions that

were taken. In addition, the monthly operating reports

include the number of hours the reactor was critical, the

number of hours the generator was on line, and the net

electrical output of the plant. 

The NRC initially compiled the information from the

monthly operating reports on a monthly basis and

published it in a hard copy form as NUREG-0020,

“Licensed Operating Reactors - Status Summary

Report” (NRC 1995b). This document is referred to as

the “Gray Book.” NUREG-0020 was discontinued

after the December 1995 report. However, the data

requested in Generic Letter 97-02 is being collected and

computerized as part of the NRC Performance Indicator

Project. 

In summary, the monthly operating reports provide

information on the number of scrams, the time spent at

full power, and the time spent in shutdown. This

information can be used in identifying and grouping

initiating events and in calculating the exposure time in

which they occurred. It is important to note that this

same information is generally available from the control

room logs and other sources. Thus, in general, the

monthly operating reports can be used to supplement or

verify other data sources.

4.1.2.2 Internal Plant Failure Reports

Different plants have different means of recording

initiating events and component failures. For each

automatic and manual scram, most plants generate an

internal scram report. Scram reports generally cover

the same information provided in LERs and monthly

operating reports. Thus, they can be used as the

primary or supplementary source for evaluating plant

scrams.

Most plants have a means of recording component

failures, records that are for the licensee’s own use

rather than for a regulatory use. Reports are generally

created when significant component failures or

degraded states occur during plant operation or are

identified during plant surveillance tests. These reports

may be called Unusual Occurrence Reports, Action

Reports, Failure Reports, Discrepancy Reports, or

Trouble Reports. Some of the events documented in

these reports may lead to an LER. However, these

reports may not identify all component failures and

generally are not exhaustive. Thus, these reports are

useful for supplemental information but are not a good

source of component reliability data.

4.1.2.3 Maintenance Records

At all plants, some form of written authorization form

is required to initiate corrective or preventative

maintenance work, or design changes. These authorization forms are known under different names at

various plants including work request/completion

records, maintenance work orders, clearance requests,

work requests, or tag-out orders. Maintenance records

are a primary source of component failure data since
Data Sources

4-4

they usually identify the component being maintained,

whether the component has failed or is degraded, the

corrective action taken, and the duration of the

maintenance action. The time of the failure is also

available but maintenance records generally contain

limited information on the impact, cause, and method of

discovery of the component failure.

4.1.2.4 Plant Logs

At each plant, a control room log is typically completed

for each shift and contains a record of all important

events at a plant. Control room logs identify power

level and plant mode changes, essential equipment

status changes, major system and equipment tests, and

entry and exit of Technical Specification Limiting

Conditions of Operation (LCOs). When properly

maintained, a control room log is a good source of

information on major equipment and unit availability.

However, the amount of information entered can vary

from shift to shift. Furthermore, the entries tend to be

brief. 

The control room logs are difficult to use as a source of

maintenance data since the tag-out and tag-in for a

maintenance event may span days or even months and

may not be dutifully recorded. The control room logs

are also limited in value as a source of component

failure data since not all failures may be recorded by the

operators. Component maintenance and failure information is generally found more easily in maintenance

work orders. All plant trips are likely to be recorded on

control room logs, but likely will not include a

description of the cause of the trip or the subsequent

transient behavior. LERs or plant scram reports must

be reviewed to obtain this additional information. 

In summary, control room logs are good supplementary

sources of information but there are usually more

convenient and complete sources of information

available such as maintenance records. However, the

control room logs are probably the best source of data

for indicating when redundant system trains are

switched from operating to standby status.

There may be other logs at a plant that contain essential

data. One example is a component history log. These

logs typically contain data on every failure and

maintenance and test action for a given component. As

such, component history logs are good sources for

identifying not only the number of component failures,

but also the number of demands a component

experiences.

4.1.2.5 Component Exposure Data Sources

Calculation of plant-specific failure rates requires

determination of the number of failures and the

corresponding number of demands or operating time.

As indicated in the previous subsections, some of the

data sources used to establish the number of failures

also contain information on the number of demands and

operating time. However, these sources do not contain

all component demands or the operating time for all

components. Additional documents that must be

reviewed for information about component demands

and operating hours include test procedures.

In addition to demands presented by automatic

initiations and maintenance activities (obtained from

sources such as control room logs and maintenance

records), periodic testing is an important source of

demands especially for safety-related equipment. To

establish the number of demands due to testing, testing

procedures pertinent to a component must be reviewed.

In addition to the actual test demands, additional test

demands may be imposed by technical specifications

following failure of a component. A typical example

where this is imposed is when a diesel generator is

unavailable for operation. Test logs or similar records

can be examined to obtain an estimate of the number of

tests carried out during the time period of interest.

It should also be noted that at some plants, some major

components may be monitored to count the number of

actuations experienced by the breakers (breaker cycle

counters). In addition, the operating hours for large

motor-driven components at some plants may be

automatically registered on running time meters at the

electrical switchgear. Such counters and logs can be

used to supplement the demand and operating time

information obtained from other sources. 

4.1.3 Plant-Specific Data Bases

The Institute of Nuclear Power Operations (INPO) has

maintained several databases of component failure data

provided by each nuclear power plant since 1984. The

first, Nuclear Plant Reliability Data System (NPRDS),

was a proprietary computer-based collection of

engineering, operational, and failure data on systems

and components in U.S. nuclear power plants through

1996. The second, the Equipment Performance and

Information Exchange (EPIX) System, replaced

NPRDS and includes data reported since 1987. Both

data bases are discussed in the following sections.
Data Sources

4-5

4.1.3.1 Nuclear Plant Reliability Data System

(NPRDS)

In the early 1970s, industry committees of the American

National Standards Institute (ANSI) and the Edison

Electric Institute (EEI) recognized the need for failure

data on nuclear plant components. As a result, a data

collection system was developed whose objective was

to make available reliability statistics (e.g., failure rates,

mean-time-between-failures, mean-time-to-restore) for

safety related systems and components.

This system, the Nuclear Plant Reliability Data System

(Tashjian 1982), was developed by Southwest Research

Institute (SwRI). Plants began reporting data on a

voluntary basis in 1974, and continued reporting to

SwRI until 1982. In January 1982, the INPO assumed

management responsibility for the system until

reporting was terminated at the end of 1996.

Originally the scope of the NPRDS covered the systems

and components classified by ANSI standards as Safety

Class 1, 2, or 1E, with a few exceptions such as reactor

vessel internals and spent fuel storage. However, later

the scope was expanded to cover any system important

to safety and any system for which a loss of function

can initiate significant plant transients (Simard 1983).

By the end of 1984, 86 nuclear power plant units were

supplying detailed design data and failure reports on

some 4,000 to 5,000 plant components from 30 systems

(Simard 1985).

Data reported to NPRDS consisted of two kinds:

engineering reports and failure reports. The

engineering reports provided detailed design and

operating characteristics for each reportable

component. The failure reports provided information

on each reportable component whenever the component

was unable to perform its intended function. The same

operational data contained in NUREG-0200 was also

included in the system. The NPRDS failure reports

provided to INPO were generally generated by plant

licensees utilizing maintenance records such as

maintenance work orders. These reports utilized a

standard set of component boundaries and failure mode

definitions. 

4.1.3.1.1 Limitations in the Data Available from

the NPRDS

Several issues regarding the quality and utility of the

NPRDS data have been observed, including: 

1. Input to NPRDS was discontinued on December

31, 1996.

2. The number of component demands is provided by

estimation.

3. The exposure time is estimated.

4. The amount of time needed to repair components

out for corrective maintenance is not provided.

5. Maintenance rates are not provided.

6. The voluntary nature of the reporting system

introduces uncertainty into measuring the

frequency at which a particular type of problem

occurs.

7. The final results of a problem investigation or the

ultimate corrective action taken are not always

included.

8. Report entries tend to be brief and often do not

provide enough information to identify the exact

failure mechanism.

4.1.3.2 Equipment Performance and Information

Exchange (EPIX) System

The need for high-quality, plant-specific reliability and

availability information to support risk-informed

applications was one impetus for a proposed reliability

data rule by the NRC to require utilities to provide such

information. Instead of a regulatory rule, the nuclear

industry committed to voluntarily report reliability

information for risk-significant systems and equipment

to the EPIX system. EPIX is a web-based database of

component engineering and failure data developed by

INPO to replace NPRDS. The utilities began reporting

to EPIX on January 1, 1997. 

EPIX enables sharing of engineering and failure

information on selected components within the scope of

the NRC’s Maintenance Rule (10 CFR 50.65) and on

equipment failures that cause power reductions. It also

provides failure rate and reliability information for a

limited number of risk-significant plant components.

This includes components in the systems included in the

scope of the Safety System Performance Indicator

(SSPI) program. EPIX consists of:

• a site-specific database controlled by each INPO

member site with web-based data entry and

retrieval,

• an industry database on the INPO web site where

selected parts of the site-specific database are

shared among plants, and

• a retrieval tool that provides access to the vast

historical equipment performance information

available in the NPRDS. 
Data Sources

4-6

Events reported to EPIX include both complete failures

of components and degraded component operation.

The number of demands and operating hours (i.e.,

reliability data) and the unavailability are required to be

collected for key components in the SSPI safety

systems for each plant. In addition, contributors to

EPIX are also to include one-time estimates of the

number of demands and run hours for other risksignificant components not included in SSPI systems. 

4.1.3.3 Reliability and Availability Data

System (RADS)

The NRC has developed the Reliability and Availability

Data System (RADS) to provide the reliability and

availability data needed by the NRC to perform generic

and plant-specific assessments and to support PRA and

risk-informed regulatory applications. The NRC is

incorporating data from EPIX and INPO’s SSPI system

along with information from other data sources (e.g.,

LERs and monthly operating reports) into RADS. Data

are available for the major components in the most riskimportant systems in both boiling water reactors

(BWRs) and pressurized water reactors ( PWRs).

The reliability parameters that can be estimated using

RADS are:

 • probability of failure on demand,

 • failure rate during operation (used to calculate

probability of failure to continue operation),

 • maintenance out-of-service unavailability (planned

and unplanned), and

 • time trends in reliability parameters.

The statistical methods available in RADS include

classical statistical methods (maximum likelihood

estimates and confidence intervals), Bayesian methods,

tests for homogeneity of the data for deciding whether

to pool the data or not, Empirical Bayes methods, and

methods for trending the reliability parameters over

time. 

4.2 Generic Data Sources

Several generic data sources currently available and

used throughout the nuclear power PRA industry are

identified in this section. Several of these data bases

are discussed with regard to their attributes, strengths,

and weaknesses. Data bases for both initiating events

and component failure rates are included. Some data

sources represent compilations of raw data which have

been collected directly from various facilities and

processed and statistically analyzed. Other data sources

utilize the results of the statistical analyzes of other data

bases to derive estimates for component probabilities.

Section 4.2.1 contains discussions and summaries of

generic data bases sponsored by the NRC for use in

both government and industry PRAs. Section 4.2.2

contains discussions and summaries of generic data

bases sponsored by the Department of Energy (DOE)

for use in PRAs. Section 4.2.3 contains discussions and

summaries of generic data bases developed by nuclear

power industry related organizations. Section 4.2.4

contains a summary of a foreign data base, the Swedish

T-book. Section 4.2.5 contains a discussion of several

non-nuclear data bases which could be useful for some

data issues in nuclear power PRA. Section 4.2.6

describes a process for selecting a generic data value

from these sources.

4.2.1 NRC-Sponsored Generic Data Bases

The discussion of NRC-sponsored generic data bases is

presented in two sections. The first discusses current

data bases. These data sources are deemed appropriate

for current and future use. The second section briefly

summarizes some historical data bases that have been

used or referenced in past analyses. While useful at

the time, these data bases are no longer considered

appropriate sources of information. 

4.2.1.1 Current Data Bases

Current NRC-sponsored data bases are discussed in the

following subsections. Major attributes for each data

base are identified, and limitations associated with each

data base are provided.

As a reminder, these data bases are considered to be

appropriate sources of information for use in PRAs or

other risk assessments. However, it is the user’s

responsibility to ensure that any information from these

data bases used in their analysis is appropriate for their

analysis.

4.2.1.1.1 Severe Accident Risks Study Generic

Data Base (NUREG-1150)

The generic data base developed for the NRC’s Severe

Accident Risks study (NUREG-1150) (NRC 1990) is

documented in NUREG/CR-4550 as supporting

documentation (Drouin et al. 1990). This data base was

developed from a broad base of information, including:
Data Sources

4-7

• WASH 1400 (NRC 1975b),

• the IREP data base (Carlson et al. 1983),

• Zion (ComEd 1981), Limerick (PECO 1982), Big

Rock Point (CPC 1981), and the Reactor Safety

Study Methodology Application Program

(RSSMAP) PRAs (Hatch et al. 1981),

• NRC LER summaries (Hubble and Miller 1980,

Appendices O through Y), and 

• the NRC’s Station Blackout Accident Analysis

(Kolaczkowski and Payne 1983).

Component failure probabilities, failure rates, and

initiating event frequencies typically modeled in the

NUREG-1150 plant analyses are included in the data

base. A mean value and an error factor on a log normal

distribution are provided for each entry into the data

base.

Limitations in the Data Available from 

NUREG-1150

The basis of the NUREG-1150 data base is from a

broad group of prior PRA analyses and generic data

bases. Thus, it does not directly represent the results of

the analysis of actual operational data. Furthermore,

the data upon which those previous analyses are based

suffer from limitations similar to those for older NRC

data sources and the NPRDS data base (Sections

4.2.1.2 and 4.2.3.1).

4.2.1.1.2 Evaluation of Loss of Offsite Power

Events at Nuclear Power Plants: 

1980 - 1996

The report Evaluation of Loss of Offsite Power Events

at Nuclear Power Plants: 1980 - 1996, NUREG/CR5496 (Atwood et al. 1998), presents an analysis of loss

of offsite power (LOSP) initiating event frequency and

recovery times for power and shutdown operations at

commercial nuclear power plants. The evaluation is

based on LERs for events that occurred during 1980

through 1996. The primary objective of the study was

to provide mean and uncertainty information for LOSP

initiating event frequencies and recovery times. A

secondary objective was to re-examine engineering

insights from NUREG-1032 (a LOSP study covering

the years 1968 through 1985) using the more recent

data.

The major findings of the report are:

• Not all LOSP events that occur at power result in

a plant trip.

• Plant-centered events clearly dominate the LOSP

frequency during both power and non-power

operational modes.

• Plant-centered LOSP frequency is significantly

higher during shutdown modes than during power

operation.

• No statistically significant variation among units

was found for plant-centered sustained initiating

events.

• During shutdown, statistically significant variation

among plants was found for plant-centered

sustained initiating events.

• Equipment faults were the main contributor (58%)

to plant-centered LOSP initiating events that

occurred during power operations. Human error

accounted for a smaller contribution (23%).

• During shutdown conditions, human error was the

dominant contributor (58%).

• A clear downward trend can be seen for the plantcentered initiating event frequency.

• Grid-related LOSP frequency is small.

• For severe weather, statistically significant site-tosite variability exists for sustained shutdown LOSP

frequencies.

• Severe weather events had significantly longer

sustained recovery times.

• For sustained recovery times, no pattern was found

correlating unit design class with longer recovery

times.

• Longer recovery times were observed for sustained

plant-centered LOSP events that did not result in a

plant trip or that occurred during shutdown.

Nominal frequencies and upper and lower bounds are

given in the report.

Limitations in the Data Available from

NUREG/CR-5496

The generic data base developed in this NRC-sponsored

data study is based on raw data from LERs. LERs

constitute data involving only reportable events at

nuclear power plants, and the degree of detail provided

in the LERs varies. Some information needed in the

data analysis had to be estimated (e.g., allocation of

1980 time into critical and shutdown time), and the

analysis ended with events that occurred in 1996. Thus,

the data base does not contain events that occurred after

1996, and may not be representative of actual current

operational experience.
Data Sources

4-8

4.2.1.1.3 Rates of Initiating Events at U.S.

Nuclear Power Plants: 1987 - 1995

The report Rates of Initiating Events at U.S. Nuclear

Power Plants: 1987 - 1995, NUREG/CR-5750 (Poloski

et al. 1999a), presents an analysis of initiating event

frequencies at domestic nuclear power plants. The

evaluation is based primarily on the operational

experience from 1987 through 1995 as reported in

LERs. The objectives of the study were to:

• provide revised frequencies for initiation events in

domestic nuclear plants, 

• compare these estimates to estimates used in PRAs

and Individual Plant Evaluations (IPEs), and 

• determine trends and patterns of plant

performance.

Major findings of the report are:

• Combined initiating event frequencies for all

initiators from 1987 through 1995 are lower than

the frequencies used in NUREG-1150 (NRC 1990)

and industry IPEs by a factor of five and four,

respectively.

• General transients constitute 77% of all initiating

events, while events that pose a more severe

challenge to mitigation systems constitute 23%.

• Over the time period of the study, either a

decreasing or constant time trend was observed for

all categories of events.

• Loss of coolant accident (LOCA) frequencies are

lower than those used in NUREG-1150 and

industry IPEs.

Nominal frequencies and upper and lower bounds are

given in the report.

Limitations in the Data Available from

NUREG/CR-5750

The generic data base developed in this NRC-sponsored

data study is primarily based on raw LER data from

1987 through 1995. For some events (e.g., LOCAs)

information from additional operating experience, both

domestic and foreign, was used with other sources of

information (e.g., engineering analyses) to estimate the

initiating event frequencies. Since the analysis ended

with events that occurred in 1995 and made use of other

sources of information, the data base may not be

representative of actual current operational experience.

4.2.1.1.4 System Reliability Studies

A series of system reliability studies, documented in the

multi-volume NUREG/CR-5500 report,1

 presents an

analysis of system unreliability for various systems.2

The following volumes comprise the systems that will

be studied::

• Volume 1: auxiliary/emergency feedwater system

(Poloski et al. 1998),

• Volume 2: Westinghouse reactor protection

system (Eide et al. 1999a),

• Volume 3: General Electric reactor protection

system (Eide et al. 1999b),

• Volume 4: high-pressure coolant injection system

(Grant et al. 1999a),

• Volume 5: emergency diesel generator power

system (Grant et al. 1999b),

• Volume 6: isolation condenser system (Grant et al.

1999c),

• Volume 7: reactor core isolation cooling system

(Poloski et al. 1999b),

• Volume 8: high-pressure core spray system

(Poloski et al. 1999c),

• Volume 9: high pressure safety injection system

(Poloski et al. 2000),

• Volume 10: CE reactor protection system

(Wierman et al. 2002a), and

• Volume 11: B&W reactor protection system

(Wierman et al. 2002b).

With the exception of the reactor protection system

volumes, the analyses of the other systems are based on

information obtained from LERs. For the reactor

protection system volumes, the analyses are based on

information obtained from NPRDS and LERs.

The analyses: (1) estimate the system unreliability

based on operating experience, (2) compare the

estimates with estimates using data from PRAs and

IPEs, (3) determine trends and patterns in the data, and

(4) provide insights into the failures and failure

mechanisms associated with the system.

1

 Currently, it is expected that some of these reports will be

updated with new information.

2

 Train, subsystem or system data can be combined with basic

event failure data to obtain improved estimates of component

failure rates. A Bayesian method for doing this is described in

Martz and Almond 1997.
Data Sources

4-9

Unreliability estimates (means and distributions) are

provided for the entire system for each plant. In

addition, unreliability estimates for major train

segments failure modes (e.g., failure to start – pump,

driver, valves, and associated piping) are provided.

Common cause failure estimates are also provided. 

Limitations in the Data Available from

NUREG/CR-5500

The information available from this NRC-sponsored

data study is based on that available from LERs and

NPRDS. LERs constitute data only involving

reportable events at nuclear power plants, and the

degree of detail provided in the LERs varies. The

limitations associated with NPRDS are provided in

Section 4.2.3.1. The information used in the studies

spans various time frames, with the most up-to-date

information coming from 1997. Thus, the results of the

studies may not be representative of actual current

operational experience.

4.2.1.1.5 Component Performance Studies

A series of component performance studies,

documented in the multi-volume NUREG-1715 report,

presents an analysis of component performance for

various components. The following volumes comprise

the components that have been studied:

• Volume 1: turbine-driven pumps (Houghton and

Hamzehee 2000a),

• Volume 2: motor-driven pumps (Houghton and

Hamzehee 2000b),

• Volume 3: air-operated valves (Houghton 2001a),

and

• Volume 4: motor-operated valves (Houghton

2001b).

The analyses are based on information obtained from

NPRDS and LERs. The data included in the studies

cover the period 1987 through 1995.

The analyses: (1) estimate the system-dependent

unreliability of selected components, (2) compare the

estimates with estimates from PRAs and IPEs, (3)

determine trends and patterns in the data, and (4)

provide insights into component performance, including

component failure mechanisms.

System-dependent unreliability estimates (means and

distributions) for various failure mechanisms are

provided for each component. Trends in component

failure rates were also evaluated in these studies.

Limitations in the Data Available from 

NUREG-1715

The information available from this NRC-sponsored

data study is based on that available from LERs and

NPRDS. LERs constitute data only involving

reportable events at nuclear power plants, and the

degree of detail provided in the LERs varies. The

limitations associated with NPRDS are provided in

Section 4.2.3.1. The information used in the studies

spans various time frames, with the most up-to-date

information coming from 1998. Thus, the results of the

studies may not be representative of actual current

operational experience.

4.2.1.2 Historical Data Bases

In the past, NRC sponsored several programs to

develop data bases on nuclear power plant component

reliability and initiating event frequencies. These

programs included:

• In-Plant Reliability Data Base for Nuclear Power

Plant Components (IPRDS) (Drago et al. 1982) –

established at Oak Ridge National Laboratory to

establish methods for data collection and analysis.

• Nuclear Reliability Evaluation Program (NREP) –

generic data base developed to support the

Probabilistic Safety Analysis Procedures Guide,

NUREG/CR-2815 (Papazoglou et al. 1984).

• Interim Reliability Evaluation Program (IREP)

Generic Data Base – developed to support the

performance of five PRAs in the 1980s and

documented in the IREP procedures guide (Carlson

et al. 1983).

• Nuclear Computerized Library for Assessing

Reactor Reliability (NUCLARR) – developed as a

repository of human error and hardware failure

information that could be used to support a variety

of analytical techniques for assessing risk.

NUCLARR was documented in five volumes as

NUREG/CR-4639 (Gertman et al. 1990).

Major attributes for each program and the resulting data

bases are documented in the cited references. 

4.2.2 DOE-Sponsored Generic Data Bases

Several data bases have been developed to support

DOE-sponsored projects. Two of these data bases are

discussed in the following sections.
Data Sources

4-10

4.2.2.1 Component External Leakage and

Rupture Frequency Estimates

Estimates of external leakage and rupture frequencies

for components such as piping, valves, pumps, and

flanges are necessary for detailed risk analysis of

internal flooding. These estimates have been developed

and documented in EGG-SSRE-9639 (Eide et al. 1991).

The estimates are based on an analysis of data gathered

from a comprehensive search of LERs contained in

Nuclear Power Experience (NPE) (Hagler-Bailly 1972).

The NPE data base was searched for data covering the

period September 1960 through June 1990. The

external leakage and rupture events collected from the

data were converted to component leakage and rupture

frequencies in a three-step process:

1. The ratios of external rupture events to external

leakage and rupture events were examined for

various components by size and system to decide

how to group the data.

2. The final probabilities of an external rupture, given

an external leakage or rupture event, were

determined.

3. Lastly, the external leakage and rupture

frequencies were obtained by estimating

component populations and exposure times.

Limitations in the Data Available from 

EGG-SSRE-9639

The generic data base developed in this DOE-sponsored

data study is based on raw LER data from 1960 through

1990. LERs constitute data only involving reportable

events at nuclear power plants, and the degree of detail

provided in the LERs varies. Since the analysis ended

with events that occurred in 1990, the data base may not

be representative of actual current operational

experience.

4.2.2.2 Generic Component Failure Data Base

for Light Water and Liquid Sodium

Reactor PRAs

A generic component failure data base was developed

by the Idaho National Engineering Laboratory (INEL)

for light water and liquid sodium reactor PRAs. This

data base is documented in EGG-SSRE-8875 (Eide et

al. 1990). The intent of this project was to base the

component failure rates on available plant data as much

as possible rather than on estimates or data from other

types of facilities. The NUCLARR data base and the

Centralized Reliability Data Organization (CREDO)

(Manning et al. 1986) were used as the primary sources

of component failure data. If specific components and

failure modes were not covered in those two sources,

then other standard sources such as IEEE STD-500

(IEEE 1983) (for electrical components) and WASH1400 (NRC 1975b) were used. The data base is

organized into four categories according to the working

fluid of the component:

• mechanical components (water or steam),

• mechanical components (liquid sodium),

• mechanical components (air or gas), and

• electrical components.

Limitations in the Data Available from 

EGG-SSRE-8875

The generic data base developed in this DOE-sponsored

data study is based on information from multiple

sources. Since the analysis ended with events that

occurred in 1990, the data base may not be

representative of actual current operational experience.

4.2.3 Industry Data Bases

Several data bases developed within the nuclear power

industry for both risk assessment and for plant

operations are summarized here. Data bases discussed

in this section were developed by the Electric Power

Research Institute (EPRI) and the consulting firms of

EQE, International and Science Applications

International Corporation (SAIC). 

Although the NPRDS and EPIX data bases (described

in Section 4.1.3) contain plant-specific data, they can be

used to generate generic failure rates for components.

Methods for aggregating individual plant data to

estimate failure rates are described in Section 8.2 of this

handbook. Aggregation of data from EPIX can be

performed using the RADS software developed under

the NRC auspice.

4.2.3.1 EQE, International

The EQE, International generic data base (formerly

known as the Pickard, Lowe, and Garrick or PLG data

base) for light water reactors is set up to support PRA

and reliability analysis for which both point estimates
Data Sources

4-11

and uncertainty distributions are developed.3

 The data

base contains information on:

• Component failure rates,

• Common cause failures,

• Component maintenance frequencies and mean

durations,

• Initiating events,

• Fire and flood events at nuclear sites, and

• Shutdown events involving loss of residual heat

removal (RHR) cooling and loss of inventory.

The fire, flood, and shutdown events are a compendium

of experience event summaries from all U.S. nuclear

sites. The common cause data are presented as event

description and have been classified according to the

methodology of NUREG/CR-4780 (Mosleh et al.

1989). The fire, flood, shutdown and common cause

events have, in addition to the description, information

in various fields making them convenient for sorting

and for use in plant-specific screening analysis. 

All other data are in the form of distributions and are

compatible with the PLG risk assessment software,

RISKMAN® . These distributions are generated using

the data analysis module of RISKMAN® which can be

used as a stand-alone software. The distributions

developed are available to the other modules of

RISKMAN® used for fault-tree quantification and core

damage sequence quantification.

The actuarial data are from over 20 nuclear sites in the

U.S. and in Europe. Other sources of generic

information also used are:

• EPRI reports on components, shutdown accident

events, initiating events, loss of offsite power;

• Special NUREG reports on components such as

pumps, valves, diesel/generators;

• Compiled data bases such as Nuclear Power

Experience, NUCLARR, IEEE-500 (IEEE 1983),

NPRDS, etc.; and

• Insurance company databases for loss events.

The database includes statistics for components that

cover population, demands, operating times, failures,

and maintenance outages and durations at specific

plants. It also includes event-by-event analyses for

initiating events, common cause failures, and fires and

floods over the whole U.S. plant population. In addition

to this factual information, parameter estimates from

published sources of generic reliability data are also

provided. 

The actuarial data and the other generic data are

combined using a two-stage Bayesian updating

technique. The generic distributions maintain what is

referred to as plant-to-plant variability. Since the data

are developed specifically to be used for Monte Carlo

sampling, they are defined with a minimum of 20

discrete bins with special attention given to the tails of

the distributions.

The database is available in a format compatible with

RISKMAN® and also as ASCII files. 

Limitations in the Data Available from EQE,

International

The EQE data base is proprietary, so the adequacy and

comprehensiveness of the underlying data have not

been evaluated for this document. As noted above,

several of the sources of generic information

incorporated into the data base are discussed previously

in this chapter (e.g., NUCLARR, NPRDS); thus, it is

possible that some of the data from the EQE data base

may have limitations similar to other data bases

discussed in this chapter. However, it should be noted

that the proprietary nature of the EQE data base

precludes any definitive judgment as to how data bases

such as NUCLARR and NPRDS were utilized in the

development of the EQE database.

4.2.3.2 Science Applications International

Corporation

Science Applications International Corporation (SAIC)

has developed a generic, proprietary data base for

application to PRAs on commercial nuclear power

plants.4

 

The scope of the data base for components and their

failure modes was established by a review and

tabulation of all basic events and component failures in

SAIC-conducted PRAs. Components were grouped

3

 The information on the EQE/PLG data base is based on personal

correspondence from Shabha Rao, PLG, Newport Beach,

California, to Timothy Wheeler, Sandia National Laboratories,

September 16, 1999, and to Donnie Whitehead, Sandia National

Laboratories, April 4, 2001.

4

 The information on the SAIC data base is based on a personal

correspondence from Alan Kolaczkowski, Vice President, SAIC,

to Donnie Whitehead, Sandia National Laboratories, April 18,

2001.
Data Sources

4-12

into generic categories rather than specifically by

system or application. Thus, all basic events for motordriven pumps were categorized into a single “motordriven-pump” category rather than delineated size or by

system. Some component failure modes were merged

to reflect the available data (e.g., air-operated valves

fail-to-open and fail-to-close were combined into a

single failure mode – fail-to-operate. Component

boundary definitions are given for all components in the

SAIC generic data base. 

The data base was developed by collecting all sources

of available parameter estimates relevant to the

component failures defined by the scoping process.

Each data source was evaluated against a set of

acceptance criteria, including availability (no

proprietary sources were included), compatibility of

data to being fit to a lognormal distribution, and

Bayesian updating. Any source which used Bayesian

parameter estimation methods to develop estimates for

component failure modes was rejected. Such data

sources were considered to be too plant-specific for

inclusion into a generic data base.

Each individual data source selected against the

acceptance criteria was fitted to a lognormal

distribution. Then, all data sources for each particular

component failure were aggregated through a weighted

sum approach (each source was weighted equally).

Each aggregated distribution was fitted to a lognormal

distribution.

Limitations in the Data Available from the SAIC

Data Base

The SAIC data base is proprietary, so the adequacy and

comprehensiveness of the underlying data have not

been evaluated for this document.

4.2.3.3 Advanced Light Water Reactor Data

Base

EPRI’s Advanced Light Water Reactor (ALWR) Utility

Requirements Document (EPRI 1989) contains a

reliability data base for use in ALWR PRAs. Several

data sources were reviewed and representative failure

rates and event probabilities were compiled from these

data sources. A best estimate value was selected for

each component type and failure mode based on

judgment regarding the applicability of the data source

to the expected ALWR design. The primary sources

used in the data survey were the Oconee PRA (Duke

1984), the Seabrook Probabilistic Safety Study (PLG

1983), parameter estimates from licensee-event reports

documented in NUREG/CR-1363 (Battle 1983) for

valves, NUREG/CR-1205 (Trojovsky 1982) for pumps,

and NUREG/CR-1362 for diesel generators (Poloski

and Sullivan 1980).

Limitations in the Data Available from the ALWR

Data Base

The ALWR data base lists only best estimates for each

initiating event, failure rate, and event probability. The

survey is well documented in that all estimates collected

for each parameter estimate are shown. However, only

a cursory statement of rationale for deriving the best

estimate value is given. No uncertainty bounds or

probability density functions are given.

4.2.4 Foreign Sources

Two sources of data from Nordic nuclear power plants

are available. The I-Book documents initiating event

frequency data and the T-Book documents component

failure data.

4.2.4.1 Sweden’s T-Book for Nordic Nuclear

Power Plants

Since the early 1980s a Reliability Data Handbook, the

T-Book (ATV 1992), has been developed and used for

nuclear power plant of Swedish design. The T-Book

provides failure data for the calculation of component

reliability for use in regulatory safety analyses of

Nordic nuclear power plants. The 3rd edition is based

on operation statistics from 12 Swedish and 2 Finnish

nuclear power plants, including approximately 110

reactor years of experience. 

The failure characteristics incorporated into the

parameter estimations in the T-Book are based on

Licensee Event Reports for Nordic plants delivered to

the Swedish Nuclear Power Inspectorate (SKi) and

from failure reports in ATV’s central data base. Only

critical failures, those that actually caused a

component’s function to stop or fail, are incorporated

into the parameter estimations. A multistage empirical

Bayesian approach is used to develop the component

parameter estimates from the raw data (Po@rn 1996).

Limitations in the Data Available from the T-Book

Data for the T-Book are collected from LERs delivered

to the SKi; thus, the parameter estimates derived from

the data are based only on data of reportable incidents.

It is not understood how representative such data may

be of actual operational experience.
Data Sources

4-13

4.2.4.2 Sweden’s I-Book for Nordic Nuclear

Power Plants

The I-Book (Po@rn et al. 1994) contains a compilation of

initiating events that have occurred in Nordic nuclear

power plants. The data reflects 215 reactor years of

operating experience prior to 1994. In the first edition

of the I-Book, issued in 1993 (Po@rn et al. 1993),

initiating event groups were identified and frequencies

generated. The operating experience from two

additional plants in Finland were included in the second

edition (Po@rn et al. 1994).

The I-Book includes the development of a statistical

model for performing a trend analysis. The model is

based on nonhomogeneous Poisson (Power Law)

processes and includes a complete treatment of

parametric uncertainty using Bayesian methods.

Limitations in the Data Available from the I-Book

Data for the I-Book are collected from operating

experience at Nordic plants. It is not understood how

representative such data may be of operational

experience in nuclear power plants in the United States.

4.2.5 Non-Nuclear Power Data Bases

There are many non-nuclear data bases that contain

failure data that can potentially be used in nuclear

power plant PRAs. Several of these data bases are

described below. When using data from noncommercial nuclear sources, care must be taken to

ensure that the data are for components and conditions

representative of those that exist in nuclear power

plants. 

4.2.5.1 Reliability Analysis Center

The Reliability Analysis Center (RAC) in Rome, New

York, maintains two data bases on electronic and nonelectronic component reliability. The data bases are:

• Electronic Parts Reliability Data (Denson et al.

1996), and 

• Non-Electronic Parts Reliability Data (Denson et

al. 1995).

These RAC databases provide empirical field failure

rate data on a wide range of electronic components and

electrical, mechanical, and electro-mechanical parts and

assemblies. The failure rate data contained in these

documents represent cumulative compilation from the

early 1970s up to the publication year for each

document. Data are collected from sources such as:

• published reports and papers,

• government-sponsored studies,

• military maintenance data collection systems,

• commercial/industrial maintenance databases, and

• direct submittals to the RAC from military or

commercial organizations that maintain failure data

bases.

Limitations in the Data Available from the RAC

Handbooks

The RAC handbooks provide point estimate parameter

estimations for failure rates (or demand probabilities).

No treatment of uncertainty is provided. 

4.2.5.2 Offshore Reliability Data Project

The Offshore Reliability Data (OREDA) project has

collected and processed data from offshore oil

platforms operated by 10 different companies off the

coasts of the U.K., Italy, and Norway. Reliability data

collected and processed by OREDA has been published

in the Offshore Reliability Data Handbook (OREDA

1997). The main objective of OREDA is to collect

reliability data for safety important equipment in the

offshore oil industry.

Components and systems for which data are collected

are:

• Machinery

– Compressors

– Gas turbines

– Pumps

• Electric generators

• Mechanical Equipment

– Heat exchangers

– Vessels

• Control and Safety Equipment

– Control Logic Units

– Fire and Gas Detectors

– Process sensors

• Valves

• Subsea Equipment

– Control Systems

– Well completions

Data have been collected from 7,629 individual

equipment units (e.g., individual pumps, valves,

motors) over a total observation period of 22,373 years.

The records include 11,154 failures. 
Data Sources

4-14

Under each category of equipment (e.g., machinery)

information is collected on each type of component

(e.g., centrifugal compressors). Data are further sorted

by a component’s driving mechanism (e.g., electric

motor-driven), by failure mode (e.g., fails-to-start, failswhile-running), and by the criticality of each failure

(e.g., critical - terminates the operation of the

component, degraded - component still operates).

The OREDA-97 handbook presents failure rate and

demand failure probability estimates for various

combinations of component function, application,

capacity, operating fluid, and size. 

Limitations in the Data Available from the OREDA

Data Base

Certain data quality issues have arisen in the

development of OREDA (Sandtorv et al. 1996). The

quality and availability of data can vary significantly

among the 10 participating companies. Interpretations

of equipment definitions and failure mode

specifications can vary among the participants as well,

affecting the quality of data. The effect of preventive

maintenance on equipment reliability is difficult to

measure. Since preventive maintenance practices vary

among the participating companies it is unclear as to

what would be the baseline rate of a generic type of

equipment.

4.2.5.3 IEEE-500 Standard

The Institute of Electrical and Electronics Engineers

(IEEE), Inc. Standard 500-1984 (IEEE 1983) contains

failure estimates for various electrical, electronic,

sensing, and mechanical components. Delphi

procedures (an elicitation process) were used in

producing component failure estimates. Multiple

sources of information, including nuclear, fossil fuel,

and industrial, were considered by the experts as part of

the Delphi process.

Limitations in the IEEE-500 Data Base

The major limitations associated with the IEEE-500

data base are (1) the data base contains dated material

(i.e., the latest information used to develop the data

base comes from the early 1980s), and (2) the process

used to support development of the failure estimates

was an uncontrolled process. (A survey was sent to

various individuals requesting them to provide

information on selected issues. No inherent controls

were placed on the individuals, and no training on how

to estimate failure probabilities was provided to the

individuals filling out the survey forms.) In addition, it

should be noted that IEEE Standard 500-1984 has been

withdrawn and is no longer available from IEEE. 

4.2.6 Selection of Parameter Estimates

from Existing Data Bases

The need to select parameter estimates from existing

generic data bases may arise when performing a PRA.

This can occur when a PRA is being performed on a

new plant that has no operating history or it may occur

when no plant-specific information exists for a specific

component. Whatever the reason, when it becomes

necessary to select parameter estimates from generic

data bases, certain cautions should be observed:

1. The generic data base should contain failure

probability estimates for components that are

identical or comparable to the ones in the PRA

model in terms of size, component boundary

definition, intended operational history (e.g.,

normally operating versus standby), and expected

or postulated operating environment.

2. The generic data base should contain a

recommended point estimate and an uncertainty

distribution for each identified failure.

3. If possible, the primary sources of information

used to develop the generic data base’s failure

probabilities and distributions should be

information from other nuclear power plants.

Supplemental information from non-nuclear

sources should be used only when necessary to

provide failure probabilities and distributions for

components that cannot be obtained from nuclear

power plant generic data sources.

4. Where possible, the generic data base’s failure

probabilities and distributions should be derived

from actual failure events. If such information is

not available, then failure probabilities and

distributions generated by other techniques (e.g.,

expert elicitation) are acceptable.

5. Generic data base failure probabilities and

distributions should reflect current trends. If

significant trends exist within the failure data

indicating either an increase or decrease in the

failure probabilities, the underlying event failure

information used to generate the failure

probabilities should represent these recent events.
Data Sources

4-15

However, if no significant trends exist, then data

from all years can be used to estimate the failure

probabilities.

6. The failure probability estimates contained within

the generic data base should not be based on

incestuous sources, i.e., the estimates should not be

derived from two different sources that employed

similar or different analysis techniques to the same

ultimate set of failure information.
Data Sources

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5-1

5. PLANT-SPECIFIC DATA COLLECTION AND INTERPRETATION

The incorporation of plant-specific data in the

parameter estimates used in a PRA produces risk

estimates that reflect the actual plant experience. A

plant-specific data analysis also allows comparison of

plant equipment performance relative to an industry

average (the generic value). A plant-specific data

analysis will identify those components or systems

whose performance is worse than the industry average.

It may also identify components or systems with betterthan-average performance.

As indicated in Chapter 4, the raw failure data needed

for a plant-specific data analysis is dependent upon the

scope the analysis. The scope can include accident

initiating events, component failure events and

unavailablilities due to maintenance or testing, and

recovery events. Typical sources of raw data available

at nuclear power plants for each of these type of events

are identified in Section 4.1. The information needed

may have to come from multiple sources. 

Interpretation and reduction of the raw data is required

to obtain the reduced data used in the parameter

estimation models described in Chapters 2 and 6. The

reduction of the raw data includes consideration of

issues such as pooling of identical component data, the

mode of operation the plant was in when a failure

occurred, and the severity of the event. Additional

issues concerning data reduction, such as aging and

time impacts, are addressed in Chapter 7.

This chapter describes a process for collecting and

reducing raw data for the purpose of generating plantspecific data for use in a PRA. Because nuclear power

plants collect and record raw data in different ways, the

process described is general in nature but, sufficient to

successfully collect and reduce available date for use in

a PRA. Some practical concerns and issues related to

the scope and performance of plant-specific data

analysis are also presented. 

A process for reducing the data necessary to calculate

initiating event frequencies, component failure data, and

recovery event data are presented in Sections 5.1, 5.2,

and 5.3, respectively. The reduced data obtained in this

process are combined according to the guidance

provided in Chapters 2 and 6 to obtain the parameters

necessary to quantify PRA models. 

5.1 Initiating Event Data

The methods for evaluating plant-specific initiating

event frequencies provided in Chapter 6 require the

number of initiating events of interest and the time

period over which these events occurred. Guidance is

provided in this section for collecting and interpreting

this required data.

5.1.1 Initiating Event Categories

The initiating events of interest in nuclear power plant

PRAs are dependent upon the mode of operation that

the plant is in. For power operation, the events of

interest are generally reactor scrams but can also

include unplanned forced shutdowns. Typical initiating

events during power operation include multiple

categories of plant transients and loss-of-coolant

accidents (LOCAs). Trips from zero power or low

power may be excluded as valid initiating events in a

full power PRA if their occurrence is precluded during

full power operation. However, low power events

should be considered as valid initiating events at full

power if they can occur during full power. For

shutdown modes of operation, the reactor is already

subcritical and thus the events of interest are somewhat

different. Typical initiating events modeled in

shutdown PRAs include loss of decay heat removal

events, reactivity insertion events, and LOCAs or draindown events.

It is a standard practice in PRAs to group initiating

events into categories based on their impact on certain

plant systems, and according to the demands they make

on other plant systems needed for accident mitigation.

Examples of typical initiating event categories include

loss of offsite power, loss of feedwater, main steam

isolation valve (MSIV) closure, and large, medium, and

small LOCAs. Lists of typical transients that have

occurred at nuclear power plants while at full power

have been categorized by EPRI (1982) and the INEEL

(Mackowiak et al. 1985 and Poloski et al. 1999a).

Typical initiating events to consider during low power

and shutdown conditions have also been established for

both boiling water reactors (BWRs) (Staple et al. 1999)

and pressurized water reactors (PWRs) (Chu et al.

1993).

5.1.2 Data Window

The time period for collecting initiating event data

should be as broad as possible. In general, data from all

of the years of plant operation should be considered.

However, screening of the data can be performed to

eliminate unrepresentative events (see the next section).

One screening technique used in general practice is to

eliminate the first year of operational data as

unrepresentative. 
Plant-Specific Data Collection and Interpretation

5-2

Since the number of plant events can decrease over time

due to improvements in the design and operation of the

plant, it is desirable to have the data reflect the most

recent operating experience. This can be accomplished

by considering only the data from the most recent years

of operation. Alternatively, an analyst could perform a

trend analysis of the data (see Chapter 7). 

5.1.3 Initiating Event Data Allocation and

Screening

To allocate plant-specific event data to the initiating

event categories modeled in the plant PRA, it is

necessary to establish the status of the plant, including

its power level at the time of the event and the impact of

the event on the plant systems. Such information is

generally available in the raw data sources discussed in

Section 4.1 that are available to identify initiating

events (i.e., LERs, scram reports, and monthly

operating reports). 

For initiating events during power operation, the events

of concern are those that result in a reactor trip or

forced shutdown. To allocate these events to the

appropriate initiating event category, a data analyst

must examine the sequence of events prior to and

immediately following the reactor trip/shutdown. The

initial plant fault leading to a sequence of events that

eventually result in an automatic or manul reactor trip

or unplanned shutdown is used in categorizing the

event. For example, one plant trip may have been

initiated by spurious closure of the MSIVs and be

identified as an MSIV closure transient. Another event

may be initiated by a loss of condenser vacuum which

produces a closure of the MSIVs. This event may also

be placed in the MSIV closure transient category,

unless some significant difference in the plant response

is identified.

The initiating event data analysis can also be used to

help establish the conditional probability of events

subsequent to the event actually leading to the plant

trip. Examples of this include the failure of the reactor

protection system leading to an anticipated transient

without scram (ATWS), and the occurrence of a relief

valve sticking open leading to a transient-induced

LOCA. 

It is possible that some events leading to plant scrams

(or loss of heat removal during a shutdown mode of

operation) can be eliminated from the data analysis.

One acceptable reason for eliminating initiating event

data involves design or operational changes that may

have been made to reduce the frequency of reactor

scrams. Such changes to the plant design or operation

can eliminate the occurrence of failures that have

occurred in the past. For example, a plant may have

experienced a significant number of loss of feedwater

events due to the design of the feedwater control

system. As a result, a utility may have replaced the

feedwater controller with a new, more reliable design

that eliminated the occurrence of loss of feedwater due

to controller faults. The data analyst can thus eliminate

past events initiated by faults in the old feedwater

controller from consideration.

Changes in the plant design or operation can also affect

the classification of events. The following example,

provided in EPRI TR-100381 (EPRI 1992), illustrates

this point. The MSIV vessel level closure set point at

some BWRs has been lowered from Level 2 to Level 1.

As a result, the fraction of initiating events that lead to

MSIV closure may be different before and after the

design change implementation and the total historical

count of MSIV closure events may not be valid for the

current condition of the plant. One approach for dealing

with such a design change is to eliminate all events

prior to the design change that result in MSIV closure

due to the generation of a low vessel level. This

approach has the undesirable impact of reducing the

sample size. An alternative is to review the past events

to determine if the MSIVs would have closed with the

revised closure set point in place. However, this may

be difficult to determine from the available information.

5.1.4 Selection of Exposure Time

For estimating the frequencies of initiating events that

occur during any plant operating mode, the appropriate

exposure time is the number of calendar years of

operation corresponding to the period of time the

initiating event data is collected. Expressing the

frequency of initiating events on a calender year basis

allows for evaluation of risk in each mode on a

consistent and average basis. 

However, it may be necessary to generate the initiating

event frequencies based on the time the plant is in the

particular mode of operation. For example, initiating

events during power operation are often expressed in

terms of events per critical year (one critical year

represents 8760 hours of reactor criticality). Since

generic initiating event frequencies are often expressed

in events per critical year (Poloski 1999a), calculation

of the plant-specific frequencies in this same unit is

required for combining the two values using Bayesian

techniques (see Section 6.2.2). To determine at-power

initiating event frequencies, the plant-specific

frequencies expressed as events per calender year have
Plant-Specific Data Collection and Interpretation

5-3

to be increased by dividing by the fraction of time the

plant was at power. This fraction is called the

criticality factor and may be determined from the

control room logs or the Grey Books where the

residence times in each of the operational modes are

recorded. Criticality factors for each plant are provided

in Appendix H of NUREG/CR-5750 (Poloski 1999a)

for the years 1987 through 1995. Alternatively, the

generic frequencies may be divided by the average

criticality factor (0.75 for the data reported in

NUREG/CR-5750) to obtain generic data expressed in

the same units as the plant-specific data (i.e., events per

calender year. For example, suppose an event is

expected to occur 1.6 times every calender year, on

average, and that the criticality factor for a specific

plant is 0.8 (i.e., the reactor has been critical 80% of the

time). Then, the same event correlated to units of

critical years is 2 events per critical year (1.6

events/calender year divided by 0.8 critical

years/calender year).

5.2 Component Failure Data

The raw data sources containing equipment operating

records in a nuclear power plant typically document

tens of thousands of component malfunctions over the

plant’s lifetime. The records may be kept in various

forms including hard copies of maintenance work

orders or a computerized file. The most useful raw data

sources provide information on the specific component

affected, the observed problem, and the action taken.

To calculate plant-specific component failure rates and

unavailability from the data in these records, the data

analyst must identify those malfunctions that cause

component functional failures and also determine the

corresponding number of demands or operating time.

This section describes this process and some of the

practical concerns required to extract the necessary

data. 

5.2.1 Component Data Identification

The first step in evaluating plant-specific component

failure rates is to identify the components and their

failure modes that will be analyzed. This step is usually

done in coordination with other PRA analysts (typically

those analysts that generate system models such as fault

trees). This coordination is critical because it focuses

the component data analysis on only those components

and their failure modes that appear in the PRA models

and establishes the definitions of the component

boundaries.

It should be noted that extremely reliable components

may never have failed in the history of the plant. This

lack of failure history makes it difficult to estimate the

true failure rate or probability. Reliable components

can generally be identified by reviewing failure rates in

generic data bases. However, the analyst is cautioned

in the use of this data since a usually reliable

component may not be reliable at a particular plant. In

addition, it is often impossible to identify the number of

demands or run times for certain components (for

example, the number of demands placed on a relay)

using the existing plant records.

5.2.1.1 Data Window

Plant-specific data is selected over a sufficient time

period to provide statistically meaningful results. Use

of data from throughout the plant history is preferred

since they will be less subject to random variability.

The following examples from EPRI TR-100381 (EPRI

1992) illustrates the amount of data required to achieve

an acceptable sample size. 

“With no failures, the statistical significance can

be measured by the 95th upper confidence limit.

To establish a 95th confidence limit on a failure

rate of 1E-3/hr, the required cumulative run time

for the population is 3,000 hours, to establish a

95th confidence limit of 1E-4/hr requires 30,000

hours. Thus, if a failure rate is believed from

generic data to be relatively low, one should

expect to have to collect a significant amount of

run time before making an impact on the generic

values.

“When failures are recorded the statistical

significance can be measured by the range from

the 5th to the 95th percentile confidence bounds.

This decreases with the number of failures. For

a Poisson distribution, the range from the 5th to

the 95th percentile is on the order of 10, with 2

failures. Thus, for greater than 2 failures the

sample is very loosely comparable to the

lognormal with an error factor of 3. Thus, for a

population of components, a total number of

failures of 2 or more is a reasonable sample

when compared with typical generic data bases.

This is true for the binomial distribution also, as

it approximates the Poisson distribution when

the parameter, p, is on the order of 10-3. These

considerations can be used to establish a

reasonable time frame for data collection.

Suppose, the generic data is on the order of l0-

3

per demand, and there are four components in
Plant-Specific Data Collection and Interpretation

5-4

the population with approximately one demand

per component per month per ISI tests. To get 2

failures, we would expect to require about 2/p

demands, or 2,000 demands. There are 48

demands per year, therefore data from 41 years

would be required to produce this statistically

meaningful data. This illustrates the importance

of making sure that all the demands are counted

and also of increasing the size of the population

if at all possible.”

5.2.1.2 Data Collection

For the list of components and their failure modes

selected for data analysis, the system analyst must

retrieve all failure, maintenance, and test records for

each component from the raw data sources generated

during the data window. The required records are

generally obtained based on the component

identification number. Because the component

boundary can include multiple piece parts, the required

records may be kept under multiple identification

numbers. However, for some components, the data

records for the different piece parts may all be kept

under the same identification number. Thus, it is

necessary to list the identification numbers for all the

piece parts included in the component boundary

definition.

Because component failures are generally infrequent, it

is preferable to pool the data from several components

to obtain a larger data base. For example, it is common

to group like pumps within a single system into one

population, but less common to group the pumps of

different systems (although it can be acceptable to

group pumps of different systems with similar

characteristics together into one population). Any

grouping of components requires careful consideration

of the similarity of their design (e.g., size or

manufacturer), the frequency of operation, their

environmental operating conditions (e.g., temperature,

humidity, and radiation), operating modes (e.g., standby

versus normally operating or intermittently operating),

and the medium they carry (e.g., air, pure water, or

borated water). Tests for poolability of data are

described in Chapter 6. 

5.2.2 Event Screening and Severity

Classification

The raw data for a specific component will contain

some events that are not relevant to the component

failure modes being analyzed. These events can be

screened from further analysis. Some of the events will

be component failures that should be included in the

data assessment. The type of component failures will

determine how they are classified and subsequently

used to generate the required component failure data.

Guidance for both event screening and classification is

provided below.

5.2.2.1 Event Screening

One consideration in the identification of plant-specific

data is whether design changes have been made to the

plant or its components that invalidate some of the

historical data. For example, changing the type of flow

controller could impact the operation of a particular

turbine-driven pump. Thus, the total historical count of

the turbine-driven pump events is not valid for the

current condition of the plant. Typically, the turbinedriven pump data prior to the design change would be

deleted from the data analysis. However, this has the

undesirable impact of reducing sample size. Another

approach is to investigate whether there is indeed a

significant difference in the fraction of events before

and after the design change. Not all the failures may be

invalidated by the design change and so the historical

data prior to the design change implementation may

have partial validity and could be included in the data

analysis. 

Consideration of design changes is one example of

where censoring of data can and should be performed.

Other reasons can be used for data censoring if they are

well supported and valid. For example, it is not

uncommon to eliminate data from the first year of plant

operation since it represents failures that occurred

during the plant break-in period. However, any data

censoring should be approached carefully to avoid

losing important information and biasing results

(eliminating the first year of data actually makes the

results less biased).

5.2.2.2 Event Severity Classification

As discussed in Chapter 3, component malfunction

events are commonly classified into one of the

following three event severity categories:

• catastrophic failures,

• degraded failures, and

• incipient failures.

Catastrophic failures require some kind of repair or

replacement action on the component in order to restore

the component to operability. Events that are classified

as catastrophic failures are used in calculating plantspecific component failure rates and probabilities of
Plant-Specific Data Collection and Interpretation

5-5

failure on demand. Information on catastrophic failures

occurring during critical operation is also used in

calculating maintenance outage unavailabilities.

Degraded failures can prevent a system or train from

meeting the success criteria modeled in the PRA. An

incipient failure is such that there is no significant

degradation in performance but there are indications of

a developing fault. The difference between the two is

generally a matter of severity. Events classified as

incipient or degraded failures are generally used in

calculating plant-specific maintenance unavailabilities.

Although both degraded and incipient failures will

typically lead to a corrective action, the corrective

action may or may not make the component unavailable

to perform its function. For example, maintenance on

the operator of a valve that is normally open will not

lead to the unavailability of the valve if is required to

open for system operation. This illustrates the

importance of ascertaining from event records the

modes of a component operation that a corrective action

would prevent. 

Sometimes the event information is so unclear and

incomplete that a definite classification of the severity

of a component malfunction event is not possible. The

data analyst in this situation is faced with the difficult

task of deciding whether to call a malfunction a failure

or not. The inability to distinguish between severity

levels of failures is particularly important. The

difference between the probabilities of catastrophic and

degraded modes of failures can be significant especially

when dealing with highly reliable components that

rarely fail. The difference between no failures and one

failure in estimating the failure rate is much more than

the difference between 10 and 11 failures. Thus, the

data analyst must be careful when classifying the few

failures that may have occurred. In the absence of

sufficient information, the tendency is to conservatively

record such events as catastrophic failures. This is

reasonable as long as the impact on the final PRA

results is not significant. For cases where the

judgement of the data analyst is important to the PRA

results, it could be incorporated explicitly into the PRA

quantification as a source of uncertainty. This issue is

discussed further in Section 6.1.2.2.

5.2.3 Component Data Allocation

This section gives guidelines on the allocation of plant

specific events to each component failure mode of

interest. This includes the allocation of events

contributing to the unavailability of components or

systems due to test and maintenance actions. The goal

of this allocation process is to correlate each event

report with one or more basic events of the PRA model.

This requires that the event report be identified with a

specific component, and that the severity of the event be

determined and associated with the proper component

failure mode(s).

The use of component identification numbers in event

reports is generally sufficient to allocate the event to a

particular component. The description of the event can

also guide the data analyst to a particular component

failure mode (i.e., a basic event in a fault tree), or in

some cases, to a particular gate in a fault tree.

However, a thorough review of the cause of the event

together with a knowledge of the boundaries of the

basic events of the fault trees is generally needed for a

correct allocation to be made. For example, an event

report identified with a specific motor-operated valve

(MOV) that involves the deenergization of a 480V bus

should be associated with the bus unavailability and not

the MOV. If the event is a local fault of the MOV or its

breaker, it is associated with MOV itself. 

As discussed previously, the severity of the event is

important in allocating the event to specific component

failure modes. A catastrophic component failure will

generally result in an extended period during which the

component is unavailable while it is being repaired.

Thus, an event involving a catastrophic failure must be

counted in estimating the failure of the component to

operate and in estimating its unavailability due to

maintenance. Degraded and incipient failures are used

in calculating plant-specific maintenance

unavailabilities. Some degraded failures may result in

sufficient degradation that it can not meet its required

success criteria (e.g., the flow rate for a pump is

reduced to 300 gpm when 500 gpm is required for

success). In such cases, a degraded failure is also

included as a component failure to operate. 

5.2.3.1 Component Failure Event Allocation

Because of the variability in the level of reporting

associated with maintenance events, the allocation of

event reports to specific PRA model events can be a

subjective process. The following are some ground

rules to help perform the component failure event

allocation. The majority of these ground rules have

been identified and published in EPRI TR-100381

(EPRI 1992). Additional guidelines are based on the

experience of PRA vendors and NRC data analysts. 
Plant-Specific Data Collection and Interpretation

5-6

1. For standby components such as pumps, diesel

generators, and fans, PRA models generally

distinguish between failure to start and failure to

run modes. It is important to understand the

definition of each failure mode in order to

associate historical maintenance events with the

different basic event types. For example, if a fault

tree basic event represents a failure of a pump to

start, it usually means exactly that. However, it is

not unusual in PRAs to define “diesel generator

fails to start” as encompassing a failure to start or

a failure during the first hour given that the start

was successful. Whatever definitions are used, the

event allocation must be performed to match them.

2. As indicated in Chapter 2, there are two ways to

model failures to start: the demand failure and

standby failure models. In the demand failure

model, the equipment is ready to operate but for

some reason, does not start or change state when

demanded. In the standby failure model, the

equipment has developed an unannounced

condition that will prevent it from starting when

demanded. When reviewing raw data, it can be

difficult to identify whether a component failed on

the demand or prior to the demand. Thus, as

indicated in Section 2.3.4, either model could be

used in this situation. The demand failure model

provides the higher failure probability.

3. A catastrophic or degraded failure that is revealed

while a component is in the standby mode, and that

results in a maintenance action, is accounted for in

the unavailability due to maintenance event for that

component. If the failure is such that it could also

occur while the component is performing its

mission, it should also be counted as a component

failure. For example, external leakage above

allowable amounts from a standby pump that

requires isolation of the pump to repair it,

contributes to the unavailability of the pump due to

maintenance. Since such leakage could occur

during pump operation, the event should also be

used to determine the failure rate for pump

leakage. The amount of leakage would have to be

sufficient to prevent the pump train from delivering

the required flow. 

4. Catastrophic failures of standby equipment to start

(or run) that occur during an actual component

demand, contribute to that failure mode. Similarly,

failures to start (or run) during tests that closely

mimics the conditions that the component would be

subjected to during an unplanned demand should

also be included in the evaluation for the

component failure mode. 

5. Degraded failures that are not serious enough to

prevent the component from performing its

function are not included as failures of the

component. Expressed in another way, the failure

of the component must match the definition of the

failure in the PRA model. For example, vibration

in a pump that results in the pump only delivering

500 gpm instead of the rated flow of 600 gpm is

not a failure event if 500 gpm is sufficient to meet

its function and the pump continued to supply that

flow for a period at least equal to the mission time

required in the PRA model. However, such

failures would be included in the unavailability due

to maintenance since their effect is to induce

maintenance activity.

There is a caveat to this guideline to consider. If

the degraded failure is revealed in a short test

duration, an analyst cannot be sure the component

would have succeeded over its mission time. In

this case, the analyst can attempt to extrapolate the

rate of degradation to determine if the component

would meet its failure criteria sometime during its

mission time. For example: a pump develops a

slow oil leak during a test. If the rate of leakage is

such that the pump would run out of lubricating oil

during the required pump mission time as modeled

in the PRA, than the event is considered as a pump

failure to continue to run.

6. Degraded conditions for which a failure would

have occurred if the system had been demanded are

considered a failure. For example, if an operator

discovers that a pump had no oil in its lubrication

reservoir, the pump may have started (unless there

was an interlock preventing a pump start on low oil

level) but likely would not have run long. In either

case, this event would be counted as a failure to

start.

7. If the event report identifies that the failure of

component A is the result of the failure of another

component B that is modeled explicitly in the

PRA, the event is associated with component B

and not with component A. For example, failures

of a pump from plugged suction screens should not

be allocated as pump failures if the screens are

modeled separately. 

The clear identification of the component boundary

is an important factor in these situations. For
Plant-Specific Data Collection and Interpretation

5-7

example, the allocation of an event that identifies

the failure of an emergency pump due to the failure

of a safety actuation signal is dependent upon

whether the actuation logic is included in the pump

boundary or is treated as a separate event in the

model. Typically, the components related to the

safety actuation signal are not included in the pump

boundary definition and this event should not be

counted as a pump failure. However, if the safety

actuation signal is included in the pump boundary,

then the command fault should be included as a

failure mode of the pump.

8. An event reporting a degraded or failed state of a

redundant piece part should be excluded from the

failure events if the component boundary includes

the redundant piece parts. For example, if a diesel

generator has two redundant air start motors that

are included in the diesel generator boundary

definition, failure of one air start motor would not

be counted as a failure of the diesel generator.

This example illustrates how a coarse definition of

a component boundary can result in the failure to

account for some degraded component states. 

9. If a documented failure during a test or actual

demand could not be repeated on subsequent tries,

it may not have be included as a potential failure.

Similarly, events which are very quickly

recoverable may also not be considered potential

failures (the recovery should not be included in the

PRA model) . Whether an event meeting either of

these situations should be considered a failure is a

function of the success criterion for the component

in terms of the time window within which it has to

operate. For example, the spurious closure of an

MOV may prevent the injection of coolant into the

core from a particular system. However, the event

records may indicate that in all such occurrences,

the valve was quickly reopened before coolant

levels dropped to unacceptable levels. In such

cases, the events should not be considered as

failure events for the MOV.

10. Successive failures of the same components over

short time intervals should be counted as a single

failure. Similarly, failures of a component during

post-maintenance testing where the failure is

related to the maintenance or to an earlier failure

that the maintenance was trying to correct should

be considered as a continuation of the original

failure and should be disregarded. The successive

failures are because proper maintenance was not

performed to fix the initial problem, and the

component is still in the failed state.

11. If failures resulting from human errors after testing,

maintenance, and instrument miscalibrations are

explicitly included in system models, these events

should not be included as component hardware

failure events. Such events are typically quantified

using human reliability analysis methods.

However, some PRAs have not explicitly included

these human errors in the models. In such cases,

the contribution from human-related failures

should be incorporated into the appropriate

component failure rate or probability. 

12. An event reported as a failure to meet technical

specifications, but which would not result in a

catastrophic failure in the PRA sense should not be

included, but it may lead to a maintenance

unavailability. For example, the failure of a diesel

generator to start and pick up loads within 10

seconds might be a reportable failure for regulatory

purposes. However, in the PRA sense it is not a

failure if the diesel did not pick up loads in 10

seconds and the “failure” did not have a discernible

effect on the ability of the plant to mitigate an

initiating event. However, this failure would

require maintenance to alleviate the fast loading

failure.

13. Failures that occur under abnormal environmental

conditions should be segregated from failures that

occur under normal conditions. These failures can

identify important interactions between systems

and thresholds for failure that should be accounted

for in the PRA. In general, PRAs assume

components fail under harsh conditions. Under

this assumption, actual failure events in harsh

environments can be eliminated from

consideration. For example, actual failures of

electrical components following a loss of a heating,

ventilation, or air-conditioning (HVAC) system

should be eliminated from the data analysis if the

HVAC dependency is modeled explicitly in the

PRA model and the component is always assumed

to fail under those conditions. However, if there

are also many component successes under the same

harsh environments, than a component failure

probability under those conditions can be

calculated and used in the PRA model conditional

on the occurrence of the harsh environment.

5.2.3.2 Allocation of Unavailability Data

Unavailability occurs primarily due to maintenance

activities but some minor contributions can also result

from testing performed during periodic surveillance

activities. These unavailability contributions can be
Plant-Specific Data Collection and Interpretation

5-8

included in a system model at a component, segment, or

train level. In addition, separate basic events for

maintenance and testing unavailabilities, or for planned

and unplanned unavailabilities can be included in

system models. In a data analysis, the allocation of

unavailability data must be performed to match the

basic events in the system models. The following

guidelines are useful in allocating events for

determining unavailabilities due to test and

maintenance. These ground rules have been extracted

from EPRI TR-100381 (EPRI 1992) and from the

experience of PRA vendors and NRC data analysts. 

1. A maintenance event must result in the component

not being capable of performing its function, as

modeled in the PRA, in order to contribute to the

component or train unavailability. For example,

maintenance performed on a normally open MOV

(that is required to stay open during its mission

time) with the valve locked in the open position is

not an event of interest. Similarly, a maintenance

event involving some electrical repairs on an MOV

that do not necessitate moving it from the position

required for successful system operation is also not

an event of interest. However, in either case, if the

valve were required to close for any reason, then

both events would be of interest. 

2. Some testing procedures may result in component,

train, or system unavailability. For example, a full

flow test of a system through a test path could

require that a normally closed injection valve be

disabled in order to prevent inadvertent injection.

The injection valve would be unavailable during

the test period. However, systems often have logic

which would actuate the system even if it was

being tested. In this situation, there would be no

system unavailability due to the test. A review of

testing procedures coupled with knowledge of

system actuation logic is required to determine if

testing can result in component, train, or system

unavailability.

3. If a maintenance report indicates that one or more

trains of front line systems are unavailable due to

maintenance activities of a support system, the

unavailability is associated only with the support

system.

4. If while performing maintenance on a support

system, maintenance is also performed on the front

line system it supports, the unavailability of the

front line system should be counted if the two

maintenance activities are not always performed

together. 

5. If an unavailability on one component is actually

due to maintenance activity on another component

that is included in the PRA model, the

unavailability is associated with the second

component only. For example, a declared

unavailability of a pump due to maintenance on a

room cooler should be included only as a

maintenance on the room cooler if the dependence

of the pump on the room cooler was modeled

explicitly. As another example, if the maintenance

results in the unavailability of a source of suction

to a pump (e.g., maintenance on a supply tank),

then it is better to model this as an unavailability of

the source rather than the pump. Assigning the

event to the source unavailability is absolutely

required if the source is shared with other pumps.

In general, maintenance unavailability should be

allocated consistent with the component

boundaries and system modeling.

6. There may be events where the unavailability of a

component in a system model is due to

maintenance on a component that is not included in

any system model. In such cases, the event should

be included as an unavailability of all the modeled

components removed from service. For example,

the contribution of maintenance on a drain valve

for a pump train will likely not be modeled in the

PRA but should be included as a contributor to the

unavailability of the entire pump train since it

would likely result in isolation of the train.

7. Coincident outage times for redundant equipment

(both intra- and inter-system) should reflect actual

plant experience. For some systems, the available

redundancy may be higher than that limited by

technical specifications. In this case, maintenance

may be performed on two out of three trains at the

same time. The modeling of dual component

maintenance events in the PRA should be

consistent with the actual plant experience. Note

that because of the allowed outage time limitations

in technical specifications, the maintenance

unavailability may be lower when two trains are

taken out for maintenance.

8. The maintenance data at the plant most likely will

contain planned and forced maintenance. Most of

the maintenance events will be forced type. If the

PRA models the two types of maintenance

separately and it is possible to distinguish between

the two types in the data, these should be recorded

separately.
Plant-Specific Data Collection and Interpretation

5-9

9. In some cases, more than one maintenance activity

may be recorded on an event report. When this

occurs, each separate maintenance activity must be

considered at the highest possible component level.

For example, if the suction or discharge valve of a

pump requires maintenance, the pump would be

tagged out for the duration of the work. As

previously discussed, the maintenance

unavailability should be associated with the valve.

If during this maintenance outage, some minor

maintenance was performed on the pump, than the

entire maintenance outage can be recorded as a

pump maintenance event. The duration of the

maintenance would be the time between when the

first component is tagged out and when the last

component is tagged in.

However, if the maintenance unavailability is being

modeled in the PRA at the train level, all

maintenance activities on any component are

included. In this situation, each maintenance event

on any component in the train is included. If

multiple components are tagged out during the

maintenance event, the duration of the maintenance

would be the time between when the first

component is tagged out and when the last

component is tagged in.

10. Functional dependencies represented in the PRA

models must be considered in the allocation of

maintenance events. For example, if a chilled

water pump is taken out for maintenance, together

with an HVAC chiller that it supports, only the

chilled water pump is counted as being unavailable

for maintenance. The functional dependency

between the two components in the PRA model

will account for the chiller being unavailable when

the chilled water pump is under maintenance.

11. The cold shutdown periods in the time window

over which data are being collected should be

defined. The maintenance performed during

shutdown is not included in the determination of

component unavailability during power operation.

12. Special attention is required when allocating

maintenance events for systems or components

shared between units at a site. The technical

specifications pertaining to shared systems can be

different depending on the status of both units.

The PRA model may include basic events to

account for the dependence of the system

unavailability on the mode of operation for each

unit. In such cases, the maintenance events should

be allocated to match those event definitions.

5.2.4 Component Exposure Evaluation

The data identification and allocation process discussed

in the previous sections results in the identification of

the number of events associated with each component

failure mode. To generate component failure

probabilities and rates, it is also necessary to estimate

the operational exposure of the components. The term

“exposure” refers to the amount of component

operating time when considering failure rates and to the

number of demands (or cycles) when considering

failure probabilities. 

Exposure data are normally developed by reviewing

plant documents; e.g., test procedures and the

knowledge of component function (standby, normally

operating, etc.), and systems lineup. In some cases, an

operation time meter provides information about the

cumulative hours of operation of a component. 

Development of exposure data involves many

judgments and assumptions. The guidance provided in

this section sometimes leads to an approximate value

for the exposure data, which may differ substantially

from the actual experience. Although typically the

range of uncertainties associated with the exposure data

are much smaller than those for the failure data, there

may be cases where the combined effect of uncertainty

about the exposure and failure has a significant impact

on the estimate of the failure rate or probability. The

issue of uncertainty in the data (both in the failure and

exposure data) is addressed in Section 6.1.2.2 of this

handbook. 

The following sections outline the process for

estimating the number of demands and the operating

time for each component. Much of this guidance is

taken from EPRI TR-100381 (EPRI 1992).

5.2.4.1 Time-Related Exposures

The operating or exposure time for a component is

dependent upon whether the component is normally

operating or is in standby. For components that are

required to continuously operate during a particular

plant mode, the operating time can be easily established

by directly relating it to the time spent in that plant

mode. 

Some plant systems, sometimes called alternating or

intermittently operated systems, have multiple

redundant trains where only a subset of those trains are

required to operate at any one time. A standard practice

at nuclear power plants is to alternate the trains that are
Plant-Specific Data Collection and Interpretation

5-10

operating and in standby at specified intervals. The

times of operation and changeover from one train to

another are typically recorded in the control room or

some other log book. However, since the pumps in

different trains of a system are usually grouped together

for data analysis, it is not necessary to have an accurate

log of how long an individual pump was in operation.

Instead, it is only necessary to evaluate the exposure

time for the pumps as a group. For example, if two of

three pumps are normally operating in a particular plant

mode, the total operating time for that pump group is

twice the calendar time spent by the plant in that mode.

For a component in a standby system, the operating

time is generally given by the time the system is

operated during testing. Note that an important

criterion for including test data when evaluating both

the failure and exposure data is that the test should

mimic the component operation that would be required

in an unplanned demand. The testing period may be

recorded in control room logs or other logs. The

operating time during testing for a population of

components may also be estimated by summing the

product of the component population, test frequency,

and test duration for each test during the period where

failure data was collected. It should be noted that for

most plants, and most components, the cumulative run

time during testing is relatively short. 

Some systems that are in standby during normal power

operation are also used during other modes of

operation. For example, the residual heat removal

(RHR) system in both BWRs and PWRs is used during

shutdown. Similarly, a standby system may be used

during power operation for a special purpose. For

example, the RHR system in a BWR may be used to

increase or decrease the suppression pool level. Thus

the operating times during these modes of operation

should be included, in addition to the run times during

testing, if any failures during these modes are pertinent

to the safety function of the system (e.g., the entire

RHR pump operating history may be pertinent since the

pump must operate when the RHR system is used to

respond to an accident). In such situations, the times of

startup and shutdown of the standby system may be

recorded in the control room logs. Alternatively, if the

component is required to continuously operate during

shutdown , the operating time can be easily established

by directly relating it to the time spent in that plant

mode.

5.2.4.2 Demand-Related Exposures

To evaluate the probability of the failure of a

component to start or change states, the number of

demands experienced by the component must be

evaluated. Although this would seem to be a simple

process, in practice the number of demands is often one

of the most difficult parameters to calculate accurately.

Component demands from all contributors should be

included. This can include contributions from testing,

automatic and manual actuations, and corrective

maintenance. The methods of calculating the number

of demands from each of these types of demands are

explained below.

5.2.4.2.1 Test Demands

Periodic testing is an important source of demands for

components in standby systems. The surveillance

testing and required frequency for the plant is

performed in accordance with the technical

specifications. However, some plants may choose to

perform testing more frequently than required by the

technical specifications. 

An important criterion for including test data in

evaluating both the failure and exposure data is that the

test should mimic the component operation that would

be required in an unplanned demand. 

Surveillance procedures identify the components that

must change state at each test. For each surveillance

test pertinent to the system, it is important to identify

which components are operated, the unavailability of

the system during the test (if applicable), and the

frequency and duration of the test. A functional test of

a pump often requires the operation of valves as well as

the pump and is an important source of information on

valve demands. Neglecting demands on components

from tests on other components can lead to a significant

underestimation of the total number of demands. The

number of test demands for individual components may

be determined from the actual number of tests as

recorded in a control room or test logs or be estimated

based on the test frequencies. 

It should be noted that the test may not be a valid test

for all the components within the component boundary.

For example: the automatic initiation portion of a

component circuit will not be tested during a test where

the component is manually initiated. For components

such as diesel generators, tests which start the engine,

but do not close the breaker onto the bus are not true

tests of the capability of the diesel generator to provide

the necessary load. Note that if there is a

subcomponent that is included in a component’s

boundary which is not tested along with the rest of the

component, it is desirable to analyze it as a separate

component.
Plant-Specific Data Collection and Interpretation

5-11

5.2.4.2.2 Automatic and Manual Initiation

Actual unplanned demands on components should be

included in the demand count. For standby safety

system components, some unplanned demands can be

traced back to the occurrence of automatic initiation

signals (both actual and spurious signals). These

signals include emergency core cooling system (ECCS)

initiating signals, turbine trip signals, losses of offsite

power, and reactor scrams. Different groups of

component may be initiated by different signals or sets

of signals, depending on the functions and the system

they are in. Information on the components that can be

initiated by each signal can be identified through

knowledge of the plant. For example, all low-pressure

ECCS pumps in a BWR could be initiated by an ECCS

signal but the motor-operated valves in the ECCS

injection paths would require an additional low vessel

pressure signal before they would open. Information on

the historical number of occurrences of actual or

spurious signals should be available from the plant

records such as the monthly operating reports or control

room logs. 

In addition, manual actuation of systems or components

may occur during plant operation. Two examples cited

above in the discussion of operating time contributors

are also pertinent here. The first is the case where

alternating trains are placed in operation and standby.

The act of switching operating trains results in demands

on components. The second case involves the use of

standby systems to perform special functions. For

example, the RHR system in a BWR may be used to

increase or decrease the suppression pool level. These

special uses also result in component demands. In both

cases, the times of startup and shutdown of the standby

system may be recorded in the control room or other

types of logs.

Finally, manual actuation of systems to respond to

adverse plant conditions is another source of unplanned

demands that needs to be accounted for in the exposure

evaluation. The occurrences of such demands are

generally recorded in LERs, control room logs, and

monthly operating reports.

 

5.2.4.2.3 Corrective Maintenance

Maintenance can result in demands on components in

several ways. Before the maintenance activities are

begun, the operating and maintenance staff make the

maintenance action safe for both personnel and the

system by disabling and tagging out appropriate

components. This then requires some components to

change state resulting in a demand.

In many instances, demands are placed on components

that are not the subject of the corrective maintenance.

The most obvious demands occur when a component is

returned to service. Before restoring the component to

service following maintenance, a complete functional

checkout is usually performed on the component and

other components in the functional loop. The number of

demands on the components resulting from corrective

maintenance is obtained from the number of

maintenance acts on specific components and an

identification of what other components may have to

change state to complete the functional test. Note that

per the guidance in the ASME PRA Standard

(ASME 2002), demands from post-maintenance

testing should be excluded from the exposure

evaluation for the component under maintenance.

Another example of a demand resulting from

maintenance involves testing of redundant trains. If

equipment fails in some systems, the technical

specifications may require that redundant components

be checked for operability before maintenance to ensure

that they are available for service. In many cases, an

increased frequency of surveillance testing of such

redundant components is required. A typical example

of this is reflected in the technical specifications for

emergency diesel generators. These demands need to

be included in the data analysis. 

As indicated in the discussions presented above,

development of exposure data involves many judgments

and assumptions. Although typically the magnitude of

error or the range of uncertainties associated with the

exposure data are small compared with those of the

failure data, there are cases where the combined effect

of uncertainty about the exposure and failure has a

significant impact on the estimate of the failure rate.

The data analyst should consider some level of

uncertainty in using such estimates. 

5.2.5 Determination of Unavailable Time

Following the identification of the maintenance events

contributing to the unavailability of a component, train,

or system, the time the component is unavailable during

each event is determined. The unavailability time is the

time between when the component is removed from

service until it is actually restored to service. In many

cases, maintenance work orders will provide this

information by identifying one or more tag-ins and tagouts for equipment with the date and time of day that

both occur. Using these times to determine the

unavailability time may be a little conservative because

the repair may be completed before the component is

declared tagged in. 
Plant-Specific Data Collection and Interpretation

5-12

Some maintenance work orders may contain multiple

tag-outs and tag-ins for a given component. If the

component was operable between these periods, than

the unavailability is the sum of the individual

unavailability times for each period. However, if the

component was inoperable between the periods, than

the unavailability time starts at the first tag-out and ends

at the last tag-out.

Unfortunately, the actual time of unavailability may not

be recorded in maintenance work order forms. In many

cases, the time recorded may reflect a prior estimate of

how long the maintenance activity will take, may

represent the man-hours taken to complete the task

rather than calendar time, or may include time to

complete paperwork.

When the unavailability time is not specified in a

maintenance work order, other plant documents should

be examined for that information. Maintenance activity

information may be recorded in other documents such

as operator logs or component operating logs. For

example, a maintenance activity on a safety-related

component will start the clock for a limiting condition

of operation (LCO) specified in the technical

specifications, and this should be recorded in some

place, usually the control room log. The time when the

function is restored should also be recorded.

Unfortunately, not all maintenance events result in an

LCO and thus timing information may not be available.

When reliable estimates of the start and finish times for

a maintenance event are not available, one recourse is

to ask plant maintenance and operations staff to provide

estimates of the ranges in the unavailable time per

maintenance act for the components. Another recourse

is to use data provided from some maintenance events

to estimate the unavailability for other events. 

5.3 Recovery Event Data

In PRA, there is a clear distinction between actions to

repair components or systems and actions to recover

components or systems. Recovery actions involve the

use of alternate equipment or means to perform a

function when primary equipment fails, or the use of

alternate means to utilize equipment that has not

responded as required. Examples of recovery actions

include opening doors to promote room cooling when

an HVAC system fails, recovering grid-related losses of

offsite power by rerouting power, manually initiating a

system when the automatic actuation signal fails,

bypassing trip logic using jumper cables, and using a

handwheel to manually open an MOV when the motor

fails to operate. Repair actions involve the actual repair

of the mechanism which caused a component or system

to fail. Examples of repair actions include repairing

weather-related losses of offsite power, repair of a

pump that failed to start, or replacement of a failed

circuit breaker.

PRA models typically include a number of recovery

actions of the type identified above. However, because

recovery actions can involve complicated actions that

are governed by procedures, most are typically

evaluated using HRA methods. A general exception is

the treatment of offsite power recovery where the

required recovery actions are often not within the

jurisdiction of the plant personnel. Thus, offsite power

recovery data is collected and reduced for use in PRAs.

The repair of components is generally not modeled in

PRAs since:

• the time available to repair most components is

generally too limited (i.e., core damage would

occur before the repair is completed), 

• because repair is an action that is not always

governed by procedures and thus difficult to

justify, 

• the availability of spare parts can not always be

certain, and 

• because abnormal procedures generally direct

operators to use alternative equipment as a first

priority. 

There are always exceptions to these general

observations. For example, the replacement of fuses is

an action identified in some fire abnormal procedures

and can be accomplished rather quickly since spare

fuses are available. As with a recovery action, either an

HRA or data reduction approach could be utilized to

generate a failure probability for a repair action. 

The modeling of recovery and repair actions in PRA

reflects the need to accomplish the action within some

time frame (e.g., before core damage occurs). Thus, the

collected data must include both the time of failure and

recovery to be utilized in the PRA. This section

provides guidance on the process for collecting and

reducing recovery and repair data. A description of the

type of data that is reviewed in this effort and guidelines

for allocating that data.

5.3.1 Recovery Data Identification

Recovery and repair information can generally be

extracted from maintenance records and LERs that
Plant-Specific Data Collection and Interpretation

5-13

identify component and system failures. Thus, the

evaluation of recovery and repair information is an

offshoot of the component failure data review. In

general, only data from actual component and system

demands should be included in the recovery/repair data

evaluation. When failures occur during actual

demands, operators should be strongly motivated to try

to recover the component or system. 

However, if a component or system fails to start during

a surveillance test, the need for repair is not as pressing

and thus not reflective of accident conditions. For this

reason, recovery and repair information for failures

during surveillance tests should be excluded from

recovery/repair probability evaluation.

5.3.2 Recovery Data Allocation

Since component recovery data evaluation should be

performed in conjunction with the component data

allocation, the general rules provided in Section 5.2.3

apply. In addition, the following guidelines are

provided to address allocating recovery data for other

events modeled in the PRA (e.g., restoring offsite

power or reopening main steam isolation valves):

1. Only failures during actual demands are included.

Failures during surveillance tests are excluded as

being nonrepresentative of accident conditions.

For the failures during actual demands, the data

analyst should assess whether the recovery/repair

action was performed under similar stresses that

would occur under accident conditions. Atypical

events should be eliminated or considered to be

sources of uncertainty.

2. For each failure event, the recovery/repair time is

the time between when the failure first occurs and

the time when it is returned to service. Using these

times ensures that the time of the failure, the time

required to recognize it has occurred, the time to

obtain spare parts if required, the actual time to

repair the component or system, and the time to

return the component to service are reflected in the

recovery/repair time. Events that do not include

either time should be excluded from the evaluation.

3. Recovery information on systems or components

resulting from an initiating event can be extracted

from LERs or scram reports. For example,

reopening MSIVs after their consequential closure

(i.e., they are signaled to close following some

other failure) may be included in a PRA for some

initiators. The recovery time for such events are

evaluated from the time the initial failure occurs

leading to MSIV closure to until the closure signal

is removed (by either fixing the original failure or

by bypassing the signal) and the MSIVs in one hot

leg are reopened. The time to perform other

actions that may be required to maintain the

MSIVs open (e.g., starting vacuum pumps) are also

included in establishing the recovery time.

4. Recovery information on systems or components

causing an initiating event can also be extracted

from LERs or scram reports. For example, the

time to recover offsite power initiating events can

be extracted from LERs. However, LERs should

also be searched for occurrences of offsite power

failure following other initiating events. Recovery

information should also be extracted for these

events.
Plant-Specific Data Collection and Interpretation

5-14
6-1

6.7 Unavailability

Portions of Chapter 2

6.3 Failure to Change State:

Failure on Demand

6.3.1 Frequentist Estimation

6.3.2 Bayesian Estimation

(many priors considered)

6.3.3 Model Validation

6.6 Durations

6.6.1 Characterization of

Distributions

6.6.2 Model Validation

6.6.3 Nonparametric Density

Estimation

6.2 Initiating Events

6.2.1 Frequentist Estimation

6.2.2 Bayesian Estimation

(many priors considered)

6.2.3 Model Validation

6.1 Overview

6.4 Failure to Change State:

Standby Failure

6.5 Failure to Run

Figure 6.1. Schematic outline of Chapter 6.

6. PARAMETER ESTIMATION AND MODEL VALIDATION

6.1 Overview

6.1.1 Chapter Contents

This chapter is the heart of the parameter-estimation

portion of this handbook. Section 6.1 gives an important discussion of Bayesian and frequentist inference,

and also a brief discussion of some topics outside the

scope of the handbook. The rest of Chapter 6 presents

statistical techniques for analyzing data for various

parameters. Sections 6.2 through 6.7 cover exactly the

same types of data as Sections 2.2 through 2.6, in the

same order. The two kinds of failure to start in Section

2.3 are split into two sections here, 6.3 and 6.4. The

three most extensive and fundamental sections are 6.2

(initiating events), 6.3 (failures on demand), and 6.6

(recovery times and other durations). The remaining

sections draw on material from these three. Figure 6.1

shows the contents in a schematic way, with arrows

indicating the logical dependencies. For example,

Section 6.4 uses material presented in Sections 6.2 and

6.3.

Each section considers both parameter estimation and

model validation. These two topics are considered

together because checking the assumptions of the model

(model validation) is a necessary part of any analysis.

Separating the model validation from the parameter

estimation might give the erroneous impression that it

is all right to estimate parameters without checking the

assumptions, or that the checks can be performed as an

afterthought.

Under parameter estimation, both Bayesian and

frequentist methods are presented. Under model validation, both graphical methods and formal statistical tests

are given.

Much thought was given to the order of presentation: do

we present the Bayesian estimates first or the

frequentist estimates? In Chapter 6, the frequentist

estimates are typically given first, not because they are

more important or more highly recommended, but only

because the frequentist point estimates are very simple,

the simplest most natural estimates that someone might

try. We cover them quickly before moving on to the

more sophisticated Bayesian estimates. In the cases

where the frequentist estimates are not simple (such as

certain distribution models for durations), Bayesian

estimation is discussed first.
Parameter Estimation and Model Validation

6-2

As in much of this handbook, general explanations are

given in Roman typeface, with boldface used for new

terms where they are introduced or defined. Arial font

is used for examples, and for any extended discussion that applies only to a particular example.

6.1.2 Bayesian and Frequentist Inference

Probabilistic risk assessment (PRA) analyzes accident

sequences in terms of initiating events, basic events,

and occasionally recovery events. 

This handbook is concerned with estimating the frequencies of initiating events, the probabilities of basic

events, and the distributions of recovery times and other

durations. These estimates are propagated through

logical relations to produce an estimated frequency of

the undesirable end state, such as core damage. Moreover, the uncertainties in the parameter estimates must

be quantified, and this must be done in a way that

allows the uncertainty in the final estimate to be quantified.

Two approaches to estimating parameters are the

Bayesian method and the frequentist, or classical,

method. The two approaches are summarized here, and

also in Appendix B.

Both approaches use probability distributions to describe the behavior of random outcomes, such as a

random number of initiating events or a random number

of failures to start. The two approaches differ in the

way they treat uncertainty of unknown parameters.

In the Bayesian setting, probability is a measure of

uncertainty, a quantification of degree of belief. The

Bayesian methodology is used to modify uncertainty in

a logically coherent way, so that “degree of belief” is

rational, not merely personal opinion. In this methodology, each unknown parameter is assigned an initial

prior probability distribution. This does not mean that

the parameter varies randomly, but only that it is

unknown, with the probability distribution modeling

belief concerning the true value. Based on data, the

analyst’s prior belief about the parameter is updated,

using Bayes’ Theorem. The final inference statement

uses the posterior distribution of the parameter to

quantify the final uncertainty about the parameter. It is

conditional on the observed data. Siu and Kelly (1998)

give a simple but thorough introduction to Bayesian

estimation in the PRA context.

The frequentist approach is quite different. The probability of a random event is defined as the long-term

fraction of times that the event would occur, in a large

number of trials. Probabilities are used only for random

quantities, the possible data values. Probability distributions are never used to describe parameters, because

the parameters are not random. When quantifying

uncertainty in an estimate, a frequentist asks questions

such as, “Under similar conditions, what other data sets

might have been generated? From data set to data set,

how much variation would be seen in the parameter

estimate? For any one data set, how far might the

estimated parameter be from the true parameter?” Any

prior or external information about the parameter value

is ignored.

Statisticians have argued vigorously over which approach is preferable. When estimating parameters for

PRA, the Bayesian approach clearly works better, for

two reasons. First, data from reliable equipment are

typically sparse, with few or even zero observed failures. In such cases, it is reasonable to draw on other

sources of information. The Bayesian approach provides a mechanism for incorporating such information

as prior belief. Second, the Bayesian framework allows

straightforward propagation of basic event uncertainties through a logical model, to produce an uncertainty

on the frequency of the undesirable end state. To do

this, it assigns a probability distribution to each of the

unknown parameters, draws a random sample from

each, and constructs the corresponding sample for the

frequency of the undesirable end state. The frequentist

approach cannot handle such complicated propagation

of uncertainties except by rough approximations.

Frequentist methods have their uses, however, even in

PRA. Box (1980) writes “sampling theory [the

frequentist approach] is needed for exploration and

ultimate criticism of an entertained model in the light of

current data, while Bayes’ theory is needed for estimation of parameters conditional on the adequacy of the

entertained model.” This viewpoint agrees with current

PRA practice. The primary use of the frequentist

approach is in preliminary examination of the data, to

check the correctness of model assumptions, and to

decide which model to use. For example, frequentist

methods can help the analyst decide whether data sets

may be pooled or whether a trend is present.

Goodness-of-fit tests and calculation of statistical

significance are commonly used frequentist tools in this

context. Then Bayesian methods are used for estimating the parameters. In addition, frequentist estimates

are often simpler to calculate than Bayesian estimates,

and therefore are useful for rough approximate calculations.

Table 6.1 summarizes the above points.
Parameter Estimation and Model Validation

6-3

Table 6.1 Comparison of Bayesian and frequentist approaches in PRA.

Frequentist Bayesian

Interpretation of probability Long-term frequency after many 

hypothetical repetitions.

Measure of uncertainty, 

quantification of degree of belief.

Unknown parameter Constant, fixed. Constant, but assigned probability distribution, measuring current state of

belief.

Data Random (before being observed). Random for intermediate 

calculations. Fixed (after being 

observed) for the final conclusions.

Typical estimators Maximum likelihood estimator (MLE),

confidence interval.

Bayes posterior mean, credible

interval.

Interpretation of 90%

interval for a parameter

If many data sets are generated, 90% of

the resulting confidence intervals will

contain the true parameter. We do not

know if our interval is one of the

unlucky ones.

We believe, and would give 9 to 1

odds in a wager, that the parameter is

in the interval.

Primary uses in PRA 1. Check model assumptions.

2. Provide quick estimates, without

work of determining and justifying

prior distribution.

1. Incorporate evidence from various

sources, as prior distribution.

2. Propagate uncertainties through

fault-tree and event-tree models.

6.1.3 Uncertainties Other Than 

Parametric Uncertainty

The above discussion might suggest that uncertainty in

the value of parameters is the only uncertainty there is.

That is not the case. Parameter uncertainty, stemming

from having only a relatively small set of randomly

generated data, is the simplest uncertainty to address.

It is the primary uncertainty considered in this

handbook of parameter estimation. However, the

following kinds of uncertainty can also be considered.

Because these subsections discuss material that is

outside the scope of the handbook, first-time readers

may wish to skip immediately to Section 6.2.

6.1.3.1 Uncertainty from Nonrepresentativeness 

of the Data Sources

One issue to consider is that the data come from settings

that do not perfectly match the problem of interest. In

general, this is a difficult issue. For example, suppose

one situation is of interest, but the data come from

equipment with a different manufacturer or different

design, or from equipment operated under different

conditions, or maintained with different practices. Then

it is difficult to quantify the relationship between the

data and the problem of interest. Engineering judgment

is used, and to be conservative the uncertainty

distribution is often assigned a larger variance than the

data alone would call for.

One tractable case is uncertainty of the value of a

parameter for one data source (such as one nuclear

power plant), when data are available from many

similar but not identical data sources (other nuclear

power plants). This case can be formulated in terms of

a hierarchical model, and analyzed by empirical Bayes

or hierarchical Bayes methods, as discussed in Chapter

8 of this handbook.

6.1.3.2 Uncertainty in the Data Counts 

Themselves

There can be uncertainty in the data counts themselves.

For example: it may be unclear whether a particular

event should be counted as a failure, or the number of

demands may not be known exactly. A Bayesian
Parameter Estimation and Model Validation

6-4

method for dealing with uncertainty in PRA data was

first proposed by Siu and Apostolakis (1984, 1986), and

has been used by several authors, including Mosleh

(1986), Mosleh et al. (1988, Section 3.3.4.4), and Martz

and Picard (1995). As outlined by Atwood and Gentillon (1996), uncertainty in classifying the data yields a

number of possible data sets, each of which can be

assigned a subjective probability. The simple approach

is to use an “average” data set, a “best estimate” of the

data, and analyze it. The uncertainty in the data is

ignored, lost, at that point. A better approach is to

analyze each data set, and combine the results. Each

analysis produces a Bayesian distribution for the

unknown parameter(s), and the final result is a mixture

of these distributions. This approach includes the data

uncertainty in the analysis, and results in wider

uncertainty intervals than the simple approach. The two

approaches are diagramed in Figure 6.2.

Data uncertainty has become the subject of recent

journal articles, such as the by Martz and Hamada

(2003), who develop a fully Bayesian method. Also,

this topic is closely related to a statistical technique

called “multiple imputation” (see Rubin 1996), in which

a moderate number of data sets are randomly generated

and then treated according to the left path in Figure 6.2.

Further treatment of this topic is beyond the scope of

this handbook, but the reader can find additional

guidance in the references cited above.

6.1.3.3 Uncertainty in the Correct Model to Use

There can be uncertainty in which probability model to

use. For example, there may be a slight trend, but it is

borderline. Should a trend be modeled? Chapters 6

and 7 of this handbook discuss model validation

extensively. However, model validation, which

concludes that the model is either “adequate” or “not

adequate,” is only a first step toward addressing this

issue.

A more ambitious approach would be to quantify the

degree of belief in each of a number of models, and

propagate uncertainty in the models into the overall

conclusions. This approach can use the predictions of

various models as evidence in a formal Bayesian

estimation procedure. See Mosleh et al. (1994) for a

number of thoughtful papers on the definition and

treatment of model uncertainties in the context of PRA

applications. The topic is also discussed and debated in

a tutorial article by Hoeting et al. (1999). Bernardo and

Smith (1994) also work out this approach in their

Chapter 6 on “remodelling.” Drougett (1999) includes

a discussion on the role of information concerning the

models themselves (for example, their structure and

past performance) in the estimation process.

Further consideration of such issues is beyond the scope

of this handbook. The parameter uncertainties given

here all assume that the model is a perfect description

of the real world.

6.2 Initiating Events

This section and Section 6.3 are fundamental. The

methods introduced here are used throughout the rest of

the handbook. The most important topics for a firsttime reader are:

• Maximum likelihood estimation (6.2.1.1),

• Bayesian estimation, especially with a discrete

prior or a conjugate prior (6.2.1-6.2.2.5), and

• Model validation, especially using graphical tools

(portions of 6.2.3).

Initiating events here use the broad definition of the

examples in Section 2.2, events that occur randomly in

time and that initiate a quick response to restore the

system to normal.

The event frequency is denoted 8, with units events per

unit time. The data consist of x observed events in time

t, where x is an integer $ 0 and t is some time > 0.

Note, t is considered nonrandom, and x is randomly

generated. This can be expressed using the notation

given in Appendix A, with upper case letters denoting

random variables and lower case letters denoting

numbers. Before data had been generated, the random

number of initiating events would have been denoted by

X. For any particular number x, the probability of x

initiating events in time t is

Pr(X = x) = e!8t

(8t)

x

/x! . (6.1)

Many possible data sets,

with subjective probabilities

Many analysis results Mean data set

Mean of results Results from analysis

 of one data set

Averaging the Analyzing the

analyses accounts for average accounts for

more uncertainty. less uncertainty.

Figure 6.2 Two possible analysis paths for uncertain

data.
Parameter Estimation and Model Validation

6-5

0e0 2e-5 4e-5 6e-5 8e-5 1e-4

λ (1/hrs)

0.0

0.1

0.2

0.3

0.4

likelihood

Figure 6.3 Likelihood as a function of 8, for data of

Example 6.1.

This formula for the Poisson distribution is a

restatement of Equation 2.1, and will be used throughout this section.

The methods of parameter estimation will be illustrated

by the following hypothetical data set.

Example 6.1 Initiating events with loss of

heat sink.

In the last six years (during which the reactor was

critical for 42800 hr.) a hypothetical PWR has had

one initiating event that involved a loss of heat

sink. The parameter to estimate is 8, the

frequency of such events while the reactor is

critical.

6.2.1 Frequentist or Classical Estimation

As explained in Section 6.1, Bayesian estimation

methods are more important in PRA, but the classical

estimator has a simpler form. Also, the comparison

among estimators flows somewhat better if the short

presentation of frequentist estimators precedes the

lengthier presentation of Bayesian estimators. For these

reasons, frequentist methods are given first in this

section.

6.2.1.1 Point Estimate

The most commonly used frequentist estimate is the

maximum likelihood estimate (MLE). It is found by

taking the likelihood, given by Equation 6.1, and

treating it as a function of 8. The value of 8 that

maximizes the likelihood is called the MLE. It can be

shown (as a calculus exercise) that the maximum

likelihood estimate (MLE) of 8 is

 . (6.2) $

λ = x t /

This formula is simple and intuitively natural: the

observed number of events divided by the observed

time period. This simplicity is part of the appeal of the

MLE. The hat notation is used to indicate that the MLE

is an estimate calculated from the data, not the true,

unknown 8.

Example 6.1 has x = 1 and t = 42800 hrs. The

likelihood is plotted on Figure 6.3 as a function of 8.

The likelihood function is maximized when 8 =

1/42800 = 2.3E-5. Therefore, the estimated event

rate for the plant is

= 1/42800 = 2.3E!5 events per critical-hour . $

λ

Converting the hours to 42800/8760 = 4.89 criticalyears yields

= 1/4.89 = 0.20 events per critical-year. $λ

In the above example, and in general throughout this

handbook, the final answer is presented with few

significant digits. This reflects the uncertainty inherent

in all estimates. Indeed, sometimes not even the first

significant digit is known precisely. During

intermediate calculations, however, more significant

digits will be shown, and used. This prevents roundoff

errors from accumulating during the calculations.

It is also possible to combine, or pool, data from

several independent processes, each having the same

rate 8. In particular, suppose that the ith Poisson

process is observed for time ti

, yielding the observed

count xi

. The total number of event occurrences is x =

Gixi, where the sum is taken over all of the processes,

and the exposure time is t = Gi

ti. The rate 8 is estimated

by

. $

λ = = xt x t / / i i ii Σ Σ

For example, if counts obtained for different years are

used to estimate the rate, the estimate is the ratio of the

total count to the total exposure time during these years.

6.2.1.2 Standard Deviation of the Estimator

The event count is random. In other words, if an

identical plant could be observed during the same years,

a different number of events might occur due to

randomness. Similarly, the same plant might yield a

different count over a different six-year period.
Parameter Estimation and Model Validation

6-6

Because the event count is random, the estimator is also

random, and the estimate is simply the observed value

for this plant during these years. Note the distinction in

the terms: an estimator is a random variable, and an

estimate is the particular value of the estimator after the

data have been generated.

For a Poisson distributed random variable X, the mean

and variance are the same, E(X) = var(X) = 8t, as stated

in Appendix A.6.2. Consequently, the standard

deviation of X is (8t)

1⁄2, and the estimated standard

deviation of the estimator / X/t is $

λ

( . $ )/ ( $ /) / / // λ λ t t t xt 12 12 12 = =

The estimated standard deviation of is also called the $

λ

standard error for 8.

Thus, the standard error for 8 in Example 6.1 is

1/4.89 = 0.20 events per reactor-year.

A standard error is sometimes used for quick approximations when the data set is large. In that case, the

MLE is approximately normal, and an approximate

95% confidence interval is given by MLE ±

2×(standard error). This approximation holds for

maximum likelihood estimation of virtually any

parameter, when the date set is large. For event

frequencies, however, the following exact confidence

interval can be found.

6.2.1.3 Confidence Interval for 8

Frequentist estimation is presented before Bayesian

estimation because the MLE is so simple, simpler in

form than the Bayes estimates. The same cannot be

said for confidence intervals; the confidence-interval

formulas are somewhat more complicated than the

formulas for Bayesian interval estimates, and the

interpretation of confidence intervals is more subtle.

Confidence intervals are used in two ways in this

handbook: they give a standard of comparison, when

Bayes credible intervals are found based on so-called

noninformative priors, and they can be used (but are not

required) in some plots for validating model

assumptions. Therefore, readers may wish to skim the

present section quickly on the first reading.

The confidence interval is given in many reference

books, such as Johnson, Kotz, and Kemp (1992, Sec.

7.3), Bain and Engelhardt (1992, Section 11.4), or

Martz and Waller (1991, Table 4.4). It is based on the

chi-squared (or in symbols, P2

) distribution, which is

tabulated in Appendix C, and which can be found easily

by many software packages. As used below, P2

p(d) is

the pth quantile, or (100p)th percentile, of the chisquared distribution with d degrees of freedom. Do not

misread P2

p(d) as involving multiplication.

For a (1 ! ") confidence interval, or equivalently a

100(1 ! ")% confidence interval, the lower limit is

λ χ

α

α

conf, /2 = / ( ) 2

2 2

2

x

t

If x = 0, this formula is undefined, but then simply set

8conf, "/2 = 0.

Similarly, the upper limit is

λ . χ

α

α

conf, 1−

− = +

/

/ ( )

2

1 2

2 2 2

2

x

t

Notice that an upper confidence limit is defined in the

case x = 0. It is reasonable that observing no

occurrences of the event would provide some

information about how large 8 might be, but not about

how small it might be.

The above formulas are in terms of ". Setting " = 0.1,

for example, gives the formulas for a 90% confidence

interval. These formulas involves the 5th percentile of

a chi-squared distribution with 2x degrees of freedom,

and the 95th percentile of a chi-squared distribution

with (2x+2) degrees of freedom.

The resulting confidence interval is conservative in the

sense that the actual confidence level is no smaller than

the nominal level of 100(1 ! ")%, but it could be

larger. This conservatism is inherent in confidence

intervals based on discrete data.

In Example 6.1, Table C.2 shows that 90%

confidence limits are

λ χ

conf, 0.05 = × = = 0 05

2 2

2 4.89

0103

9.78 0 010 . ( ) . .

λ χ

conf, 0.95 = × = = 0 95

2 4

2 4.89

9.488

9.78 0 97 . ( ) .

with units events per critical-year.
Parameter Estimation and Model Validation

6-7

λ (events/year) GC00 0351

MLE and 90% confidence interval

Event

10

5

10

6

10

10

7

10

9

2

6

14

7

9

9

9

6

6

3

5

6

7

3

10

5

7

9

5

8

6

5

3

5

4

16

9

9

4

9

11

Count

0 1 2 3 4

Figure 6.4 Confidence intervals from random data, all

generated from the same process.

The interpretation of confidence intervals is given in

Appendix B. This interpretation deserves emphasis, so

we elaborate on the topic here. In the frequentist

approach, 8 is fixed and the data are random.

Therefore, the maximum likelihood estimator and the

confidence limits are all random. For most data sets the

MLE, , will be close to the true value of 8, and the $

λ

confidence interval will contain 8. Sometimes,

however, the MLE will be rather far from 8, and

sometimes (less than 10% of the time) the 90% confidence interval will not contain 8. The procedure is

good in the sense that most of the time it gives good

answers, but the analyst never knows if the current data

set is one of the unlucky ones.

To illustrate this, consider the following example with

many hypothetical data sets from the same process.

Example 6.2 Confidence intervals from 

computer-generated data.

A computer was used to generate Poisson data,

assuming an event rate 8 = 1.2 events per year

and assuming that 6 years were observed. Thus,

the event count followed a Poisson distribution

with mean 8t = 7.2. This was repeated, and 40

event counts were generated in all. These may

be interpreted as counts from 40 identical plants,

each observed for 6 years, or from 40 possible

six-year periods at the same plant.

Figure 6.4 shows that the first randomly generated

event count was 10, the next was 5, the next was

again 10, and so on. Some of the event counts were

less than the long-term mean of 7.2, and some were

greater. The maximum likelihood estimates of 8 are

plotted as dots in Figure 6.4. The corresponding

90% confidence intervals for 8 are also plotted.

In Figure 6.4, the vertical dashed line shows the true

value of 8, 1.2. Two of the 40 intervals (5%) are to

the right of the true 8. These resulted from observing

event counts of 14 and 16. One of the 40 intervals

(2.5%) is to the left of the true 8. This interval was

computed from an observed event count of two.

Ideally, the error rates should both be 5%. They are

not, for two reasons. First, 40 is not a very large

number, so the random data do not exactly follow the

long-run averages. Second, confidence intervals

with discrete data are inherently conservative: a 90%

confidence interval is defined so that the probability

of containing the true 8 is at least 90%, and the error

probabilities at each end are each at most 5%.

The data analyst will normally have data from just

one plant for the six-year period. The resulting confidence interval will contain the true value of 8, unless

the data happen to deviate greatly from the mean.

Unfortunately, the analyst does not know when this

has happened, only that it does not happen often.

6.2.2 Bayesian Estimation

6.2.2.1 Overview

Bayesian estimation of 8 involves several steps. The

prior belief about 8 is quantified by a probability distribution, the prior distribution. This distribution will be

restricted to the positive real line, because 8 must be

positive, and it will assign the most probability to the

values of 8 that are deemed most plausible. The data

are then collected, and the likelihood function is

constructed. This is given by Equation 6.1 for initiating

events. It is the probability of the observed data,

written as a function of 8. Finally, the posterior
Parameter Estimation and Model Validation

6-8

Figure 6.5 Prior distribution and posterior distributions

corresponding to three hypothetical data sets.

distribution is constructed, by combining the prior

distribution and the likelihood function through Bayes’

theorem. (For background on Bayes’ theorem and

Bayesian estimation, see Appendix A.5 and B.5.1.)

This theorem says that

fpost(8) % likelihood(8) × fprior(8) .

Here, the symbol % denotes “is proportional to.” The

posterior distribution shows the updated belief about

the values of 8. It is a modification of the prior belief

that accounts for the observed data. 

Figure 6.5, adapted from a tutorial article by Siu and

Kelly (1998), shows how the posterior distribution

changes as the data set changes. The figure is based on

a diffuse prior, and on three hypothetical data sets, with

x = 1 event in t = 10,000 hours, x = 10 events in t =

100,000 hours, and x = 50 events in t = 500,000 hours,

respectively. Note, each of these data sets has = x/t $

λ

= 1.E!4 events per hour. The figure shows the prior

distribution, and the three posterior distributions

corresponding to the three data sets.

For a small data set, the posterior distribution resembles

the prior to some extent. As the data set becomes

larger, several patterns are evident:

• the posterior distribution departs more and more

from the prior distribution, because the data

contribute the dominant information,

• the posterior distribution becomes more

concentrated, indicating better knowledge of the

parameter, less uncertainty, and 

• the posterior distribution becomes approximately

centered around the MLE, . $

λ

To be consistent with the notation for random variables,

upper case letters would be used for uncertain

parameters that have probability distributions. Such

notation is not customary in the Bayesian literature, and

will not be used here. The reader must judge from

context whether the letter 8 denotes a particular value,

or the uncertain parameter with an associated

distribution.

6.2.2.2 Choosing a Prior

The subsections below consider estimation of 8 using

various possible prior distributions. The simplest prior

distribution is discrete. The posterior can be calculated

easily, for example, by a spreadsheet. The next

simplest prior is called conjugate; this prior combines

neatly with the likelihood to give a posterior that can be

evaluated by simple formulas. Finally, the most general

priors are considered; the posterior distribution in such

a case can only be found by numerical integration or by

random sampling.

The prior distribution should accurately reflect prior

knowledge or belief about the unknown parameter.

However, quantifying belief is not easy. Raiffa and

Schlaifer (1961, Sections 3.3.3-3.3.5) point out that

most people can think more easily in terms of

percentiles of a distribution than in terms of moments.

They also give advice on looking at the situation from

many directions, to make sure that the prior belief is

internally consistent and has been accurately quantified.

Siu and Kelly (1998, Sec. 5.1.4) present seven warnings

in connection with developing a prior distribution,

which are summarized here:

• Beware of zero values. If the prior says that a

value of 8 is impossible, no amount of data can

overcome this.

• Beware of cognitive biases, caused by the way

people tend to think.

• Beware of generating overly narrow prior distributions.

• Ensure that the evidence used to generate the prior

distribution is relevant to the estimation problem.

• Be careful when assessing parameters that are not

directly observable.

• Beware of conservatism. Realism is the ideal, not

conservatism.

• Be careful when using discrete probability distributions.

For a fuller discussion of these points, see Siu and

Kelly.
Parameter Estimation and Model Validation

6-9

Some priors are chosen to be “noninformative,” that is,

diffuse enough that they correspond to very little prior

information. The Jeffreys noninformative prior is

often used in this way. If information is available, it is

more realistic to build that information into the prior,

but sometimes the information is difficult to find and

not worth the trouble. In such a case, the Jeffreys

noninformative prior can be used. It is one of the priors

discussed below.

6.2.2.3 Estimation with a Discrete Prior

When the prior distribution is discrete, the calculations

can easily be performed in a spreadsheet. Newcomers

to Bayesian estimation are strongly encouraged to work

through some examples of the type given here, to

develop a sense of how the process works and how the

posterior distribution depends on the prior and on the

data.

The parameter is assumed to take one of m possible

values, 81, ... , 8m. Let the probability distribution

function (p.d.f.) be denoted by f, so f(8i

) = Pr(8i). This

probability quantifies the analyst’s prior belief that each

of the possible values is the one operating in nature. 

Then, some evidence is observed, denoted conceptually

by E. Bayes’ theorem says that

f E (6.3) f LE

LE f i

i i

j j j

m (|) ( )(| )

(| )( )

λ λ λ

λ λ =

∑ =1

where

f(8i

 | E) = the probability of 8i given evidence E

(posterior distribution),

f(8i

) = the probability of 8i prior to having

evidence E ( prior distribution), and

L(E | 8i

) = the likelihood function (probability of the

evidence given 8i

) .

Note that the denominator in Equation 6.3, the total

probability of the evidence E, is simply a normalizing

constant. Therefore, a more abbreviated form of Bayes’

theorem is

f E f LE . ii i i ( | ) ( )(| ) λ λλ ∝

This is the form of the theorem that was given in the

overview of Section 6.2.2.1.

When the evidence is in the form of x failures generated

by a Poisson process over an operational time t, the

likelihood function is given by Equation 6.1:

LE e . t

x i

it i

x

(|) ( )

! λ λ λ = −

The above equations are illustrated here with several

prior distributions. For data, they all use the first

sample in Example 6.2, 10 events in six years. They

all use simple, flat, prior distributions over a

moderately wide range, but with different degrees of

discreteness. One could argue that this prior is not

very informative, but the real reason we choose it is

to make the impact of the Bayesian updating process

easy to see.

Given the ease of calculation with current computers,

a finely discretized prior (say, at 0, 0.01, 0.02,...6.00)

would give the most accurate results, and we will

provide that calculation in a moment. First, however,

let us use a very coarse prior at 0, 0.5, 1.0, ...6.0.

With only 13 bins, the reader can perform hand

calculations quite easily. The results are given in

Table 6.2. The prior is discrete, and is shown in

Figure 6.6. The posterior distribution is also discrete,

and is shown in Figure 6.7.

Table 6.2 Example 6.2, first sample (10 events

in 6 years) with coarse prior.

Event 

Rate

8i

Prior

Probability

pi

Likelihood

Li

pi x Li

Posterior

Probability

Pr(8i

|E)

Cumulative

Probability

E Pr(8j|E)

0.0 0.077 0.00E+0 0.00E+0 0.00E+0 0.00E+0

0.5 0.077 8.10E-4 6.23E-5 2.43E-3 2.43E-3

1.0 0.077 4.13E-2 3.18E-3 1.24E-1 1.26E-1

1.5 0.077 1.19E-1 9.12E-3 3.56E-1 4.82E-1

2.0 0.077 1.05E-1 8.06E-3 3.14E-1 7.96E-1

2.5 0.077 4.86E-2 3.74E-3 1.46E-1 9.42E-1

3.0 0.077 1.50E-2 1.15E-3 4.49E-2 9.87E-1

3.5 0.077 3.49E-3 2.68E-4 1.05E-2 9.98E-1

4.0 0.077 6.60E-4 5.07E-5 1.98E-3 1.00E+0

4.5 0.077 1.07E-4 8.20E-6 3.20E-4 1.00E+0

5.0 0.077 1.52E-5 1.17E-6 4.57E-5 1.00E+0

5.5 0.077 1.97E-6 1.51E-7 5.90E-6 1.00E+0

6.0 0.077 2.34E-7 1.80E-8 7.01E-7 1.00E+0

There is some value in plotting both distributions in

the same graph, so they can be compared easily. 

In such a plot, the vertical bars fall on top of each

other and are easily confused. Therefore, we draw

the graph by connecting the tops of the bars, in

Figure 6.8.
Parameter Estimation and Model Validation

6-10

0.0 1.0 2.0 3.0 4.0 5.0 6.0

lambda (events per year)

0.00

0.10

0.20

0.30

0.40

probability

Figure 6.6 Coarse discrete prior distribution for 8.

0.0 1.0 2.0 3.0 4.0 5.0 6.0

lambda (events per year)

0.00

0.10

0.20

0.30

0.40

probability

Figure 6.7 Discrete posterior distribution of 8 based

on 10 events in 6 years.

0123456

lambda (events per year)

0.00

0.10

0.20

0.30

0.40

probability

prior probability

posterior probability

Figure 6.8 Discrete prior and posterior distributions

plotted together.

0123456

lambda (events per year)

0.00

0.05

0.10

0.15

0.20

probability

prior probability

posterior probability

Figure 6.9 Discrete prior and posterior distributions

for 10 events in 6 years, with finely discretized prior.

0123456

lambda (events per year)

0.00

0.01

0.02

0.03

0.04

probability

prior probability

posterior probability

Figure 6.10 Discrete prior and posterior distributions

for 10 events in six years, with very finely discretized

prior.

The natural tendency is to think of these curves as

densities, but this is not quite correct because they

are not normalized to integrate to 1.0. Except for that

detail, the curves can be thought of as continuous

approximations of the discrete distribution. 

Even with such a coarse prior, the evidence is strong

and forces the distribution to peak at about 8 = 1.5

per year. There is essentially no chance that 8 is

greater than four or less than 0.5.

If we repeat the calculation with a discrete prior twice

as fine (i.e., on the points 0, 0.25, 0.50, 0.75,...6.00),

the prior now has 25 bins and the results are much

more smooth, as shown in Figure 6.9. These results

are quite smooth, and of course follow the previous

results.

Finally, let us repeat the calculation for a discrete flat

prior on the points 0, 0.05, 0.10, 0.15,...6.00, i.e., a

121-point grid. This time, the results, shown in

Figure 6.10, are detailed enough to closely

approximate a smooth, continuous distribution.
Parameter Estimation and Model Validation

6-11

The spreadsheet calculation is identical with Table

6.2, except for having 121 bins rather than 13. Some

values are summarized in Table 6.3. These

Bayesian results are also compared with the

frequentist estimates obtained from the first sample,

shown earlier in Figure 6.4. The Bayesian posterior

distribution has a mode where the probability

function is maximized, and a mean equal to E8i

 f(8i).

Table 6.3. Comparison of Bayesian and

frequentist estimates for the data in

Example 6.2. 

Quantity Bayes,

flat prior

Frequentist,

Figure 6.4

Point estimate

(Bayes mode,

 Bayes mean, 

 frequentist MLE)

1.65

1.833

1.73

Lower end of interval

(Bayes 5th percentile,

 lower confidence limit)

1.00 0.95

Upper end of interval

(Bayes 95th percentile,

 upper confidence limit)

2.80 2.85

These values are compared to the frequentist point

estimate, the MLE. The Bayesian 5th and 95th percentiles form a Bayes credible interval, which is

compared with the frequentist confidence limits

shown at the top of Figure 6.4. The Bayes 90%

interval, based on a flat, essentially noninformative

prior, is slightly more narrow than the frequentist

90% confidence interval.

This concludes the examples for this section. However,

we suggest that the reader make up a data set for

examining the way the posterior distribution responds

to growing evidence. For example, try beginning with

zero failures in year one; then adding two failures in

year two; then zero failures in year three; etc. Also try

a case that does not agree with the prior; for example

five failures in year one; then seven more in year two;

then six in year three. Such examples are given for p in

Section 6.3.2.1, but they are most valuable to someone

who constructs them and works them out, instead of

merely reading about them.

6.2.2.4 Estimation with a Conjugate Prior

We now turn from discrete to continuous prior

distributions. We begin with a very convenient family

of distributions: the conjugate priors.

6.2.2.4.1 Definitions

The conjugate family of prior distributions for Poisson

data is the family of gamma distributions. Two parameterizations of gamma distributions are given in Appendix A.7.6. For Bayesian estimation, the following

parameterization is the more convenient one:

f e ( ) . (6.4) ( ) λ β

α

λ

α

α λβ = − −

Γ

1

Here, 8 has units 1/time and $ has units of time, so the

product 8$ is unitless. For example, if 8 is the

frequency of events per critical-year, $ has units of

critical-years. The parameter $ is a kind of scale

parameter. That is, $ corresponds to the scale of 8. If

we convert 8 from events per hour to events per year by

multiplying it by 8760, we correspondingly divide $ by

8760, converting it from hours to years. The other

parameter, ", is unitless, and is called the shape

parameter. The gamma function, '("), is a standard

mathematical function, defined in Appendix A.7.6. If

" is a positive integer, '(") equals ("!1)!

Let 8 have a gamma uncertainty distribution. In the

present parameterization, the mean of the gamma

distribution, also written as the expected value E(8), is

"/$, and the variance, var(8), is "/$ 2. Note that the

units are correct, units 1/time for the mean and 1/time2

for the variance.

6.2.2.4.2 Update Formulas

As stated earlier and in Appendix B.5.1, the posterior

distribution is related to the prior distribution by

f Xx f (6.5) post prior ( ) Pr( | ) ( ) λ λλ ∝ =

This is the continuous analogue of Equation 6.3. The

probability of the data is also called the likelihood, in

which case it is considered as a function of the parameter 8 for a given x. For Poisson data, it is given by

Equation 6.1. The symbol % denotes “is proportional

to.” Probability density functions generally have

normalizing constants in front to make them integrate to

1.0. These constants can be complicated, but using

proportionality instead of equality allows us to neglect

the normalizing constants. Stripped of all the normalizing constants, the gamma p.d.f. is 

f e ( ) λ λ . α λβ ∝ − − 1
Parameter Estimation and Model Validation

6-12

The gamma distribution and the Poisson likelihood

combine in a beautifully convenient way:

f e

t

x

e

e

t

x

x t

post ( ) ( )

!

( ) ()

λ

λ

λ

λ

λ α λβ

α λβ

∝

∝

− −−

+−− +

1

1

In the final expression, everything that does not involve

8 has been absorbed into the proportionality constant.

This result is “beautifully convenient,” because the

posterior distribution of 8 is again a gamma distribution. This is the meaning of conjugate: if the prior

distribution is a member of the family (in this case, the

gamma family), the posterior distribution is a member

of the same family. The update formulas are:

"post = x + "prior

$post = t + $prior

This leads to an intuitive interpretation of the prior

parameters: a gamma("prior, $prior) distribution is equivalent, at least intuitively, to having seen "prior events in

$prior time units, prior to taking the current data.

Figure 6.5 was constructed in this way. The prior

distribution was gamma(0.2, 10,000). Therefore, the

posterior distributions were gamma(1.2, 20,000),

gamma(10.2, 110,000), and gamma(50.2, 510,000).

When using these update formulas, be sure that t and

$prior have the same units. If one is expressed in hours

and one in years, one of the two numbers must be

converted before the two are added.

The moments of the gamma distribution were mentioned previously. The posterior mean is "post/$post and

the posterior variance is "post/($post)

2

.

The percentiles of the gamma distribution are given by

many software packages. If you use such software, be

careful to check that it is using the same parameterization that is used here! Here are three ways to get the

correct answer. (1) If the software uses the other

parameterization, fool it by inverting your value of $.

Then check to make sure that the numbers appear

reasonable. (2) A safe method is to have the software

find the percentiles of the gamma("post, 1) distribution.

Then manually divide these percentiles by $post. This

ensures that the scale parameter is treated correctly. (3)

As a final alternative, the percentiles of the gamma

distribution can be found from a tabulation of the chisquared distribution, possibly interpolating the table.

To do this, denote the (100p)th percentile of the posterior distribution by 8p. For example, denote the 95th

percentile by 80.95. The (100p)th percentile is given by:

8p = P2

p(2"post)/(2$post)

where, as elsewhere, P2

p(d) is the pth quantile, or

(100p)th percentile, of a chi-squared distribution with

d degrees of freedom. Note the presence of 2 in the

numerator and denominator when the chi-squared

distribution is used.

The next section contains examples that use these

update formulas with several priors.

6.2.2.5 Possible Conjugate Priors

6.2.2.5.1 Informative Priors

The prior distribution must come from sources other

than the current data. It might be tempting to use the

data when constructing the prior distribution, but that

temptation must be resisted. Prior distributions are

named “prior” for a reason: they reflect information

that does not come from the current data. Ideally,

generic data provide the basis for prior belief. Generic

data sources are given in Section 4.2.

Consider again Example 6.1, involving initiating

events with loss of heat sink. With no special knowledge about the plant, prior belief about the plant is

reasonably based on the overall industry performance, so we use the generic industry distribution as

the prior. Poloski et al. (1999a) examined initiatingevent data from the nuclear power industry over nine

years. For PWRs, and initiating events involving loss

of heat sink, they determined that the variability of 8

across the industry can be described by a gamma

distribution with shape parameter = 1.53, and scale

parameter = 10.63 reactor-critical-years. Regrettably, Table G-1 of the report gives only a mean and a

90% interval, not the distribution and its parameters.

The distribution given here is taken from the unpublished work that formed the basis of the report. The

distribution is a gamma distribution, so the update

formulas given above can be used in the hypothetical

example of this section. The prior distribution is

shown in Figure 6.11.

Now, consider updating this prior with the data from

Example 6.1. To make the units consistent, convert

the 42800 reactor-critical-hours in the example to

42800/8760 = 4.89 reactor-critical-years. The update

formula yields:

"post = x + "prior = 1 + 1.53 = 2.53

$post = t + $prior = 4.89 + 10.63 = 15.52 reactor-criticalyears.
Parameter Estimation and Model Validation

6-13

Figure 6.11 Prior density for 8, gamma(1.53, 10.63).

Figure 6.13 Posterior cumulative distribution of 8

for Example 6.1 with industry prior. The 5th and

95th percentiles are shown.

λ (events/reactor-crit.-yr.) GC99 0292 1

Area = 0.05

Area = 0.90

Area = 0.05

Density

0.1 0.2 0.3 0.4 0.5 0.6

0

1

2

3

4

5

6

0

Figure 6.12 Posterior density of 8, gamma(2.53,

15.52), for Example 6.1 with industry prior. The 5th

and 95th percentiles are shown.

The mean, "post/$post, is 0.163 events per reactorcritical-year, the variance is 0.0105 (per reactorcritical-year squared), and the standard deviation is

the square root of the variance, 0.102 per reactorcritical-year.

A 90% credible interval is the interval from the 5th to

the 95th percentiles of the posterior distribution. A

software package finds the two percentiles of a

gamma(2.53, 1.0) distribution to be 0.5867 and

5.5817. Division by $post yields the two percentiles of

the posterior distribution: 0.038 and 0.36. Alternatively, one may interpolate Table C.2 of Appendix C

to find the percentiles of a chi-squared distribution

with 5.06 degrees of freedom, and divide these

percentiles by 2$post. Linear interpolation gives

answers that agree to three significant digits with the

exact answers, but if the degrees of freedom had not

been so close to an integer, the linear interpolation

might have introduced a small inaccuracy.

The interpretation of the above numbers is the

following. The best belief is that 8 is around 0.16,

although it could easily be somewhat larger or

smaller. Values as small as 0.038 or as large as

0.36 are possible, but are approaching the limits of

credibility.

Two graphical ways of presenting this information

are given below. Figure 6.12 shows the posterior

density. The areas to the left of the 5th percentile

and to the right of the 95th percentile are shaded.

The 90% credible interval is the interval in the middle, with probability 90%. Figure 6.13 shows the

same information using the cumulative distribution.

The 5th and 95th percentiles are the values of 8

where the cumulative distribution is 0.05 and 0.95,

respectively. These percentiles are the same values,

as shown in the plot of the density.

For PRA applications, however, the right tail is

typically of concern for risk, corresponding to high

initiating event frequency (or, in other sections of this

chapter, high probability of failure on demand, high

unavailability, or long time to recovery). The interval

given above holds the error probability for the right

tail equal to 0.05. This number is customary in much

statistical practice, and has therefore been used in

many studies for the NRC. The lower end of the

interval, on the other hand, is not of great safety

concern. It is easy to calculate, however. Therefore,

the above 90% interval, corresponding to 5% posterior probability in each tail, is commonly presented in

PRA studies.

Actually, however, the interval presents only a

portion of the information in the posterior distribution,

two summary numbers. The full distribution is used

in a PRA.
Parameter Estimation and Model Validation

6-14

Figure 6.14 Jeffreys noninformative prior distribution

for an event frequency.

6.2.2.5.2 Noninformative Prior

The Jeffreys noninformative prior is intended to

convey little prior belief or information, thus allowing

the data to speak for themselves. This is useful when

no informed consensus exists about the true value of the

unknown parameter. It is also useful when the prior

distribution may be challenged by people with various

agendas. Some authors use the term reference prior

instead of “noninformative prior,” suggesting that the

prior is a standard default, a prior that allows consistency and comparability from one study to another.

With Poisson data, the Jeffreys noninformative prior is

obtained if the shape parameter of a gamma distribution

is taken to be " = 1⁄2 and the parameter $ is taken to be

zero. (See, for example, Box and Tiao 1973.) Ignoring

the normalizing constant at the front of Equation 6.4

yields a function that is proportional to 8!1⁄2, shown in

Figure 6.14. Although this function is interpreted as a

density function, it is an improper distribution because its integral from 0 to 4 is infinite.

It is not intuitive that this prior is “noninformative.”

Simple intuition might expect a uniform distribution

instead. To better educate the intuition, suppose we had

some median prior value m; that is, the prior distribution of 8 satisfies Pr(8 < m) = Pr(8 > m). This can be

rewritten as

Pr(0 < 8 < m) = Pr(m < 8 < 4) .

The interval from 0 to m is shorter than the interval

from m to 4. Therefore, the prior density should be

larger to the left of m than to the right. The density

shown in Figure 6.14 has this property. (We ignore the

fact that the density in Figure 6.14 is improper.)

Further arguments for the prior are too complicated to

give here. For a fuller explanation of noninformative

priors, see Appendix B.5.3.1 and the references cited

there.

Suppose that the data consist of x events in time t.

Formal application of the update formulas yields

"post = x + 1⁄2

$post = t + 0 .

That is, the Bayes posterior distribution for 8 is

gamma(x + 1⁄2, t).

It is interesting to compare the interval using the

Jeffreys prior with the corresponding confidence

interval. The 90% posterior credible interval is

80.05 = P2

0.05(2x + 1)/2t

80.95 = P2

0.95(2x + 1)/2t.

These may be compared with the 90% confidence

interval:

8conf, 0.05 = P2

0.05(2x)/2t

8conf, 0.95 = P2

0.95(2x + 2)/2t.

The confidence intervals differ from the Bayes credible

intervals only in the degrees of freedom, and there only

slightly. This is the primary sense in which the Jeffreys

prior is “noninformative.” The lower and upper confidence limits have degrees of freedom 2x and 2x + 2,

respectively. The two Bayesian limits each use the

average, 2x + 1. The confidence interval is wider than

the Jeffreys credible interval, a reflection of the conservatism of confidence limits with discrete data. However the similarity between the confidence limits and

the Jeffreys limits shows that the result using the

Jeffreys prior will resemble the result using frequentist

methods, that is, using no prior information at all.

Consider again Example 6.1, with one event in 4.89

critical-years, and use the Jeffreys noninformative

prior. The resulting posterior distribution has

"post = 1.5

$post = 4.89 critical-years .

The mean of this distribution is 1.5/4.89 = 0.31

events per critical-year. A 90% Bayes credible

interval can be obtained from a chi-squared table

without any need for interpolation, because the

degrees of freedom parameter is 3, an integer. The

5th and 95th percentiles of the chi-squared distribution are 0.352 and 7.815. Division by 2×4.89 yields

the percentiles of the posterior distribution, 0.036 and

0.80.
Parameter Estimation and Model Validation

6-15

This posterior distribution has a larger mean and

larger percentiles than the posterior distribution in

Section 6.2.2.5.1. The data set is the same, but the

different prior distribution results in a different posterior distribution. The results will be compared in

Section 6.2.2.5.4.

6.2.2.5.3 Constrained Noninformative Prior

This prior is a compromise between an informative

prior and the Jeffreys noninformative prior. The mean

of the constrained noninformative prior uses prior

belief, but the dispersion is defined to correspond to

little information. These priors are described by

Atwood (1996), and by references given there. Constrained noninformative priors have not been widely

used, but they are mentioned here for the sake of

completeness.

For Poisson data, the constrained noninformative prior

is a gamma distribution, with the mean given by prior

belief and the shape parameter = 1⁄2. That is:

"prior = 1⁄2

$prior satisfies "prior/$prior = prior mean .

To illustrate the computations, consider again the

Example 6.1, with one event in 4.89 reactor-criticalyears. Suppose we knew that in the industry overall

such events occur with an average frequency of

0.144 events per reactor-critical-year. (This is

consistent with the informative prior given above in

Section 6.2.2.5.1.) Suppose further that we were

unable or unwilling to make any statement about the

dispersion around this mean — the full information

used to construct the informative prior was not

available, or the plant under consideration was

atypical in some way, so that a more diffuse prior

was appropriate.

The constrained noninformative prior with mean

0.144 has "prior = 1⁄2 and $prior = 3.47 critical-years.

The resulting posterior distribution has

"post = x + 1⁄2 = 1.5

$post = t + 3.47 = 8.36

The mean is 0.18 events per critical-year, and the

90% credible interval is (0.021, 0.47). This notation

means the interval from 0.021 to 0.47.

6.2.2.5.4 Example Comparisons Using Above 

Priors

In general, the following statements can be made:

• The Jeffreys noninformative prior results in a

posterior credible interval that is numerically

similar to a confidence interval, but slightly

shorter.

• If the prior mean exists, the posterior mean is

between the prior mean and the MLE.

• If two prior distributions have the same mean, the

more concentrated (less diffuse) prior distribution

will yield the more concentrated posterior distribution, and will pull the posterior mean closer to the

prior mean.

These statements are now illustrated by example.

The estimates found in the above sections for

Example 6.2 and the various priors are compared in

Table 6.4 and in Figure 6.15.

Table 6.4 Comparison of estimates with 1 event in 4.89 reactor-critical-years.

Method Prior mean Posterior

parameters

Point estimate

(MLE or posterior mean)

90% interval (confidence

interval or posterior credible interval)

Frequentist NA NA 0.20 (0.010, 0.97)

Bayes with Jeffreys

noninformative prior,

gamma(0.5, 0)

undefined " = 1.5

$ = 4.89

0.31 (0.036, 0.80)

Bayes with (informative)

industry prior,

gamma(1.53, 10.63)

0.144 " = 2.53

$ = 15.52

0.16 (0.038, 0.36)

Bayes with constrained

noninformative prior,

gamma(0.5, 3.47)

0.144 " = 1.5

$ = 8.36

0.18 (0.021, 0.47)
Parameter Estimation and Model Validation

6-16

Figure 6.15 Comparison of four point estimates and

interval estimates for 8.

In Table 6.4 and in Figure 6.15, the Jeffreys prior and

the frequentist approach are listed next to each other

because they give numerically similar results. The

Jeffreys prior yields a posterior credible interval that

resembles the frequentist confidence interval. It is a

little shorter, but it is neither to the right nor to the

left. This agrees with the earlier discussion of the

Jeffreys prior.

In each Bayesian case, the posterior mean falls

between the prior mean (if defined) and the MLE,

0.20. The prior distribution has more influence when

the prior distribution is more tightly concentrated

around the mean. The concentration is measured by

the shape parameter "prior, because 1/" equals the

relative variance (= variance/mean2). Therefore the

larger ", the smaller the relative variance. The

industry prior and the constrained noninformative

prior have the same mean, but the industry prior has

the larger ", that is, the smaller variance. As a

consequence, in both cases the posterior mean is

between the MLE, 0.204, and the prior mean, 0.144,

but the posterior mean based on the industry prior is

closer to 0.144, because that prior has a smaller

variance. Because the prior mean is smaller than the

MLE, the bottom two lines give smaller posterior

estimates than do the top two lines. Also, the prior

distribution with the most information (largest ")

yields the most concentrated posterior distribution,

and the shortest 90% interval.

In some situations, no conjugate prior is satisfactory.

For example, a gamma distribution is very unrealistic if

the shape parameter is very small. As a rule of thumb,

the lower percentiles of the distribution are unrealistic

if " is much smaller than 0.5. Such a posterior distribution arises with Poisson data when the prior distribution

is very skewed (" very small) and the data contain zero

events. Then, the posterior distribution also is very

skewed, and the posterior 5th percentile may be many

orders of magnitude below the posterior mean. The

subject-matter experts must look at the percentiles and

decide if they are believable. If not, a more appropriate

prior should be chosen. It will not be conjugate. This

is the subject of the next subsection. 

6.2.2.6 Estimation with a Continuous 

Nonconjugate Prior

Discrete priors and conjugate priors were updated

above with simple formulas. What remains are the

continuous nonconjugate priors. Any continuous

distribution defined on the allowed range of 8 can, in

principle, be used as a prior. The resulting posterior

distribution is a continuous distribution, with no simple

form. (Because the posterior distribution does not have

a simple analytical form, it cannot be entered directly as

an input to most PRA codes. Instead, a discrete approximation of the posterior distribution must usually

be used.)

Three approaches for obtaining the posterior are given

here. Some examples will be worked out in Section

6.2.2.7.

6.2.2.6.1 Direct Numerical Integration

If software is available for performing numerical

integration, the following approach can be used. Find

the form of the posterior distribution, using Equation

6.5. Suppose, for example, that the prior distribution

for 8 is lognormal, with : and F2

 denoting the mean and

variance of the normal distribution of ln8. As stated in

Appendix A.7.3, the lognormal density is proportional

to

f e . LN ( )

ln

λ

λ

λ μ

σ ∝ − ⎛ −

⎝

⎜ ⎞

⎠ 1 ⎟ 1

2

2

Substitute this and Equation 6.1 into Equation 6.5, to

obtain the form of the posterior density:

Cf e e . t x

post( )

ln

λ λ

λ

λ

λ μ

σ = − − ⎛ −

⎝

⎜ ⎞

⎠ 1 ⎟ 1

2

2

All terms that do not involve 8 have been absorbed into

the normalizing constant, C. The normalizing constant

can be evaluated by numerically integrating Cfpost from

0 to 4, that is, integrate the right hand side of the

equation. Unless x is unrealistically large, the function

does not need to be integrated in practice out beyond,

say, ln8 = : + 5F. C equals the integral of Cfpost,

because the integral of fpost must equal 1. Once C has

been evaluated, the mean and percentiles of fpost can be

found numerically.
Parameter Estimation and Model Validation

6-17

Numerical integration, using a technique such as the

trapezoidal rule or Simpson’s rule, can be programmed

easily, even in a spreadsheet. The ideas are found in

some calculus texts, and in books on numerical methods

such as Press et al. (1992).

6.2.2.6.2 Simple Random Sampling

A second approach, which does not directly involve

numerical integration, is to generate a large random

sample from the posterior distribution, and use the

sample to approximate the properties of the distribution.

Some people think of this as numerical integration via

random sampling. Surprisingly, the random sample can

be generated without explicitly finding the form of the

posterior distribution, as explained by Smith and

Gelfand (1992).

The algorithm, called the rejection method for sampling from a distribution, is given here in its general

form, and applied immediately to sampling from the

posterior distribution. In general, suppose that it is

possible to sample some parameter 2 from a continuous

distribution g, but that sampling from a different

distribution f is desired. Suppose also that a positive

constant M can be found such that f(2)/g(2) # M for all

2. The algorithm is:

(1) Generate 2 from g(2);

(2) Generate u from a uniform distribution, 0 # u # 1;

and

(3) If u # f(2)/[Mg(2)] accept 2 in the sample. Otherwise discard it.

Repeat Steps (1) through (3) until enough values of 2

have been accepted to form a sample of the desired

size.

This algorithm is the basis for many random-number

generation routines in software packages. It is applied

below to the generation of a sample from the posterior

distribution for 8. The equations are worked out here,

and the algorithm for the posterior distribution is

restated at the end.

Let f be the posterior density and let g be the prior

density. Then Equation 6.5 states that the ratio

f(8)/g(8) is proportional to the likelihood, which is

maximized, by definition, when 8 equals the maximum

likelihood estimate, x/t. That is, the ratio of interest is:

f(8)/g(8) = Ce!8t

(8t)

x

for some constant C. This is maximized when 8 equals

x/t. Therefore, define M = max[f(8)/g(8)] = Ce!x

xx.

The condition in Step (3) above is equivalent to:

u # [Ce!8t

(8t)

x

] / [Ce!xxx ] = [e!8t(8t)

x

] / [e!xxx ] .

The constant cancels in the numerator and denominator,

so we do not need to evaluate it! It would have been

possible to work with m = M/C, and the calculations

would have been simpler. This rewritten form of the

algorithm, for Poisson data, is given here.

If x > 0, define m = e!x

xx. If x = 0, define m = 1.

The steps of the algorithm are:

(1) Generate a random 8 from the prior distribution;

(2) Generate u from a uniform distribution, 0 # u # 1;

and

(3) If u # e!8t

(8t)

x

/m, accept 8 in the sample. Otherwise discard 8.

Repeat Steps (1) through (3) until a sample of the

desired size is found.

Intuitively, this algorithm generates possible values of

8 from the prior distribution, and discards most of those

that are not very consistent with the data. The result is

a sample from the posterior distribution.

6.2.2.6.3 More Complicated Random Sampling

All-purpose Bayesian update programs can be used for

the present simple problem. For example, the program

BUGS1

 (Bayesian inference Using Gibbs Sampling)

performs Markov chain Monte Carlo (MCMC)

sampling. This package is intended for complicated

settings, such as those described in Chapters 7 and 8.

Using it here is like using the proverbial cannon to kill

a mosquito. Nevertheless, the program is free, and very

flexible, and can be used here. It is available for

download at

http://www.mrc-bsu.cam.ac.uk/bugs/ 

and is described more fully in Sections 7.2.3 and

8.3.3.3 of this handbook. An example is given below.

1

 Mention of specific products and/or manufacturers in

this document implies neither endorsement or

preference, nor disapproval by the U.S. Government or

any of its agencies of the use of a specific product for

any purpose.
Parameter Estimation and Model Validation

6-18

model

{

 mu <- lambda*rxyrs

 x ~ dpois(mu)

 lambda ~ dlnorm(-6.908, 0.5104)

}

list(rxyrs=2102, x=0)

Figure 6.16 Script for analyzing Example 6.3 using

BUGS.

6.2.2.7 Examples Involving Nonconjugate Priors

These techniques will be illustrated with Example 6.3,

from Appendix J-4 of Poloski et al. (1999a).

Example 6.3 Small-break LOCAs.

No small-break loss-of-coolant accidents

(SBLOCAs) have occurred in 2102 reactorcalendar-years at U.S. nuclear power plants. The

WASH-1400 (NRC 1975) distribution for the

frequency of this event was lognormal with

median 1E!3 and error factor 10.

6.2.2.7.1 Example with Lognormal Prior

Poloski et al. (1999a) use the WASH-1400 distribution as a prior, and update it with the 2102 years of

data.

The resulting posterior distribution was sampled

100,000 times using the method described in Section

6.2.2.6.2 above, and the mean was found. Then, the

values were arranged in increasing order, and the

percentiles of the sample were found. This process

took less than 15 seconds in 1999 on a 166 MHz

computer. Based on the mean and percentiles of the

sample, the mean of the posterior distribution is

3.5E!4, and the 90% posterior credible interval is

(4.5E!5, 9.8E!4).

To illustrate the method of Section 6.2.2.6.3, the

distribution was also sampled using BUGS. Figure

6.16 shows the script used for running BUGS. 

The section in curly brackets defines the model.

Note that <-, intended to look like a left-pointing

arrow, is used to define quantities in terms of other

quantities, and ~ is used to generate a random

quantity from a distribution. The names of distributions begin with the letter d. Thus, X is a Poisson

random variable with mean :, with :` = 8 × rxyrs.

The prior distribution of 8 is lognormal. The parameters given in the script arise as follows. BUGS

parameterizes the normal in terms of the mean and

inverse of the variance, for reasons explained in

Section 6.6.1.2.1. It parameterizes the lognormal

distribution using the parameters of the underlying

normal. It is shown below that a lognormal with

median 1E!3 and error factor 10 corresponds to an

underlying normal with mean !6.980 and standard

deviation 1.3997. Therefore, the inverse of the

variance is 1/1.39972

 = 0.5104.

The line beginning “list” defines the data, 0 events is

2102 reactor years. BUGS also requires an initial

value for 8, but generated it randomly.

When BUGS generated 100,000 samples, the mean,

5th percentile, and 95th percentile of 8 were 3.5E!4,

4.5E!5, and 9.8E!4, just as found above.

6.2.2.7.2 Example with “Moment-Matching” 

Conjugate Prior

Conjugate priors have appeal: Some people find

algebraic formulas tidier and more convenient than

brute-force computer calculations. Also, when a

PRA program requests a distribution for a parameter,

it is usually easier to enter a distributional form and

a couple of parameters than to enter a simulated

distribution.

Therefore, a nonconjugate prior is sometimes replaced by a conjugate prior having the same mean

and variance. This method is carried out here with

the above example.

Begin by finding the gamma prior with the same

moments as the above lognormal prior. As explained

in Appendix A.7.3, the median, error factor, and

moments of the lognormal distribution are related to

: and F of the underlying normal distribution of ln7

as follows:

median(8) = exp(:)

EF(8) = exp(1.645F)

mean(8) = exp(: + F2

/2)

var(8) = [median(8)]2

@exp(F2)@[exp(F2) ! 1] .

The lognormal prior has median 1.0E!3, and error

factor 10. Solving the first two equations yields

: = !6.907755

F = 1.399748 .

Substituting these values into the second two equations yields

mean(8) = 2.6635E!3

var(8) = 4.3235E!5 .
Parameter Estimation and Model Validation

6-19

Now the gamma distribution must be found with this

mean and this variance. The formulas for the moments of a gamma distribution were given in Section

6.2.2.4.1 and in Appendix A.7.6:

mean = "/$

variance = "/$

2

 .

Therefore,

" = mean2/variance = 0.164

$ = mean/variance = 61.6 reactor-years.

Warning flags should go up, because " is considerably smaller than 0.5. Nevertheless, we carry out the

example using this gamma distribution as the prior.

The update formulas yield:

"post = 0 + 0.164 = 0.164

$post = 2102 + 61.6 = 2164 reactor-years .

The posterior mean is 7.6E!5, and a 90% credible

interval is (3.4E!12, 4.1E!4), all with units events

per reactor-year.

6.2.2.7.3 Comparison of Example Analyses

The two posterior distributions do not agree closely,

as will be discussed below. If the shape parameter

" of the gamma prior had been larger, the two prior

distributions would have had more similar percentiles, and the two posterior distributions likewise

would have agreed better. As it is, however, the two

analyses are summarized in Table 6.5.

Table 6.5 Posterior distributions from two 

analyses.

Prior Mean 90% Interval

Lognormal 3.5E!4 (4.5E!5, 9.8E!4)

Gamma 7.6E!5 (3.4E!12, 4.1E!4)

The most notable difference between the two posterior distributions is in the lower endpoints, the 5th

percentiles, which differ by many orders of magnitude. This is explained, to some extent, by graphical

comparisons. Figures 6.17 and 6.18 show the prior

cumulative distributions. When plotted on an ordinary scale in Figure 6.17, the two prior distributions

look fairly similar, although the gamma distribution

seems to put more probability near zero. The differences become much more obvious when the two

prior distributions are plotted on a logarithmic scale

in Figure 6.18. These differences between the two

prior distributions are present in spite of the fact that

the two priors have equal means and equal variances.

Figure 6.17 Two prior distributions having the same

means and variances.

Figure 6.18 The same prior distributions as in the

previous figure, with 8 plotted on a logarithmic scale.

The two resulting posterior distributions are also

quite different in the lower tail, as shown in Figure

6.19, and this difference is especially clear when the

distributions are plotted on a log scale, as shown in

Figure 6.20.

Figure 6.19 Two posterior distributions, from priors

in previous figures.
Parameter Estimation and Model Validation

6-20

Figure 6.20 The same posterior distributions as in

the previous figure, with 8 plotted on logarithmic

scale.

Incidentally, these illustrations use cumulative distributions instead of densities, for an important reason.

Cumulative distributions simply show probabilities,

and so can be plotted with the horizontal scale either

linear or logarithmic. Alternatively, the density of

ln(8) could be plotted against ln(8), but take care to

calculate the density of ln(8) correctly, as explained

in Appendix A.4.7.

6.2.2.8 Analysis with Fixed Count and Random

Time

Sometimes it is useful to consider a fixed number of

events in a random time, a waiting time. For example,

if the event frequency is believed to change over time,

only the most recent history may represent current

behavior. In such a situation, one might decide to use

only the most recent few events, such as x = 3, and to

treat the corresponding time t as random. Here t is the

time measured backwards from the present to the xth

event in the past. Earlier events could be used to

construct a prior distribution, but the dispersion of the

prior distribution should be set large because the earlier

events are not considered fully relevant to the present.

The above data consist of x exponential(8) durations.

The analysis techniques are given in Section 6.6.1.2.2.

It turns out that Bayesian analysis with an informative

prior is exactly the same whether the data are regarded

as coming from a Poisson count x in fixed time t or a

sum t of x exponential durations. The two likelihoods

are proportional to each other, and the posterior distributions are identical.

6.2.3 Model Validation

Model validation should go hand in hand with parameter estimation. Philosophically, it would seem natural

first to confirm the form of the model, and second to

estimate the parameters of that model. However,

typically one can perform goodness-of-fit tests, and

other validations of a model, only after the model has

been fully specified, that is, only after the form of the

model has been assumed and the corresponding parameters have been estimated. Because parameter-estimation is built into most model-validation procedures, it

was presented first.

It is usually wise not to stop the analysis with just

estimating the parameters. Foolish results have been

presented by analysts who estimated the parameters but

did not thoroughly check that the assumptions of the

model were correct. This section presents ways to

check the model assumptions.

That being said, there is more in this section than will

be needed on any one analysis. Often, a simple plot is

sufficient to show that the model appears adequate.

When the data are very sparse, perhaps not even that is

needed, because the data set is too small to invalidate

any model; in such a case, the simplest model is normally accepted. The methods here are offered for

possible use, and the analyst should select the appropriate ones.

The Poisson process was introduced in Section 2.2.2.

The three assumptions were listed there: constant event

occurrence rate, no simultaneous events, and independent time periods. These assumptions are considered

here. Much of the following material is taken from an

INEEL report by Engelhardt (1994).

The assumption of constant rate is considered in the

next two sections, first, where the alternative possibility

is that different data sources may have different values

of 8, but in no particular order, and then, where the

alternative possibility is that a time trend exists. Both

graphical methods and formal statistical hypothesis tests

are given for addressing the issues. The assumption of

no exactly simultaneous events is then discussed from

the viewpoint of examining the data for common-cause

events. Finally, the assumption of independent time

intervals is considered, and some statistical tests of the

assumption are given.

When Bayesian methods are used, one must also

examine whether the data and the prior distribution are

consistent. It makes little sense to update a prior with

data, if the data make it clear that the prior belief was

incorrect. That topic constitutes the final subsection of

the present section.

6.2.3.1 Poolability of Data Subsets

Assumption 1 in Section 2.2.2 implies that there is one

rate 8 for the entire process. The correctness of such an

assumption can be investigated by analyzing subsets of
Parameter Estimation and Model Validation

6-21

0.0 0.5 1.0 1.5 2.0 2.5

Pooled (8/22.5)

CR3 (5/5.2)

TP3 (2/5.8)

TP4 (1/5.6)

SL2 (0/2.1)

SL1 (0/3.9)

λ (events/reactor-shutdown-yr.) GC99 0292 8

Figure 6.21 MLEs and 90% confidence intervals for

8, based on each plant’s data and based on pooled

data from all the plants.

the data and comparing the estimates of 8 for the

various subsets.

Example 2.2 described LOSP events during shutdown.

For this section, consider a portion of that example.

The entire data set could be used, but to keep the

example from being too cumbersome we arbitrarily

restrict it to five plants at three sites, all located in one

state. 

An obvious question concerns the possibility of different rates for different plants. A general term used in

this handbook will be data subsets. In Example 6.4,

five subsets are shown, corresponding to plants. In

other examples, the subsets could correspond to years,

or systems, or any other way of splitting the data. For

initiating events, each subset corresponds to one cell in

the table, with an event count and an exposure time.

Example 6.4 Shutdown LOSP events at five

plants, 1980-96.

During 1980-1996, five plants experienced eight

LOSP events while in shutdown. These were

events from plant-centered causes rather than

external causes. The data are given here.

Plant

code

Events Plant shutdown

years

CR3 5 5.224

SL1 0 3.871

SL2 0 2.064

TP3 2 5.763

TP4 1 5.586

Totals 8 22.508

Sometimes, data subsets can be split or combined in

reasonable ways. For example, if the subsets were time

periods, the data could be partitioned into decades,

years, or months. The finer the division of the cells, the

more sparse the data become within the cells. Too fine

a partition allows random variation to dominate within

each cell, but too coarse a partition may hide variation

that is present within individual cells. In the present

simple example, the most reasonable partition is into

plants. Analysis of more complicated data sets may

require examination of many partitionings. 

First, a graphical technique is given to help the analyst

understand what the data set shows. Then, a formal

statistical procedure is presented to help quantify the

strength of the evidence for patterns seen in the graphical investigation.

6.2.3.1.1 Graphical Technique

To explore the relations between cells, identify the cells

on one axis. Then, for each cell, plot a point estimate

of 8 and an interval estimate of 8 against the other axis.

Patterns such as trends, outliers, or large scatter are then

made visible.

In Example 6.4, the cells are plants. The data set

from each plant was analyzed separately, using the

tools of Section 6.2.1. The graph in Figure 6.21

shows the maximum likelihood estimate and a

confidence interval for each plant, plotted side by

side. For this handbook, the plot was produced with

a graphics software package, although a hand-drawn

sketch would be adequate to show the results.

The confidence interval for the pooled data is also

shown. Take care, however: this interval is only

valid if all the plants have the same 8, which is what

must be decided. Nevertheless, the interval and

point estimate for the pooled data give a useful

reference for comparisons with the individual plants.

For this reason, a vertical dotted line is drawn

through the mean of the pooled data.

Note that the plants are not displayed in alphabetical

order, which is a meaningless order for the event

rate, but in order of decreasing . (When two plants $

λ

have the same MLE, as do SL1 and SL2, the upper

confidence limit is used to determine the order.)

Experience has shown that such a descending order

assists the eye in making comparisons.

CR3 appears somewhat high compared to the

others. Although there is considerable overlap of the

intervals, the lower confidence limit for CR3 is just
Parameter Estimation and Model Validation

6-22

barely higher than the MLE for the utility as a whole.

Of course, the picture might give a different impression if slightly different intervals were used: 95%

confidence intervals instead of 90% confidence

intervals, or Bayes intervals with the Jeffreys

noninformative prior instead of confidence intervals.

From the graph alone, it is difficult to say whether the

data can be pooled.

A graph like this should not be used to draw conclusions without also using a formal statistical test. For

example, if many confidence intervals are plotted,

based on data sets generated by the same 8, a few will

be far from the others because of randomness alone.

This was seen in Figure 6.4, where all the variation was

due to randomness of the data, and some intervals did

not overlap some others at all. Thus, an outlying

interval does not prove that the 8s are unequal. This

same statement is true if other intervals are used, such

as Bayes credible intervals based on the noninformative

prior. The issue is the random variability of data, not

the kind of interval constructed.

Conversely, if there are only a few intervals, intervals

that just barely overlap can give strong evidence for a

difference in the 8s.

To quantify the strength of the evidence against poolability, a formal statistical procedure is given in the next

subsection. The graph gives an indication of what the

test might show, and helps in the interpretation of the

test results. If the statistical test turns out to find a

statistically significant difference between plants, it is

natural then to ask what kind of difference is present.

Figure 6.21 shows that most of the plants appear

similar, with only one possible outlier. An unusually

long interval, such as that seen in Figure 6.21 for SL2,

is generally associated with a smaller exposure time.

The picture provides insight even though it does not

give a quantitative statistical test.

6.2.3.1.2 Statistical Test

The Chi-Squared Test. To study whether the rate is

the same for different cells, use a chi-squared test.

Many statistics texts, such as Bain and Engelhardt

(1992, Chapter 13), discuss this test, and many software

packages perform the chi-squared test. It is presented

here in enough detail so that the reader could perform

the calculations by hand if necessary, because it is

instructive to see how the test works.

Let the null hypothesis be:

H0: 8 is the same in all the data subsets.

In the present application, the data subsets are the five

plants. The method is to see what kind of data would

be expected when 8 really is constant, and then to see

how much the observed counts differ from the expected

counts. If the difference is small, the counts are consistent with the hypothesis H0 that the rate is constant. If,

instead, the difference is large, the counts show strong

evidence against H0.

Write xj

 and tj for the count and exposure time corresponding to the jth cell, and let x = Gxj

 and t = Gtj. If

H0 is true, that is, if 8 is really the same for all the

plants, then the estimate (MLE) of 8 is = x/t. The $

λ

estimate of the expected count is built from this quantity. Assuming the hypothesis of a single rate 8, an

estimate of the expected count for the jth cell is simply:

e t . j j = $λ

In Example 6.4, the estimate of 8 is 8/22.508 = 0.355

events per shutdown-year. Therefore, the expected

count for CR3 is the estimate of 8 times the exposure

time for CR3, 0.335 × 5.224 = 1.857 events. Table

6.6 is an extension of the original table given in

Example 6.4, showing the quantities needed for the

calculation.

Table 6.6 Quantities for calculation of 

chi-squared test.

Cell code xj tj ej

CR3 5 5.224 1.857

SL1 0 3.871 1.376

SL2 0 2.064 0.734

TP3 2 5.763 2.048

TP4 1 5.586 1.985

Totals 8 22.508 8.000

The total of the expected counts agrees with the total

of the observed counts, except possibly for small

round-off error.

The test for equality of rates that is considered here is

based on the following calculated quantity,

X2

 = Ej(xj ! ej)

2

/ej ,
Parameter Estimation and Model Validation

6-23

sometimes called the Pearson chi-squared statistic,

after its inventor, Karl Pearson, or simply the chisquared statistic. The notation became standard long

before the custom developed of using upper-case letters

for random variables and lower-case letters for numbers. In the discussion below, the context must reveal

whether X2

 refers to the random variable or the observed value.

Observe that X2

 is large if the xjs (observed counts)

differ greatly from the ej

s (expected values when H0 is

true). Conversely, X2

 is small if the observed values are

close to the expected values. This statement is made

more precise as follows. When H0 is true and the total

count is large, the distribution of X2

 has a distribution

that is approximately chi-squared with c ! 1 degrees of

freedom, where c is the number of cells. If the calculated value of X2

 is large compared to the chi-squared

distribution, there is strong evidence that H0 is false; the

larger the X2

 value, the stronger the evidence.

For the data of Table 6.4, X2 = 7.92, which is the

90.6th percentile of the chi-squared distribution with

four degrees of freedom. The next subsection

discusses the interpretation of this.

Interpretation of Test Results. Suppose, for any

example with 5 cells, that X2

 were 9.8. A table of the

chi-squared distribution shows that 9.488 is the 95th

percentile of the chi-squared distribution with 4 degrees

of freedom, and 11.14 is the 97.5th percentile. After

comparing X2

 to these values, we would conclude that

the evidence is strong against H0, but not overwhelming. The full statement is:

• If H0 is true, that is, if all the cells have the same 8,

the chance of seeing such a large X2

 is less than

0.05 but more than 0.025.

Common abbreviated ways of saying this are:

• We reject H0 at the 5% significance level, but not

at the 2.5% significance level.

• The difference between cells is statistically significant at the 0.05 level, but not at the 0.025 level.

• The p-value is between 0.05 and 0.025.

There will be some false alarms. Even if 8 is exactly

the same for all the cells, sometimes X2

 will be large,

just from randomness. It will be greater than the 95th

percentile for 5% of the data sets, and it will be greater

than the 99th percentile for 1% of the data sets. If we

observed such a value for X2

, we would probably decide

that the data could not be pooled. In that case, we

would have believed a false alarm and made the incorrect decision. Just as with confidence intervals, we

cannot be sure that this data set is not one of the rare

unlucky ones. But following the averages leads us to

the correct decision most of the time.

If, instead, X2

 were 4.1, it would be near the 60th

percentile of the chi-squared distribution, and therefore

be in the range of values that would be expected under

H0. We would say the observed counts are consistent

with the hypothesis H0, or H0 cannot be rejected, or the

evidence against H0 is weak. We would not conclude

that H0 is true, because it probably is not exactly true to

the tenth decimal place, but the conclusion would be

that H0 cannot be rejected by the data.

In fact, for the data of Table 6.6, X2 equals 7.92,

which is the 90.6th percentile of the chi-squared

distribution with 4 degrees of freedom. That means:

if all five plants have the same event rate, there is a

9.4% probability of seeing such a large value of X2

.

The evidence against H0 is not convincingly strong.

CR3 might be suspected of having a higher event

rate, but the evidence is not strong enough to prove

this.

The traditional cut-off is 5%. The difference between

cells is called statistically significant, with no qualifying phrase, if it is significant at the 0.05 level. This is

tradition only, but it is very widely followed.

In actual data analysis, do not stop with the decision

that a difference is, or is not, statistically significant.

Do not even stop after reporting the p-value. That may

be acceptable if the p-value is very small (much less

than 0.05) or very large (much larger than 0.05). In

many cases, however, statistical significance is far from

the whole story. Engineering significance is just as

important.

To illustrate this, consider a possible follow-up to the

above statistical analysis of Example 6.4. As mentioned, the statistical evidence against poolability is not

strong, but some might consider it borderline. Therefore, a thorough analysis would ask questions such as:

• Are there engineering reasons for expecting CR3

to have a different event rate than the other plants

do, either because of the hardware or because of

procedures during shutdown? (Be warned that it

is easy to find justifications in hindsight, after

seeing the data. It might be wise to hide the data

and ask these questions of a different knowledgeable person.)

• What are the consequences for the PRA analysis

if the data are pooled or if, instead, CR3 is treated

separately from the other plants? Does the decision to pool or not make any practical difference?
Parameter Estimation and Model Validation

6-24

Required Sample Size. The above considerations are

valid if the total count is “large,” or more precisely, if

the ej

s are “large.” If the ejs are small, the chi-squared

distribution is not a good approximation to the distribution of X2

. Thus, the user must ask how large a count is

necessary for the chi-squared approximation to be

adequate. An overly conservative rule is that each

expected cell-count, ej

, should be 5.0 or larger. Despite

its conservatism, this rule is still widely used, and cited

in the statistical literature and by some software packages.

A readable discussion of chi-squared tests by Moore

(1986, p.71) is applicable here. Citing the work of

Roscoe and Byars (1971), the following recommendations are made:

(1) With equiprobable cells, the average expected

frequency should be at least 1 when testing at the

0.05 level. In other words, use the chi-squared

approximation at the 5% level when x/c $ 1,

where x is the number of events and c is the

number of cells. At the 1% level, the chi-squared

approximation is recommended if x/c $ 2.

(2) When the cells are not approximately equiprobable, the average expected frequencies in (1)

should be doubled. Thus, the recommendation is

that at the 5% level x/c $ 2, and at the 1% level

x/c $ 4.

Note that in rules (1) and (2) above, the recommendation is based on the average rather than the minimum

expected cell-count. As noted by Koehler and Larntz

(1980), any rule such as (2) may be defeated by a

sufficiently skewed assignment of cell probabilities.

Roscoe and Byars also recommend when c = 2 that the

chi-squared test should be replaced by the test based on

the exact binomial distribution of X1 conditional on the

total event count. For example, if the two cells had the

same exposure times, we would expect that half of the

events would be generated in each cell. More generally, if

• the two cells have exposure times t1 and t2,

• a total of x events are observed, and

• 8 is the same for both cells,

then, conditional on x, X1 has a binomial(n, p) distribution, with p = t1/(t1 + t2). Exact binomial tests are

discussed by Bain and Engelhardt (1992, p.405).

Example 6.4 has x = 8 and c = 5. The cells are not

equiprobable, that is, ej

 is not the same for all cells,

because the plants did not all have the same exposure

time. Nevertheless, the expected cell counts differ from

each other by, at most, a factor of two. This is not a

large departure from equiprobability, as differences of

an order of magnitude would be. Because x/c = 1.6,

and the calculated significance level is about 10%, the

sample size is large enough for the chi-squared approximation to be adequate. The conclusions reached earlier

still stand. If, on the other hand, the sample size had

been considerably smaller, one would have to say that

the p-value is approximately given by the chi-squared

distribution, but that the exact p-value has not been

found.

If the expected cell-counts are so small that the chisquared approximation is not recommended, the analyst

can pool data in some “adjacent cells,” thereby increasing the expected cell-counts.

In the Example 6.4, suppose that there were engineering

reasons for thinking that the event rate is similar at units

at a single site. Then, the sister units might be pooled,

transforming the original table of Example 6.4 into

Table 6.7 here.

Table 6.7 Shutdown LOSP events at three

sites, 1980-96.

Site code Events Plant shutdown years

CR 5 5.224

SL 0 5.935

TP 3 11.349

We repeat, this pooling of cells is not required with the

actual data, but it could be useful if (a) the cell counts

were smaller and (b) there were engineering reasons for

believing that the pooled cells are relatively homogeneous, that is, the event rates are similar for both units

at a site, more similar than the event rates at different

sites.

Generally speaking, a chi-squared test based on a larger

number of cells will have better power for detecting

when rates are not equal, but this also makes it more

difficult to satisfy guidelines on expected cell-counts

for the chi-squared approximation. Thus, it is sometimes necessary to make a compromise between expected cell counts and the number of cells.

Options involving the exact distribution of X2

 are also

possible. The most widely known commercial software

for calculating the exact p-value is StatXact (1999).
Parameter Estimation and Model Validation

6-25

Events/reactor-critical-year

MLE and 90% conf. interval

0.0

0.5

1.0

1.5

2.0

87 88 89

Year

90 91 92 93

GC99 0292 11

Figure 6.22 MLEs and 90% confidence intervals

for 8, each based on data from one calendar year.

6.2.3.2 No Time Trend

The chi-squared method given above does not use any

ordering of the cells. Even if the test were for differences in years, say, the test would not use the natural

ordering by calendar year or by plant age. When there

is a meaningful order to the data subsets, it may be

useful to perform additional analyses. The analysis

given above is valid, but an additional possible analysis,

making use of time order, is considered now.

The methods will be illustrated with Example 6.5. 

6.2.3.2.1 Graphical Techniques

Confidence-Interval Plot. First, the same kind of plot

that was used in the previous subsection can be used

here. The time axis is divided into cells, or bins in the

terminology of some authors. For example, if the time

span is divided into calendar years, the counts and

reactor-critical-years for Example 6.5 are given in

Table 6.8.

Example 6.5 Unplanned HPCI demands.

Grant et al. (1995, Table B-5) list 63 unplanned

demands for the HPCI system to start at 23 BWRs

during 1987-1993. The demand dates are given

in columns below, in format MM/DD/YY.

01/05/87

01/07/87

01/26/87

02/18/87

02/24/87

03/11/87

04/03/87

04/16/87

04/22/87

07/23/87

07/26/87

07/30/87

08/03/87

08/03/87

08/16/87

08/29/87

01/10/88

04/30/88

05/27/88

08/05/88

08/25/88

08/26/88

09/04/88

11/01/88

11/16/88

12/17/88

03/05/89

03/25/89

08/26/89

09/03/89

11/05/89

11/25/89

12/20/89

01/12/90

01/28/90

03/19/90

03/19/90

06/20/90

07/27/90

08/16/90

08/19/90

09/02/90

09/27/90

10/12/90

10/17/90

11/26/90

01/18/91

01/25/91

02/27/91

04/23/91

07/18/91

07/31/91

08/25/91

09/11/91

12/17/91

02/02/92

06/25/92

08/27/92

09/30/92

10/15/92

11/18/92

04/20/93

07/30/93

Table 6.8 HPCI demands and reactor-criticalyears.

Calendar

year

HPCI 

demands

Reactor-criticalyears

1987 16 14.63

1988 10 14.15

1989 7 15.75

1990 13 17.77

1991 9 17.11

1992 6 17.19

1993 2 17.34

This table has the same form as in Example 6.4,

showing cells with events and exposure times. The

relevant exposure time is reactor-critical-years,

because the HPCI system uses a turbine-driven

pump, which can only be demanded when the

reactor is producing steam. The counts come from

the tabulated events of Example 6.5, and the criticalyears can be constructed from information in Poloski

et al. (1999a). The variation in critical-years results

from the facts that several reactors were shut down

for extended periods, and one reactor did not receive

its low power license until 1989.

This leads to a plot similar to Figure 6.21, showing

the estimated value of the demand frequency, 8, and

a confidence interval for each year. This is shown in

Figure 6.22.

Figure 6.22 seems to indicate a decreasing trend in the

frequency of HPCI demands. However, the picture

does not reveal whether the apparent trend is perhaps

merely the result of random scatter. To answer that

question, a formal statistical test is necessary, quantifying the strength of the evidence. Such tests will be

given in Section 6.2.3.2.2.

Cumulative Plot. Figure 6.22 required a choice of

how to divide the time axis into cells. A different plot,

given next, does not require any such choice, if the

dates of the events are recorded. Plot the cumulative

event count at the n event dates.

Figure 6.23 shows this for Example 6.5. The events

are arranged in chronological order, and the cumulative count of events is plotted against the event

times.

The slope of a string of plotted points is defined as the

vertical change in the string divided by the horizontal

change, )y/)x. This is the familiar definition of slope

from mathematics courses. In the plot given here, the

horizontal distance between two points is elapsed time,
Parameter Estimation and Model Validation

6-26

Figure 6.23 Cumulative number of HPCI demands,

by date.

and the vertical distance is the total number of events

that occurred during that time period. Therefore,

slope = (number of events)/(elapsed time) ,

so the slope is a graphical estimator of the event frequency, 8. A constant slope, or a straight line, indicates

a constant 8. Changes in slope indicate changes in 8:

if the slope becomes steeper, 8 is increasing, and if the

slope becomes less steep, 8 is decreasing.

In Example 6.5 the time axis represents calendar

years. Because the relevant frequency is events per

reactor-critical-year, it would be better to plot the time

axis in terms of total reactor-critical-years from the

start of 1987. However, it is somewhat difficult to

calculate the reactor-critical-years preceding any

particular event, or equivalently, the reactor-criticalyears between successive events. Therefore, simple

calendar years are used. This is adequate if the

number of reactors operating at any time is fairly

constant, because then the rate per reactor-criticalyear remains roughly proportional to the rate per

industry-calendar year. In the present case, as

shown by Table 6.8, later calendar-years correspond

to more critical-years than do early calendar- years.

The slope in Figure 6.23 is steepest on the left, and

gradually lessens, so that the plot is rising fastest on

the left and more gently on the right. More HPCI

demands are packed into a time interval on the left

than into a time interval of the same length on the

right. This indicates that the frequency of unplanned

HPCI demands was decreasing during the time

period of the study. Thus, this figure leads to the

same general conclusion as does Figure 6.22.

Figure 6.23 shows more detail, with the individual

events plotted, but it is less accurate in this example

because we have not gone through the work of

plotting events versus reactor-critical time.

It is important that the horizontal axis cover the entire

data-collection period and not stop at the final event. In

Figure 6.23, the lack of events during the last half of

1993 contributes to the overall curvature of the plot.

If the frequency is constant, the plot should follow a

roughly straight line. For comparison, it is useful to

show a straight diagonal line, going from height 0 at the

start of the data collection period to height n + 1 at the

end of the data collection period, where n is the number

of data points.

In Figure 6.23, the diagonal line is shown as a dotted

line, rising from height 0 on the left to height n + 1 =

64 on the right.

As mentioned above, the early calendar years

contain fewer reactor-critical-years than do the later

calendar years. Therefore, the time axis in Figure

6.23 would reflect reactor-critical-years more accurately if the left end of the axis were compressed

slightly or the right end were stretched slightly. The

effect would be to increase the curvature of the plot,

making it rise more quickly on the left and more

slowly on the right.

A cumulative plot contains random bounces and clusters, so it is not clear whether the observed pattern is

more than the result of randomness. As always, a

formal statistical test will be needed to measure the

strength of the evidence against the hypothesis of

constant event frequency.

6.2.3.2.2 Statistical Tests for a Trend in 8

The Chi-Squared Test. This is the same test as given

in Section 6.2.3.1.2, only now the cells are years or

similar divisions of time.

In Example 6.5, the p-value is 0.009, meaning that a

random data set with constant 8 would show this

much variability with probability only 0.9%. Two

points are worth noting.

• The chi-squared test makes no use of the order

of the cells. It would give exactly the same

conclusion if the intervals in Figure 6.22 were

scrambled in a random order instead of generally decreasing from left to right.
Parameter Estimation and Model Validation

6-27

• The calculated p-value is accurate enough to

use, by the guidelines of Section 6.2.3.1.2,

because the number of events is 63, and the

number of cells is 7, so x/c = 63/7 = 9. Even

splitting the cells into six-month periods or smaller periods would be justified.

Chapter 7 will take Figure 6.22, fit a trend, and perform

an additional test based on the fit; see Sections 7.2.3

and 7.2.4. Therefore, the chi-squared test is not discussed further here. 

The Laplace Test. This test does not use the binning

of times into cells, but instead uses the exact dates. In

the example, there are 63 occurrences of events during

a seven-year period. In general, consider a time interval

[0, L], and suppose that during this period n events

occur at successive random times T1, T2, ... , Tn. Although the number of occurrences, n, is random when

the plants are observed for a fixed length of time L, we

condition on the value of n, and so treat it as fixed.

Consider the null hypothesis:

H0: 8 is constant over time.

Consider the alternative hypothesis:

H1: 8 is either an increasing or a decreasing function of

time.

This hypothesis says that the events tend to occur more

at one end of the interval than at the other. A test that

is often used is based on the mean of the failure times,

T Tn . The intuitive basis for the test is the i i = Σ /

following. If 8 is constant, about half of the events

should occur before time L/2 and half afterwards, and

the average event time should be close to L/2. On the

other hand, if 8 is decreasing, more events are expected

early and fewer later, so the average event time should

be smaller than L/2. Similarly, if 8 is increasing, the

average event time is expected to be larger than L/2.

Therefore, the test rejects H0 if is far from T L/2.

Positive values of the difference T ! L/2 indicate an

increasing trend, and negative values indicate a decreasing trend. 

When H0 is true, has expected value T L/2 and variance

L2/(12n). The resulting test statistic is

U . T L

L n = − /

/

2

12

The statistic U is approximately standard normal for n

$ 3. A test of H0 at significance level 0.05 versus an

increasing alternative,

H1: 8 is increasing in time,

would reject H0 if U $ 1.645. A 0.05 level test versus

a decreasing alternative,

H1: 8 is decreasing in time,

would reject H0 if U # !1.645. Of course, ±1.645 are

the 95th and 5th percentiles, respectively, of the standard normal distribution. A two-sided test, that is, a test

against the original two-sided alternative hypothesis, at

the 0.10 level would reject H0 if |U| $ 1.645.

This test, generally known as the “Laplace” test, is

discussed by Cox and Lewis (1978, p. 47). The Laplace test is known to be good for detecting a wide

variety of monotonic trends, and consequently it is

recommended as a general tool for testing against such

alternatives.

Let us apply the Laplace test to the HPCI-demand

data of Example 6.5. First, the dates must be converted to times. The first event time is 0.011 years

after January 1, 1987, the final event is 6.581 years

after the starting date, and the other times are

calculated similarly. Here, a “year” is interpreted as

a 365-day year. The total number of 365-day years

is L = 7.00. The mean of the event times can be

calculated to be 2.73. Therefore, the calculated

value of U is

2 73 35

7 00 12 63

3 02 . .

. / . −× = − .

This is statistically very significant. The value 3.02 is

the 0.1th percentile of the standard normal distribution. Thus, the evidence is very strong against a

constant demand rate, in favor instead of a decreasing demand rate. Even against the two-sided hypothesis

H1: 8 is increasing or decreasing in time,

the p-value is Pr( |U| > 3.02) = 0.002.

In the example, the Laplace test statistic was calculated in terms of calendar time instead of reactorcritical-time. As remarked earlier, using reactorcritical-time would increase the curvature of the plot

in Figure 6.23. A similar argument shows that using

reactor-critical-time in computing U would increase
Parameter Estimation and Model Validation

6-28

the strength of the evidence against the hypothesis

of a constant demand rate. However, the computations would be very tedious. That is an advantage of

the chi-squared test, because it is typically easier to

find the exact relevant exposure time for blocks of

time, such as years, than for each individual event.

In the example, the result of the Laplace test agrees

with the result from the chi-squared test, but is more

conclusive. The chi-squared test gave a p-value of

0.009, meaning that if H0 is true, the cells would

appear so different from each other with probability

only 0.009. The Laplace test gives a p-value of

0.002.

The chi-squared and Laplace tests differ because they

are concerned with different alternatives to H0. The

chi-squared test is concerned with any variation from

cell to cell (from year to year in the example). If the

event rate goes up and down erratically, that is just as

much evidence against H0 as if the event rate decreases

monotonically. The Laplace test, on the other hand, is

focused on the alternative of a trend. It has more power

for detecting trends, but no power at all for detecting

erratic changes upward and downward.

Other tests exist in this setting. See Ascher and Feingold (1984, page 80) and Engelhardt (1994, p. 19) for

details.

6.2.3.3 No Multiple Failures

The second assumption of the Poisson process is that

there are no exactly simultaneous failures. In practice

this means that common-cause failures do not occur. In

most situations, common-cause failures will occur from

time to time. This was seen in some of the examples

discussed in Section 2.2. However, if common-cause

events are relatively infrequent, their effect on the

validity of the Poisson model can normally be ignored.

No statistical methods are given here to examine

whether common-cause events can occur. Instead, the

analyst should think of the engineering reasons why

common-cause events might be rare or frequent, and the

data should be examined to discover how frequent

common-cause events are in practice.

In Example 6.5, HPCI demands, it is reasonable that

common-cause events could occur only at multiple

units at a single site. There was one such pair of

events in the data, with HPCI demands at Hatch 1

and Hatch 2, both on 08/03/87. Examination of the

LERs reveals that the demands occurred from

different causes. They happened at different times,

and so were not exactly simultaneous. The conclusion is that common causes may induce exactly

simultaneous events, but they are infrequent.

If common-cause events are relatively frequent, so that

they cannot be ignored, it might be necessary to perform two analyses, one of the “independent”, or notcommon-cause, events, and one of the common-cause

occurrences. The frequency of independent events

could be estimated using the methods given here. The

common cause events would have to be analyzed by

other methods, such as methods described in the

references given in Section 1.3.

6.2.3.4 Independence of Disjoint Time Periods

This section is less important than the others, and of

interest only to truly dedicated readers. Others should

skip directly to Section 6.2.3.5.

The final assumption of the Poisson model is that event

occurrences in disjoint time periods are statistically

independent. This should first be addressed by careful

thinking, similar to that in the examples of Section 2.2.

However, the following statistical approach may also be

useful.

One possible type of dependence would be if events

tend to cluster in time: large between-event times tend

to occur in succession, or similarly small ones tend to

occur in succession. For example, suppose that a repair

is done incorrectly several times in succession, leading

to small times between failures. The occurrence of a

failure on one day would increase the probability of a

failure in the next short time period, violating the

Poisson assumption. After the problem is diagnosed,

the personnel receive training in proper repair procedures, thereafter resulting in larger times between

failures.

To illustrate the ideas, an example with no trend is

needed. The shutdown LOSP events introduced in

Section 2.2 can be used as such an example. The

data are restricted here to the years 1991-1996,

primarily to reduce any effect of the overall downward trend in total shutdown tome. Atwood et al.

(1998) report 24 plant-centered LOSP events during

shutdown in 1991-1996. They are given as Example 6.6.

The null hypothesis is that the successive times between

events are independent and exponentially distributed.

We consider the alternative hypotheses that

• the times are not exponentially distributed, possibly with more short times between events than

expected from an exponential distribution; or
Parameter Estimation and Model Validation

6-29

• successive times are correlated, that is that short

times tend to be followed by short times and long

times by long times.

Example 6.6 Dates of shutdown LOSP

events and days between

them.

The consecutive dates of shutdown LOSP

events are shown in columns below. After each

date is the time since the preceding event, in

days. For the first event, the time since the

start of the study period is shown. Also, the

time is shown from the last event to the end of

the study period, a 25th “between-event time.”

03/07/91 66

03/13/91 6

03/20/91 7

04/02/91 13

06/22/91 81

07/24/91 32

10/20/91 88

01/29/92 101

03/23/92 54

04/02/92 10

04/06/92 4

04/28/92 22

04/08/93 345

05/19/93 41

06/22/93 34

06/26/93 4

10/12/93 108

05/21/94 221

09/27/94 129

11/18/94 52

02/27/95 101

10/21/95 236

01/20/96 91

05/23/96 124

 — 223

Section 6.6.2.3 discusses ways to investigate whether

data come from a particular distribution. Therefore, the

issue of the exponential distribution is deferred to that

section. The issue of serial correlation motivates the

following procedure. Let yi

 be the ith time between

events, and let xi

 be the (i!1) time between events, xi =

yi!1. We look to see if xi

 and yi are correlated.

In the above example, the first few (x, y) pairs are

(66, 6), (6, 7), and (7, 13), and the final pair is (124,

223).

6.2.3.4.1 Graphical Method

As just mentioned, the issue of whether the distribution

is exponential is deferred to Section 6.6.2.3. Consider

here the question of serial correlation. A scatter plot of

x versus y will indicate whether the values are correlated. However, with skewed data the large values tend

to be visually dominant, distorting the overall message

of the plot. One could try an ad hoc transformation,

such as the logarithmic transformation, but a more

universally applicable approach is to use the ranks of

the variables. That is, sort the n times in increasing

order, and assign rank 1 to the smallest time and rank n

to the largest time.

In the example, the two shortest times are each

equal to 4 days. Each is assigned the average of

ranks 1 and 2, namely 1.5. The next largest time is

6 days, which is assigned rank 3, and so forth. The

17th and 18th times are each 101 days, so those two

are each assigned rank 17.5. Selected values of x,

y and their ranks are shown in Table 6.9. For compactness, not all of the values are printed.

Table 6.9 Calculations for analyzing LOSP

dates.

x rank(x) y rank(y)

 —

 66

 6

 7

 13

 81

 32

 88

101

 54

 ...

 52

101

236

 91

124

223

 —

 13

 3

 4

 6

 14

 8

 15

 17.5

 12

 ...

 11

 17.5

 24

 16

 20

 23

 66

 6

 7

 13

 81

 32

 88

101

 54

 10

 ...

101

236

 91

124

223

 —

 13

 3

 4

 6

 14

 8

 15

 17.5

 12

 5

 ...

 17.5

 24

 16

 20

 23

 —

Figure 6.24 shows a scatter plot of rank(x) versus

rank(y). The plot seems to show very little pattern,

indicating little or no correlation from one time to the

next. The barely perceptible trend from lower left to

upper right (“southwest to northeast”) is probably not

meaningful, but a hypothesis test will need to be

performed to confirm or refute that judgment.

rank(x)

rank( y) 0

5

10

15

20

25

0 5 10 15 20 25

Figure 6.24 Scatter plot of rank(x) versus rank(y).
Parameter Estimation and Model Validation

6-30

6.2.3.4.2 Statistical Tests

This section considers whether the between-event times

are serially correlated. The question of whether they

are exponentially distributed is discussed in Section

6.6.2.3, under the topic of goodness-of-fit tests.

To test for correlation, it is not appropriate to assume

normality of the data. Instead, a nonparametric test

should be used, that is, a test that does not assume any

particular distributional form. A test statistic that is

commonly produced by statistical software is Kendall’s

tau (J). Tau is defined in Conover (1999), Hollander

and Wolfe (1999), and other books on nonparametric

statistics. 

Based on the data of Table 6.9, the hypothesis of no

correlation between X and Y was tested. Kendall’s

tau gave a p-value of 0.08. This calculation indicates

that the very slight trend seen in Figure 6.24 is not

statistically significant.

Recall, from the discussion of Section 6.2.3.1.2, that

a small p-value is not the end of an analysis. The pvalue for this example, although larger than the

customary cut-off of 0.05, is fairly small. This indicates that the trend in Figure 6.24 is somewhat

unlikely under the assumption of no correlation. If

we are concerned about this fact, we must seek

possible engineering mechanisms for the trend. The

data are times between LOSP events in the industry

as a whole. Therefore, the most plausible explanation is the overall industry trend of fewer shutdown

LOSP events. This trend would produce a tendency

for the short times to occur together (primarily near

the start of the data collection period), and the long

times to occur together (primarily near the end of the

data period).

6.2.3.5 Consistency of Data and Prior

As an example, if the prior distribution has mean

Eprior(8), but the observed data show x/t very different

from the prior mean, the analyst might wonder if the

data and the prior are consistent, or if, instead, the prior

distribution was misinformed. To investigate this, one

could ask what the prior probability is of getting the

observed data. Actually, any individual x may have

small probability, so a slightly more complicated

question is appropriate.

Suppose first that x/t is in the right tail of the prior

distribution. The relevant quantity is the prior probability of observing x or more events. This is

Pr( ) Pr( | ) ( ) Xx Xx f d ≥= ≥ ∫ λ λλ (6.6) prior

where

 (6.7)

Pr( | ) ( ) / !

()/!

Xx e t k

e tk

t k

k x

t k

k

x

≥ = −

=

∞

−

=

−

∑

= − ∑

λ λ

λ

λ

λ 1

0

1

In general, Equation 6.6 does not have a direct analytical expression. However, in the special case when the

prior distribution is gamma("prior, $prior), it can be shown

that the probability in question equals

(6.8)

Pr( )

( )

!()

(/ )( / ) ( )

X x

k

k

t t k k

k

x

≥ =

+ − ∑ + − +

=

−

1 1

0

1Γ

Γ

α

α

β β α

where '(s) is the gamma function, a generalization of

the factorial function as described in Appendix A.7.6.

The distribution defined by Equation 6.8 is named the

gamma-Poisson or negative binomial distribution.

The above probability can be evaluated with the aid of

software.

When Equation 6.8 is not applicable, one method of

approximating the integral in Equation 6.6 is by Monte

Carlo sampling. Generate a large number of values of

8 from the prior distribution. For each value of 8, let y

be the value of Equation 6.7, which can be calculated

directly. The average of the y values is an approximation of the integral in Equation 6.6. Another method of

approximating the Equation 6.6 is by numerical integration.

If the probability given by Equation 6.6 is small, the

observed data are not consistent with the prior belief —

the prior belief mistakenly expected 8 to be smaller

than it apparently is. When should the probability be

considered “small”? Many people consider probabilities < 0.05 to be “small,” but there is no rigid rule.

Similarly, if x/t is in the left tail of the prior distribution,

the relevant quantity is the prior probability that X # x.

When the prior is a gamma distribution, the desired

probability is the analogue of the sum in Equation 6.8,

Pr( ) ( )

!() (/ )( / ) ( ) X x kk

t t k k

k

x

≤ =

+ ∑ + − +

−

Γ

Γ

α

α

β β α 1

0

In any case, the desired probability can be approximated by Monte Carlo sampling. If that probability is

small, the prior distribution mistakenly expected 8 to be

larger than it apparently is.
Parameter Estimation and Model Validation

6-31

In Example 6.3, we ask whether the observed zero

failures in 2102 reactor-calendar-years is consistent

with the WASH-1400 prior, lognormal with median

1E-3 per year and error factor 10. To investigate

this, 100,000 random values of 8 were generated

from the lognormal prior. (The details are given

below.) For each 8, Pr(X # 0) = exp(!21028) was

found. The mean of these probabilities was 0.245.

This is a sample mean, and it estimates the true

probability. It is not small, and therefore gives no

reason to question the applicability of the prior.

One must ask whether the sample was large enough.

The software that calculated the sample mean also

calculated the standard error to be 0.0009. Recall

from Section 6.2.1.2 that in general a 95% confidence interval can be approximated as the estimate

plus or minus 2×(standard error). In this case, this

interval becomes 0.245 ± 0.002. We conclude that

the true mean equals 0.245 except perhaps for

random error in the third digit. This shows that the

sample size was more than large enough to give an

answer to the accuracy required.

The recipe for generating 8 from a lognormal distribution is as follows:

(1) Generate z from a standard normal distribution,

using commercial software,

(2) Define loglam = : + Fz, where : and F were

found in Section 6.2.2.7.2, and then

(3) Define lambda = exp(loglam).

6.3 Failures to Change State: 

Failure on Demand

This section is similar to Section 6.2, but the details are

different. The structure of this section parallels that of

Section 6.2 almost exactly, and some admonitions from

that section are repeated here. The most important

topics for a first-time reader are:

• Maximum likelihood estimation (6.3.1.1),

• Bayesian estimation, especially with a discrete

prior or a conjugate prior (6.3.1-6.3.2.3), and

• Model validation, especially using graphical tools

(portions of 6.3.3).

This section applies to data satisfying the assumptions

of Section 2.3.2.1. The probability of a failure on

demand is denoted p, a unitless quantity. The data

consist of x failures in n demands, with 0 # x # n.

Before the data are generated, the number of failures is

random, denoted X. For any particular number x, the

probability of x failures in n demands is

Pr( ) ( ) X x , (6.9)

n

x

p p x nx == − ⎛

⎝

⎜ ⎞

⎠

⎟ − 1

where the binomial coefficient is defined as

 . n

k

n

kn k

⎛

⎝

⎜ ⎞

⎠

⎟ = −

!

!( )!

The methods will be illustrated by the following hypothetical data set.

Example 6.7 AFW turbine-train failure to start

In the last 8 demands of the turbine train of the

auxiliary feedwater (AFW) system at a PWR, the

train failed to start 1 time. Let p denote the

probability of failure to start for this train.

As in Section 6.2, frequentist methods are presented

first, followed by Bayesian methods. This choice is

made because the frequentist point estimate is so very

simple, not because frequentist estimation is preferable

to Bayesian estimation. Indeed, in PRA p is normally

estimated in a Bayesian way.

6.3.1 Frequentist or Classical Estimation

6.3.1.1 Point Estimate

The most commonly used frequentist estimate is the

maximum likelihood estimate (MLE). It is found by

taking the likelihood, given by Equation 6.9, and

treating it as a function of p. The value of p that maximizes the likelihood is called the MLE. It can be

shown, by setting a derivative to zero, that the maximum likelihood estimate (MLE) of p is . p xn $ = /

This is intuitively appealing – the observed number of

failures divided by the observed number of demands.

Figure 6.25 shows the likelihood as a function of p,

for the data of Example 6.7. The figure shows that

the likelihood is maximized at p = 1/8, as stated by

the formula.

If several subsets of data, such as data corresponding to

several plants, several types of demand, or several

years, are assumed to have the same p, data from the

various sources may be combined, or pooled, for an

overall estimate. Denoting the number of failures and

demands in data subset j by xj

 and nj, respectively, let x

= Ej

xj and n = Ejnj. The MLE is x/n.
Parameter Estimation and Model Validation

6-32

0.0 0.1 0.2 0.3 0.4 0.5

p

0.0

0.1

0.2

0.3

0.4

likelihood

Figure 6.25 Likelihood as a function of p, for the

data of Example 6.7.

As mentioned in Section 6.2.1.1, final answers will be

shown in this handbook with few significant digits, to

avoid giving the impression that the final answer

reflects precise knowledge of the parameter. Intermediate values will show more significant digits, to

prevent roundoff errors from accumulating.

6.3.1.2 Standard Deviation of Estimator

The number of failures is random. One number was observed, but if the demands were repeated a different

number of failures might be observed. Therefore, the

estimator is random, and the calculated estimate is the

value it happened to take this time. Considering the

data as random, one could write . This $

P Xn = /

notation is consistent with the use of upper case letters

for random variables, although it is customary in the

literature to write for both the random variable and p$

the calculated value. The standard deviation of the

estimator is [p(1 - p)/n]

1/2. Substitution of the

estimate for p$ p yields an estimate of the standard

deviation,

 . [ $( $)/ ] /

p pn 1 1 2 −

The estimated standard deviation of an estimator is also

called the standard error of the estimate. The handy

rule given in Section 6.2.1.2 applies here as well:

MLE ± 2×(standard error)

is an approximate 95% confidence interval for p, when

the number of demands, n, is large. However, an exact

confidence interval is given below.

In Example 6.7, the standard error for p is

[0.125 × (1 - 0.125) / 8]1/2 = 0.12.

6.3.1.3 Confidence Interval for p

Confidence intervals are used in two ways in this

handbook. They give a standard of comparison, when

Bayes credible intervals are found based on so-called

noninformative priors, and they can be used (but are not

required) in some plots for validating model assumptions. Therefore, readers may wish to skim the present

section quickly on the first reading.

The interpretation of confidence intervals is given in

Appendix B and in Section 6.2.1.3. It is so important

that it is repeated here. In the frequentist approach, p is

fixed and the data are random. Therefore the maximum

likelihood estimator and the confidence limits are all

random. For most data sets, the MLE, , will be close p$

to the true value of p, and the confidence interval will

contain p. Sometimes, however, the MLE will be rather

far from p, and sometimes (less than 10% of the time)

the 90% confidence interval will not contain p. The

procedure is good in the sense that most of the time it

gives good answers, but the analyst never knows if the

current data set is one of the unlucky ones. A figure

like Figure 6.4 could be constructed for p, to illustrate

that many data sets could be generated from the same p,

yielding many confidence intervals, most of which

contain the true value of p.

The following material is drawn from Johnson et al.

(1992, Section 3.8.3). A confidence interval for p can

be expressed in terms of percentiles of a beta distribution. Appendix A.7.8 presents the basic facts about the

beta distribution. As mentioned there, the beta family

of distributions includes many distributions that are

defined on the range from 0 to 1, including the uniform

distribution, bell-shaped distributions, and U-shaped

distributions. The beta distribution is also discussed

more fully in the section below on Bayesian estimation.

Denote the lower and upper ends of a 100(1 ! ")%

confidence interval by pconf, "/2 and pconf, 1!"/2, respectively. It can be shown that the lower limit is

pconf, "/2 = beta"/2(x, n ! x + 1)

and the upper limit is

pconf, 1!"/2 = beta1 ! "/2(x + 1, n ! x)
Parameter Estimation and Model Validation

6-33

where betaq(", $) denotes the q quantile, or 100×q

percentile, of the beta(", $) distribution. For example,

a 90% confidence interval for p is given by beta0.05(x, n

! x + 1) and beta 0.95(x + 1, n ! x). If x = 0, the beta

distribution for the lower limit is not defined; in that

case, set pconf, "/2 = 0. Similarly, if x = n, the beta distribution for the upper limit is not defined; in that case, set

pconf, 1 ! "/2 = 1. In any case, note carefully that the

parameters of the beta distribution are not quite the

same for the lower and upper endpoints. 

Appendix C tabulates selected percentiles of the beta

distribution. However, interpolation may be required.

Some software packages, including commonly used

spreadsheets such as Microsoft Excel (2001) and

Quattro Pro (2001), calculate the percentiles of the beta

distribution. Those calculations are more accurate than

interpolating tables. Finally, Appendix A.7.8 gives a

last-resort method, which allows beta percentiles to be

calculated by complicated formulas involving tabulated

percentiles of the F distribution.

In Example 6.7, with 1 AFW train failure in 8

demands, suppose that a 90% interval is to be

found. Then " = 0.10, and 1!"/2 = 0.95. For the

lower limit, beta0.05(1, 8!1+1) = 6.39E!3, from

Table C.5. Thus,

pconf, 0.05 = 0.0064.

For the upper limit, beta0.95(1+1, 8!1) = 4.71E!1,

also from Table C.5. Thus,

pconf, 0.95 = 0.47.

6.3.2 Bayesian Estimation

Section 6.2.2.1 gives an overview of Bayesian estimation, which applies here. Just as for 8 in that section,

Bayesian estimation of p involves several steps. The

prior belief about p is quantified by a probability

distribution, the prior distribution. This distribution

will be restricted to the range [0,1], because p must lie

between 0 and 1, and it will assign the most probability

to the values of p that are deemed most plausible. The

data are then collected, and the likelihood function is

constructed. The likelihood function is given by

Equation 6.9 for failures on demand. It is the probability of the observed data, written as a function of p.

Finally, the posterior distribution is constructed, by

combining the prior distribution and the likelihood

function through Bayes’ theorem. The posterior

distribution shows the updated belief about the values

of p. It is a modification of the prior belief that accounts for the observed data.

Figure 6.5, showing the effect of various data sets on

the posterior distribution, is worth studying. Although

that figure refers to 8, exactly the same idea applies

to p.

The subsections below consider estimation of p using

various possible prior distributions. The simplest prior

distribution is discrete. The posterior can be calculated

easily, for example, by a spreadsheet. The next simplest prior is called conjugate; this prior combines

neatly with the likelihood to give a posterior that can be

evaluated by simple formulas. Finally, the most general

priors are considered; the posterior distribution in such

a case can only be found by numerical integration or by

random sampling.

Section 6.2.2.2 discusses how to choose a prior, and

gives references for further reading. It applies to

estimation of p as much as to estimation of 8, and

should be read in connection with the material given

below.

6.3.2.1 Estimation with a Discrete Prior

The explanation here will be easier to follow if the

examples in Section 6.2.2.3 have also been read. The

parameter p is assumed to take one of only m possible

values, p1, ... , pm. Denote the p.d.f. by f, so f(pi

) =

Pr(pi

), the prior probability that the parameter has the

value pi

. After evidence E is observed, Bayes’ theorem

says:

f pE (6.10) f p LEp

LEp f p i

i i

j j j

m ( |) ( )(| )

(| )( ) =

∑ =1

where

f(pi

 | E) = the probability of pi given evidence E

(posterior distribution),

f(pi

) = the probability of pi prior to having evidence

E ( prior distribution), and

L(E | pi

) = the likelihood function (probability of the

evidence given pi

).

Just as in Section 6.2.2.3, the denominator in Equation

6.10, the total probability of the evidence E, is simply

a normalizing constant.

When the evidence is in the form of x failures in n

demands and the assumptions for a binomial distribu-
Parameter Estimation and Model Validation

6-34

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

p

0.00

0.02

0.04

0.06

0.08

Pr(p)

Figure 6.26 Discrete informative prior distribution

for p.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

p

0.00

0.02

0.04

0.06

0.08

Pr(p)

prior Pr(p)

posterior Pr(p)

Figure 6.27 Discrete prior and posterior distribution

for data in Example 6.7.

0.0 0.1 0.2 0.3 0.4 0.5

p

0.00

0.04

0.08

0.12

likelihood

Figure 6.28 Likelihood as a function of p, for ten

times the data of Example 6.7.

tion are satisfied, the likelihood function is the given by

Equation 6.9:

LEp .

n

x

p p i

x

i

n x (| ) ( ) = ⎛

⎝

⎜ ⎞

⎠

⎟ − − 1

As an example, let us use the data in Example 6.7.

We will use a discrete prior distribution, just as in

Section 6.2.2.3. Unlike the examples in that earlier

section, the present example uses an informed prior.

Assume that a prior distribution was developed by

plant equipment experts based on population variability data from similar systems, but adapted to

account for untested new design aspects of this

system. The prior is defined on 81 points, for p = 0,

0.01, 0.02, ..., 0.8. The most likely value is p = 0.1.

From there, the prior falls linearly until p = 0.3, then

tails off to 0 at p = 0.8. On the low end it falls

linearly to 0 at p = 0. The distribution is shown in

Figure 6.26.

This prior is discrete. We will want to compare the

prior with the posterior distribution, and the graph of

two discrete distributions is easier to read if just the

tops of the vertical bars are plotted. The resulting

plot of the prior looks like a continuous density, but it

still is intended to represent the above discrete

distribution.

The likelihood function is shown in Figure 6.25. The

posterior distribution is proportional to the product of

the prior and the likelihood, normalized so that the

total probability equals 1. Figure 6.27 shows the

prior and the posterior distributions on the same plot.

Note that the posterior follows the shape of the prior

very closely. This is because the data are consistent

with the peak area of the prior, but are not yet strong

enough to appreciably reduce the uncertainty in the

prior — there are only eight demands.

What happens to this posterior as additional data

accumulate? Suppose that ten times as much data

had been collected, 10 failures in 80 demands. The

likelihood function, given by Equation 6.9 with this

new data set, is shown in Figure 6.28.

The posterior distribution is proportional to the

product of the prior and this new likelihood. Figure

6.29 shows the prior and this new posterior.

Table 6.10 compares the results of the Bayesian

analyses with the original data and with ten times as

much data.
Parameter Estimation and Model Validation

6-35

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

p

0.00

0.05

0.10

0.15

Pr(p)

prior Pr(p)

posterior Pr(p)

Figure 6.29 Discrete prior and posterior distributions

for p, with 10 times as much data as in previous

figure.

Table 6.10 Comparison of Bayesian 

distributions.

Distribution 5th

%tile

mean 95th

%tile

Prior 0.04 0.206 0.54

Posterior, original

data

0.05 0.153 0.29

Posterior, ten times

more confirmatory

data

0.07 0.130 0.19

The difference between the two posterior distributions results from the differences between the two

likelihoods. In this hypothetical example, both data

sets have the same MLE, 0.125, but the larger data

set has a likelihood that is more concentrated. The

posterior distribution from the larger data set is

dominated by the likelihood, and closely resembles

it.

Readers are strongly encouraged to work through a few

examples like this on their own. The calculations are

easy to carry out with a spreadsheet.

6.3.2.2 Estimation with a Conjugate Prior

We now consider the use of continuous prior distributions, beginning with a very convenient family of

distributions, the conjugate priors.

6.3.2.2.1 Definitions

By far the most convenient form for the prior distribution of p is a beta("prior, $prior) distribution. The beta

distributions are the conjugate family for binomial data.

The properties of the beta distribution are therefore

summarized here, as well as in Appendix A.7.8.

If p has a beta(", $) distribution, the density is

fp p p () ( ) . ( )

()() = − + − − ΓΓ Γ

α β

α β

α β 1 1 1

For most applications the gamma functions in the front

can be ignored — they only form a normalizing constant, to ensure that the density integrates to 1. The

important feature of the density is that 

f(p) % p" ! 1(1 ! p)

$ ! 1 (6.11)

where the symbol % denotes “is proportional to.” The

parameters of the distribution, " and $, must both be

positive. The mean and variance of the distribution are

: = "/("+$), (6.12)

variance = αβ

( )( ) αβ αβ + ++ 2 1

= :(1!:)/("+$+1). (6.13)

The shape of the beta density depends on the size of the

two parameters. If " < 1, the exponent of p is negative

in Equation 6.11, and therefore the density is unbounded as p 6 0. Likewise, if $ < 1, the density is

unbounded as p 6 1. If both " > 1 and $ > 1, the

density is roughly bell shaped, with a single mode.

Appendix A.7.8 shows graphs of some beta densities.

Equation 6.13 shows that as the sum " + $ becomes

large, the variance becomes small, and the distribution

becomes more tightly concentrated around the mean.

As will be seen below, if the prior distribution is a beta

distribution, so is the posterior distribution. Therefore,

the above statements apply to both the prior and the

posterior distributions.

Appendix C tabulates selected percentiles of beta

distributions. Also, the percentiles of a beta distribution

can be found by many software packages, including

some spreadsheets. Also, the percentiles can be obtained from algebraic formulas involving percentiles of

the F distribution, as explained in Appendix A.7.8.
Parameter Estimation and Model Validation

6-36

6.3.2.2.2 Update Formulas

The beta family is conjugate to binomial data. That is,

updating a beta prior distribution with the data produces

a posterior distribution that is also a beta distribution.

This follows immediately from the derivation of the

posterior distribution. By Bayes’ theorem (Appendix

B.5), the posterior distribution is related to the prior

distribution by

f p X xp f p . (6.14) post prior ( ) Pr( | ) ( ) ∝ =

This is the continuous analogue of Equation 6.10. As

mentioned in the earlier sections, the probability of the

data is also called the “likelihood.” It is given by

Equation 6.9. Stripped of all the normalizing constants,

the beta p.d.f. is given by Equation 6.11.

Therefore, the beta distribution and the binomial

likelihood combine as:

fpp pp p

p p

x nx

x nx

post

 .

() ( ) ( )

( )

∝− −

∝ −

−− −

+ − −+ −

1 1

1

1 1

1 1

α β

α β

In the final expression, everything that does not involve

p has been absorbed into the proportionality constant.

This shows that the posterior distribution is of the form

beta("post, $post), with

"post = "prior + x

$post = $prior + (n ! x) .

The mean and variance of the prior and posterior

distributions are given by Equations 6.12 and 6.13,

using either the prior or posterior " and $.

These update formulas give intuitive meaning to the

beta parameters: "prior corresponds to a prior number of

failures and $prior to a prior number of successes.

Assuming a beta("prior , $prior ) distribution is equivalent

to having observed "prior failures and $prior successes

before the current data were observed.

6.3.2.3 Possible Conjugate Priors

A concentrated distribution (small variance, large value

of "prior + $prior) represents much presumed prior knowledge. A diffuse prior (large variance, small value of

"prior + $prior) represents very little prior knowledge of p.

6.3.2.3.1 Informative Prior

The warning given in Section 6.2.2.5.1 applies here as

well: the prior distribution must be based on information other than the data. If possible, relevant information from the industry should be used.

The calculations are now illustrated with Example

6.7, one failure to start in eight demands of the AFW

turbine train. Poloski et al. (1998) examined nine

years of data from many plants, and found a

beta(4.2, 153.1) distribution for the probability of the

AFW train failure to start.

Application of the update formulas yields

"post = "prior + x = 4.2 + 1 = 5.2

$post = $prior + (n ! x) = 153.1 + (8 !1) = 160.1 .

The mean of this distribution is

5.2/(5.2 + 160.1) = 0.031,

and the variance is

0.031×(1 ! 0.031)/(5.2 + 160.1 + 1) = 1.89E!4,

and the standard deviation is the square root of the

variance, 0.014. The 5th and 95th percentiles of the

posterior beta(", $) distribution are found from Table

C.5, except the tabulated $ values do not go above

100. A footnote to that table gives an approximation

that is valid for $ >> ". That formula applies, because 160.1 >> 5.2. According to the formula the q

quantile is approximated by

P2

q(2×5.2)/[2×160.1 + P2q(2×5.2)].

Therefore the 5th percentile of the beta distribution is

approximately

P2

0.05(10.4)/[320.2 + P20.05(10.4)] = 4.19/[320.2 + 4.19]

= 0.013,

and the 95th percentile is approximately

P2

0.95(10.4)/[320.2 + P20.95(10.4)] = 18.86/[320.2 +

18.86] = 0.056 .

All these quantities are unitless.

The prior density, posterior density, and posterior

c.d.f. of p are shown in Figures 6.30 through 6.32.

The posterior density is slightly to the right of the

prior density. It is to the right because the data, one

failure in eight demands, show worse performance

than the industry history. The posterior density is

only slightly different from the prior density because

the data set is small compared to the industry experience (eight demands in the data and an effective

157.3 demands for the industry).
Parameter Estimation and Model Validation

6-37

Figure 6.32 Posterior cumulative distribution of p.

The 5th and 95th percentiles are shown.

0

2

4

6

8

0 0.2 0.4 0.6 0.8 1

p GC00 0505

Density Figure 6.33 Jeffreys noninformative prior distribution

for p.

Figure 6.30 Prior density for p, beta(4.2, 153.1).

Area = 0.05 Area = 0.05

Area = 0.90

0

5

10

15

20

25

30

35

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

p (failures/demand) GC99 0292 17

Density Figure 6.31 Posterior density for p, beta(5.2, 160.1).

The 5th and 95th percentiles are shown.

The 5th and 95th percentiles are shown for the

posterior distribution, both in the plot of the density

and in the plot of the cumulative distribution.

6.3.2.3.2 Noninformative Prior

The Jeffreys noninformative prior is beta(1⁄2 , 1⁄2); see

Box and Tiao (1973), Sections 1.3.4-1.3.5. This

density is shown in Figure 6.33. It is not the uniform

distribution, which is a beta(1, 1) distribution, but

instead rises sharply at the two ends of the interval

(0, 1).

Although the uniform distribution is sometimes used to

model no prior information, there are theoretical

reasons for preferring the Jeffreys noninformative prior.

These reasons are given by Box and Tiao, and are

suggested by the comparison with confidence intervals

presented below. The uniform distribution would

correspond intuitively to having seen one failure in two

demands, which turns out to be too informative. The

Jeffreys noninformative prior corresponds to having

seen one-half a failure in one demand.

The Bayes posterior distribution for p, based on the

Jeffreys noninformative prior, is beta(x + 1⁄2, n ! x + 1⁄2).

The mean of the distribution is (x + 1⁄2)/( n + 1). Selected percentiles are tabulated in Appendix C.

The posterior distribution given here is very similar to

the distributions used in the formulas for confidence

intervals in Section 6.3.1.3. The only difference is in

the parameters. The parameters here are averages of

the parameters used in the confidence intervals. For

example, the first parameter for the lower confidence

limit is x, and the first parameter for the upper confidence limit is x+1. The Bayesian limits, on the other

hand, use the same parameters for the entire posterior

distribution, and the first parameter is x + 1⁄2, the average of the corresponding values for the confidence

limits.
Parameter Estimation and Model Validation

6-38

Figure 6.34 Comparison of four point estimates and

interval estimates for p.

In Example 6.7, failure to start of the turbine-driven

AFW train, the posterior distribution is beta(1.5, 7.5).

The posterior mean is 1.5/(1.5 + 7.5) = 0.17. The

posterior 90% interval is (0.023, 0.40). As is always

the case with discrete data, the confidence interval is

conservative, and so is wider than the Jeffreys

credible interval. However, the two intervals are

similar to each other, being neither to the right nor

the left of the other. Tabular and graphical comparisons are given later.

6.3.2.3.3 Constrained Noninformative Prior

This prior distribution is a compromise between an

informative prior and the Jeffreys noninformative prior.

As was the case in Section 6.2.2.5.3, the prior mean,

denoted here as p0, is based on prior belief, but the

dispersion is defined to correspond to little information.

The priors are described by Atwood (1996) and by

references given there.

For binomial data, the constrained noninformative prior

distribution is not as neat as for Poisson data. The

exact constrained noninformative prior has the form

fprior(p) % ebpp!1/2(1 ! p)

!1/2 , (6.15)

where b is a number whose value depends on the

assumed value of the mean, p0. The parameter b is

positive when p0 > 0.5 and is negative when p0 < 0.5.

Thus, in typical PRA analysis b is negative. Atwood

(1996) gives a table of values, a portion of which is

reproduced in Appendix C as Table C.8. The table

gives the parameter b of the distribution for selected

values of p0. In addition, it gives a beta distribution that

has the same mean and variance as the constrained

noninformative prior.

The beta approximation is illustrated here, and the exact

constrained noninformative distribution is treated more

fully in the section below on nonconjugate priors.

Return again to Example 6.7, the AFW turbine train

failure to start. Let us use the mean of the industry

prior found above, 4.2/157.3 = 0.0267. However,

suppose that the full information for the industry prior

is not available, or that the system under consideration is considered atypical so that the industry prior

is not fully relevant. Therefore, the beta-approximation of the constrained noninformative prior will be

used.

Interpolation of Table C.8 at p0 = 0.0267 yields " =

0.4585. Solving $ = "(1 ! p0)/p0 gives $ = 16.7138.

The resulting posterior distribution has parameters

1.4585 and 23.7138. Interpolation of Table C.5 gives

a 90% interval of (0.0068, 0.15).

6.3.2.3.4 Example Comparison of Above Methods

Just as in Section 6.2, the following general statements

can be made:

• The Jeffreys noninformative prior results in a

posterior credible interval that is numerically

similar to a confidence interval.

• If the prior mean exists, the posterior mean is

between the prior mean and the MLE.

• If two prior distributions have about the same

mean, the more concentrated (less diffuse) prior

distribution will yield the more concentrated

posterior distribution, and will pull the posterior

mean closer to the prior mean.

Figure 6.34 and Table 6.11 summarize the results of

analyzing the AFW-failure-to-start data in the four

ways given above.

As in Section 6.2.2.5.4, the Jeffreys prior and the

frequentist approach are listed next to each other

because they give numerically similar results. The

Jeffreys prior yields a posterior credible interval that

is strictly contained in the confidence interval, neither

to the right nor to the left.

In each Bayesian case, the posterior mean falls

between the prior mean and the MLE, 0.125. The

prior distribution has more influence when the prior

distribution is more tightly concentrated around the

mean. One measure of the concentration (at least

when the means are similar) is the sum "prior + $prior,

because it corresponds to the total number of prior

demands, and it is in the denominator of the variance

in Equation 6.13. In the present example, when the

prior distributions in Table 6.11 are ordered by

increasing values of "prior + $prior, the order is the

noninformative prior, then the approximate constrained noninformative prior, and finally the industry

prior. The three 90% intervals for the corresponding

posterior distributions have decreasing length in the

same order.
Parameter Estimation and Model Validation

6-39

Table 6.11 Comparison of estimates with one failure in eight demands.

Method Prior mean Posterior

parameters

Point estimate

(MLE or 

posterior mean)

90% interval (confidence

interval or posterior credible interval)

Frequentist NA NA 0.125 (0.0064, 0.47)

Bayes with Jeffreys

noninformative prior,

beta(0.5, 0.5)

0.5 " = 1.5

$ = 7.5

0.17 (0.022, 0.40)

Bayes with industry prior,

beta(4.2, 153.1)

0.027 " = 5.2

$ = 160.1

0.031 (0.013, 0.056)

Bayes with approx. constrained noninform. prior,

beta(0.4585, 16.7138)

0.027 " = 1.4585

$ = 23.7138

0.058 (0.0068, 0.15)

6.3.2.4 Estimation with a Continuous 

Nonconjugate Prior

Just as for 8, continuous nonconjugate priors for p

cannot be updated with simple algebra. The resulting

posterior distribution does not have a simple form.

Therefore, to enter it as the distribution of a basic event

in a PRA code, a discrete approximation of the distribution must usually be used.

The posterior distribution must be obtained by numerical integration or by random sampling. Three methods

are mentioned here, and the analyst may choose whatever seems easiest.

6.3.2.4.1 Direct Numerical Integration

To use numerical integration, use Equation 6.14 and

write the posterior distribution as the product of the

likelihood and the prior distribution:

Cfpost(p) = px

(1 ! p)

n ! xfprior(p) . (6.16)

Here C is a constant of proportionality. All the normalizing constants in fprior and in the likelihood may be

absorbed into C, leaving only the parts that depend on

p on the right-hand side of the equation. Integrate

Cfpost(p) from 0 to 1. That is, integrate the right hand

side of Equation 6.16. This integral equals C, because

the integral of fpost must equal 1. Divide both sides of

Equation 6.16 by the just-found constant C, to obtain

the function fpost. Use numerical integration to find the

moments and percentiles of this distribution. Some

suggested methods of numerical integration are mentioned in Section 6.2.2.6.

6.3.2.4.2 Simple Random Sampling

To use random sampling, follow the rejection algorithm

given in Section 6.2.2.6. The general algorithm, given

in Section 6.2.2.6, can be restated for binomial data as

follows. Define

m = (x/n)

x

(1 ! x/n)

n!x

if 0 < x < n. If x = 0 or x = n, define m = 1. The steps

of the algorithm are:

(1) Generate a random p from the prior distribution.

(2) Generate u from a uniform distribution, 0 # u # 1.

(3) If u # px

(1 ! p)

n!x

/m, accept p in the sample.

Otherwise discard p.

Repeat Steps (1) through (3) until a sample of the

desired size is found.

6.3.2.4.3 More Complicated Random Sampling

All-purpose Bayesian update programs can be used for

the present simple problem, just as in Section 6.2. The

powerful program BUGS is mentioned in Section

6.2.2.6.3, and described more fully in Sections 7.2.3

and 8.3.3.3. It can be used here, although it is intended

for much more complicated problems.

6.3.2.5 Examples with Nonconjugate Priors

Several possible nonconjugate prior distributions are

discussed here.
Parameter Estimation and Model Validation

6-40

6.3.2.5.1 Lognormal Distribution 

The lognormal distribution is by far the most commonly

used nonconjugate distribution. The parameter p has a

lognormal distribution if ln(p) is normally distributed

with some mean : and variance F2

.

Facts about the lognormal distribution are given in

Appendix A.7.3. One important fact is that the range of

the lognormal distribution is from 0 to 4. Thus, the

distribution of p cannot be exactly lognormal, because

p cannot be greater than 1. When using a lognormal

prior, one must immediately calculate the prior Pr(p >

1). If this probability is very small, the error can be

neglected. (When generating values p from the lognormal distribution, either throw away any values

greater than 1 or set them equal to 1. In either case,

such values hardly ever occur and do not affect the

analysis greatly.) On the other hand, if the prior Pr(p >

1) is too large to be negligible, then the lognormal

distribution cannot possibly be used. Even if the

software accepts the lognormal distribution, and hides

the problem by somehow handling the values that are

greater than 1, the actual distribution used is not

lognormal. It is truncated lognormal, or lognormal with

a spike at 1, with a different mean and different percentiles from the initially input lognormal distribution. The

analyst’s two options are to recognize and account for

this, or to use a different prior distribution.

To use the above sampling algorithm with a lognormal

prior, p must be generated from a lognormal distribution. The easiest way to do this is first to generate z

from a standard normal distribution, that is, a normal

distribution with mean = 0 and variance = 1. Many

software packages offer this option. Then, let y = : +

Fz, so that y has been generated from a normal(:, F2)

distribution. Finally, let p = ey

. It follows that p has

been randomly generated from the specified lognormal

distribution.

6.3.2.5.2 Logistic-Normal Distribution

This distribution is explained in Appendix A.7.9. The

parameter p has a logistic-normal distribution if

ln[p/(1 ! p)] is normally distributed with some mean :

and variance F2. The function ln [p/(1 ! p)] is called

the logit function of p. It is an analogue of the

logarithm function for quantities that must lie between

0 and 1. Using this terminology, p has a logistic-normal

distribution if logit(p) is normally distributed.

Properties of the logistic-normal distribution are given

in Appendix A.7.9, and summarized here. Let y =

ln[p/(1 ! p)]. Then p = ey

 / (1 + ey). This is the inverse

of the logit function. As p increases from 0 to 1, y

increases from !4 to +4.

Note, unlike a lognormally distributed p, a logisticnormally distributed p must be between 0 and 1.

Therefore, the logistic-normal distribution could be

used routinely by those who like the lognormal distribution, but do not know what to do when the lognormal

distribution assigns p a value that is greater than 1.

The relation between p and y = logit(p) gives a way to

quantify prior belief about p in terms of a logisticnormal distribution. First, decide on two values, such

as lower and upper plausible bounds on p or a median

and plausible upper bound, equate them to percentiles

of p, translate those percentiles to the corresponding

two percentiles of the normal random variable Y, and

finally, solve those two equations for : and F.

To generate a random value from a logistic-normal

distribution, first generate y from a normal (:, F2

)

distribution, exactly as in the section above on the

lognormal distribution. Then let p = ey

 / (1 + ey). This

p has been randomly generated from the specified

logistic-normal distribution.

6.3.2.5.3 Exact Constrained Noninformative 

 Distribution

The prior distribution has the form of Equation 6.15,

and the posterior distribution is

fpost(p) = C1ebppx ! 1/2(1 ! p)

n ! x !1/2 ,

where C1 is a normalizing constant to make the density

integrate to 1.0. Except for the normalizing constant,

this is ebp times a beta(x+1⁄2, n!x+1⁄2) distribution.

Numerical integration is straightforward, and will not

be explained here. To generate a sample from the

posterior distribution, the rejection method algorithm

originally given in Section 6.2.2.6 takes the following

form.

Write the beta(x+1⁄2, n!x+1⁄2) density as

fbeta(p) = C2px ! 1/2(1 ! p)

n ! x !1/2 .

Typically, the desired mean of p is less than 0.5; if it is

not, reverse the roles of p and 1 ! p. The algorithm

first defines M to be the maximum possible value of the

ratio fpost(p) / fbeta(p). Because b < 0 in Table C.8, we

have ebp # 1, making M equal to C1/C2. Therefore, the

condition in Step (3) of the algorithm reduces to 

u # ebp .
Parameter Estimation and Model Validation

6-41

Therefore, the algorithm simplifies to the following:

(1) Generate a random p from the beta(x+1⁄2, n!x+1⁄2)

distribution. Ways to do this are discussed below.

(2) Generate u from a uniform distribution, 0 # u # 1.

(3) If u # ebp, accept p in the sample. Otherwise

discard p.

Repeat Steps (1) through (3) until a sample of the

desired size is found.

Not all standard software packages give the option of

generating random numbers from a beta distribution,

although many more allow random number generation

from a gamma distribution or from a chi squared

distribution. When working with such software, let y1

be randomly generated from a gamma(x+1⁄2, 1) distribution and let y2 be randomly generated from a

gamma(n!x+1⁄2, 1) distribution. Alternatively, let y1 be

randomly generated from a chi-squared(2x+1) distribution and let y2 be randomly generated from a chisquared(2n!2x+1) distribution. In either case, define p

= y1/(y1+y2). Then, p has been generated from the

specified beta(x+1⁄2, n!x+1⁄2) distribution. (See Chapter

25 of Johnson et al. 1995.)

6.3.2.5.4 Maximum Entropy Prior

The maximum entropy prior and the constrained noninformative prior were developed with the same goal: to

produce a diffuse distribution with a specified plausible

mean. The diffuseness of the maximum entropy distribution is obtained by maximizing the entropy, defined

as 

− =− E f p f p f p dp [ln ( )] [ln ( )] ( ) ∫ .

When p is restricted to the range from 0 to 1, it can be

shown that the density f maximizing the entropy is

uniform,

f(p) = 1 for 0 # p # 1

and f(p) = 0 elsewhere. More interesting is the case

when the mean of the distribution is required to equal

some prespecified value p0. In this case the maximum

entropy distribution has the form of a truncated exponential distribution,

f(p) = Cebp for 0 # p # 1

and f(p) = 0 elsewhere. In this form, b is negative when

p0 < 0.5 and b is positive when p0 > 0.5. The value of

b corresponding to a particular mean must be found by

numerical iteration. Some authors write e!bp instead of

ebp; this simply reverses the sign of the parameter b.

The maximum entropy distribution and the uniform

distribution are related — if the constraint on the mean

is removed, the maximum entropy distribution equals

the uniform distribution. In this sense, the maximum

entropy distribution is a generalization of the uniform

distribution. The constrained noninformative distribution is the same sort of generalization of the Jeffreys

noninformative distribution — if the constraint is

removed, the constrained noninformative prior becomes

the Jeffreys noninformative prior. Atwood (1996)

reviews the reasons why the Jeffreys prior is superior to

the uniform prior, and uses the same reasoning to argue

that the constrained noninformative prior is superior to

the maximum entropy prior.

In practice, it may make little difference which distribution is used. Both distributions are intended to be used

when little prior knowledge is available, and quantifying “little prior knowledge” is not something that can be

done precisely.

Sampling from the posterior distribution is similar to

the other sampling procedures given above, so most of

the details are not given. The only point deserving

discussion is how to generate a random sample from the

maximum entropy prior. The most convenient method

is the inverse c.d.f. algorithm. This algorithm is

simple in cases when the c.d.f. and its inverse can be

calculated easily.

For example, let the random variable P have c.d.f. F.

Let F!1

 be the inverse function, defined by u = F(p) if

and only if p = F!1

(u). Let U be defined as F(P). What

is the distribution of U? The c.d.f. of U is found by

Pr(U # u) = Pr[ F(P) # u ]

= Pr[ P # F!1

(u) ]

= F[ F!1

(u) ] because F is the c.d.f. of P

= u .

Therefore, U has a uniform distribution. The letter U

was not chosen by accident, but in anticipation of the

uniform distribution.

To generate a random value p from the distribution F,

generate a random u from the uniform (0, 1) distribution, something that many software packages allow.

Then define p = F!1

(u). This is the inverse c.d.f.

method of random number generation.

To apply this to the maximum entropy distribution, first

integrate the maximum entropy density to yield the

c.d.f.

F(p) = (1 ! ebp)/(1 ! eb

) .
Parameter Estimation and Model Validation

6-42

0

10

20

30

40

50

60

70

0.00 0.05 0.10

Prior

Posterior

0.15 0.20

p (failures/demand) GC99 0292 19

Density Figure 6.35 Lognormal prior density and posterior 

density for p.

Generate u from a uniform(0, 1) distribution, and set

u = (1 ! ebp)/(1 ! eb

) .

Solve this equation for p,

p = !ln[ 1 ! (1 ! eb

)u ]/b .

Then, p has been randomly generated from the maximum entropy distribution. Repeat this with new values

of u until enough values of p have been obtained.

6.3.2.5.5 Example Calculation

These techniques will be illustrated with the

Example 6.7, one failure to start in eight demands of

the AFW turbine train. Two prior distributions will be

assumed, the lognormal prior used by the Accident

Sequence Evaluation Program (ASEP), as presented

by Drouin et al. (1987), and a logistic-normal distribution having the same 50th and 95th percentiles.

The ASEP distribution for turbine-driven pump failure

to start is lognormal with mean 3E!2 per demand

and error factor 10. The three relevant equations

from Appendix A.7.3 are

EF(p) = exp(1.645F)

mean(p) = exp(: + F2

/2)

pq = exp(: + Fzq)

where the subscript q denotes the qth quantile, and

zq is the qth quantile of the standard normal distribution.

Solving the first equation yields F = 1.3997. Substitution of this into the second equation yields : =

!4.4862.

The percentiles are not needed yet, but the third

equation gives the median, p0.50 = exp(:) = 0.01126,

and the 95th percentile, p0.95 = exp(: + 1.645F) =

0.1126. (The relation of these two percentiles can

also be derived from the fact that the error factor

equals 10.)

The prior Pr(p > 1) is 6.75E!4, a very small number.

In the calculations of this section, the lognormal

distribution is truncated at 1.0. That is, integrals are

renormalized to make the integral of the density from

0 to 1 equal to exactly 1.0. If random sampling is

performed, any sampled values that are greater than

1 are discarded. 

The prior and posterior densities of p are shown in

Figure 6.35. The densities were calculated using

software for numerical integration.

As a second example, consider the logistic-normal

prior distribution having the same 50th and 95th

percentiles as the above lognormal prior. These

percentiles are 0.01126 and 0.1126. To find the

parameters of the underlying normal distribution, set

Y = ln[p/(1 ! p)]. By the properties of the logisticnormal distribution given in Appendix A.7.9, the 50th

and 95th percentiles of Y are

y0.50 = ln[0.01126/(1 ! 0.01126)] = !4.475

y0.95 = ln[0.1126/(1 ! 0.1126)] = !2.064 .

Because Y has a normal(:, F2

) distribution, it follows

that

: = !4.475

: + 1.645F = !2.064

so F = 1.466 .

Monte Carlo simulation shows that the truncatedlognormal and logistic-normal prior densities are

virtually the same, with means, medians, 5th and

95th percentiles agreeing to two significant digits. As

a consequence, the posterior distributions from the

two priors are also nearly the same, although the

means and percentiles may differ slightly in the

second significant digit.

Numerical integration was used, but BUGS could

have been used. As an illustration, the script for

using BUGS is given in Figure 6.36.

This script assigns a logistic-normal prior distribution

to p. If a lognormal prior is used instead, BUGS

returns an error message during the simulation,

presumably because it has generated a value of p

greater than 1. The script assigns Y a normal distribution with mean !4.475. The second parameter is

1/F2

, because that is how BUGS parameterizes a

normal distribution. The entered value, 0.4653,
Parameter Estimation and Model Validation

6-43

model

{

 y ~ dnorm(-4.475, 0.4653)

 p <- exp(y)/( 1 + exp(y))

 x ~ dbin(p, 8)

}

list(x = 1)

Figure 6.36 Script for analyzing Example 6.7 with

BUGS.

equals 1/1.4662

. The script then gives X a binomial(8, p) distribution. Finally, the line beginning “list”

contains the data, the single observed value 1 in this

example. BUGS also wants an initial value for p, but

it is willing to generate it randomly.

For the present example, the difference between the

lognormal and logistic-normal priors is very small,

having no effect on the posterior. The difference

between the two priors can be important if the probability of failure is larger and/or the uncertainty is larger.

That can be the case with some human errors, with

hardware failures in unusually stressful situations, and

with recovery from failure if recovery is modeled as an

event separate from the original failure. For example,

the NUREG 1150 PRA for Surry (Bertucio and Julius

1990) uses the lognormal distribution for most failure

probabilities. However, some failure probabilities are

large, considerably larger than 3E!2. In nearly all of

those cases, the PRA does not use a lognormal distribution. Instead, the maximum entropy distribution is the

PRA’s distribution of choice. Other possible distributions, which were not widely known in the PRA community in 1990, would be the constrained noninformative distribution or a logistic-normal distribution.

6.3.2.6 Estimation with Fixed Number of Failures

and Random Number of Demands

Sometimes it is useful to consider a random number of

demands, a waiting time, to achieve a fixed number of

failures x. For example, if the failure probability p is

believed to change over time, only the most recent

history may represent current behavior. In such a

situation, one might decide to use only the most recent

few failures, such as x = 3, and to treat the corresponding number of demands n as random. Here n is the

number of demands counted backwards from the

present to the xth failure in the past. Earlier failures and

demands could be used to construct a prior distribution,

but the dispersion of the prior distribution should be set

large because the earlier events are not considered fully

relevant to the present.

With such waiting-time data, the likelihood is proportional to 

px

(1 ! p)

n ! x .

Therefore, except for the normalizing constant the

likelihood is the same as for binomial data. Therefore,

it works out that Bayesian analysis with an informative

prior is exactly the same whether the data are regarded

as coming from a random count of x failures in a fixed

number of demands, n, or a random number of demands, n, for a fixed number of failures, x. The posterior distributions are identical.

6.3.3 Model Validation

All the methods in this section are analogues of methods considered for failure rates, but the details are

somewhat different. Some repetition is inevitable, but

the examples in this section are chosen to complement

the examples of Section 6.2.3, not to duplicate them.

For a more complete appreciation of the model validation techniques, both this section and Section 6.2.3

should be read.

The comments at the start of Section 6.2.3 apply

equally to this section, and must not be ignored. In

particular, an analyst who estimates parameters should

check the assumptions of the model. However, this

section contains more than will be needed on any one

analysis. The methods here are offered for possible

use, and the analyst should select the appropriate ones.

The first assumption of the binomial model, given in

Section 2.3.2, is that the probability of failure is the

same on any demand. This assumption will be examined against two possible alternative assumptions: (1)

different subsets of the data have different values of p,

but in no special order; and (2) a time trend exists. The

second assumption of the binomial model is that the

outcome on one demand is statistically independent of

the outcome on a different demand. This will be

examined against the alternatives of common-cause

failures and of clustering in time of the failures. Finally, the consistency of the prior distribution and the

data will be considered.

One need not worry about whether n is really constant.

If n is not constant, we may treat it as constant by

conditioning on n, as explained in Section 2.3.2.4.2.
Parameter Estimation and Model Validation

6-44

Figure 6.37 MLEs and 90% confidence intervals for

p, for three types of demand and for the pooled data.

6.3.3.1 Poolability of Data Sources

The methods will be illustrated by data from diesel

generator failures to start, shown in Example 6.8.

Example 6.8 EDG failures to start on demand.

Emergency diesel generator (EDG) failures to

start on demand were recorded for three kinds of

demands: unplanned demands, the tests performed once per operating cycle (approximately

every 18 months), and the monthly tests. The

counts are given below.

Type of

demand

Failures to

start

Number of

demands

Unplanned 2 181

Cyclic test 17 1364

Monthly test 56 15000

Table C.1 of Grant et al. (1996) gives the data for the

first two rows, at plants reporting under Regulatory

Guide RG-1.108 during 1987-1993. The failures

were those reported in LERs. The number of failures

on monthly tests at those plants comes from the

unpublished database used for that report, and the

number of monthly demands was estimated in a very

crude way for use in this example.

6.3.3.1.1 Graphical Technique

To explore the relations between subsets of the data,

mark the subsets on one axis. For each of these subsets

of the data, plot an estimate of p and a confidence

interval for p against the other axis. Patterns such as

trends, outliers, or large scatter are then visible.

In Example 6.8, the subsets are types of demand.

The data set from each demand type is analyzed

separately, and the graph shows an estimate and a

confidence interval for each year, plotted side by

side. This is shown in Figure 6.37. The plot was

produced with a graphics package, although a handdrawn plot would be adequate to show the results.

The plot shows that the unplanned demands and the

cyclic tests appear to have similar values of p, but

the monthly tests appear to have a lower value.

Several reasons for the difference could be conjectured: the monthly tests may be less stressful, the

failures may not all be reported in LERs, or the

estimated number of demands may be badly incorrect.

Figure 6.21, which is the corresponding plot in Section

6.2.3.1.1, has the cells (plants, in that example) arranged in order of decreasing . Figure 6.37 does not $

λ

order the cells by decreasing , because the number of p$

cells is small, only three, and because the cells already

have a natural order. The analyst must decide what

order makes the most sense and is easiest for the user to

interpret.

The interval for the pooled data is also shown, not

because the data justify pooling, but simply as a reference for comparison. A dotted reference line is drawn

through the point estimate based on the pooled data. If

only a few data subsets need to be compared, as in

Figure 6.37, these embellishments are unnecessary.

With many subsets, however, the eye tends to get lost

without the reference line. The reference line has the

added advantage of focusing the eye on the confidence

intervals rather than the point estimates.

The graph is only a picture. Pictures like these are

useful, but cannot always be used in an easy way to

draw conclusions about differences between data

subsets. The warnings given in Section 6.2.3.1.1

deserve repetition:

C If many confidence intervals are plotted, all

based on data with the same p, a few will be far

from the others because of randomness alone.

An outlying interval does not prove that the ps

are unequal.

C This same statement is true if other intervals are

used, such as Bayes credible intervals based on

the noninformative prior. The issue is the random variability of data, not the kind of interval

constructed. 

C If there are few intervals, on the other hand,

intervals that just barely overlap can give strong

evidence for a difference in the ps.
Parameter Estimation and Model Validation

6-45

To quantify the strength of the evidence seen in the

picture, a formal statistical procedure is given in the

next subsection. The picture gives a preview, and helps

in the interpretation of the formal statistical quantifica

tion. In the present example, if the statistical test finds

a statistically significant difference between data

subsets, it is natural to then ask what kind of difference

exists. The picture shows that p seems to be similar for

the unplanned demands and for the cyclic tests, but

smaller for the monthly tests. In this way, the picture

provides insight, even though it does not provide a

quantitative statistical test.

6.3.3.1.2 Statistical Tests

Simple Contingency Tables (2 × J). The natural

format for the data is a “contingency table.” An introductory reference to this subject is Everitt (1992), and

many general statistics texts also have a chapter on the

topic. In a two-way table, two attributes of the events

are used to define rows and columns, and the numbers

in the table are counts. In the present example, two

attributes of any event are the type of demand and

whether it is a failure or success. One way to build a

contingency table is to let the first row show system

failures and the second row system successes. Then let

the columns correspond to the demand types. (Of

course, the roles of rows and columns can be reversed

if that fits better on the sheet of paper.) The table

entries are the counts of the events for each cell, shown

in Table 6.12 for Example 6.8. 

Table 6.12 Contingency table for Example 6.8.

Unplanned Cyclic Monthly Total

Failure 2 17 56 75

Success 179 1347 14944 16470

Total 181 1364 15000 16545

The essence of this table is a 2 × 3 table, because the

basic data counts occupy two rows and three columns.

The row totals, column totals, and grand total are shown

in the right and bottom margins. A general, two-way

contingency table has I rows and J columns. (Although

this discussion considers only 2 × J tables, it does no

harm to give the general formulas, keeping in mind that

the examples of this section have I = 2.) The count in

the ith row and jth column is denoted nij, for i any

number from 1 to I and j from 1 to J. The total count in

row i is denoted ni+ and the total count in column j is

denoted n+j

. The grand total is denoted n++.

For example, Table 6.12 has n1,3 = 56 and n2,1 = 179.

It has n2+ = 16470 and n+2 = 1364. The grand total,

n++, equals 16545 in the example.

Let the null hypothesis be

H0: p is the same for all the data subsets.

The alternative hypothesis is

H1: p is not the same for all the data subsets.

In the example, the data subsets are the three demand

types. The analyst must investigate whether H0 is true.

The method used is to see what kind of data would be

expected when p really is the same, and then to see how

much the observed counts differ from the expected. If

the differences are small, the counts are consistent with

the hypothesis H0. If, instead, the differences are large,

the counts show strong evidence against H0.

If H0 is true, that is, if p is really the same for all the

demand types, the natural estimate of p is

pn n $ = / . 1+ ++

Then for column j, one would have expected n p + j $

failures on average. This reasoning leads to the formula

for the expected count in cell ij:

eij = ni+n+j

 / n++.

In Table 6.12, for unplanned demands one would

have expected 181×(75/16545) = 0.82 failures on

average, for cyclic tests 1364×(75/16545) = 6.19

failures, and so forth.

The difference between the observed count and the

expected count for any cell is nij ! eij. There are many

cells, and therefore many ways of combining the

differences to yield an overall number. One useful way

is to construct

X2

 = EiEj (nij ! eij)

2

/eij.

X2

 is called the chi-squared statistic, or sometimes the

Pearson chi-squared statistic. Note, X2

 as defined here

is slightly different from the chi-squared statistic for

constant event rate in Section 6.2.3.1.2. In that section,

the cells had one index, whereas in this section, the cells

have two indices, and the expected counts are calcu-
Parameter Estimation and Model Validation

6-46

lated differently. Other than that, the statistics are the

same. Table 6.13 expands Table 6.12 to show the

quantities needed to calculate X2

. The observed counts

and the expected counts have the same totals, except for

roundoff.

Table 6.13 Counts, expected counts, and 

contributions to X2

, for Example 6.8.

Unplanned Cyclic Monthly Total

Failure 2

0.82

1.70

17

6.19

18.92

56

68.00

2.12

75

Success 179

180.18

0.01

1347

1357.80

0.09

14944

14932

0.01

16470

Total 181 1364 15000 16545

For example, there were 2 failures on unplanned

demands. The expected number of failures on unplanned demands, if H0 is true, is 181×75/16545 =

0.82. And the contribution of that cell to X2

 is

(2 ! 0.82)2/0.82 = 1.70 .

When H0 is true and the total count is large, the distribution of X2

 has a distribution that is approximately chisquared with (I-1)×(J-1) degrees of freedom. In Table

6.12, the number of degrees of freedom is (2!1)×(3!1)

= 2. If X2

 is large compared to the chi-squared distribution, the evidence is strong that H0 is false; the larger

X2

, the stronger the evidence.

Interpretation of Test Results. Based on any 2×3

contingency table, such as Table 6.12, suppose that X2

were 6.4. A table of the chi-squared distribution shows

that 5.991 is the 95th percentile of the chi-squared

distribution with 2 degrees of freedom, and 7.378 is the

97.5th percentile. After comparing X2

 to these values,

an analyst would conclude that the evidence is strong

against H0, but not overwhelming. Quantitatively, the

analyst would “reject H0 at the 5% significance level,

but not at the 2.5% significance level.” This is sometimes phrased as “the p-value is between 0.05 and

0.025.” See the bulleted list in Section 6.2.3.1.2, in the

interpretation following Table 6.6, for other phrases

that are sometimes used.

If instead X2

 were 1.5, it would lie between the 50th and

the 60th percentiles of the chi-squared distribution, and

therefore would be in the range of values that would be

expected under H0. The analyst could say “the observed counts are consistent with the hypothesis H0,” or

“H0 cannot be rejected,” or “the evidence against H0 is

very weak.” The analyst would not conclude that H0 is

true, because it probably is not exactly true to the tenth

decimal place, but would conclude that it cannot be

rejected by the data.

In fact, in Example 6.8 X2 equals 22.8, as found by

totaling the six contributions in Table 6.13. This

number is far beyond the 99.5th percentile of the chisquared distribution, so the evidence is overwhelming against H0. Such an analysis contributed to the

decision of Grant et al. (1999b) not to consider

monthly tests in their report.

This example was chosen to illustrate that subsets of the

data can correspond not only to different locations or

different hardware (for example, different plants or

systems), but also to different conditions, in this case

different types of demands. In reality, the data analyst

should consider various kinds of subsets; in this example, with data coming from many plants, the analyst

should consider possible between-plant differences.

The plots and chi-squared tests are exactly the same as

given above.

This brings up a difficulty with the present example that

has been carefully hidden until now. The hypothesis H0

is that all the subsets of the data have the same p. A

hidden hypothesis, never even proposed for testing, is

that within each data subset, every demand has the same

p. In fact, this turns out not to be the case. Based on

only the unplanned demands and cyclic tests, Grant et

al. (1999b) report that the difference between plants is

statistically significant — the evidence is strong that p

differs from plant to plant. This means that the above

analysis must be refined to account for possible differences between plants. Such variation is discussed in

Chapter 8 of this handbook.

Thus, the data set has two sources of variation, differences between demand types and also differences

between plants. In such a situation, consideration of

only one variable at a time can throw off the results if

the data set is “unbalanced,” for example, if the worst

few plants also happen to have the most unplanned

demands and the fewest monthly demands. If such

between-plant differences are contaminating the EDG

data in Example 6.8, the observed difference might not

reflect anything about the nature of the demands, but

only that the plants with EDG problems were

underrepresented on the monthly tests. Example 6.9

shows hypothetical data under such a scenario. 

If only the good plants are considered, or if only the

bad plants are considered, the data of Example 6.9
Parameter Estimation and Model Validation

6-47

show no difference between unplanned demands

and tests. The estimated p is the same for unplanned demands and for tests, 0.2 from the bad

plants’ data and 0.02 from the good plants’ data.

However, if the data from good plants and bad plants

are combined, the unplanned demands appear to

have a much higher failure probability than do the

tests, 0.07 versus 0.03. This erroneous conclusion

is a result of ignoring differences in the data, the

existence of two kinds of plants, when the data are

unbalanced because the bad plants have a much

higher percentage of unplanned demands. Such a

situation is known as Simpson’s paradox.

Example 6.9 Hypothetical unbalanced data.

Suppose that the industry consists of “bad” plants

and “good” plants. The bad plants have a

relatively high probability of failure to start, and

also have relatively many unplanned demands.

Suppose that the tests perfectly mimic unplanned

demands, so that at either kind of plant p is the

same on an unplanned demand and on a test.

Data from such an industry might be given in the

table below. The tables entries show failures/

demands.

Unplanned Tests

Bad plants 4/20 = 0.2 4/20 = 0.2

Good plants 1/50 = 0.02 8/400 = 0.02

Totals 5/70 = 0.07 12/420 = 0.03

In fact, this scenario cannot be greatly influencing

the data in Example 6.8, because most of the demands are periodic. Therefore, every plant must

have approximately the same fraction of monthly

tests and of cyclic tests. In conclusion, although

between-plant variation must be considered, it is

hard to imagine that it affects the outcome in Example 6.8.

As mentioned in Section 6.2.3.1.2, a full data analysis

must not stop with the calculation of a p-value. In the

present example, with a very large number of demands,

it may be that the statistically significant difference is

not very important from an engineering viewpoint. In

other words, a large data set can detect differences in

the second decimal place — differences that are not

worth worrying about in practice. 

This concern is addressed in the example by Figure

6.37, which shows that the probability of FTS is

about 1/3 as large on monthly tests as on other

demands, at least according to the reported data.

Therefore, the difference is substantial in engineering terms, and the engineering portion of the data

analysis can investigate reasons for the difference.

Required Sample Size. The above approach is valid

if the values of nij are “large.” If they are small, X2

 has

a discrete distribution, and so cannot have a chi-squared

distribution. As a rather extreme example, if n++, the

total number of demands, were equal to four in the

framework of Example 6.8, there would only be a few

ways that the four demands (and the number of failures,

at least zero and at most four) could be arranged among

the three demand types. Therefore X2

 could only take

a few possible values.

Therefore, the user must ask how large a count is

necessary for the chi-squared approximation to be

adequate. An overly conservative rule is that all the

expected cell counts, eij, be 5.0 or larger. Despite its

conservatism, this rule is still widely used, and cited in

the outputs of some current statistics packages. For a

2×J table, Everitt (1992, Sec. 3.3), citing work by

Lewontin and Felsenstein (1965), states that the chisquared approximation is adequate if all the values of eij

are 1.0 or greater, and that in “the majority of cases” it

is sufficient for the eij values to be 0.5 or greater. For a

2×2 table, however, it is generally best not to use the

chi-squared approximation at all, but to use the p-value

from “Fisher’s exact two-sided test,” discussed below.

If the expected cell counts are so small that the chisquared approximation appears untrustworthy, the

analyst has two choices: (a) Pool some columns,

thereby combining cells and increasing the expected

cell counts. For example, in an investigation of differences between years, with few failures, it might be

necessary to combine adjacent years so that the expected number of failures in each time-bin is at least

0.5; or (b) Some statistical software packages can

compute the “exact distribution” of X2

 in some cases

(typically for small tables). Conditional on the ni+

values and n+j values, this exact distribution is the finite

set of values that X2

 can possibly take, together with

their associated probabilities. If the analyst is willing to

base the decision on this conditional distribution, the

exact distribution can be used. The commercial package StatXact performs such calculations using modern,

fast algorithms, even for large tables, subject only to the

memory available in the machine. In the special case of

a 2×2 contingency table, many software packages

compute this p-value, calling it the p-value from

“Fisher’s exact two-sided test.” In general, the p-value

from Fisher’s exact test is preferable to the p-value
Parameter Estimation and Model Validation

6-48

Figure 6.39 Cumulative number of failures versus

cumulative number of demands.

from the chi-squared approximation, and should be

used whenever the software produces it. This, and

other considerations for a 2×2 table, are discussed by

Everitt (1992) and Atwood (1994).

In Table 6.13, the smallest expected count is e11 =

0.82. All the other expected counts are larger than

1.0. This indicates that the sample size is large

enough.

6.3.3.2 No Time Trend

This section uses the unplanned HPCI demands from

Example 6.5, with the failures indicated. To make a

data set with a moderate number of failures, all types of

failures are counted together, including failure to start,

failure to run, failure of the injection valve to reopen

after operating successfully earlier in the mission, and

unavailability because of maintenance. For the example, no credit is taken for failures that were recovered.

The data are given as Example 6.10.

Example 6.10 Dates of HPCI failures and 

unplanned demands, 1987-1993.

The HPCI demands of Example 6.5 are listed here

with an asterisk marking demands on which some

kind of failure occurred. The demands dates are

given in columns, in format MM/DD/YY.

01/05/87*

01/07/87

01/26/87

02/18/87

02/24/87

03/11/87*

04/03/87

04/16/87

04/22/87

07/23/87

07/26/87

07/30/87

08/03/87*

08/03/87*

08/16/87

08/29/87

01/10/88

04/30/88

05/27/88

08/05/88

08/25/88

08/26/88

09/04/88*

11/01/88

11/16/88*

12/17/88

03/05/89

03/25/89

08/26/89

09/03/89

11/05/89*

11/25/89

12/20/89

01/12/90*

01/28/90

03/19/90*

03/19/90

06/20/90

07/27/90

08/16/90*

08/19/90

09/02/90

09/27/90

10/12/90

10/17/90

11/26/90

01/18/91*

01/25/91

02/27/91

04/23/91

07/18/91*

07/31/91

08/25/91

09/11/91

12/17/91

02/02/92

06/25/92

08/27/92

09/30/92

10/15/92

11/18/92

04/20/93

07/30/93

6.3.3.2.1 Graphical Techniques

Just as elsewhere in this chapter, the time axis can be

divided into bins, and the data can be analyzed separately for each bin and compared graphically.

For Example 6.10, defining the bins to be years

leads to Table 6.14. This leads to a plot similar to

Figures 6.21 and 6.22, shown in Figure 6.38. The

plot with the example data shows no evidence of a

trend.

Table 6.14 HPCI failures on demand, by year.

Calendar year Failures Demands

1987 4 16

1988 2 10

1989 1 7

1990 3 13

1991 2 9

1992 0 6

1993 0 2

Figure 6.38 Point and interval estimates of p, each

based on one year’s data.

A plot that does not require a choice of how to construct bins is given in Figure 6.39, the analogue of

Figure 6.23. It can be constructed when the demands

can be ordered sequentially, as is the case for Example

6.10. In this plot, the cumulative number of failures is

plotted against the cumulative number of demands. To

help the eye judge curvature, a straight line is drawn,

connecting the origin with the dot at the upper right.
Parameter Estimation and Model Validation

6-49

The slope of any part of the graph is the vertical distance divided by the horizontal distance, )y/)x. In the

present figure the horizontal distance is the number of

demands that have occurred, and the vertical distance is

the corresponding number of failures. Therefore,

slope = (number of failures)/(number of demands) ,

so the slope is a visual estimator of p. A roughly

constant slope, that is, a roughly straight line, indicates

a constant p. A changing slope indicates changes in p.

In Figure 6.39, the slope is relatively constant,

indicating that p does not seem to change with time.

This agrees with Figure 6.38. It is not clear whether

the slight departure from the diagonal line in the right

half of the figure is more than can be attributed to

random variation. Such questions must be addressed by statistical tests, given below.

The details of the diagonal line probably do not matter.

The line shown is the maximum likelihood estimate of

the expected height of the plot at any horizontal point,

assuming constant p. Other lines, slightly different,

could also be justified.

6.3.3.2.2 Statistical Tests for a Trend in p

In this section, the null hypothesis remains

H0: p is the same for all the data subsets.

but the alternative is now

H1: p is either increasing or decreasing over time .

The Chi-Squared Test. This is the same test as given

in Section 6.3.3.1.2, except now the data subsets are

years or similar bins of time.

The data of Table 6.14 can be written as a 2×7

contingency table. The smallest expected cell count

corresponds to failures in 1993, with the expected

count = 2×12/63 = 0.4. This is too small to justify

calculating a p-value from the chi-squared distribution. The problem can be remedied by pooling the

two adjacent years with the smallest number of

demands, 1992 and 1993. (Note, the decision of

which subsets to pool is based on the numbers of

demands only, not on whether or not those demands

resulted in failures. Pooling based on demand

counts is legitimate. Pooling based on the failure

counts is not.)

When this 2 × 6 contingency is analyzed by the chisquared test, the p-value is 0.77, indicating no

evidence at all of differences between years. This is

no surprise.

The Wilcoxon-Mann-Whitney Test. This test is

similar in spirit to the Laplace test for a trend in 8. The

null hypothesis is that p is the same for all demands.

Suppose that the individual demands are in a known

sequence. Against the alternative hypothesis that the

failures tend to occur more at one end of the sequence

— that is, p is either an increasing or a decreasing

function of the sequence number — use the WilcoxonMann-Whitney test, described in texts that cover

nonparametric statistics. Two good sources of standard

nonparametric methods are Conover (1999), and

Hollander and Wolfe (1999). Hollander and Wolfe call

this test the Wilcoxon rank sum test.

The test is based on the sum of the ranks of the failures.

For example, in the sequence of failures and successes

failure, success, failure, failure, success,

the three failures have ranks 1, 3, and 4, and the sum of

their ranks is 8. Let W denote the sum of the ranks of x

failures in n trials. If x and n ! x are both large and if

the probability of a failure is the same for the entire

sequence, W is approximately normal with mean :W =

x(n+1)/2 and variance F2

W = x(n!x)(n+1)/12. If Z = (W

! :W)/FW is in either tail of the distribution, the null

hypothesis should be rejected. If x or n ! x is small,

statistics books give tables, or statistical computer

packages calculate the exact tail probability.

The data of Example 6.10 show 12 failures in 63

demands. The first failure was on the first demand

(01/05/87), so that failure has rank 1. The next was

on the sixth demand, so that failure has rank 6. Two

demands occurred on 03/19/90, the 36th and 37th

demands. One of the two demands resulted in

failure, so that failure was assigned rank 36.5, as is

usual in case of ties. The sum of the ranks of the

failures is 321.5, and Z can be calculated to equal

!1.09. This is the 13.8th percentile of the normal

distribution. Because Z is not in either tail, H0 is not

rejected.

6.3.3.3 Independence of Outcomes

The second assumption for binomial data is that the

outcomes of different demands be independent — a

success or failure on one demand does not influence the

probability of failure on a subsequent demand.

Outcomes can be dependent in many ways, and some of

them must be addressed by careful thinking rather than

by statistical data analysis. The analyst or the study
Parameter Estimation and Model Validation

6-50

team should consider possible common-cause mechanisms, and examine the data to see if many commoncause failures occurred. If common-cause failures form

a noticeable fraction of all the failures, the analyst

should probably divide the independent failures and the

common-cause failures into separate data sets, and

separately estimate the probabilities of each kind of

failure.

The rest of this section is less important on the first

reading than other sections. Some readers may wish to

skip directly to Section 6.3.3.4.

If demands occur in sequence, it is natural to consider

serial dependence, in which the occurrence of a failure

on one demand influences the probability of a failure on

the next demand. Some people believe that hits in

baseball occur this way, that a slump or streak can

persist because of a batter’s attitude, which is influenced by how successful he has been recently. In the

context of hardware failures, suppose that failures are

sometimes diagnosed incorrectly, and therefore repaired

incorrectly. Immediately after any failure, the probability of failure on the next demand is higher, because the

first failure cause may not have been truly corrected. In

such a case, the failures would tend to cluster, rather

than being uniformly scattered among the successes. A

cumulative plot, such as that in Figure 6.39, can be

inspected for such clusters.

If the question of independence is restricted to successive outcomes — outcome i!1 versus outcome i — the

data can be analyzed by a 2×2 contingency table. Let

yi

 be the outcome on demand i, either success or failure.

Let xi

 be the outcome on demand i ! 1. The possible

values of successive outcomes (xi

, yi) are (S, S), (S, F),

(F, S), and (F, F).

To put this in more familiar language, let p denote the

probability of a failure, and consider two kinds of

demands, those when the previous outcome (x) was a

failure and those when the previous outcome was a

success. The null hypothesis is

H0: p is the same on both kinds of demands .

Perform the usual chi-squared test of H0 based on a

contingency table.

Example 6.10 results in the contingency table shown

in Table 6.15. Although the chi-squared approximation should be acceptable, it is preferable to use

Fisher’s exact test for a 2×2 table. The p-value

reported by SAS for Fisher’s exact test is 0.67. This

large p-value shows that the data are very consistent

with the hypothesis of independence of successive

outcomes. Because the data come from the entire

industry, independence is entirely reasonable.

Table 6.15 Contingency table for successive

outcomes in Example 6.10.

x = F x = S Total

y = F 1 10 11

y = S 11 40 51

Total 12 50 62

6.3.3.4 Consistency of Data and Prior

If the prior distribution has mean Eprior(p), but the

observed data show x/n very different from the prior

mean, the analyst must ask if the data and the prior are

inconsistent, or if the prior distribution was misinformed. The investigation is similar to that in Section

6.2.3.5.

Suppose first that x/n is in the right tail of the prior

distribution. The relevant quantity is the prior probability of observing x or fewer events. This is

Pr( ) Pr( | ) ( ) X x X x p f p dp ≥= ≥ ∫ prior (6.17)

where

Pr( | ) ( ) X xp . (6.18) n

x p p k nx

k x

n

≥ = ⎛

⎝

⎜ ⎞

⎠ ∑ ⎟ − −

=

1

If the prior distribution is beta(", $), it can be shown

that Equation 6.17 equals

Pr( )

( )

( )

( )

( )

( )

( )

X x

k nk

n

n

k k x

n

≥

=

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

+ +− +

+ + =

∑ Γ

Γ

Γ

Γ

Γ

Γ

α

α

β

β

α β

α β

where '(s) is the gamma function, a generalization of

the factorial function as described in Appendix A.7.6.

The name of this distribution is beta-binomial. This

probability can be evaluated with the aid of software.

If the prior probability is any distribution other than a

beta distribution, Equation 6.17 does not have a direct

analytical expression.
Parameter Estimation and Model Validation

6-51

Just as in Sec. 6.2.3.5, one method of approximating the

integral in Equation 6.17 is by Monte Carlo sampling.

Generate a large number of values of p from the prior

distribution. For each value of p, let y be the value of

Equation 6.18, which can be calculated directly. The

average of the y values is an approximation of the

integral in Equation 6.17. Another method of approximating the Equation 6.17 is by numerical integration.

If the probability given by Equation 6.17 is small, the

observed data are not consistent with the prior belief —

the prior belief mistakenly expected p to be smaller than

it apparently is.

Similarly, if x/n is in the left tail of the prior distribution

of the prior distribution, the relevant quantity is the

prior Pr( X # x ). It is the analogue of Equation 6.17

with the limits of the summation in Equation 6.18 going

from 0 to x. If that probability is small, the prior

distribution mistakenly expected p to be larger than it

apparently is.

Again consider Example 6.7, one AFW failure to start

in eight demands, and consider the industry prior,

beta(4.2, 153.1). One easy approach is Monte Carlo

simulation. Therefore, values of p were generated

from the beta distribution, using the technique

mentioned at the end of Section 6.3.2.5.3. That is,

y1 was generated from a gamma(4.2, 1) distribution,

y2 was generated from a gamma(153.1, 1) distribution, and p was set to y1/(y1 + y2).

The industry-prior mean of p is 0.027, Because the

observed number of failures, one, is larger than the

prior expected number, 8×0.027 = 0.21, we ask

whether such a large failure count is consistent with

the prior. The probability in question is Pr(X $ 1).

For each randomly generated p, Pr(X $ 1 | p) was

found, equal to 1 ! Pr(X = 0 | p) = 1 ! (1 ! p)

8

. The

average of these probabilities, calculated for 100,000

random values of p, was 0.192, with a standard error

of 0.0003. This means that the true probability is

0.192, with negligible random error. Because this

probability is not small, the data appear consistent

with the prior distribution.

6.4 Failure to Change State: 

Standby Failure

As explained in Sec. 2.3.3, this type of failure is modeled as a failure condition that occurs at an unknown

time between the most recent previous inspection, test,

or demand and the present one.

Each demand corresponds to a standby time. The only

thing that can be observed is whether the system is

failed or not at the end of the standby period. From

Equation 2.3, the probability that the system is failed at

time t is

p = 1 ! e!8t

 . (6.19)

Suppose that x failures are observed on n demands. For

any one of the failures, denote the corresponding

standby time by ti

, i = 1, ..., x. For any one of the

successes, denote the corresponding standby time by sj

,

j = 1, ..., n ! x. All these numbers are observable in

principle. Therefore, the likelihood is proportional to

e e ( ). (6.20) s

j

n x

t

i

x

j i −

=

−

−

=

∏ ∏ − λ λ

1 1

1

This likelihood will be treated in three distinct ways

below. First, a simple special case will be considered.

Second, an approximation of the likelihood will be

developed and used. Finally, a way to use the exact

likelihood in Bayesian analysis will be given.

First, consider a simple special case, when all the standby times are equal, say, to some number t. This can

happen if all the demands are test demands at equally

spaced intervals. In this case, the probability of failure

on demand is the same for each demand, the quantity p

given by Equation 6.19. Therefore, the number of

failures in n demands is binomial(n, p). The analysis

methods of Section 6.3 can all be used – Bayesian or

frequentist estimation of p and all the methods of model

validation. At the very end of the analysis, the conclusions in terms of p should be translated into conclusions

in terms of 8, by solving Equation 6.19 for

8 = !ln(1 ! p)/t .

This equation for 8 can be approximated as

8 . p/t

if p is small (say, < 0.1).

This last equation shows that the MLE of 8 is approximated by

p t x nt $ / / = . 

Here x is the number of failures and nt is the total

standby time. This total standby time is approximately

the total calendar time, so a simple estimate of 8 is the

number of failures divided by the total calendar time.
Parameter Estimation and Model Validation

6-52

model

{ for (i in 1:n) {

 p[i] <- 1 - exp(-lambda*t[i])

 x[i] ~ dbern(p[i])

 }

 lambda ~ dgamma(0.5, 0.00001)

}

Figure 6.40 Script for analyzing standby failure data

exactly.

The above simple approach assumes that all the standby

times are equal. If the standby times are approximately

equal, or nearly all equal, it is very appealing to use the

above technique, calling it an adequate approximation.

If, instead, the standby times differ greatly, one of the

two approaches given below can be used. The first uses

an approximation of the likelihood, and the second is an

exact Bayesian method.

An approximation of the exact likelihood given in

Equation 6.20 can be developed as follows. It is well

known that

1 ! exp(!8ti

) . 8ti .

This is the first order Taylor-series approximation, and

is valid when 8ti

 is small. The error is on the order of

(8ti

)

2

. A second-order approximation is less well

known, but it is not hard to show that

1 ! exp(!8ti

) . 8tiexp(!8ti/2) .

That is, the two quantities on the left and right of the .

have the same second order Taylor expansions, and

they differ only by a term of order (8ti

)

3

. Therefore, the

likelihood in Equation 6.20 is approximately equal to

exp exp / − .

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟ −

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟ =

−

= =

∑ λλ λ st t j ∏ ∑

j

n x

i

i

x

i

i

x

1 1 1

2

This is proportional to

e!8t

8x

where 

ts t . j

j

n x

i

i

x

= +

⎛

⎝

⎜ ⎞

⎠

⎟ =

−

=

∑ ∑ 1 1

2

Compare this approximation of the likelihood with

Equation 6.1, and see that the approximate likelihood

here is proportional to the likelihood of x Poisson

events in time t, where t equals the total standby time

for the successes plus half the standby time for the

failures.

Therefore, all the likelihood-based methods for Poisson

data are approximately valid, treating the data as

showing x failures in time t. The likelihood-based

methods consist of maximum-likelihood estimation and

all the Bayesian techniques.

The graphical methods for model validation from

Section 6.2 remain valid, because they give qualitative

indications and do not require a rigorous justification.

The above argument also suggests that the chi-squared

test of poolability in Section 6.2 can be used with the

present data, because the chi-squared test is only an

approximation in any case. However, no simulations

to confirm this have been carried out for this handbook.

Finally, we give a different approach, an exact Bayesian

method that can be used if the standby times have been

recorded, based on Equation 6.20. Figure 6.40 gives a

portion of a script for analyzing this type of data with

BUGS, based on the exact likelihood. (See Figures

6.16 and 6.36 for similar scripts in other situations.)

In this script, pi

 is defined as 1 ! exp(!8ti). The

random variable Xi

 is assigned a Bernoulli(pi) distribution. This means that Xi

 equals 1 with probability pi and

equals 0 with probability 1 ! pi

. It is the same as a

binomial distribution with n = 1. Finally, 8 is assigned

a prior distribution. In Figure 6.40, the prior distribution is chosen to be close to the Jeffreys noninformative

prior for Poisson data, but any proper prior distribution

could be used. BUGS requires a proper distribution, so

the second parameter of the gamma distribution cannot

be exactly zero. An additional required portion of the

script, giving the data, is not shown in Figure 6.40.

6.5 Failures to Run during 

Mission

6.5.1 Estimates and Tests

This type of data can be analyzed using almost exactly

the same tools as for event rates in Section 6.2. Certain

tools carry over exactly, and others are approximately

correct.
Parameter Estimation and Model Validation

6-53

∏ [ ] λλ λ exp( ) exp( ) − ∏ − = =

−

t s i

i

x

j j

n x

1 1

= − λ λ

x exp( )t

=−+

⎛

⎝

⎜ ⎞

⎠

⎟ ⎡

⎣

⎢

⎢

⎤

⎦

⎥ = = ⎥

−

λ λ ∑ ∑ x

i

i

x

j

j

n x

exp t s

1 1

6.5.1.1 Likelihood-Based Methods: MLEs and

Bayesian Methods

Suppose that n systems are run for their missions.

(Equivalently, we might assume that a system is run for

n missions.) Suppose that x of the runs result in failure,

at times t1, ..., tx. The remaining n ! x runs are completed successfully, and the systems are turned off at

times s1, ..., sn!x. Observe the notation: t for a failure

time and s for a completed mission time. The likelihood is the product of the densities of times to failure,

for the systems that fail, times the probability of no

failure, for the systems that did not fail:

(no failure by sj f ti ).

j

n x

i

x

( ) Pr

=

−

=

∏ ∏ 1 1

(As elsewhere, the capital pi denotes a product, analogous to a capital sigma for a sum.) Under the model

introduced in Section 2.4, the failure rate is assumed to

be constant, 8, the same for all the systems. Therefore,

the time to failure has an exponential distribution. As

stated in Appendix A.7.4, the density of an exponential(8) distribution is

f(t) = 8e!8t

and the cumulative distribution function (c.d.f.) is

F(t) = 1 ! e!8t

.

In particular, the probability of no failure by time s is

1 ! F(s). Substitution of these values into the general

expression for the likelihood results in 

where t is defined as Eti

 + Esj, the total running time.

Except for a normalizer that does not depend on 8, this

is the Poisson probability of x failures in time t,

exp(!8t)8x

t

x

/x! .

Recall that Section 6.2 dealt with x failures in time t.

Therefore, any statistical analysis that requires only a

multiple of the likelihood is the same in Section 6.2 and

here. In particular, the maximum likelihood estimate of

8 is x/t. The gamma distributions form the family of

conjugate priors, and any Bayesian analysis is carried

out the same way for the data here, and the data in

Section 6.2.

The subtle difference is that Eti

 is randomly generated

here, so t is randomly generated (although if most of the

systems do not fail during their missions, the random

portion of t is relatively small). Also, the likelihood

here is not a probability, but a combination of densities

and probabilities, explaining the missing normalizer in

the likelihood. These differences between this section

and Section 6.2 result in small differences in the confidence intervals and the tests for poolability.

6.5.1.2 Confidence Intervals

Engelhardt (1995) recommends the following method

when all the mission times equal the same value, s. The

probability of a system failure before time s is 

p = F(s) = 1 ! exp(!8s) . (6.21)

Based on x failures in n trials, find a confidence interval

for p, using the methods of Sec. 6.3. Translate this into

a confidence interval for 8, using Equation 6.21

8conf, 0.05 = !ln(1 ! pconf, 0.05)/s

8conf, 0.95 = !ln(1 ! pconf, 0.95)/s .

This method does not use all of the information in the

data, because it ignores the times of any failures, using

only the fact that there was a failure at some time before

the mission time s. However, if failures are few, the

loss of information is small.

Similarly, to perform tests when all the mission times

are the same, for example to test whether two data

subsets can be pooled, one can work with p, defined by

Equation 6.19, and use the tests given in Section 6.3.

The translation to 8 needs to be made only at the very

end of the analysis. 

When the mission times are not all equal, no exact

confidence interval method exists. However, Bayesian

intervals can still be found, and are suggested.

6.5.1.3 Jeffreys Noninformative Prior

The Jeffreys prior can be worked out exactly, following

the process given in Appendix B.5.3.1. If 8×(typical

mission time) is small (say, < 0.1), then the Jeffreys
Parameter Estimation and Model Validation

6-54

prior is approximately the same as in Section 6.2, an

improper distribution proportional to 8-1/2.

6.5.1.4 Tests for Poolability

The above arguments suggest that it is adequate to

ignore the random element of t, and use the methods of

Sec. 6.2, when estimating 8. For testing whether

subsets of the data can be pooled, the same arguments

suggest that the chi-squared test of Sec. 6.2 can be used.

The chi-squared distribution is only an asymptotic

approximation in any case, and can probably be used

even when t has a small amount of randomness, although no simulations to confirm this have been carried

out for this handbook.

The rest of this section considers a diagnostic plot that

was not introduced earlier.

6.5.2 Hazard Function Plot

One plot that is especially useful for failures to run is

the hazard function plot. It is used to investigate whether 8 is constant during the entire mission. As explained

in Appendix A.4.4 for a nonrepairable system, 8)t is

the approximate probability that the system will fail

during a time interval of length )t, given that it has not

yet failed. The precise name for 8 is the hazard rate,

or hazard function, although it is often also called the

failure rate.

Suppose that the system must run for some mission

time, and the data value for that mission is either the

mission time, if the system runs to the end without

failing, or the failure time, if the system fails during the

mission. The outcome, failure or success, is also

recorded. The total data set consists of the data from a

number of missions.

Now consider the possibility that 8 is not constant.

Therefore, we write it as 8(t). An estimate of 8(t))t at

some time t is the number of systems that failed during

the interval (t, t + )t) divided by the number of systems

that had not yet failed by time t. This leads to the

following rather unsatisfactory estimate of 8(t). Divide

the mission time into small intervals, each of length )t,

with the intervals so short that hardly any of them

contain more than one failure time. In an interval with

no recorded failures, estimate 8(t) by 0. In an interval

(t, t + )t) with one failure, estimate 8(t))t by 1/nt

,

where nt

 is the number of systems that had not yet failed

by time t. Therefore, the estimate of 8(t) there is

1/(nt

)t). For intervals with more than one failure, set

the numerator to the number of failures.

This estimate consists of a number of spikes, at times

when failures were observed. Because it is so unsmooth, this estimate is not at all attractive. However,

it motivates a very simple estimate of the cumulative

hazard function, defined as

Λ () ( ) t u du . t= ∫ λ

0

In this definition, the argument t of 7 is the upper limit

of integration. Here 7 and 8 are related in the same

way that a c.d.f. and a density are related. In particular,

8(t) is the derivative of 7(t).

A natural and simple estimate of 7(t) is a step function,

which is flat except at times when failures occurred. At

a time t when a failure occurred, the estimate of 7

jumps by 1/nt, where nt is defined, just as above, as the

number of systems that had not yet failed by time t. If

exactly simultaneous failures occur, for example

because of roundoff in reporting the failure times, the

estimate of 7 jumps by the number of failures divided

by nt

. This plot is due to Nelson (1982). The full name

of the plot is the cumulative hazard function plot.

This technique is illustrated with the following example.

Example 6.11 EDG failure-to-run times.

Grant et al. (1999b) state that 23 failures to run

occurred during the EDG tests performed approximately once every 18 months. All these failures

were reported by plants subject to Regulatory

Guide RG1.108, and there were approximately

665 such tests performed at these plants during

the study period. These tests require the EDG to

run for 24 hours. Of the 23 failure reports, 19

reported the times to failure. The 19 reported

times are given below, in hours.

0.17

0.23

0.25

0.33

0.33 

0.35

0.93 

1.18

2.67

3.00 

4.00

5.50

 6.00 

 8.00 

10.00

10.00 

11.50 

13.00 

17.78

Grant et al. (1999b) assume that the lack of a reported time is statistically independent of the time at

failure, so that the 19 reported times are representative of all 23 times.

There were approximately 665 such tests. Therefore, the cumulative hazard plot jumps by 1/665 at

time 0.17 hours, by 1/664 at time 0.23 hours, and so

forth, until it jumps by 1/647 at time 17.78. It is

important that the duration of all the tests is known to
Parameter Estimation and Model Validation

6-55

Failure time (hours)

Cumulative failures

GC99 0292 23

0

5

10

15

20

0 4 8 12 16 20 24

Figure 6.41 Plot of cumulative failure count, a close

approximation of plot of cumulative hazard function

when only a small fraction of the systems fail.

be 24 hours. This fact guarantees that none of the

EDGs drop out early, so that after 18 failures 647

EDGs are still running. Actually, this is only approximate, because it ignores the four failures with unreported times.

The jumps are almost the same height, because

1/665 equals 1/647 to two significant digits. Therefore Grant et al. (1999b) plot the cumulative number

of failures (a jump of 1 at each failure), instead of the

estimated cumulative hazard function. The two

graphs make the same visual impression, and the

cumulative failure plot was easier to explain in the

report. This plot is shown here, as Figure 6.41.

The cumulative hazard plot would differ only in that

the vertical scale would be different, and the jumps

would not be exactly the same size, though the

jumps would be almost the same size in this example.

As explained in introductory calculus courses, when a

function is graphed as a curve, the derivative of the

function is the slope of the curve. Therefore, the slope

of a cumulative hazard plot near time t estimates the

derivative of 7 at time t. But the derivative of 7(t) is

8(t). Therefore, a constant slope indicates constant

8(t), and a changing slope indicates changing 8(t).

Grant et al. (1999b) note that for times less than

about one half hour the slope is approximately

constant, and steep. It is again constant, but less

steep, from about 1/2 hour until about 14 hours, and

it is smaller yet after 14 hours. Therefore, Grant et

al. (1999b) estimate three values for 8, corresponding to these three time periods. They comment that

the early, middle, and late failures seem to correspond in part to different failure mechanisms. 

6.6 Recovery Times and Other

Random Duration Times

The previous analyses have all involved a single

parameter, 8 or p. The analysis of duration times is

different because now a distribution must be estimated,

not just a single parameter.

A distribution can be estimated in many ways. If the

form of the distribution is assumed, such as exponential

or lognormal, it is enough to estimate one or two

parameters; the parameter or parameters determine the

distribution. If the form of the distribution is not

assumed, the distribution can be estimated nonparametrically, or characteristics of the distribution, such as

moments or percentiles, can be estimated.

To test whether data sets can be combined (pooled),

both parametric tests and nonparametric tests exist.

The parametric tests typically test whether the means or

variances of two distributions are equal, when the

distributions have an assumed form. The most common

nonparametric tests test equality of the distributions

against the alternative that one distribution is shifted

sideways from the other.

This section is long, because so many distribution

models can be assumed and because the model assumptions can be violated in so many ways. A brief outline

of the section is as follows:

6.6.1 Characterization of a single distribution

Estimation of moments, percentiles, c.d.f.s

Fitting of four parametric models (frequentist

and Bayesian parameter estimates)

6.6.2 Model validation (graphs and hypothesis tests)

Poolability, trend

Goodness of fit to assumed parametric models

Consistency of data with prior for Bayesian

parameter estimates

6.6.3 Nonparametric density estimation

Many of the methods will be illustrated using the data

of Example 6.12, taken from Atwood et al. (1998).

This example shows the times when power could

have been recovered, for plant-centered LOSP

events, that is, for events not caused by grid problems or by widespread severe weather. (Real life is

complicated: sometimes a plant does not restore

power as quickly as it could, and the event report

states when power was actually restored, and usually also when it could have been restored. The

times given by Atwood et al. (1998) as “recovery

times” show when power could have been restored,
Parameter Estimation and Model Validation

6-56

if that time was reported and different from the actual

recovery time.) Discussions of this example will use

the terms recovery time and duration interchangeably. Momentary events (duration less than two

minutes) and events with no reported duration have

been excluded. For common-cause events that

affected multiple units at a site, the average recovery

time is used.

Example 6.12 LOSP recovery times.

.

Atwood et al. (1998) report 115 times of recovery of

lost offsite power. The data are categorized into

three possible values for plant status: T, S, and P,

with meanings explained in the table below. The

durations in minutes and the dates (MM/DD/YY) are

shown.

P: Plant remained at power throughout LOSP event

(8 times)

 6 03/01/80

 45 07/25/85

 65 07/16/88

 113 01/18/96

 147 06/03/80

 355 11/12/90

 385 04/11/94

1138 01/03/89

S: Plant was shut down before and during LOSP event

(62 times)

 2 06/04/84

 2 08/17/87

 2 06/29/89

 2 05/21/94

 3 06/26/93

 3 10/22/84

 3.5 11/21/85

 4 04/22/80

 4 04/04/87

 4 10/20/91

 5 05/03/84

 8 06/24/88

 9 12/26/88

 10 08/01/84

 10 04/28/92

 10 12/23/81

 11 10/04/83

 11 07/24/91

 12 06/22/93

 12 07/19/86 

 14 02/26/90

 14 11/16/84

 14 02/01/81

 15 04/27/81

 15 12/19/84

 15 10/12/93

 17 04/26/83

 17 10/14/87

 20 03/23/92

 22 08/24/84

 24 07/29/88

 24 07/29/88

 29 03/20/91

 29 09/16/87

 29 05/14/89

 35 04/02/92

 37 03/21/87

 37 05/19/93

 37 07/09/90

 43 05/07/85

 53 09/11/87

 59 10/16/87

 60 06/22/91

 60 06/16/89

 62 07/15/80

 67 03/13/91

 73 08/28/85

 77 03/29/92

 97 01/08/84

 120 06/05/84

 120 01/16/81

 127 01/20/96

 132 02/27/95

 136 04/08/93

 140 03/20/90

 155 03/05/87

 163 10/08/83

 240 11/14/83

 240 03/07/91

 335 04/29/85

 917 10/21/95

1675 11/18/94

T: Plant tripped because of LOSP event

(45 times)

 2 02/28/84

 4 11/21/85

 4 11/17/87

 5 08/16/85

 6 05/03/92

 10 09/10/93

 10 10/12/93

 11 07/26/84

 13 10/07/85

 14 08/13/88

 15 02/16/84

 15 09/14/93

 19 10/25/88

 20 12/12/85

 20 03/27/92 

 20 08/21/84

 20 07/16/84

 20 06/27/91

 24 06/15/91

 25 10/03/85

 29 06/22/82

 38 07/17/88

 40 02/11/91

 45 01/16/90

 45 03/25/89

 46 01/01/86

 57 10/19/92

 60 03/21/91

 60 10/22/85

 62 07/15/80

 90 02/12/84

 90 03/29/89

 90 06/17/89

 95 12/31/92

 95 12/31/92

 95 10/16/88

 96 12/27/93

 100 01/28/86

 106 06/03/80

 118 07/23/87

 118 07/23/87

 277 04/23/91

 330 02/06/96

 388 07/14/87

 454 08/22/92

The group P exists because some plants are permitted to remain at power during certain LOSP events.

Throughout this section, the random variable is denoted

by T, because typically the random quantity is a duration time, such as time to recovery of the system.

Several examples were given in Section 2.5.1: time

until restoration of offsite power, duration of a repair

time, and others. Let F denote the c.d.f. of T, F(t) =

Pr(T # t). It is assumed that n times will be observed,

T1, T2, ... , Tn. The assumptions of Section 2.5.2 are

repeated here.

C The Ti s are independent,

C Each Ti has the c.d.f. F(t).

A data set satisfying these assumptions is called a

random sample from the distribution. Sometimes the

Ti s are called independent identically distributed

(i.i.d.). The term random sample can refer either to the

random variables (Ti

s) or to the observed values, t1, t2,

... , tn. The data are used to estimate properties of the

distribution. This can also be described as estimating

properties of the population, where the population is

the infinite set of values that could be randomly generated from the distribution.

6.6.1 Characterization of Distribution

6.6.1.1 Nonparametric Description

The tools in this subsection are called nonparametric

because they do not require any assumption about the

form of the distribution. For example, the distribution

is not assumed to be lognormal, exponential, or any

other particular form.

6.6.1.1.1 Moments

To estimate the population mean : or a population

variance F2

, two simple estimators are the sample

mean, defined as

T n Ti

i

n

=

=

∑

1

1

and the sample variance, defined as

S . n T T i

i

n

2 2

1

1

1 = − −

=

∑ ( )

The sample mean and sample variance are known to be

unbiased for the population mean and variance, respec-
Parameter Estimation and Model Validation

6-57

tively. In other words, E(T

_

) = : and E(S2) = F2. These

statements are true regardless of the distribution F,

requiring only the assumptions of a random sample.

The sample standard deviation, S, is the square root

of the sample variance. When defining S2

 some authors

use n in the denominator instead of n ! 1, with corresponding adjustment of formulas that involve S, but this

handbook uses the above definition consistently, both

here and in Appendix B. In applications with computer

packages, note which definition is used and make any

necessary adjustments to formulas in this handbook.

These are all-purpose estimators, but they are not the

only possible estimators. For example, the variance of

an exponential distribution is the square of the mean.

Therefore, a good estimator of the variance would be

the square of the estimator of the mean. This estimator

relies heavily on the assumption of exponentiality,

whereas the above estimators make no such assumptions. General principles of estimation are discussed in

Appendix B.4.1.

6.6.1.1.2 Percentiles

To estimate percentiles of a distribution, it is useful to

put the data in ascending order from the smallest to the

largest observation. The recovery times in Example 6.12 have been arranged this way. The variables

obtained by ordering the random sample are called the

order statistics, and are denoted by T(1) # T(2) # AAA #

T(n). The observed values are written t(1) # t(2) # AAA # t(n).

Some important estimates based on the order statistics

are the sample median, other sample percentiles, and

the sample range. The general definition of the 100qth

sample percentile, where 0 < q < 1, is a number t such

that the fraction of observations that are # t is at least q

and the fraction of observations that are $ t is at least

1 ! q.

For example, the sample median is defined to be t such

that at least half (because q = 0.5) of the observations

are # t and at least half (because 1 ! q = 0.5) are $ t.

This boils down to the following. If n is odd, the

sample median is the “middle” order statistic, t(m)

 with

m = (n + 1)/2. If n is even, with m = n/2, there is no

unique “middle” order statistic. Any number between

the two middle order statistics, t(m) # t # t(m+1), could be

used. However, nearly everyone uses the average of the

two middle order statistics (t(m)

 + t(m+1))/2 as “the”

sample median.

The other sample percentiles are defined similarly, with

some averaging of two order statistics if necessary.

Note that the sample 90th percentile is t(n)

 if n < 10, the

sample 95th percentile is t(n)

 if n < 20, and so forth.

Order statistics that are sometimes used are: the lower

and upper quartile, defined as the 25th and 75th

percentiles; percentiles that include most of the distribution, such as the 5th and 95th percentiles; and the

extremes, t(1) and t(n)

. The interquartile range is the

upper quartile minus the lower quartile. The sample

range is the difference between the largest and smallest

ordered observations, t(n) ! t(1). Be careful with interpretation. As data continue to be collected, the sample

interquartile range stabilizes at the interquartile range of

the distribution, but the sample range does not stabilize

at all — it just grows every time a new t is observed

that is outside the former observations.

The sample median has the advantage of not being

strongly influenced by extreme observations. The

sample mean, on the other hand, can be strongly influenced by even one extreme data value. The sample

variance is even more sensitive to extreme values,

because it is based on squared terms. Therefore, the

sample standard deviation, defined as the square root of

the sample variance, is also sensitive to extreme terms.

Other measures of dispersion, such as the interquartile

range, are much less sensitive to extreme values. In

general, sample percentiles are much less sensitive to

extreme observations than are sample moments.

The recovery times of Example 6.12 have sample

moments and percentiles given in Table 6.16.

Table 6.16 Statistics based on the recovery

times (minutes) of Example 6.12.

PST

n 8 62 45

Stand. deviation 373.2 241.4 99.9

95th %ile 1138 240 330

75th %ile

(upper quartile)

 370 73 95

Mean 281.75 92.3 73.4

50th %ile

(median)

 130 24 40

25th %ile

(lower quartile)

 55 10 15

5th %ile 6 2 4

For the P group, the sample median is taken as the

average of the two middle numbers. Even though
Parameter Estimation and Model Validation

6-58

the S group has an even number of observations, its

sample median is unique, because t(31) and t(32)

happen to be equal. The T group has an odd number of observations, so its sample median is unique,

t(23).

The S group has one very extreme value, which

influences the moments. The sample mean for this

group is larger than the upper quartile — someone

who considers the mean to be “the” average could

say that more than 75% of the observed times are

below average. Such a thing can happen with a

skewed distribution. This is one reason why many

people prefer percentiles to moments for describing

a skewed distribution.

There are situations in which some of the times are not

observed. Section 6.5 dealt with such a situation, when

the times of interest were times of EDG failure to run,

and not all these times were reported. In the present

section, nearly all the times are assumed to be observed,

with no systematic bias in which times fail to be observable.

6.6.1.1.3 The Empirical Distribution Function

An estimate of F(t) called the empirical distribution

function (EDF) is defined as follows: For an arbitrary

value of t > 0, define

= (Number of observations # t) / n. $

F t( )

The EDF is a step function. It increases by 1/n at each

observed time if all observations are distinct. More

generally, if there are m times equal to t, has a $

F t( )

positive jump of m/n at t.

In some settings the function 

1 ! F(t) = Pr(T > t)

is of interest. If T is the time until failure, 1 ! F(t) is

called the reliability function, R(t), in engineering

contexts, and the survival function, S(t), in medical

contexts. A suitable word remains to be coined when T

is the time until recovery or repair. The empirical

reliability function, or the empirical survival function,

is defined as 1 minus the EDF. Anything that can be

done with F can be translated in terms of 1 ! F, so this

discussion will only consider F.

With a little mental exercise, the EDF can be expressed

in familiar terms. For any particular t, let p denote F(t)

= Pr(T # t). In the data, define a “demand” to be the

generation of an observed time, and define the ith

observation ti

 to be a “failure” if ti # t. By the assumptions for a random sample, any observation has probability p of being a failure, and the outcomes (failures or

successes) are statistically independent of each other.

By its definition, is the number of failures di- $

F t( )

vided by the number of demands, which is , as p$

indicated in Section 6.3.1. Therefore, is an $

F t( )

unbiased estimator of F(t) at any t. It is close to F(t)

when the number of observations is large, and a confidence interval for F(t) can be constructed, the familiar

confidence interval for p.

Figure 6.42 shows the EDF based on the data in

Example 6.12 for group T.

Recovery time, t (min.)

Estimated 

F(t ) 0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250 300 350 400 450 500

Figure 6.42 Empirical distribution function (EDF) for

the data from group T in Example 6.12.

6.6.1.1.4 Histogram Estimate of the Density

The eye smooths the EDF, compensating for its jagged

form. To accomplish the same sort of smoothing for a

density estimate, group the observed times into bins of

equal width, count the number of observations in each

bin, and plot the histogram, a form of bar chart with the

height of each bin equal to the number of observations

in the bin. The histogram is proportional to an estimate

of the density. Some software packages can rescale the

height of the histogram so that the total area equals 1,

making it a true density estimate. Books and Ph. D.

theses have been written on density estimation, and

some modern density estimators are quite sophisticated.

A few such are given in Section 6.6.3. Nevertheless,

the lowly histogram is often adequate for PRA purposes.

Figures 6.43 and 6.44 show two histograms for the

data from the above EDF, using two different bin

widths. The analyst must decide what bin width

gives the most reasonable results, based on belief

about how smooth or ragged the true density might
Parameter Estimation and Model Validation

6-59

Recovery times (min.)

Nimber of observations

0

1

2

3

4

5

6

7

8

9

10

11

0 50 100 150 200 250 300 350 400 450 500

Figure 6.44 Histogram of same data, with bin

width 10.

be. Most people would judge Figure 6.44 to be too

rough, and would therefore choose wider bins.

Recovery times (min.)

Number of observations

0

4

8

12

16

20

24

28

0 50 100 150 200 250 300 350 400 450 500

Figure 6.43 Histogram of the data from group T in

Table 6.16, with bin width 50.

6.6.1.2 Fitting a Parametric Distribution

Sometimes it is desirable to fit some assumed distributional form to data. This subsection gives estimators if

the assumed distribution is lognormal, exponential,

gamma, or Weibull. Bayesian and non-Bayesian

estimates are given, with much of the latter taken from

an INEEL report by Engelhardt (1996).

6.6.1.2.1 Lognormal Distribution

This model assumes that T has a lognormal distribution,

or equivalently, that ln T has a normal(:, F2

) distribution. Define X = lnT. 

Frequentist Estimates. The usual estimates of : and

F2 are:

x n xi i = 1Σ

and

s . n x x X ii

2 2 1

1 = − Σ ( ) −

These estimates have the same form as those given in

Section 6.6.1.1.1 for the mean and variance of T, but

these are for lnT. Calculate the estimates of : and F2

to determine the estimated normal distribution of lnT,

which determines the estimated lognormal distribution

of T. Note that the sample variance is defined with n !

1 in the denominator, although some authors use n in

the definition and slightly different formulas below.

The material below is presented in many statistics

books, based on the fact that lnT has a normal distribution. The distribution of is chi-squared ( )/ n S − 1 X

2 2 σ

with n ! 1 degrees of freedom. It follows that a twosided 100(1 ! ")% confidence interval for F2

 is

( ) ( ) / ( ),( ) / ( ) . ns n ns n X X / / − −− − 1 11 1 −

2

1 2

2 2

2

2 χ χ α α

Here is the χ q quantile, that is, the 100q q n 2( ) − 1

percentile, of the chi-squared distribution with n!1

degrees of freedom.

The distribution of is normal( :, F2

/n). If F2 X is

known, it follows that a 100(1!")% confidence interval

for : is 

xz n ± , 1 2 −α / σ /

where z1!"/2 is the 100(1!"/2) percentile of the standard

normal distribution. For example, z0.95 gives a twosided 90% confidence interval.

In the more common case that both : and F2

 are unknown, use the fact that

( )/( / ) X Sn − μ X

has a Student’s t distribution with n!1 degrees of

freedom. It follows that a 100(1 ! ")% confidence

interval for : is

xt n s n , X ± − 1 2 −α / ( )/ 1

where is the 1 t n !"/2 quantile of the Stu- 1 2 −α / ( ) − 1

dent’s t distribution with n!1 degrees of freedom. For

example, t0.95(n!1) gives a two-sided 90% confidence

interval. Percentiles of Student’s t distribution are

tabulated in Appendix C.
Parameter Estimation and Model Validation

6-60

Bayesian Estimation. Bayesian estimates are given

here.

Conjugate Priors. The conjugate priors and

update formulas are presented by Lee (1997, Sec. 2.13).

They depend on four prior parameters, denoted here as

d0, F0

2

, n0, and :0. The notation here follows the

notation used elsewhere in this handbook. It is not the

same as Lee’s. Quantities with subscripts, such as F0

2

or d1, are numbers. Quantities without subscripts, F2

and :, have uncertainty distributions.

It is useful to think of having d0 degrees of freedom,

corresponding to d0 + 1 prior observations for estimating the variance, and a prior estimate F0

2

. More precisely, let the prior distribution for F2

/(d0F0

2

) be inverted chi-squared with d0 degrees of freedom. That is,

d0F0

2

/F2 has a chi-squared distribution with d0 degrees

of freedom. Therefore it has mean d0, and therefore the

prior mean of 1/F2

 is 1/F0

2

. (See Appendix A.7.7 for

more information on the inverted chi-squared distribution.)

An alternative notation for the above paragraph would

define the precision J = 1/F2

, and the prior precision J0

= 1/F0

2

. Then the prior distribution of d0J/J0 is chisquared with d0 degrees of freedom. Although we shall

not use this parameterization, it has adherents. In

particular, BUGS (1995) uses J instead of F2

 as the

second parameter of the normal distribution; see

Spiegelhalter et al. (1995).

Conditional on F2

, let the prior distribution for : be

normal with mean :0 and variance F2

/n0. This says that

the prior knowledge of : is equivalent to n0 observations with variance F2

. It is not necessary for n0 to

have any relation to d0. 

The Bayes update formulas are

d1 = d0 + n,

n1 = n0 + n,

μ μ and 1 00 1 = + ( )/ n nx n

σσ μ . 1

2

0 0

2 2 0 1

0 1

0

2 1 1 = +− +

+ − ⎡

⎣

⎢ ⎤

⎦ d ns ⎥ n n

n n

( ) ( )/ X x d

Here the subscript 1 identifies the posterior parameters.

The posterior distributions are given as follows. First,

F 2/(d1F1

2

) has an inverted chi-squared distribution with

d1 degrees of freedom. That is, the posterior mean of

1/F 2 is 1/F1

2

, and a two-sided 100(1!") credible

interval for F 2 is

( ) d dd d . 1 1

2

1 2

2

1 11

2

2

2 σχ σχ α α 1 / ( ), / ( ) − / /

Conditional on F 2, the posterior distribution of : is

normal(:1, F2

/n1). Therefore, conditional on F 2, a twosided 100(1!")% credible interval for : is

μ σ . 1 12 1 α ± − z n / /

The marginal posterior distribution of :, that is, the

distribution that is not conditional on F 2, is as follows.

The expression

( )/( / ) μμ σ − 1 11 n

has a Student’s t distribution with d1 degrees of freedom. It follows that a 100(1 ! ")% credible interval

for : is

μ σ . 1 121 1 1 α ± − td n / () /

Noninformative Prior. The joint noninformative prior for (:, F 2) is proportional to 1/F 2. Lee

(1997, Sec. 2.13) presents this prior, as do Box and

Tiao (1973, Sec. 2.4). Lee points out that when d0 =

!1, n0 = 0, and F0

2

 = 0, the conjugate prior distribution

reduces to the noninformative prior. In the formulas

just given above, n1 = n, d1 = n ! 1, :1 = , and F1 x =

sX. The credible intervals then agree numerically with

the confidence intervals given above.

Possible Further Analyses. Some data

analyses require only the posterior distribution of one or

both parameters. In that case, use the above posterior

distributions, with either an informative or noninformative prior. Other analyses require more, such as simulation of a set of lognormal times T or a credible

interval for the mean of T. If so, simulation of the

quantity of interest is a useful technique. Begin each

case of the simulation by generating a value of F2

 from

its posterior distribution. Then generate a value of :

from its distribution conditional on F 2. Then do

whatever is required next to obtain the quantity of

interest: generate a random value of T from the lognormal(:, F) distribution, or calculate E(T) = exp(: +

F2/2), or calculate whatever else is needed. Save the

quantity of interest produced in this way. Repeat this

process as many times as needed to obtain a sample that

accurately represents the distribution of interest.

Model Validation. Model validation is discussed in

general in Section 6.6.2. Many of the methods given

there are applicable to any assumed distribution. Some

methods, however, have been developed just for the

normal and lognormal distributions. They are contained in Sections 6.6.2.1.2, 6.6.2.2.2, and 6.6.2.3.2.
Parameter Estimation and Model Validation

6-61

λ χ α

β 0 95

0 95

2

1

1

2

2 .

. ( ) =

6.6.1.2.2 Exponential Distribution

The exponential distribution is related to a Poisson

process, because the times between successive events in

a Poisson process have an exponential distribution.

The exponential distribution is presented in Appendix

A.7.4, with two possible parameterizations. The first

uses 8 = 1/E(T), and the second uses : = 1/8 = E(T). In

data analysis, sometimes one parameter seems more

natural and convenient than the other. In the two

parameterizations, the likelihood function is

8nexp(!8Gti)

or

:!nexp(!Gti/:) .

As a function of 8, the likelihood function here is

proportional to the likelihood function given by Equation 6.1 for Poisson data. (Replace x in Equation 6.1 by

n and t by Eiti.) Therefore, many of the results below

are similar to or identical to the results in Section 6.2

for Poisson data.

Frequentist Estimation. It can be shown that the MLE

of : is the sample mean, . Therefore, to estimate the t

distribution, estimate : by . This determines the t

estimated exponential distribution. The corresponding

estimate of 8 / 1/: is . 1/ t

For a (1 ! ") confidence interval, or equivalently a

100(1 ! ")% confidence interval, the lower limit for 8

is

8conf, "/2 = χ α / ( ) 2

2 2

2

n

t Σ i

and the upper limit is

8conf, 1 ! "/2 = . χ1 2 α

2 2

2

− / ( ) n

t Σ i

(See Martz and Waller 1991.) Confidence limits for :

= 1/8 are obtained by inverting the confidence limits for

8. For example, the lower confidence limit for : equals

1 divided by the upper confidence limit for 8.

Bayesian Estimation. Now consider Bayesian estimation.

Conjugate Prior. The gamma distribution is

a conjugate prior for 8. That is, let t1, ... , tn be independent observations from an exponential(8) distribution. Let the prior distribution of 8 be gamma("0, $0).

This uses the same parameterization as when 8 is a

Poisson parameter (Section 6.2.2.4), so that $0 has units

of time and the prior mean of 8 is "0/$0. A direct

calculation shows that the posterior distribution of 8 is

also gamma, with posterior parameters

"1 = "0 + n

$1 = $0 + E ti .

The subscript 1 identifies the posterior parameters. The

prior parameters have a simple intuitive interpretation

– the prior information is “as if” "0 duration times had

been observed with total value $0. 

The percentiles of the posterior distribution are given

by

λ . χ α

β p

p

=

2

1

1

2

2

( )

Therefore, for example, a two-sided 90% credible

interval has end points

λ χ α

β 0 05

0 05

2

1

1

2

2 .

. ( ) =

and 

 .

There are two possible ways to perform the corresponding analysis in terms of :. (a) One way is to perform

the above analysis in terms of 8, and then translate the

answer into answers for : = 1/8. Be careful when

doing this. The percentiles translate directly, with the

100p percentile :p = 1/81!p. For example, :0.95 = 1/80.05.

The moments do not translate directly, however. For

example, the posterior mean of : is $1/("1 ! 1), not 1

divided by the mean of 8. (b) The other way is to let :

have an inverted gamma distribution. This distribution

is defined in Appendix A.7.7.

Either analysis gives exactly the same results. The

second approach is just a disguised version of the first

approach, using a different distribution to avoid introduction of the symbol 8.

Noninformative Prior. The Jeffreys noninformative prior for 8 can be expressed as a gamma(0, 0)

distribution. This is an improper distribution, that is, it

does not integrate to 1, but it does result in proper
Parameter Estimation and Model Validation

6-62

posterior distributions as long as some data have been

observed. Note, this prior is slightly different from the

Jeffreys prior when the data have a Poisson distribution.

When the gamma(0, 0) prior is used with exponential

data, the posterior parameters reduce to

"post = n

$post = E ti .

Then the Bayes posterior credible intervals are numerically equal to the confidence intervals. If the purpose

of a “noninformative” prior is to produce intervals that

match confidence intervals, this purpose has been

perfectly accomplished.

Discussion. The above work has illustrated some facts

that are generally true. When the observations have a

discrete distribution, such as Poisson or binomial, the

so-called noninformative priors do not produce credible

intervals that exactly match confidence intervals. This

is related to the fact that confidence intervals from

discrete data do not have exactly the desired confidence coefficient. Instead, they are constructed to have

at least the desired long-run coverage probability. The

situation is different when the observations are continuously distributed, as in the present case with exponentially distributed times. In this case, the confidence

intervals have exactly the desired long-run coverage

probability, and posterior credible intervals, based on

the noninformative prior, are numerically equal to the

confidence intervals.

Nonconjugate priors can also be used. The procedure

is similar to that in Section 6.2.2.6, but now uses the

exponential likelihood given above. Therefore, it is not

discussed here.

Model Validation. Model validation is discussed in

general in Section 6.6.2. Many of the methods given

there are applicable to any assumed distribution. A few

methods, however, have been developed just for the

exponential distribution. They are mentioned in Sections 6.6.2.3.1 and 6.6.2.4.1.

6.6.1.2.3 Gamma Distribution

The distribution of T is gamma(", J) if the density is

ft t e . t ( )

( )

/ = − − 1 1

τ α

α

α τ

Γ

Note, this is a different parameterization from the

previous section and from Equation 6.4. This parameterization is related to the earlier parameterization by

the relation J = 1/$. In the present context, t and J

both have units of time.

The MLEs of the parameters are given by Bain and

Engelhardt (1991, p. 298) or by Johnson et al. (1994,

Sec. 17.7). They are the solutions of the equations

τ α = t /

ln( ) ( ) ln( / , (6.23) ~ α ψα − = t t )

where R(u) = 'N(u)/'(u) is the digamma function,

calculated by some software packages, and

[ ], ~

t nt = exp ( / ) ln 1 Σ i

is the geometric mean of the observed times. Equation

6.23 must be solved for " by numerical iteration. Bain

and Engelhardt (1991, p. 298) give a table of approximate solutions, which may be interpolated.

The MLEs of the two parameters determine the estimated gamma distribution.

Bayes estimation is complicated because the gamma

distribution, like the lognormal distribution, has two

parameters, and these two parameters must have a joint

distribution. Martz and Waller (1991, Sec. 9.5.2) cite

Lwin and Singh (1974) for an analysis that was feasible

in the 1970s. A simpler approach today would use the

freely available software package BUGS (1995),

described in Section 6.2.2.7, Section 8.3.3.3, and

elsewhere in this handbook. BUGS is designed for

models with many unknown parameters, and should

make short work of a model with only two. The joint

prior distribution would not need to be conjugate.

6.6.1.2.4 Weibull Distribution

A three-parameter Weibull distribution is given in

Appendix A.7.5. A two-parameter form of the Weibull

distribution is given here, by setting the location parameter 2 to zero. The density is

ft t t ( ) ( / )( / ) exp ( / ) = −[ ] . − βα α α β β 1

As with the gamma distribution, the maximum likelihood equations do not have closed-form solutions. The

estimates must be found by iteratively solving

Σ

Σ

Σ

t t

t n

t i i

i

i

β

β β

ln( ) − − ln

1 1
Parameter Estimation and Model Validation

6-63

and

α . β

β

= ⎛

⎝

⎜ ⎞

⎠

⎟ 1 1

n

t Σ i

/

Zacks (1992, Section 7.5) gives the following simple

method for solving the first equation. Begin with

. Then repeatedly solve the equation $

β0 = 1

$ ln( ) ln

$

β $

β

n β

i i

i

i

t t

t n

t

n

n + = − ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ 1 1

Σ 1

Σ

Σ

with n = 0, 1, 2, ... The value of converges quickly $

βn

to the MLE . Then set $

β

$ . $ / $

α

β

β

= ⎛

⎝

⎜ ⎞

⎠

⎟ 1 1

n

t Σ i

For more information, see Zacks (1992) or Bain and

Engelhardt (1991).

Alternatively, a simple approximate graphical estimate

is based on the hazard function. Plots of the cumulative

hazard were discussed in Section 6.5.2. It can be

shown that the cumulative hazard function of the

Weibull distribution is

H(t) = (t/")

$ .

Therefore, estimate the cumulative hazard function as

explained in Section 6.5.2, by jumping at each observed

time, with the size of the jump equal to 1 divided by the

number of times that have not yet been equalled or

exceeded. The jump at t(1) is 1/n, the jump at t(2) is 1/(n

! 1), and so forth until the final jump at t(n) is 1. Call

this estimate . The equation for the Weibull $H t( )

cumulative hazard function can be rewritten as

log H(t) = $logt !$log" , (6.24)

which is linear in logt. Therefore, plot log[ $H t( )]

against logt, that is, plot against t on log-log $H t( )

paper, and fit a straight line to the plot by eye. Pick a

point on the line and substitute those values of t and

H t $ ( ) into Equation 6.24. This is one equation that $

and log" must satisfy. Pick a second point on the line

and obtain a second equation in the same way. Solve

those two equations for $ and log", thus obtaining

estimates of $ and ". In the calculations, it does not

matter whether natural logarithms or logarithms to base

10 are used, as long as the same type is used everywhere.

This plot also gives a diagnostic test of whether the

Weibull distribution is appropriate. The degree to

which the plotted data follow a straight line indicates

the degree to which the data follow a Weibull distribution.

Just as in Sections 6.6.1.2.1 and 6.6.1.2.3, Bayes

estimation is complicated here by the multiple parameters. Martz and Waller (1991, Sec. 9.1) cite a number

of early papers using various prior distributions.

However, the easiest Bayesian approach today would

be to assign convenient diffuse priors to the parameters

and use BUGS (1995), described in Section 6.2.2.7,

Section 8.3.3.3, and elsewhere in this handbook.

6.6.2 Model Validation

This section considers several topics. First, the usual

investigations of the model assumptions are considered:

whether subsets of the data all correspond to the same

distribution, whether the distribution changes with time,

and whether the times are serially correlated instead of

statistically independent. In addition, the distribution

may have been modeled by some parametric form, so

the goodness of fit is investigated. Finally, if parameters have been estimated in a Bayesian way, the consistency of the data with the prior must be investigated.

The order described above follows the actual order of

analysis. First, the analyst would check to see what

data subsets can be pooled and whether the usual

assumptions seem to be satisfied. Only then would it be

appropriate to try to fit some standard distribution to the

data.

6.6.2.1 Poolability of Data Sources

To illustrate the methods here, this subsection will

consider the three groups of data in Example 6.12,

corresponding to three conditions of the plant during

the LOSP event. As elsewhere in this chapter, graphical methods are considered first, and statistical tests

second.

6.6.2.1.1 Graphical Methods

A simple, graphical method of comparison is to overlay

the EDFs for the different data subsets on a single

graph. Then, look to see if the EDF are intertwined,

indicating that the subsets may be pooled, or if they are
Parameter Estimation and Model Validation

6-64

separated, shifted sideways from each other, indicating

that the data subsets may not be pooled. This method

is simple, but the graph can become very cluttered,

especially if a moderate or large number of subsets

must be compared. The same comment can be made

for comparing separate histograms of the data subsets.

A graph that has come into common use is the boxand-whisker plot, or box plot. The lower and upper

edges of the box are the lower and upper quartiles of

the data. Thus, the box can be thought of as containing

half the data, with 1/4 of the remaining data on each

side. The median is marked somehow. The “whiskers”

are two lines coming out of the box and going out to

cover the range of most of the data, up to 1.5 times the

interquartile range in each direction. A few outlying

points are plotted individually.

Figure 6.45 shows a box plot of the group T data

from Example 6.13 generated using the STATISTICA

(1995) software. The median is marked by a small

square in the box. The software documentation does

not give a precise definition of the difference between an outlier and an extreme point. Also, this

release of the software seems to have a small bug,

in that the maximum (excluding outliers) is labeled

as 11, when it should be 118.

Non-Outlier Max = 11

Non-Outlier Min = 4

75% = 95

25% = 17

Median = 42.5

Outliers

Extremes

Recovery time (min.)

0

100

200

300

400

500

T

Figure 6.45 One form of a box plot. The box shows

the lower and upper quartiles, with the median

marked. The whiskers show most of the range, from

4 to 118, and individual outlying points are plotted.

Figure 6.46 shows the same box plot as drawn by a

different software package, SAS/INSIGHT (1995).

As before, the box shows the lower and upper quartiles, and the median is marked, this time with a

stripe. Points beyond the whiskers are shown as

individual dots.

Box plots were invented by Tukey (1977), and are still

being modified according to individual taste. Any form

of the plot that is produced by a convenient software

package is probably adequate.

0

100

200

300

400

500

 T

Figure 6.46 A different style box plot of the same

data. The box shows the upper and lower quartiles,

with the median indicated by a stripe. The whiskers

show much of the range, with dots marking outliers.

The example here is typical, in that the data are skewed,

and the most obvious feature of the box plots given here

is the long distance from the box to the largest value.

Box plots are supposed to focus on the bulk of the data,

with only moderate attention given to the extremes.

Therefore, there are visual advantages to transforming

skewed data by taking logarithms. Therefore, all the

remaining box plots shown in this section will use

log10(recovery time) instead of the raw times. 

Figure 6.47 shows side-by-side box plots of the three

data subsets in Example 6.12. Incidentally, the box

plot of log(time) is different from the box plot of time

plotted on a logarithmic axis — the logarithms of

large times tend not to be considered as outliers.

This can be seen by comparing Figure 6.45 with the

group-T portion of Figure 6.47.

Non-Outlier Max

Non-Outlier Min

75%

25%

Median

Outliers

Log10(recovery time)

0

1

2

3

4

PST

Figure 6.47 Side-by-side box plots of the three

groups of data from Table 6.16, based on

log10(recovery time).

Figure 6.47 shows that group P seems to have

somewhat longer recovery times than the other

groups. There seems to be little difference between

groups S and T. Tests will be given below to investigate whether this visual impression is correct.
Parameter Estimation and Model Validation

6-65

6.6.2.1.2 Statistical Tests

Tests Based on Normality. Warning: these tests are

only valid if normality or lognormality can be assumed.

If each data subset corresponds to a lognormal distribution, work with X = log(T). Either natural logs or base10 logs can be used, because log10(T) = ln(T)/ln(10), so

both are normally distributed if either is. 

When X has a normal distribution, standard tests based

on normal theory can be used, as given in many statistic

books. These tests investigate whether :, the mean of

X, is the same in each data subset, under the assumption

that the variances are the same. For added sophistication, tests of equality of the variances can also be

performed:

• To compare the means of two data subsets, perform a Student’s t test.

• To simultaneously compare the means of two or

more data subsets, perform a one-way analysis of

variance test.

• To compare the variances of two data subsets,

perform an F test.

• To compare variances of two or more data subsets,

use some version of a likelihood ratio test, such as

Bartlett’s test or a Pearson-Hartley test, as discussed by Bain and Engelhardt (1992, p. 426).

These tests are not considered further here, because

they rely heavily on the assumption of normality. This

is especially true of the tests later in the list. Most

statistical software packages will perform these tests.

The analyst must ask whether the assumption of normality is established well enough to justify the use of

the tests.

Nonparametric Tests Based on Ranks. For general

use when normality or lognormality is not well established, nonparametric tests are preferable. The books

by Conover (1999) and Hollander and Wolfe (1999) are

excellent summaries of standard tests. As before, let X

= log(T), but do not assume that X has a normal

distribution or any other particular distribution. Tests

for location assume that various data subsets have

distributions that are shifted sideways from each other.

The shapes are the same, but the medians may be

different. This is the nonparametric analogue of

assuming that the distributions are normal with a

common variance, but possibly different means. Tests

for dispersion assume that the shapes are the same, but

possibly with different location and scale parameters.

This is the nonparametric analogue of assuming normal

distributions with possibly different means and variances.

To test equality of two medians against a shift alternative, use the Wilcoxon-Mann-Whitney test. This test

was introduced in Section 6.3.3.2.2. In the present

context, let W denote the sum of the ranks of times for

the first data subset, when all the times are considered

together. The ranks are the same whether or not the

logarithmic transformation is performed.

For example, to compare group P to group S in

Example 6.12, arrange all 70 times from the two

groups in ascending order, and mark the times

corresponding to group P. The smallest time from

group P is 6 minutes. This has rank 12, because it

is preceded by 11 values in group S from 2 to 5

minutes. The other ranks are found similarly. Ties

are handled by assigning the average rank to all tied

values. The rest of the test was explained in Section

6.3.3.2.2. It is not detailed here, because the test is

normally performed by a computer.

To test whether two or more data subsets can be

pooled, the test of choice is the Kruskal-Wallis test. It

tests whether the distribution of T is the same in all the

data subsets, against the alternative that the distributions have the same shape but different medians. The

test is based on a sum of ranks for each data subset.

Those who want details can look in Conover (1999) or

Hollander and Wolfe (1999); everyone else can just let

the computer do the test.

When the Kruskal-Wallis test is applied to all three

groups in the data of Example 6.12, it rejects equality

of the distributions with p-value 0.026. This is

consistent with the graphical comparison in Figure

6.47 — clear evidence of a difference, though not

extreme overwhelming evidence. Based on these

analyses, Atwood et al. (1998) dropped group P from

the analysis of durations, and combined groups S

and T. Group P consists of LOSP durations when

the plant remained at power throughout the event.

The authors comment on reasons why plant personnel might be very deliberate in restoring offsite power

while the plant is still up and running.

To test for equality of dispersion of two data subsets,

the rank-like test of Moses is recommended. This

requires splitting each data subset into two or more

parts, and so is not suitable for very small data sets.

See Hollander and Wolfe or documentation of a statistical software package for details for applying this test.

A well-known nonparametric test has not been developed for testing equality of dispersion of more than two

data subsets. Therefore, graphical comparisons, such as

side-by-side box plots, should be an important part of

the analysis.
Parameter Estimation and Model Validation

6-66

Nonparametric Test Based on EDFs. A well-known

test for comparing two data subsets is the two-sample

Kolmogorov-Smirnov test. It is based on comparing

the empirical distribution functions for the two data

sets. The test statistic is 

D Ft Gt = − max [| t $( ) $( )|]

where and are the empirical distribution $F t( ) $

G t( )

functions from the two data sets. Many software

packages can perform this test.

6.6.2.2 No Time Trend

This section will be illustrated by an extension of

Example 6.12, taken directly from Atwood et

al. (1998).

Based on the above type of analysis of Example

6.12, the LOSP study (Atwood et al. 1998) pooled

the data from groups S and T, but excluded group P.

That report also combined common-cause pairs of

events at multiple units into single site-events (one

pair of shutdown events, two pairs of trip events, and

two pairs that involved a shutdown reactor and a

reactor that tripped). This gave a total of 102 site

events instead of the 107 in Example 6.12. They are

sorted by event date and listed as Example 6.13.

Times are in minutes, and dates are MM/DD/YY.

6.6.2.2.1 Graphical Methods

One natural way to examine the data for a trend is

through a scatter plot of the observed values against

calendar time. Often, as in Example 6.13, a few large

values are outliers. They will determine the scale of the

vertical axis. Compared to those large values most of

the other values are very small, hugging the horizontal

axis. In such a case, the observed values should be

transformed, typically by taking logs.

Figure 6.48, from the LOSP study (Atwood et al.

1998), shows a plot of log10(recovery time), for the

data of Example 6.13. Visually, any trend in time

appears to be very slight. The section below, which

considers statistical tests, will re-examine this example.

A potentially more helpful plot is a cumulative plot of

recovery time against chronological sequence. The

vertical axis shows cumulative recovery time, that is,

cumulative duration of LOSP events. No logarithmic

transformation is made, because a sum of durations is

easy to interpret, but a sum of log(duration) is harder to

interpret. Also, logarithms can be negative, so a cumulative plot of logarithms would not necessarily be

monotone.

Example 6.13 LOSP recovery times and event

dates.

 4 04/22/80

 106 06/03/80

 62 07/15/80

 120 01/16/81

 14 02/01/81

 15 04/27/81

 10 12/23/81

 29 06/22/82

 17 04/26/83

 11 10/04/83

 163 10/08/83

 240 11/14/83

 97 01/08/84

 90 02/12/84

 15 02/16/84

 2 02/28/84

 5 05/03/84

 2 06/04/84

 120 06/05/84

 20 07/16/84

 11 07/26/84

 10 08/01/84

 20 08/21/84

 22 08/24/84

 3 10/22/84

 14 11/16/84

 15 12/19/84

 335 04/29/85

 43 05/07/85

 5 08/16/85

 73 08/28/85

 25 10/03/85

 13 10/07/85

 60 10/22/85

 3.5 11/21/85

 4 11/21/85

 20 12/12/85

 46 01/01/86

 100 01/28/86

 12 07/19/86

 155 03/05/87

 37 03/21/87

 4 04/04/87

 388 07/14/87

 118 07/23/87

 2 08/17/87

 53 09/11/87

 29 09/16/87

 17 10/14/87

 59 10/16/87

 4 11/17/87

 8 06/24/88

 38 07/17/88

 24 07/29/88

 14 08/13/88

 95 10/16/88

 19 10/25/88

 9 12/26/88

 45 03/25/89

 90 03/29/89

 29 05/14/89

 60 06/16/89

 90 06/17/89

 2 06/29/89

 45 01/16/90

 14 02/26/90

 140 03/20/90

 37 07/09/90

 40 02/11/91

 240 03/07/91

 67 03/13/91

 29 03/20/91

 60 03/21/91

 277 04/23/91

 24 06/15/91

 60 06/22/91

 20 06/27/91

 11 07/24/91

 4 10/20/91

 77 01/29/92

 20 03/23/92

 20 03/27/92

 35 04/02/92

 10 04/28/92

 6 05/03/92

 454 08/22/92

 57 10/19/92

 95 12/31/92

 136 04/08/93

 37 05/19/93

 12 06/22/93

 3 06/26/93

 10 09/10/93

 15 09/14/93

 12.5 10/12/93

 96 12/27/93

 2 05/21/94

1675 11/18/94

 132 02/27/95

 917 10/21/95

 127 01/20/96

 330 02/06/96

 1/1/80 1/1/84 1/1/88 1/1/92 1/1/96

Date

0

1

2

3

L

o

g

1

0

D

u

r

Figure 6.48 Log10(recovery time) plotted against

event date, for data from groups S and T in Example

6.13.

If the horizontal axis shows event date, the slope of the

curve represents average LOSP duration per calendar

time. If, instead, the horizontal axis shows event

sequence number, that is, the cumulative number of

events, then the slope represents average LOSP duration per event. The latter is more meaningful in a study

of durations.
Parameter Estimation and Model Validation

6-67

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 10 20 30 40 50 60 70 80 90 100 110

Cumulative number of events

Cumulative duration (min.)

GC99 0292 24

Figure 6.49 Cumulative duration of LOSP events

versus cumulative number of events.

Finally, a diagonal line, connecting the origin to the

final point, provides a reference guide, so that the eye

can better judge the straightness of the plot.

Figure 6.49 shows the cumulative duration plot for

the data of Example 6.13.

The cumulative plot clearly departs from the diagonal

straight line, because of two large duration times

near the right of the plot. The LOSP report mentions

that one of those two times is conservatively large.

The LER narrative states that recovery could have

been performed earlier, but it does not give an

estimated possible recovery time. The LOSP report

used times when recovery would have been possible, when such times were available, but for this

event the report was forced to use the actual recovery time.

In Figure 6.49, a second dashed line connects the

origin (0, 0) to the 97th point, just before the first of

the two large jumps. The cumulative plot stays close

to this line until the large recovery times occur.

Thus, any “trend” is the result, not of a gradual

increase in recovery time, but of a couple of outlying

values, one of which is conservatively large. Figures

6.48 and 6.49 both reveal the two large recovery

times. In this example, however, the cumulative plot

seems more informative than the scatter plot, because the log-transformation in Figure 6.48 makes

the large times appear less dramatic.

6.6.2.2.2 Statistical Tests

Test Based on Normality. Using the method of least

squares fitting, data from a scatter plot may be fitted

with a straight line. Most software packages then test

of the hypothesis that the slope is zero, assuming

normally distributed scatter around the line.

The cited LOSP report fitted a straight line to the

data in Figure 6.48 using the least squares method.

The trend was reported as statistically significant at

the 0.03 level.

This conclusion of a statistically significant trend

seems surprising, based on the minimal apparent

trend in the figure. The report authors did not have

the insights given by the cumulative plot, but they

critiqued the calculation in several ways:

• The calculation assumes that log(T) is normally

distributed around the trend line. The lognormal

distribution (without modeling a trend) was found

to fit the data well, and the scatter plot appears

consistent with normality. Therefore, the calculated p-value of 0.03 is close to correct.

• The evidence for the trend was very sensitive to

the two values in the upper right of the figure.

Dropping either value raised the p-value to 0.08.

Further, one of those values was known to be

conservatively high, as discussed above. This

means that the trend may in part be an artifact of

the data coding.

• The magnitude of the trend is small. A linear

trend in the mean of log(T) corresponds to an

exponential trend in the median of T. The magnitude of this trend is a factor of 3.6 over the 17

years of the study. This is fairly small from an

engineering viewpoint.

• No solid engineering reasons were found to

explain the trend.

Section 6.2.3.1.2 of this handbook discusses how test

results should be interpreted. It states that calculation

of a p-value is only part of the analysis, and should be

followed by critical thinking. The above bulleted list of

considerations illustrates that kind of thinking. Use of

a cumulative plot would have helped the report authors

even more, revealing that a smooth trend of any kind is

inappropriate. The authors of the LOSP study chose

not to model a trend, but recognized that additional data

might change this decision.

Nonparametric Test. A test for a trend that does not

assume normality is easy to construct. Such a test is

necessary if normality cannot be assumed. If normality

can be assumed, the nonparametric test is less powerful

for detecting a trend, because it ignores available information, that the data are normally distributed.
Parameter Estimation and Model Validation

6-68

lognorm (x, 3.388836, 1.434476)

Recovery time (min.)

Number of observations

0

20

40

60

80

100

120

0 200 400 600 800 1000 1200 1400 1600

Figure 6.50 Histogram of data from Table 6.19, with

multiple of lognormal density overlaid. The skewness

makes goodness of fit difficult to assess.

normal (x, 3.388836, 1.434476)

Ln(recovery time)

Number of observations

0

5

10

15

20

25

30

35

-1 0 1 2 3 4 5 6 7 8 9

Figure 6.51 Histogram of ln(time), with a multiple of

a normal density overlaid. Fit appears as good as

achievable without using a bimodal distribution.

The test is the Wilcoxon-Mann-Whitney test, first

introduced in Section 6.3.3.2.2. To apply it here,

arrange the times sequentially, in order of their event

dates. Count an event as A if it is above the median and

as B if it is below the median. Discard any values that

equal the median. Now carry out the Wilcoxon-MannWhitney test based on the ranks of the As in the sequence of all the events. Because this test is based only

on comparisons to the median, it is the same whether or

not logarithmic transformations are used.

When this was done with the data from Example

6.13, the median duration was 29. The first duration

in Example 6.13 was a B, the next three were A, and

so forth. In all, there were 48 As and 50 Bs. The As

had higher average rank than the Bs, suggesting an

upward trend, but the p-value was 0.09, not quite

statistically significant. The nonparametric test is not

as sensitive as the parametric test for detecting the

small trend, in part because it does not make as

much use of the two extreme values seen in Figure

6.49. If the normality assumption were not satisfied,

only the nonparametric test would be valid.

6.6.2.3 Goodness of Fit to Parametric Models

One way to model recovery times and other durations

is to model the distribution of the durations by some

parametric distribution, such as lognormal, Weibull,

etc. One must then check to see if the data fit this

proposed model well. This section gives graphical

methods and statistical tests for such investigations.

6.6.2.3.1 Graphical Methods

The basic idea is to compare nonparametric estimates,

which come directly from the data, with estimates based

on the fitted model under consideration. For example:

• Compare the histogram to the density from the

fitted model.

• Compare the EDF to the c.d.f. of the fitted parametric model. Equivalently, compare the empirical

reliability function (1 minus the EDF) to the fitted

reliability function.

• Compare the quantiles of the data to the quantiles

of the fitted distribution. This plot is called a

quantile-quantile plot, or a Q-Q plot. Q-Q plots

have become very popular for assessing goodness

of fit, although they take getting used to.

These three comparisons are illustrated below, using

the data of Example 6.13, and an assumed lognormal distribution. First, the fitted distribution is found

by taking natural logarithms of the recovery times,

and estimating the mean and variance of their

distribution. The estimated mean is 3.389 and the

estimated standard deviation is 1.434. The ln(time)

values are modeled as normally distributed with this

mean and variance. The raw times have the corresponding lognormal distribution.

Figure 6.50 shows the histogram density with a fitted

lognormal density overlaid. Because this distribution

is concentrated at small values, the goodness of fit is

difficult to judge. Therefore, the histogram of the

ln(time) values are also plotted, with a normal density

overlaid, in Figure 6.51. Actually, the area under the

histogram equals the number of observations, and

the density has been rescaled to have the same

area.

Figure 6.52, from the LOSP report, shows a plot of

the reliability function, 1 minus the EDF, with the

corresponding fitted function, 1 minus the lognormal

c.d.f. The plot in this form is useful for interpreting

the degree to which the fitted c.d.f. differs from the

empirical c.d.f., because the horizontal axis is in

units of time. A plot in terms of log(time) would not

hug the axes so closely. Therefore, discrepancies
Parameter Estimation and Model Validation

6-69

Theoretical normal quantile

Observed recovery time

.01 .05 .1 .25 .5 .75 .9 .95 .99

-500

0

500

1000

1500

2000

-3 -2 -1 0 1 2 3

Figure 6.54 Quantile-quantile plot of raw recovery

times against fitted normal distribution. The strong

curvature indicates bad fit.

between the curves would be more visible, but their

magnitudes would be harder to interpret in real-world

terms.

Figure 6.52 Empirical and theoretical reliability

functions, where the reliability function is defined as

1 minus the c.d.f.

Finally, Figure 6.53 gives a quantile-quantile (Q-Q)

plot, described by Wilk and Gnanadesikan (1968). If

only one plot could be used, a Q-Q plot would be a

strong contender. A Q-Q plot compares two distributions by plotting the quantiles of one against the corresponding quantiles of the other. If X is a linear function

of Y, X = a + bY, then a Q-Q plot of X versus Y will be

linear. The parameters a and b do not need to be

known or estimated; linearity of the plot tells the analyst

whether the two distributions are the same except for a

linear transformation. Users of probability paper will

recognize that a plot on probability paper is a form of a

Q-Q plot. 

Theoretical normal quantile

Observed ln(recovery time)

.01 .05 .1 .25 .5 .75 .9 .95 .99

0

1

2

3

4

5

6

7

8

9

-3 -2 -1 0 1 2 3

Figure 6.53 Quantile-quantile plot of ln(recovery

time) and fitted normal distribution. The points fall

nearly on a straight line, indicating good fit.

In Figure 6.53, the software package implemented

the Q-Q plot by plotting the ordered values of ln(time)

against the theoretical expected values of the corresponding order statistics assuming normality. For

example, denote ln(t) by y. In the implementation of

this particular software package, the ith ordered

value, y(i)

, is plotted against the expected value of Z(i),

assuming that 102 values of Z are randomly sampled

from a standard normal distribution. The points

follow a straight line. This indicates that the data are,

apparently, normally distributed.

The parameters, : and F, can be ignored in a Q-Q plot

based on the normal distribution, because a normal

random variable Y with mean : and standard deviation

F is related to Z by Y = : + FZ. This is a linear transformation, and so does not change the linearity or

nonlinearity of the plot. In fact, it is not even necessary

to obtain estimates of : and F. For distributions other

than normal, the parameters may need to be estimated

before the Q-Q plot can be constructed.

The expected values of the order statistics cannot be

constructed without tables or a computer program.

Users of probability paper may construct a simpler

version, plotting y(i)

 against the i/(n+1) quantile of a

standard normal distribution. Here n is the total number

of observations, 102 in the present example. This

simpler version gave its name to the plot, a quantilequantile plot.

For the purpose of illustration, Figure 6.54 gives a QQ plot of the same example data, assuming that the

raw recovery times have a normal distribution. Of

course the fit is horrible — no one expects the raw

times to have a normal distribution. This lack of fit is

shown by strong curvature in the plot. The two

largest times show the lack of fit most emphatically,

but even without them the plot appears to show a

curvature that indicates non-normality.
Parameter Estimation and Model Validation

6-70

t

(i)

-ln[1 - i/(n+1)]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 50 100 150 200 250 300 350 400

Figure 6.55 Q-Q plot for checking exponential

distribution in Example 6.6.

The particular form of the distribution can sometimes

allow special tricks. Let us leave the present example,

and consider investigating whether data t1, ..., tn come

from an exponential distribution. Example 6.6, which

was deferred from Section 6.2.3.4, will be used to

illustrate the method.

The idea of the Q-Q plot is that, when the data come

from the assumed distribution, then 

t(i) . F!1[i/(n+1)],

where F!1

 is the inverse of the assumed c.d.f. Let us

find the inverse of the exponential c.d.f. Set

y = F(t) = 1 ! e!8t .

To find the inverse, solve for t = F!1

(y):

e!8t = 1 ! y

and therefore

t = !ln(1 ! y)/8 .

The right-hand side is F!1(y), so the defining relation of

the Q-Q plot is 

t(i) . !ln[1 ! i/(n+1)]/8 .

Thus, a plot of the ordered times against !ln[1 !

i/(n+1)] should be approximately linear, regardless of

the value of 8. The linearity or nonlinearity of the plot

does not depend on whether 8 has been estimated well.

Nonlinearity is evidence against the assumed exponential distribution. 

Example 6.6 contains times between LOSP events,

which should be exponentially distributed. A plot of

the ordered times against !ln[1 ! i/(n+1)] is shown in

Figure 6.55. Because the plot does not show much

curvature, it indicates good fit to the exponential

distribution.

6.6.2.3.2 Statistical Tests

The previous section used graphs to investigate whether

data followed a certain kind of distribution. The

present section gives statistical tests of hypotheses, for

investigating the same question. The tests here are

called goodness-of-fit tests, because they are intended

to test whether the data fit the assumed model well.

The null hypothesis is that the data come from a distribution of the assumed form, for example, from a

lognormal distribution. The null hypothesis does not

specify the parameters. Therefore, the null hypothesis

includes a family of distributions. The alternative

hypothesis is that the data come from some other

distribution.

As always, remember that “acceptance” of the null

hypothesis does not provide evidence that the null

hypothesis is true. It merely indicates a lack of evidence that the null hypothesis is false. For example, the

data may be consistent with a lognormal distribution,

and also consistent with a gamma distribution and a

Weibull distribution. In such a case, the analyst should

not make assertions that are highly dependent on the

form of the distribution. For example, a sample of 10

observations may be consistent with many possible

distributions. An estimate of the 99.9th percentile of

the distribution would be a large extrapolation from the

actual data, highly dependent on the assumed form of

the distribution. A confidence interval on this percentile would be even worse, because it would give an

appearance of quantified precision, when in reality the

distribution could have practically any form out in the

tail.

In summary, even though a model has been “accepted,”

it is only an approximation. The analyst should not

make assertions that are sensitive to small departures

from the model.

Chi-Squared Test. The chi-squared test, seen in

Sections 6.2 and 6.3, is also an all-purpose goodnessof-fit test. To apply it in the present context, estimate

any unknown parameters of the hypothesized distribution of T. Based on these parameter estimates, divide

the time axis into c bins of equal probability. The letter

c stands for cell, another term for a bin in this context.

Based on the recommendations of Moore (1986),

choose the number of bins so that n/c is at least 1, and

preferably at least 2. Let xi

 be the observed number of
Parameter Estimation and Model Validation

6-71

values of T in the ith bin. Because the bins have equal

probability, the expected number of values of T that will

fall in any bin is n/c, the number of observations

divided by the number of bins. The Pearson chisquared statistic is 

X2 = Ej(xj ! ej)

2

/ej ,

where each ej

 equals n/c and each xj is an observed

count.

If the null hypothesis is true, the distribution of X2

 is

approximately chi-squared. The commonly quoted rule

is that the degrees of freedom is c ! 1 ! p, where p is

the number of estimated parameters. For example,

suppose the null hypothesis is that the distribution of T

is lognormal, or equivalently, that ln(T) is normal.

Then two parameters must be estimated, : and F. Thus,

the commonly quoted rule for the degrees of freedom is

c ! 3. In fact, researchers have found that this is not

quite correct, for subtle reasons described by Moore

(1986, Section 3.2.2.1). The correct degrees of freedom are somewhere between c ! 1 ! p and c ! 1. The

exact value depends on the form of the distribution in

the null hypothesis.

Let us apply this to the LOSP-recovery data from

Example 6.13, and use X = ln(T) for convenience.

Let H0 be the hypothesis that X is normally distributed. As mentioned above, the estimates of : and

F are 3.389 and 1.434. With 102 observations, it is

convenient to take 50 bins, so that each expected

count is 102/50 = 2.04. The bin boundaries are the

0.02, 0.04, ..., 0.98 quantiles of the distribution.

These are estimated as

yq = 3.389 + 1.434zq ,

where q is 0.02, 0.04, etc., and zq is a quantile

interpolated from a table of the standard normal

distribution. For example, z0.02 = !2.054.

When this is carried out, using a computer to perform

the calculations, the value of X2

 is 63.69. The

distribution under H0 is chi-squared with degrees of

freedom between 47 and 49. Therefore, the p-value

is between 0.053 and 0.077. The test almost rejects

normality of ln(T) at the 0.05 level, in spite of the

graphical evidence to the contrary.

Upon examination, the test is revealed to be too

powerful for its own good. It notices that the values

tend to cluster, five occurrences of 2 minutes, six

values of 20 minutes but no values of 21 minutes,

etc. With 50 cells, each observed time is commonly

the sole occupant of a cell. The test notices that the

numbers have been rounded to convenient times,

such as 20 minutes, and uses this as evidence

against normality. In fact, such clustering is a departure from normality, and from any other continuous

distribution. But it is not the kind of departure that is

of interest to most analysts.

A coarser binning, into fewer cells, would not be

distracted by fine clustering, and would search for

more global departures from the null hypothesis.

We conclude this discussion of the chi-squared test by

considering again the exponential example that was

deferred from Section 6.2.3.4.

Example 6.6 consists of 25 times. The null hypothesis is that the data come from an exponential distribution. The unknown 8 is estimated as the number

of events divided by the total observation period,

25/(2192 days) = 0.0114 events per day. This MLE

is justified based on the Poisson count of events, as

in Section 6.2.1.1. To obtain a moderate expected

count in each bin, let us use ten bins. They have

equal estimated probabilities, 0.10 each, if they run

from

0 days to [!ln(0.9)]/0.0114 = 9.24 days

9.24 days to [!ln(0.8)]/0.0114 = 19.57 days

...

201.89 days to infinity.

These calculations are all based on the exponential

c.d.f., F(t) = 1 - exp(!8t). Setting F(t) to 0.1, 0.2, and

so forth gives the bin boundaries.

There are four observed times in the first bin, two in

the second, and so on. The expected count in each

bin is 25/10 = 2.5. The calculated value of X 2 is

9.00. This must be compared with the percentiles of

the chi-squared distribution. There are c = 10 bins,

and p = 1 estimated parameter. Therefore, the

degrees of freedom are between 10 ! 1 = 9 and 10

! 2 = 8. The value 9.00 is in the middle of both of

these distributions, the 56th percentile of one and the

66th percentile of the other. Therefore, the chisquared test finds no evidence against the exponential distribution. This agrees with the earlier graphical

analysis.

Shapiro-Wilk Test for Normality. Many software

packages offer the Shapiro-Wilk test for normality. It

is based on obseving how closely the order statistics

follow theoretical normal values, as displayed for

example in Figure 6.53. For testing the normal distribution, the Shapiro-Wilk test is one of the most powerful

tests against a wide variety of alternatives. Details are

not given here, because all the calculations are carried

out by the computer.
Parameter Estimation and Model Validation

6-72

With the logarithms of the data of Example 6.13, the

Shapiro-Wilk test does not reject normality of ln(T),

giving a p-value of 0.34. This agrees with the visual

evidence of Figure 6.53.

Tests Based on the EDF. Several families of tests

have been proposed based on the empirical distribution

function (EDF, defined in Section 6.6.1.1.3). The idea

is to reject the null hypothesis if the EDF is not “close

to” the theoretical c.d.f. Closeness can be measured in

various ways, giving rise to a variety of tests. EDFbased tests are appealing because they do not require a

choice of bins, but simply use the data as they come.

The most famous such test is the Kolmogorov test, also

known as the Kolmogorov-Smirnov test. It is described

in Appendix B.3.4. This test differs from the similarlynamed test in Section 6.6.2.1.2 because the present test

asks whether a random variable has a certain distribution, and the earlier test asks if two random variables

have the same distribution. These are slightly different

questions. The test here rejects H0 if

max | $

Ft Ft ( ) ( )| −

is large, where the maximum is over all values of t.

Here, any unknown parameters in F must be estimated;

the effect of this estimation is typically ignored.

When SAS (SAS Version 8, 2000) performs the

Kolmogorov test of lognormality on the times in

Example 6.13, it gives a p-value > 0.15. That is, it

does not calculate the exact p-value, but it does

report that the departure from lognormality is not

statistically significant.

The Cramér-von Mises test and the Anderson-Darling

test are other EDF-based tests, designed to remedy

perceived weaknesses in the Kolmogorov test. The

Cramér-von Mises test is based on

[ . $F t F t f t dt ( ) ( )] ( ) − ∫ 2

Here, F is the distribution that is assumed under the null

hypothesis, and f is the corresponding density. Thus,

the Kolmogorov test looks at the maximum difference

between and F, while the Cramér-von Mises test F$

looks at an average squared difference. The AndersonDarling test is based on 

{ } . [ $

F t F t F t F t dt ( ) ( )] / { ( )[ ( )]} − − ∫ 2 1

This division by F(t)[1 ! F(t)] gives greater weight to

the tails of the distribution, where departures from F is

most likely to occur. Thus, this test is intended to be

more powerful than the Cramér-von Mises test against

common alternative hypotheses. Many statistical

packages perform one or more of these tests.

When testing lognormality of the data in Example

6.13, SAS reports a p-value of >0.25 for the Cramérvon Mises test and also for the Anderson-Darling

test. Just as for the Kolmogorov test, SAS does not

compute the exact p-value, but it does report that the

departure from lognormality is not statistically significant.

6.6.2.4 Consistency of Data with Prior in

Bayesian Parametric Estimation

The issue here is whether the data are consistent with an

assumed informative prior distribution for the unknown

parameters. If a noninformative prior distribution is

used, then the question does not arise, because the

noninformative distribution is supposed to be consistent

with anything.

6.6.2.4.1 Exponential Durations

A quantitative approach is possible when T has an

exponential(8) distribution. In this case all the information of interest about 8 is contained in Gti

, as seen in

Section 6.6.1.2.2. Therefore, we can compare Gti

 to

what would be expected based on prior belief about 8.

If Gti

 is surprisingly large or surprisingly small, that is,

if Gti

 is in either tail of the distribution of GTi, then the

prior distribution is questionable. The value Gti

 is in

the lower tail if Pr(GTi

 < Gti) is a small probability, and

in the upper tail if Pr(GTi

 > Gti) is a small. To be

specific, consider the upper tail. The relevant probability is 

Pr(GTi

 > Gti) = IPr(GTi > Gti | 8) fprior(8) d8 . (6.25)

The inner conditional probability can be evaluated by

using the fact that the distribution of GTi

 , given 8, is

gamma(n, 8). If the prior distribution of 8 is not conjugate, the integral in Equation 6.25 must be evaluated

numerically, just as in Sections 6.2.3.5 and 6.3.3.4:

either (a) compute the integral using numerical integration, or (b) generate a random sample of 8 values from

the prior distribution, find Pr(GTi

 > Gti | 8) for each

such 8, and find the average of these probabilities as the

overall probability. 

Treatment of the lower tail follows the same pattern.
Parameter Estimation and Model Validation

6-73

If the prior distribution of 8 is conjugate, that is,

gamma(", $), then Equation 6.25 simplifies. By

working out the integrals it can be shown that GTi

/(GTi

+ $) has a beta(n, ") distribution. Equivalently, $/(GTi

+ $) has a beta(", n) distribution. These are marginal

distributions corresponding to Equation 6.25, from

which 8 has been integrated out. Therefore, if Gti

 /(Gti

+$) is in either extreme tail of a beta(n, ") distribution,

or equivalently, if $ /(Gti

 +$) is in either extreme tail of

a beta(", n) distribution, then the gamma(", $) prior

distribution is questioned.

In Example 6.12, suppose that the only events of

interest are those in the group of S (shutdown)

events. Suppose also that the times are assumed to

be exponential(8) – the realism of that assumption is

not the subject of the present investigation. Finally,

suppose that 8 is assigned a gamma(2, 30) prior

distribution, roughly equivalent to two prior observed

times with total duration of 30 minutes. The shape

parameter of only 2 means that the prior is not very

informative, so we expect the data to be consistent

with it, unless 30 minutes is very unrealistic.

From Table 6.15, we find n = 62 and the total of the

durations is 62×92.3 = 5722.6. The beta tables in

Appendix C assume that the first beta parameter is

smaller than the second, so it is convenient to work

with the beta(2, 62) distribution rather than the

beta(62, 2) distribution. Therefore, we ask if

30/(5722.6 + 30) = 5.2E!3 

is in either tail of a beta(2, 62) distribution. Table C.5

shows that the 5th percentile of the beta(2, 62)

distribution is roughly 6E!3 (it is an interpolation of

7.01E!3 and 3.53E!3 in the table). Table C.6 shows

that the 2.5th percentile is roughly 4E!3. So the

observed value is somewhere between the 2.5th and

5th percentiles of the predictive distribution. This

means that the prior may need rethinking. It should

either be modified or it should be justified more

carefully. (In the present example the prior came out

of thin air, but the real difficulty is that the durations

are not really exponential – the whole exercise is

only for illustration.)

6.6.2.4.2 Distributions Having Two or More 

Parameters

When the topic of comparing the data to the prior arose

in connection with estimating 8 or p, there was a single

parameter of interest, and a single observed random

variable that contained all the information of interest for

that parameter. This random variable was the total

count of initiating events, the count of failures on

demand, or, in the previous section, the total duration.

However, the present subsection considers a distribution with (at least) two parameters, such as : and F or

" and $. No single random variable contains all the

information of interest. Therefore, in such cases it is

simplest to compare the data with the prior by constructing:

1. a prior credible region for the two parameters, and

2. a posterior credible region based on noninformative priors.

The first case shows what the prior distribution says,

and the second case shows what the data say. Compare

the answers from 1 and 2 to see if the prior distribution

and the data seem consistent, that is, if the prior region

contains most of the posterior region. Instead of twodimensional credible regions, one might calculate

credible intervals for the individual parameters. This is

simpler, but ignores the possible correlation of the two

parameters. Because this is such an advanced topic, no

examples are worked out here.

6.6.3 Nonparametric Density Estimation

The most prevalent methods of estimating a density

function are parametric methods. As described in

Section 6.6.1.2, the density is specified in terms of a

functional form, such as lognormal or Weibull, with

unknown parameters. The parameters are then estimated from the data. However, there also exist

nonparametric methods for estimation of a density

function, some of which are described here. 

The simplest and best known method of estimating a

density function is to construct a frequency table, and

then to plot the histogram. This method was discussed

in Section 6.6.1.1.4. Two illustrations are given there,

Figures 6.43 and 6.44. Both use the 45 recovery times

from part T of Example 6.12. The methods discussed

below are illustrated with the same set of 45 recovery

times.

6.6.3.1 Smoothing Techniques and Kernel 

Estimators

 

Smoothing techniques can be motivated by recalling

that the density function, f(t), is the derivative of the

c.d.f., F(t). The EDF, discussed in Section 6.6.1.1.3

and denoted by , is a natural estimator of F(t). $

F t( )

Thus, a natural estimator of the density is the differential quotient using the EDF in place of the c.d.f.,

, (6.26) $ ( )

$( ) $( ) f t

Ft h Ft h

h n = +− −

2
Parameter Estimation and Model Validation

6-74

Figure 6.56 Density estimate of the data from group

T in Example 6.12, with rectangular kernel and

bandwidth 25.

Figure 6.57 Density estimate of the data from group

T in Example 6.12 with rectangular kernel and

bandwidth 50.

where h is an increment of the variable t. The main

problem in applying such an estimator is to choose h

small enough so that the differential quotient adequately

approximates the derivative, but large enough so that

the interval with limits t ± h contains a sufficient

amount of data.

Recall that equals the number of observations $

F t( )

having a value less than or equal to t divided by the

total number of observations, n. Therefore, Equation 6.26 can also be written as

, (6.27) $

f t( ) nhK

t t

h n

i

i

n

= − ∑ ⎛

⎝

⎜ ⎞

⎠

⎟ =

1

1

where K is a function defined as K(u) = 1/2 if u is

between ± 1, and zero otherwise, and ti

 is the ith observation. Notice that an observation ti

 only enters into

this calculation if (ti ! t)/h is between ± 1, or in other

words if ti

 is near t; specifically if ti is within h units of

t. Thus, the estimate is based on averaging values of

1/2 when observations are near t. This is a special case

of a general type of estimator known as a kernel

density estimator. The function K(u) is called the

kernel and the increment h is called the bandwidth.

The bandwidth defines a “window”, centered at t and

having width 2h, which contains the data involved in

the estimate at the point t.

6.6.3.1.1 The Rectangular Kernel

When graphed, the kernel corresponding to Equation 6.27 is a rectangle of height 1/2 and width 2h. The

resulting estimator is illustrated here with group T of

Example 6.12 and two bandwidths.

Figure 6.56 shows a graph of the estimate of the

density when the bandwidth is h = 25 minutes.

Notice that the estimated density is zero in the

interval roughly from 150 to 250 minutes. This

corresponds to the fourth and fifth bins of the histogram of Figure 6.43, both of which were empty.

It is also evident that the graph is somewhat jagged,

indicating that the bandwidth may be so small that

not enough data are being captured in the window. 

The vertical dashed line marks the point t = 0, to be

discussed later.

Consider now a rectangular kernel estimate with the

same data but with a larger bandwidth, h = 50

minutes. The results are shown in Figure 6.57.

There is still some jaggedness, but it is somewhat

less than in Figure 6.56. There is still a noticeable

low point in the vicinity of 200 minutes, but it is

narrower than in Figure 6.56. 

It is clear that by smoothing over a very wide window,

any features can be smoothed out. For this reason, it is

desirable to give some thought to whether there is some

explanation for low density. In other words, are these

real effects or are they just due to randomness? If the

low estimates can be explained by something other than

random fluctuation, smoothing would tend to hide this

fact, but if they are due to randomness, then smoothing

should be helpful.
Parameter Estimation and Model Validation

6-75

Figure 6.58 Density estimate from group T of

Example 6.13, with rectangular kernel and bandwidth

50, forced to be nonzero on positive axis only.

This issue was also seen with histograms. Choosing too

narrow bins for the size of the data set caused the shape

to be influenced too much by random variation. Choosing too wide bins smoothed out nearly all the variation.

The question of how much to smooth and how much

roughness to allow is inherent in all forms of density

estimation.

6.6.3.1.2 Boundary Problems

Notice, that as the bandwidth is increased, the interval

over which the estimated density is positive becomes

wider. This is because the window is picking up more

data as it gets wider. This causes the anomaly that the

estimated density is positive over negative values of the

t axis, even though t represents a positive variable,

namely recovery time. The vertical dashed line marks

the point t = 0 in each figure, and the portion of the

density to the left of this line is substantial. In addition,

although many values of ti

 are close to zero, the density

estimate decreases as t moves leftward to zero. Various

methods have been suggested for correcting the estimate at the boundary of the possible region.

Silverman (1986) gives a method that is very easy to

implement. If the density is allowed to be positive only

for t $ 0, augment the data by reflecting it around 0.

That is, create a new data set that consists of

{..., !t2, !t1, t1, t2, ... } .

Estimate the density based on this data set. Call this

estimate . The integral from !4 to 4 of is ~

f t( ) ~f t( )

1.0, because is a density. Also, if the kernel is a ~

f

symmetrical function, then is symmetrical around ~

f

zero, that is, . Now, define the real ~

( ) ~ f t ft − = ( )

density estimate by

for t < 0 $

f t( ) = 0

for t $ 0 . $( ) ~ ft ft = 2 ( )

Then, is a density that is zero for negative f

$ t and

nonnegative for positive t. It estimates the unknown

true density.

Figure 6.58 shows the resulting estimate with the

data of this section, when the kernel is rectangular

and the bandwidth h = 50. This estimate can be

compared with Figure 6.57.

For large t, this estimate is very similar to that of

Figure 6.57. However, it is quite different for t near

zero. The density is not plotted for t < 0, but it equals

zero there.

The simple method just given forces the density estimate to have slope zero at the boundary. Those who

want to allow a density estimate with nonzero slope at

the boundary can see Hart (1997, Sec. 2.5). Technically, Hart’s book deals with smoothing a scatter plot,

but the method given there can be adapted as follows to

smoothing a density estimate: construct a rough histogram density estimate, place a dot at the top of each

histogram bar (including the bars with height zero!),

and treat those dots as a scatter plot.

6.6.3.1.3 The Triangular Kernel

It may also be desirable in some cases to give less

weight to the data in the extremes of the window and to

produce a smoother graph. This can be accomplished

by choosing a different function for the kernel. A very

simple one which does this is the function K(u) = 1!|u|

if u is between ± 1, and zero otherwise. The graph of

K(u) is an isosceles triangle with a base two units in

width. This kernel gives more weight to the data in the

middle of the window and less to data at the sides of the

window. It is also possible, by choosing a kernel function with a smoother graph, to produce a kernel estimate which is also smoother. The normal kernel, given

next, is such a smooth kernel.
Parameter Estimation and Model Validation

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Figure 6.59 Density estimate of the data from group

T in Example 6.12, with standard normal kernel and

bandwidth 25.

Figure 6.60 Density estimate of the data from group

T in Example 6.12, with standard normal kernel and

bandwidth 50.

6.6.3.1.4 The Standard Normal Kernel

A kernel function that is often used is the standard

normal kernel, equal to the standard normal p.d.f.,

which is given in Appendix A.7.2.

Figure 6.59 shows the density estimate for the same

recovery time data, but using the standard normal

kernel and bandwidth 25. The density has been

made positive on the positive time axis only, by the

technique of Section 6.6.3.1.2.

The resulting plot is clearly much smoother than the

ones obtained using the rectangular kernel. The

increased smoothness is provided by the standard

normal kernel, which is differentiable everywhere.

The low estimate of density near 200 is still present,

but the low spot does not drop to zero as it did in

Figure 6.58. This is because the standard normal

kernel is always positive. Even though this kernel

gives less weight to data which are farther from the

center of the kernel, it makes use of every observation in the data set. Consequently, with the standard

normal kernel, all terms in the density estimate of

Equation 6.27 are positive, although the extreme

ones will tend to be relatively small.

For the sake of comparison, Figure 6.60 shows the

standard normal kernel estimates for bandwidth h =

50.

Although the graphs shown in Figures 6.59 and 6.60

retain some general features of the graphs in Figures

6.56 through 6.58, they are somewhat smoother. As

mentioned in the case of the rectangular kernel in

Section 6.6.3.1.1, this type of smoothing is desirable if

the sparsity of data in these intervals is due to randomness, but possibly undesirable if there is an explanation

for the sparseness.

6.6.3.2 Choosing the Bandwidth

General guidelines for choosing a kernel and bandwidth

are difficult to formulate. The choice of a bandwidth

always involves a trade-off between bias and variability. An attempt to reduce bias generally requires a

small bandwidth, but this tends to result in a large

variance. On the other hand, choosing a large bandwidth will reduce the variance, but at the expense of

increasing the bias. A criterion which accounts for

both the bias and variance is based on a quantity called

the mean squared error (MSE) which is equal to the

mean squared difference between the unknown parameter and its estimator. It is easy to show that 

MSE = (bias)2

 + variance of estimator

so that as the MSE approaches zero, both the bias and

the variance of the estimator also approach zero.

A reasonable choice of bandwidth should take into

account the amount of data, and so the solution must

depend on n. Thus, we consider a sequence, h = h(n).

The sequence should converge to zero, but not too

quickly or too slowly. It is known, for example, that

under certain fairly modest assumptions, a desirable

form for the bandwidth is

h(n) = cn!1 / 5 .
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The main problem is that calculation of the constant c

requires more than is typically known about the p.d.f. to

be estimated, and it also depends on the choice of a

kernel. For example, according to p. 45 of Silverman

(1986), for the standard normal kernel, and assuming

the distribution of the data to be normal with standard

deviation F, the bandwidth which minimizes the integrated MSE asymptotically is 

h(n) = 1.06Fn!1 / 5 .

Notice that the constant c in this case requires that the

standard deviation be known or at least estimated.

For example, with the recovery time data, the sample

standard deviation, given in Table 6.16, is 99.9

minutes. If this is used to estimate F, then the

optimal bandwidth is h(n) = 105.9n!1/5. Using the

sample size n = 45 yields h = 49.5. This result is

very nearly the bandwidth of 50 minutes that was

used in Figure 6.60.

Keep in mind that this choice of bandwidth was derived

for the case where both the distribution being estimated

and the kernel are normal, so the result would be good

with these assumptions. However, this might be a good

place to start if trial and error is used to determine what

bandwidth to use. In other words, if it is not clear what

to assume, then it would be best to try a few different

bandwidths and choose one which provides some

smoothing, but does not obscure basic features. As

Silverman says, “There is a case for undersmoothing

somewhat; the reader can do further smoothing ‘by eye’

but cannot easily unsmooth.”

Another problem that often occurs in practice is that the

data will be plentiful in some parts of the range, but

sparse in others. This is typical with data from highly

skewed distributions. For example, with a positively

skewed distribution, such as any of the distributions in

Section 6.6.1.2, there will tend to be more data in the

lower end than in the upper tail. This would suggest the

desirability of having a bandwidth that varies with t, so

that a shorter increment can be used in the lower end

where the data points are more abundant, and a larger

increment used in the upper tail where there are not as

many points. This idea is not developed here, but such

methods exist. For additional reading on this topic, see

the discussions of the nearest neighbor method and the

variable kernel method in Silverman (1986). 

6.7 Unavailability

Data analysis methods for unavailability do not have the

long history that they have for other parameters. Most

of the material here is taken directly from a paper by

Atwood and Engelhardt (2003). Users of the methods

described below should refer to the final peer-reviewed

and published version of the article.

As in Section 2.6, the discussion here is presented in

terms of trains, although other hardware configurations,

such as individual components, could be considered.

The terminology of Section 2.6 will be used: outage

times are random and the unavailability is a parameter,

an unknown constant, denoted here by q. Subscripts,

such as “planned” and “unplanned,” can be attached to

q for clarity if needed.

Two possible kinds of data are considered:

• Detailed data: the onset time and duration of each

individual outage are recorded, as well as the total

time when the train should have been available.

• Summary data: the history is grouped into reporting periods, such as calendar months, and the total

outage time and total exposure time are recorded

for each reporting period.

The methods of this section are illustrated with

Example 2.16, taken from Atwood and Engelhardt

(2003). This example concerns the unavailability of

a train in a two-train CVC system. The 21 outages

are given in Table 6.17, and monthly summary data

are shown in Table 6.18. All times are in hours.

6.7.1 Analysis of Detailed Data

Denote the mean duration of an outage by MTTR, an

acronym for mean time to repair. Similarly, denote

the mean duration of the period between outages (the up

time) by MTTF, an acronym for mean time to failure.

Both of these can be estimated when the individual

outage times and exposure times are recorded.

Under the model assumptions given in Section 2.6.2, it

can be shown (see Ross 1983, pp. 66-67) that the

unavailability equals

q . (6.28)

MTTR

MTTF MTTR = +

This provides a basis for estimating availability and

unavailability when the full duration data are recorded.

Upon reflection, this striking result is not surprising. If

the data consist of n up times, with n down times

interspersed, MTTR would be estimated by (total down

time)/n, and MTTF would be estimated by (total up

time)/n. Based on Equation 6.28, the natural estimate

of unavailability would be
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[down-time/n]/[up-time/n + down-time/n]

= down-time/(up-time + down-time) ,

which is just the observed fraction of time when the

system is down.

Table 6.17 Detailed data for Example 2.16.

Month

Train 

Exposure

Train 1 

Outages

Train 2 

Outages

1 364 0 0

2 720 25.23 24.88

75.08

3 744 0 0

4 711 23.45 0

5 621 9.75

0.49

1.24

15.15

1.02

0.49

6 502 0.34

2.90

4.43

4.43

0.34

7 0 0 0

8 637 18.02 12.47

9.48

18.02

9 676 0 0

10 595 0 0

11 600 11.05 0

12 546 0 0

13 745 0 52.25

14 720 0 0

15 744 0 0

Total 8,925 96.90 213.61

In addition, MTTF + MTTR is the mean time from one

outage onset to the next, which can be interpreted as the

reciprocal of the outage frequency. Therefore, the

unavailability can be rewritten as

q = (outage frequency)

× (mean outage duration) . (6.29)

This equation is the foundation of all the methods for

using detailed data to estimate q.

6.7.1.1 Frequentist Point Estimate

Suppose that n outages are observed in exposure time

texpos, with total outage duration tdur. Based on Equation

6.29, the simple frequentist point estimate of q is

(n/texpos)×(tdur/n) = tdur/texpos .

Table 6.18 Summary data for Example 2.16.

Month

Train 

Exposure

Train 1 

Outage

Time

Train 2 

Outage

Time

Total 

Outage

Time

1 364 0 0 0

2 720 25.23 99.96 125.19

3 744 0 0 0

4 711 23.45 0 23.45

5 621 11.48 16.66 28.14

6 502 7.67 4.77 12.44

7 0 0 0 0

8 637 18.02 39.97 57.99

9 676 0 0 0

10 595 0 0 0

11 600 11.05 0 11.05

12 546 0 0 0

13 745 0 52.25 52.25

14 720 0 0 0

15 744 0 0 0

Total 8,925 96.90 213.61 310.51

This is the moment estimate, replacing the means in

Equation 6.28 by the sample means. It is a natural,

intuitively appealing point estimate.

For the data of Example 2.16, either Table 6.17 or

6.18 shows that the estimate of q is 310.51/17850 =

1.74E!2. (Note, the exposure time of 8925 hours for

a single train must be doubled to get the total trainexposure-hours for the full data set.)

A confidence interval is not easy to construct. Therefore, we leave frequentist estimation and proceed at

once to Bayesian estimation.

6.7.1.2 Bayesian Estimation under Simple

Assumptions

The simplest assumptions are that:

• the outages occur according to a Poisson process

with frequency 8freq, and

• the outage durations have an exponential(8dur)

distribution.

The first assumption is essentially true if the up-times

have an exponential distribution and MTTF >> MTTR.

Then the time from one outage onset to the next is

dominated by the time up, which is exponentially
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6-79

0 20 40 60 80

t

(i)

0

1

2

3

-ln[1 - i/(n+1)]

Figure 6.61 Q-Q plot for checking whether durations

have exponential distribution in Example 2.16.

distributed. The parameter 8dur can be interpreted as the

reciprocal of the mean outage duration. Here, both

parameters have units 1/time. 

The conjugate prior for 8freq can be denoted

gamma("freq,0, $freq,0), as stated in Section 6.2.2.4.1. The

conjugate prior for 8dur can be denoted gamma("dur,0,

$dur,0), as stated in Section 6.6.1.2.2. As above, suppose

that n outages are observed in exposure time texpos, with

total outage duration tdur. The posterior distributions

are

8freq ~ gamma("freq,1, $freq,1)

8dur ~ gamma("dur,1, $dur,1) ,

where

"freq,1 = "freq,0 + n

$freq,1 = $freq,0 + texpos

"dur,1 = "dur,0 + n

$dur,1 = $dur,0 + tdur .

By Equation 6.29 we have

q = 8freq/8dur ,

which is proportional to the ratio of two chi-squared

variables, because the chi-squared distribution is a reexpression of the gamma distribution. In addition, it is

reasonable to think that the posterior distributions are

independent; that is, if the outage frequency were to

increase or decrease, this would provide no information

about whether the mean outage duration increases or

decreases. Therefore, q is proportional to the ratio of

two independent chi-squared variables. However, the

ratio of two independent chi-squared distributions, each

divided by its degrees of freedom, has an F distribution,

as is shown in many books on statistics and stated in

Appendix A.7.11. It follows that q is distributed as

($dur,1/$freq,1)("freq,1/"dur,1)F(2"freq,1, 2"dur,1) .

The two quantities in parentheses following the F are

the two parameters of the distribution. Selected percentiles of the F distribution are tabulated in many statistics

books. They are not tabulated in this handbook because

they are calculated by many software packages, including spreadsheets such as Microsoft Excel (2001) and

Quattro Pro (2001). Facts about the F distribution are

given in Appendix A.7.11.

If the Jeffreys noninformative priors are used, the

posterior distribution of q is

(tdur/tfreq)[(n+0.5)/n]F(2n+1, 2n) . (6.30)

Table 6.17 shows 21 outages, with a total duration

tdur = 310.51 train-hours, and a total exposure time for

the two trains texpos = 17,850 train-hours. First, let us

examine the assumptions, and then perform the

calculations.

Section 2.6.2 lists an assumption about the independence of durations. Atwood and Engelhardt

(2003) point out that Table 6.17 shows two instances

when both trains were out for exactly the same

amount of time in the same month. This is probably

not coincidence, but indication that both trains were

briefly taken out of service together, violating the

independence assumption. However, the affected

outage time is less than 1% of the total outage time,

so Atwood and Engelhardt feel that the violation of

the assumption is not serious.

Section 6.6.2.3.1 presents a Q-Q plot for checking

whether durations follow an exponential distribution.

The resulting plot for the present outage-duration

data is given as Figure 6.61.

The line is not perfectly straight. The two largest

times are a bit too large, and there are too many very

small times. Thus, the true distribution is apparently

more skewed than an exponential distribution.

Nevertheless, we will assume that the exponential

distribution is approximately correct, while recognizing that the resulting uncertainty intervals are not

exact.

Atwood and Engelhardt use a similar plot to check

the Poisson assumption for the outage count. More

precisely, they investigate whether the times between outage onsets are exponential, which they

should be if the outages follow a Poisson process.

The outage onset times are not given in Table 6.17,

but the authors assume that outages are approximately uniformly spaced in the month in which they
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0 1000 2000 3000

Observed times between outages, t

(i)

0.0

0.5

1.0

1.5

2.0

2.5

-ln[1-i/(n+1)]

Figure 6.62 Q-Q plot for examining whether times

between outages are exponential.

occur. Under this approximation, the times between

outage onsets result in Figure 6.62.

This figure shows the same general curvature as

does Figure 6.61, but with even more curvature,

suggesting that the distribution is more skewed than

an exponential distribution. Atwood and Engelhardt

recognize this fact but choose to press on with the

analysis anyway.

Informative priors would be easy to use, but for this

illustration we use the Jeffreys noninformative priors,

resulting in a posterior distribution given by Expression 6.30. This says that the unavailability q has a

distribution

(310.51/17850)(21.5/21) F(43, 42) . 

The 5th and 95th percentiles of the F distribution with

43 and 42 degrees of freedom are 0.601 and 1.667.

These percentiles are given by many software

packages. Therefore, the 5th and 95th percentiles of

q are 1.07E!2 and 2.97E!2. The mean is

(310.51/17850)(21.5/21) (42/40) = 1.87E!2,

using the formula for the mean of an F distribution

from Appendix A.7.11. As a sanity check, this can

be compared with the simple estimate given in

Section 6.7.1.1, 310.51/17850 = 1.74E!2.

6.7.1.3 Model Validation

If the durations appear to be exponentially distributed,

and the counts appear to have a Poisson distribution,

one may then investigate the other assumptions listed in

Section 2.6.2. Independence is difficult to verify from

data, and is more likely verified by careful thinking. As

for the assumption of a common distribution for all the

down times and a common distribution for all the up

times, the methods of Section 6.6.2 can be used: tools

for investigating whether different portions of the data

have different distributions, or whether a trend is

present. The analyst must decide whether the data set

is large enough to deserve this effort.

6.7.1.4 Bayesian Estimation under Other 

Assumptions

If the outages do not follow a Poisson distribution or if

the durations do not follow an exponential distribution,

Equation 6.29 can still be used, but it is much more

difficult to estimate the two pieces of the equation.

Atwood and Engelhardt (2003) discuss this issue very

briefly. It will not be considered further here. Instead,

we proceed now to methods that do not require

assumptions on the forms of the distributions.

6.7.2 Analysis of Summary Data

The task now is to use the summary data only, not the

data from individual outages, to obtain a Bayesian

distribution for q. The fundamental technique is data

aggregation to yield quantities Xi

 that are approximately

independent and identically normally distributed, as

described next. Atwood and Engelhardt (2003) are

unable to prove theoretically that aggregation must

result in normally distributed Xi. However, they perform simulations of representative cases and show that

asymptotic normality occurs in those cases.

For ease of wording, we assume that the reporting

periods are months, as they are in Example 2.16.

6.7.2.1 Data Aggregation

Denote the exposure time for the ith train-month by ei

and denote the corresponding outage time by oi. The

corresponding simple point estimate of the unavailability q is the ratio xi

 = oi/ei. This gives one such estimate

of q for each train-month of data. The estimate from

any one train-month is not very good, because it is

based on only a small data set. Indeed, if ei

 = 0 the

estimate is undefined.

The data may contain many zeros, as seen in Table

6.18. As a result of the many zeros and few relatively

large outage times, the data can be quite skewed. To

eliminate some of the zeros and make the data less

skewed, the data can be pooled in various ways. For

example, the rightmost column of Table 6.18 shows the

total outage time for the two trains. Similarly, the data

could be aggregated by time periods longer than one

month, such as by calendar quarter or calendar year.
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This aggregation over time could be done separately for

each train or for the pooled data from the trains.

This aggregation groups the data into subsets, for

example train-months (the least aggregation), or trainquarters, or system-months, etc. Note, we are still

trying to estimate the train unavailability, not system

unavailability, even if we pool the data from both trains

in the system. Let oi

 and ei now denote the outage time

and exposure time for the ith subset. The simple

moment estimate of q based on the ith subset is xi

 =

oi/ei.

In the data of Table 6.18, if the two trains are pooled

the total train exposure time for month 2 is e2 = 720

+ 720 = 1440 hrs, and the total train outage time is o2

= 125.19 hrs. The estimate based on this one month

is 125.19/1440 = 8.69E-2. If calendar quarters are

pooled but trains are not pooled, the total train

exposure time for Train 1 in quarter 3 is e3 = 0 + 637

+ 676 = 1313 hrs, and the corresponding train outage

time is o3 = 0 + 18.02 + 0 = 18.02 hrs. The estimate

of q based on this one train-quarter is 18.02/1313 =

1.37E-2.

Whatever level of aggregation is used, this approach

pools the numerators and denominators separately

within each subset and then calculates the ratio.

The purpose of this aggregation is to produce multiple

estimates oi

/ei that we denote generically as xi. The xi

values must all come from a single distribution. Therefore, the pooling assumes that the parameter q does not

change within the data set, and that the various subsets

of the data (calendar quarters or years, etc.) have

similar exposure times, so that the random xi

s all come

from close to the same distribution.

In addition, the distribution of the xi

s should be approximately normal. A normal distribution would not

generate repeated values, such as multiple observed

values of zero, nor would it produce strongly skewed

data. Therefore, we must aggregate enough to obtain

data that are not skewed and do not have repeated

values.

How much aggregation is necessary? To investigate

this question Atwood and Engelhardt (2003) perform

some simulations, which indicate that minimal aggregation is not enough. In fact, the example data set

should be many times larger than it is to make the

method work really well. If an analyst has a large

enough data set so that there is a choice between little

aggregation into many subsets or much aggregation into

few subsets, the second choice is the better one.

Table 6.19, from Atwood and Engelhardt (2003)

gives some sample statistics for x, based on various

amounts of aggregation of the data of Table 6.18.

The skewness is a measure of asymmetry. Positive

skewness corresponds to a long tail on the right.

Zero skewness corresponds to a symmetrical distribution.

Table 6.19 Sample statistics for estimates of q,

with different levels of aggregation.

TrainmonthSystemmonthTrainquarterSystemquarter

n 28 14 10 5

Mean 1.63E!2 1.63E!2 1.78E!2 1.78E!2

Median 0.00E+0 4.60E!3 1.38E!2 1.75E!2

St. dev., s 3.07E!2 2.51E!2 1.64E!2 1.16E!2

s/n1/2 5.81E!3 6.70E!3 5.18E!3 5.19E!3

Skewness 2.79 2.02 1.25 0.33

No. zeros 17 7 2 0

The 28 values of x corresponding to train-months do

not come from a normal distribution. They are too

skewed, as is seen by the fact that the mean

(1.63E-2) is very different from the median (0), and

the skewness (2.79) is far from zero. Also, they have

many occurrences of a single value, 0. Pooling the

two trains into 14 subsets somewhat reduces the

skewness and the percentage of zeros.

Pooling the three months for each train makes the

distribution still more symmetrical: the mean and

median are within 30% of each other, and the skewness is down to 1.25. When the data are aggregated

by pooling trains and by pooling months into quarters, multiple values of zero are finally eliminated,

and the distribution appears to be more nearly

symmetrical: the mean and median are within 2% of

each other, and the skewness is moderately small.

This suggests that the five values of xi

 may be

treated as a random sample from a normal distribution.

To investigate this further, a Q-Q plot was constructed, as in Section 6.6.2.3.1. The plot given by

Atwood and Engelhardt (2003) does not use the

expected order statistics as in Figure 6.53, but

instead is a different version of the plot. The authors

found it easier to plot the ith order statistic against

M-1[i/(n+1)], where M is the standard normal cumulative distribution function given in Table C.1. The plot

is shown here as Figure 6.63.
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6-82

0E0 1E-2 2E-2 3E-2

x(i)

-1.0

-0.5

0.0

0.5

1.0

Φ-1(i/(n+1))

Figure 6.63 Q-Q plot for investigating normality of x

when trains are pooled and months withing quarters

are pooled.

The points lie almost on a straight line, indicating

consistency with the normal distribution. For this

reason, the authors treat the x values as coming from

a normal distribution, when the data are aggregated

to this degree. A goodness-of-fit test could be

performed, but it would not be able to detect nonnormality based on only five points.

As mentioned above, more aggregation would make

the method work better. However, with the present

very small data set, little more aggregation is possible.

There is a problem with the third quarter, because it

has smaller exposure time than the other quarters.

That means that x corresponding to this quarter has

larger variance than the others. This is ignored here,

but if the exposure time for quarter 3 had been even

smaller, we might have dropped that quarter from the

analysis, or pooled differently.

We repeat: The only purpose of the data aggregation is

to eliminate the skewness and eliminate multiple values,

thus permitting the use of normal methods. To the

extent possible, we pool so that the aggregated subsets

have similar exposure times, in order to have x values

that come from a common distribution.

6.7.2.2 Frequentist Estimation

The same frequentist estimate of q can be used as with

detailed data, the sum of the outage times divided by

the sum of the exposure times. This ratio of the sums is

not quite the same as the average of the ratios from the

data subsets, because the various subsets are not based

on the same exposure times. For example, in Example

2.16 quarter 3 has fewer exposure hours. Averaging the

ratios for the subsets would treat the data from quarter

3 with as much weight as the data from the other

quarters. Summing the outage times and exposure

times first, before taking the ratio, gives quarter 3 only

the weight that it should have.

A method to obtain a confidence interval for q uses

facts about normally distributed random variables that

are presented in many statistics books, and summarized

here.

When {x1, ..., xn} is a random sample from a normal(:,

F 2) distribution, the usual estimates of : and F2 are:

x

n

xi i = ∑

1

and

s . nx x X i i

2 2 1

1 = − ∑ ( ) −

(Note the n ! 1 in the denominator, although some

authors use n, and therefore use slightly different

formulas below.) 

The distribution of is normal(:, F 2 X /n). When both

: and F 2 are unknown, a 100(1!")% confidence

interval for : is

xt n s n , (6.32) X ± − 1 2 −α / ( )/ 1

where t1!"/2(n ! 1) is the 1!"/2 quantile of the Student’s

t distribution with n ! 1 degrees of freedom. For

example, t0.95(n!1) gives a two-sided 90% confidence

interval. Student’s t distribution is tabulated in Appendix C, and is calculated by some software packages.

Do not misread the (n ! 1) as a multiplier; it is a

parameter, the degrees of freedom, of the Student’s t

distribution. In Table C.3 each row of the table corresponds to one value of the degrees of freedom.

In Example 2.16, we aggregate by system and

calendar quarter, and use the resulting five values of

x as if they are a random sample from a normal

distribution. In the formulas above, the mean : is the

unavailability, q. From Expression 6.32 and Table

6.19, we obtain that a 90% confidence interval for q

is

1.78E!2 ± 2.132×5.19E!3

 = 1.78E!2 ± 1.11E!2

because 2.132 is the 95th percentile of the Student’s

t distribution with 4 degrees of freedom. Thus, the
Parameter Estimation and Model Validation

6-83

lower and upper confidence limits are

qconf,0.05 = 7.E!3

qconf,0.95 = 2.9E!2 .

This interval is approximate because the x values

come from an approximately normal distribution.

6.7.2.3 Bayesian Estimation

Bayesian estimates are given here. Examples are

worked out after the general formulas are given. As

above, assume that the data have been aggregated

enough so that {x1, ..., xn} is a random sample from a

normal(:, F2 ) distribution.

6.7.2.3.1 Noninformative Prior

The joint noninformative prior for (:, F2

) is proportional to 1/F 2, as stated in Section 6.6.1.2.1 (in the

context of lognormal distributions.) As stated in that

section, the posterior distribution then results in

( )/( / ) μ − xs n X

having a Student’s t distribution with n!1 degrees of

freedom. Here : is the quantity with the Bayesian

uncertainty distribution, and everything else in the

expression is a calculated number. In the present

context, : is the unavailability, q. It follows that the

credible intervals agree exactly with the confidence

intervals given by Expression 6.32. For example, a

90% credible interval for q is 

xt n s n . X ± − 0 95 . ( )/ 1

In Example 2.16, the mean : is interpreted as the

unavailability q. Based on the values of xi

, the

expression

(q ! 1.78E!2)/5.19E!3

has a Student’s t distribution with 4 degrees of

freedom. A 90% credible interval is 

1.78E!2 ± 2.132×5.19E!3

= 1.78E!2 ± 1.11E!2

= (7.E!3, 2.9E!2)

which agrees exactly with the 90% confidence

interval found above.

Not all PRA software packages contain Student’s t

distribution. Sometimes it is necessarily to temporarily

adjust the method to match the available software.

Analysts who are working without Student’s t distribution in their software package may be forced to use a

normal distribution with the same 90% interval as the

one given by the above calculation. (To be more

conservative, match the 95% intervals or the 99% intervals.) If the degrees of freedom are not too small ($ 3

as a bare minimum) the approximation of a Student’s t

by a normal is probably acceptable.

In the above example, a normal distribution with the

same 90% interval would have 1.645F = 1.11E!2.

Therefore, a normal approximation for the posterior

distribution of q is normal with mean 1.78E!2 and

standard deviation 6.75E!3.

6.7.2.3.2 Informative Priors

Informative conjugate priors for : and F2 are presented

in Section 6.6.1.2.1, along with the Bayesian update

formulas. The prior parameters are:

d0 = degrees of freedom for prior estimate of F2

F0

2

 = prior estimate of F2 (more precisely, 1/F0

2

 is prior

mean of 1/F2

) 

n0 = prior number of observations (with variance F2)

for estimating :

:0 = prior mean of :

The update formulas are given in Section 6.6.1.2.1,

resulting in four corresponding parameter values

identified with subscript 1. The final result is that the

posterior distribution of

( )/( / ) μμ σ − 1 11 n

is Student’s t with d1 degrees of freedom. Therefore, a

90% credible posterior interval is

μ σ . 1 0 95 1 1 1 ± td n . () /

To develop an informative prior, Atwood and Engelhardt (2003) use some generic data from seven

plants, with a mean unavailability for CVC trains of

6E!3 and a between-plant standard deviation

3.5E!3. Therefore, they set :0 = 6E!3 and F0 =

3.5E!3. They point out that this data set does not

perfectly match the (much older) data of Example

2.16. Therefore, they set n0 = 3, not 7. They also

argue that the between-plant variance has little or

nothing to do with the between-calendar-quarter

variance of Example 2.16. Therefore, they set d0 =

!1, corresponding to no prior information about the

variance.
Parameter Estimation and Model Validation

6-84

The update formulas of Section 6.6.1.2.1 yield:

d1 = !1 + 5 = 4

n1 = 3 + 5 = 8

:1 = (3×6E!3 + 5×1.78E!2)/8 = 1.34E!2

F1

2

 = { !1×(3.5E!3)2 + (5 ! 1)×(1.16E!2)2 +

 [3×8/(3+8)](6E!3 ! 1.78E!2)2 }/4

 = 2.07E!4

 = (1.44E!2)2

Using the notation q instead of :, it follows that

(q ! 1.34E!2)/[1.44E!2) / 2.83]

has a posterior Student’s t distribution with 4 degrees

of freedom. A 90% posterior credible interval for

unavailability is

1.34E!2 ± 2.132×(1.44E!2)/2.83

= (3E!3, 2.4E!2).

6.7.2.4 Model Validation

A crucial feature of the simple method proposed above

is aggregation of data, to reduce skewness and achieve

approximate normality. The example analysis given

above used a Q-Q plot, Figure 6.63, to check whether

normality was achieved. In addition, other data plots

and goodness-of-fit tests can be used to check the

normality, as described in Section 6.6.2.3. However, if

the data have been aggregated into a small number of

sets, these tests and plots will probably not discover any

departure from normality — there are too few data

points. As mentioned above, to make the method work

well, one should over-aggregate, beyond what the

above analysis tools suggest is minimally acceptable.

An implicit assumption when pooling data subsets is

that the various subsets correspond to the same distribution. Therefore, one may try to check this assumption,

as follows.

The methods discussed in detail in Section 6.6.2.1 may

be used, although the data may not be of good enough

quality to show much. In particular, box plots may be

used to suggest whether subsets can be pooled. The

Kruskal-Wallis test, the nonparametric analogue of the

analysis-of-variance test for equal means of normally

distributed random variables, can be used to test equality of the unavailability in the data subsets. However,

the many observations tied at zero make these methods

difficult to apply.

6.7.3 Comparison of the Analyses with

the Two Types of Data

The approach with detailed data works most easily if

the number of outages follows a Poisson distribution

and the outage durations can be assumed to have an

exponential distribution. In that case, the posterior

distribution of unavailability is a rescaled F distribution. The assumptions of Poisson counts and exponential outage durations must be checked.

The approach with summary data uses aggregation of

the data into subsets until the estimated unavailabilities

for the various subsets appear to be approximately

normal. The resulting posterior distribution is a

rescaled Student’s t. Data aggregation eliminates the

need to assume any particular distribution in the underlying process. However, one must confirm that approximate normality has been achieved.

Both methods work easily with either noninformative

and informative priors, although the formulas for

updating an informative prior with summary data are

more intricate than when using detailed data.

The posterior distributions from the two analyses

given above for Example 2.16 are compared in Table

6.20. It can be seen that in this example the two

posterior distributions are quite consistent with each

other.

Table 6.20 Comparison of Bayesian results

from two approaches, using

noninformative priors.

Data Mean 90% Interval

Detailed 1.87E!2 (1.1E!2, 3.0E!2)

Summary 1.78E!2 (7.E!3, 2.9E!2)
7-1

7. TRENDS AND AGING

7.1 Overview

The material in this chapter is needed only if the modelvalidation tools in Section 6.2 or 6.3 have discovered

the existence of a trend in an initiating-event rate 8 or

in a probability p. This chapter is more advanced than

Sections 6.2 and 6.3, because it actually models the

trend.

Such a trend might be in terms of calendar time, or in

terms of system age. Section 7.2 considers trends in 8

when the events are grouped into bins, such as counts of

events in calendar years. Section 7.3 briefly considers

trends in 8 when the individual event times are used as

the data. Section 7.4 considers trends in p. These

sections all model 8 or p as a parametric function of

time. The final section of this chapter, Section 7.5, ties

together some of the methods that have been presented

in different specific applications in this chapter and in

Chapter 6.

Modeling a trend normally involves some rather detailed mathematics. A Bayesian analysis must construct

a sample from the posterior distribution, and a

frequentist analysis must calculate the fitting equations

and estimate the uncertainties in any unknown parameters. The viewpoint taken here is that the computer

software will do those calculations. This chapter avoids

any equations that the user will not need. Instead, this

chapter presents the various approaches that are possible, and shows how to interpret the computer output

and translate it, if necessary, into the quantities needed

for probabilistic risk assessment (PRA).

It turns out that the Bayesian analysis is no harder to

present than the frequentist analysis. Therefore, it is

given first in Chapter 7.

Some of this material is drawn from an INEEL report

by Atwood (1995), and indirectly from more advanced

sources. The INEEL report gives additional examples,

including definitions of somewhat more complex

models and discussion of pitfalls in constructing such

models.

7.2 Binned Poisson Data

7.2.1 Examples

Two examples are given here. The first example was

introduced as Example 6.5, unplanned demands for the

high pressure coolant injection (HPCI) system during

1987-1993. Table 6.8, summarizing that example, is

repeated here for convenience as Table 7.1.

Table 7.1 HPCI demands and reactor-criticalyears (from Example 6.5).

Calendar

year

HPCI

demands

Reactor-criticalyears

1987 16 14.63

1988 10 14.15

1989 7 15.75

1990 13 17.77

1991 9 17.11

1992 6 17.19

1993 2 17.34

The second example, Example 7.1, groups events not

by year of occurrence, but by age of the reactor.

Example 7.1 Thermal-fatigue leak events, by

plant age.

Thermal-fatigue leaks in PWR stainless-steel

primary-coolant-system piping are tabulated by

Shah et al. (1998).

Age (years

from initial

criticality)

Number

of leaks

Reactoryears

0.0 – 5.0 2 1052

5.0 – 10.0 1 982.5

10.0 – 15.0 4 756.9

15.0 – 20.0 4 442.4

20.0 – 25.0 2 230.9

25.0 – 30.0 0 43.9

The PWR plants considered here include all the

western-designed PWRs, 217 reactors in all, from

initial criticality until May 31, 1998, or until decommissioning. For details, see Shah et al. 1998. Plant

age is summarized in 5-year bins, shown in the first

column of the table. Other bins, such as 1-year bins

or 10-year bins, could have been constructed. For

each thermal-fatigue leak event, the age of the plant

at the time of the event was calculated. The number

of such events for each age bin is given in the

second column of the table.
Trends and Aging

7-2

To count the reactor-years for each age, the number

of reactors that experienced 1 year, 2 years, etc.

were totaled. For example, Three Mile Island 2 had

its initial criticality in December 1978, and was shut

down in March 1979. It was counted as contributing

1/4 of a reactor-year (= 3 months) to the first bin (age

0.0 – 5.0). At the other extreme, Yankee Rowe

operated from July 1961 to September 1991. It was

counted as contributing 5 reactor-years in each of

the six age bins. These counts of reactor-years are

totaled in the third column of the table in Example

7.1.

The two examples have identical form. Each bin

corresponds to a time, which can be coded numerically.

This time will be called clock time here. In Example

6.5, the clock time is calendar time. Each bin corresponds to a year, and could run from 1987 through

1993, or from 87 to 93, or from 0 to 6, or even !3 to

+3. Any coding works, as long as it is numerical.

Denote the coded clock time for bin i by ti

. The units of

ti

 are calendar years, and the different possible codes

correspond to different definitions of year 0. In Example 7.1, clock time is age. The bins are identified most

simply by the midpoints of the age bins: 2.5, 7.5, 12.5,

etc. The units of ti

 are age in years. They could also be

identified by other numerical codes, corresponding to

the smallest age in the bin or the largest age in the bin

or some other code.

In both examples, each bin has a clock time, an event

count, and an exposure time. When considering possible trends, we must distinguish between clock time and

exposure time, paying attention to a distinction that was

not so important in the earlier chapters. In this chapter

the symbols ti

, xi, and si denote the coded clock time,

the event count, and the exposure time, respectively,

for bin i. Suppose that 8 has units of events per reactoryear. Then the units of exposure time must be reactoryears. During any short time interval from t to t + )t,

the exposure time )s equals )t times the number of

reactors operating during the time period.

To avoid redundancy, this section will use only the

Example 6.5 for illustrating the methods, although

either example could be used.

7.2.2 Model

7.2.2.1 General Model

The assumed model is an extension of the model for a

Poisson process given in Section 2.2.2. The following

assumptions are made. These are a simplification of the

slightly weaker assumptions given by Thompson

(1981):

1. The probability that an event will occur in any

specified interval with short exposure time approaches zero as the exposure time approaches

zero.

2. Exactly simultaneous events do not occur.

3. Occurrences of events in disjoint time periods are

statistically independent.

This model is a nonhomogeneous Poisson process

(NHPP). The model in Section 2.2 is a homogeneous

Poisson process (HPP), a special case of the model

given here. Consider now a single operating system, so

that exposure time equals elapsed clock time. In the

HPP, the probability of an event in the time interval (t,

t + )t) is approximately 8)t. In the NHPP, 8 is not

constant, but is a function of t. The function 8(t) is

called the time-dependent event occurrence rate. Some

authors call it the Poisson intensity function. The

probability of an event in the time interval (t, t + )t) is

approximately 8(t))t. In a longer interval, from a to b,

the random number of events is Poisson with mean

λ( )t dt . a

b

∫

Four special cases are mentioned here.

1. The HPP has 8(t) = a constant > 0.

2. The loglinear model has ln8(t) = a + bt, or equivalently, 8(t) = Aebt, with A = ea

. This is also called

the exponential event rate model. Here, a and b

are unknown parameters, which must be estimated

from the data.

3. The Weibull process, or power-law event rate

model has 8(t) = (b/c)(t/c)

b!1

, or equivalently 8(t)

= Atb!1

. Both b and c are unknown parameters

(with b > 0, c > 0), to be estimated from the data.

This model can be rewritten as 

ln8(t) = ln(A) + (b!1)ln(t) ,

which is a linear function of ln(t). Several parameterizations are found in the literature.

4. An extended power-law process has 8(t) = Atb!1

+ 80, for 80 $ 0. The Swedish I-Book (Pörn et al.

1994) uses a Bayesian analysis of this model. The

method is developed by Pörn (1990) and briefly

explained by Pörn et al. (1993) and Lehtinen et al.

(1997).

An occurrence rate must be nonnegative. Note that all

four of these models have this requirement built in —

they force 8(t) to be positive for all t. The number of

possible models is infinite, because any nonnegative

function of t gives a different model.
Trends and Aging

7-3

Figure 7.1 Loglinear model, ln8(t) = a + bt, for

a = 0.1 and three possible values of b. The vertical

axis shows ln8(t).

Figure 7.2 Same model as in previous figure,

showing 8(t) instead of ln8(t).

In the first bulleted case, 8(t) is constant, and in the

other three cases 8(t) is monotone in t, either increasing

forever or decreasing forever. An event frequency has

a bathtub shape if 8(t) is decreasing when t is small,

then roughly constant, and finally increasing when t is

large. Models of a bathtub shape require more parameters. Therefore, bathtub curves are commonly used to

describe models qualitatively, but have not been implemented in widely-used quantitative models.

As noted, the loglinear model satisfies

ln8(t) = ln(A) + bt

and the power-law model satisfies

ln8(t) = ln(A) + (b!1)u

where u = ln(t). Therefore, the power-law model can be

expressed as a loglinear model in ln(t), as long as t stays

away from zero. Therefore, the illustrations of this

section will use the loglinear model. If they so desire,

readers can translate this material into the power-law

model by redefining b and replacing t with u / ln(t).

The extended power-law model will not be considered

further here.

When multiple systems are observed simultaneously,

the total number of events is again a Poisson random

variable, and the mean count is the sum of the means

for the individual systems. This fact will be used to

account for the exposure times in the examples.

The loglinear and simple power-law models given

above are now discussed in detail.

7.2.2.2 Loglinear Model

An occurrence rate must have units of events per

exposure time unit. Therefore, in the loglinear model,

A has units of events per exposure time unit, and b has

the inverse units of the time axis. In Example 6.5, A

has units 1/reactor-critical-year, and b has units

1/calendar-year. In Example 7.1, A has units 1/reactorcalendar-year, and b has units 1/year-of-age. 

The loglinear model is illustrated here, for a = 0.1

and b = +1.0, 0.0, and !1.0. Figure 7.1 shows ln8(t)

as a function of t, and Figure 7.2 shows 8(t) as a

function of t.

The interpretation of b is the slope of ln8(t). If b is

negative, ln8(t) is decreasing, and therefore the event

occurrence rate 8(t) is also decreasing. If b is positive,

8(t) is increasing, and if b = 0, 8(t) is constant. Tests

for trend will be formulated as tests of whether b = 0.

The interpretation of a is the intercept of ln8(t), that is,

the value of ln8(t) at t = 0. The meaning of a depends

on how time is coded. In Example 6.5, with HPCI

demands, if t runs from 1997 to 2003, the value t = 0

corresponds to about 2000 years ago, and a is the value

of ln8(t) at that time. This is an enormous extrapolation, and a can be estimated only with great uncertainty.

Coding t as running from 97 to 103 involves less

extrapolation, because now t = 0 corresponds to the

year 1900, only some 100 years before the data. Other

possibilities are to let t run from 0 to 6, or from !3 to

+3. These coding schemes involve no extrapolation at

all, because 0 is included in the range of the observed

data.

In theory, it makes no difference which coding system

is used. The different codings for t and the different

meanings of a compensate for each other. For any

particular time, such as 1996 or 2001, the different

coding systems give exactly the same estimate of 8 at

that time. In practice, however, no computed value is

exact, and roundoff errors can accumulate. Use of large
Trends and Aging

7-4

Figure 7.4 Same model as in previous figure, with

8(t) shown as function of t.

extrapolations can introduce large errors. Well-programmed software should protect against this problem

automatically, no matter how the times are entered.

Nevertheless, the analyst can do no harm by choosing

a coding system with zero reasonably close to the data.

7.2.2.3 Power-Law Model

In both parameterizations given above for the powerlaw model, b is a unitless shape parameter. As will be

seen below, b ! 1 is the slope of ln8(t) as a function of

ln(t). In the first parameterization, c is a scale parameter with units of t. It does not have a simple interpretation. In the second parameterization, A has strange

units, but a simple interpretation as the numerical value

of 8(t) at t = 1.

Figures 7.3 and 7.4 illustrate the power-law model for

A = 1.0, and b = 0.5, 1.0, 2.0, and 3.0.

Figure 7.3 Power-law model, showing ln8(t) as a

linear function of ln(t), with A = 1.0 and several

values of b.

In Figure 7.3, the parameter b ! 1 is the slope of ln8(t)

as a function of ln(t). In Figure 7.4, b is a shape parameter, defining the shape of the curve. In either figure,

the interpretation of A is the numerical value of 8(t) at

t = 1.

This model requires t $ 0, so a coding system should be

chosen so that all the observed times correspond to

nonnegative values of t. Also, if the occurrence rate is

decreasing, the modeled occurrence rate becomes

infinite at t = 0. 

The loglinear and power-law models are widely used,

but they are chosen for their simplicity and convenience, not their theoretical validity. Any model must

be checked for goodness of fit. Moreover, no model

should be extrapolated far into the future — even if

some convenient algebraic formula fits a trend well in

the past, that is no guarantee that the data will continue

to follow that formula in the future.

7.2.3 Bayesian Estimation with 

Loglinear Model

The first few paragraphs here describe the big picture in

very general terms. Following that, the section carries

out the Bayesian estimation when the occurrence rate

satisfies the equation ln8(t) = a + bt.

A large-sample approximation is applicable. As the

observed event counts become large, the form of the

likelihood function approaches the form of a normal

density in the unknown parameters. That is, if the

likelihood function were treated as a probability density, it would be approximately a normal density. This

is a general fact for large data sets that is exploited by

advanced statistics texts, such as Cox and Hinkley

(1974, Section 10.6). Therefore, with large data sets

the conjugate prior is normal: if the unknown parameters are given a normal prior distribution, the posterior

will be approximately normal, with very good approximation as the data set becomes large. The corresponding noninformative prior for a and b is the limiting

case as the variance approaches infinity, which is a

constant density.

For the work here, it will be assumed that a computer

program produces a sample from the posterior distribution. The theory sketched above then leads us to the

conclusion that the posterior distributions not only

appear normal, they really are normal, or very close to

normal.
Trends and Aging

7-5

si

μi

a b

λi

xi

Year i

i

Figure 7.5 Directed graph for analysis of Poisson

trend in Example 6.5.

model

{

 for (i in 1:N) {

 lambda[i] <- exp(a + i*b)

 mu[i] <- lambda[i]*s[i]

 x[i] ~ dpois(mu[i])

 }

 a ~ dnorm(0.0, 0.0001)

 b ~ dnorm(0.0, 0.0001)

}

Figure 7.6 BUGS script for analyzing data of 

Example 6.6.

Now let us move to the specific case at hand, with

ln8(t) = a + bt, and with a and b as the unknown

parameters. For this case it happens that the above

normal approximation is valid when the event counts

are only moderate in size.

The bins must be small enough that 8(t) is approximately a straight line function within each bin, not

strongly curved within the bin. Denote the midpoint of

the ith bin by ti

. Then the expected number of events in

the bin is well approximated by 8(ti

)si, where si is the

exposure time for the bin. The method is to fit the

observed Poisson counts to 8(ti

)si, while assuming that

8(ti) has the form a + bti. 

A convenient software package is BUGS (1995),

Bayesian inference Using Gibbs Sampling. The Windows version is called WinBUGS. It is also described

in Section 8.2.3, and documented by Spiegelhalter et al.

(1995). It is currently available for free download at

http://www.mrc-bsu.cam.ac.uk/bugs/ .

WinBUGS is a high-powered research tool, capable of

analyzing very complex models. It does this by not

trying to obtain a simple random sample from the

posterior distribution. Instead, it tries for something

more restricted, a Markov chain Monte Carlo

(MCMC) model. Here a chain, or sequence, of

numbers is generated, starting at an arbitrary point but

eventually sampling from the posterior distribution.

The values in the sequence are not independent, but this

does not matter. After the initial value has been essentially forgotten, the remaining values form a sample

from the posterior distribution. They can be used to

approximate the moments, percentiles, and shape of the

distribution.

WinBUGS can be used either with a graphical description of the model, called a “directed graph,” or with a

text script. Example 6.5 will be analyzed using

WinBUGS here, assuming a loglinear model.

With the data of Example 6.5 (Table 7.1), BUGS was

used to model 8(i) = exp(a + bi), for i from 1 to 7.

Then X(i) was modeled as having a Poisson distribution with mean :(i) = 8(i)×s(i). Finally, a and b were

given very diffuse prior normal distributions. Figure

7.5 is the logical diagram showing the relations.

In Figure 7.5, deterministic relations are shown by

dashed arrows, and stochastic relations (random

number generation) are shown by solid arrows. The

parts of the model that depend on i are enclosed in

the box.

Figure 7.6 shows the BUGS script that was used.

Many users find the text script easier to manipulate

than the graph.

One idiosyncrasy of BUGS is that it parameterizes

the normal distribution in terms of the precision J =

1/F2

. The reasons are explained in Section 6.6.1.2.1.

Therefore, a precision of 0.0001 in the script corresponds to a standard deviation of 100. That gives a

very diffuse distribution.

The script was executed with four separate starting

values of a and b, generating four chains of values,

each 10,000 elements long. The first 200 elements

of each chain were discarded, and the remaining

39,200 elements were used to estimate the posterior

distributions. Table 7.2 summarizes the posterior

distributions for a and b. When interpreting these

summaries, be aware that a and b are not independently distributed.

Even though the numbers are not necessarily accurate to three places, the table shows that the mean

and median are nearly equal in each case, and the

5th and 95th percentiles are approximately 1.645
Trends and Aging

7-6

87 88 89 90 91 92 93

Year

0.0

0.5

1.0

1.5

2.0

λ

lambda hat

median

5th and 95th %iles

Figure 7.7 Posterior distribution of 8, assuming

exponential trend in Example 6.5.

standard deviations from the mean. That is, the

distributions appear to be approximately normal.

(BUGS supplies graphical estimates of the densities,

which also appear normal.) This is consistent with

the theory mentioned earlier.

Table 7.2 Posterior statistics for a and b, for

loglinear model of Example 6.5.

a b

mean 0.264 !0.237

median 0.27 !0.237

st. dev. 0.251 0.067

5th percentile !0.157 !0.348

95th percentile 0.662 !0.129

The mean of b is negative, and 3.5 standard deviations away from the mean. This is very strong evidence of a downward trend. The posterior belief in

a flat or rising trend is only 2.3E!4. (This is M(!3.5),

from Table C.1.)

The posterior distribution of 8 is lognormal in any

year. It is shown in Figure 7.7. The median is

plotted as a solid line and the 5th and 95th percentiles are shown as dashed lines. The simple point

estimates

$

λ = x s/ i i

are plotted as dots.

For any particular year, the value of 8(t) = exp(a + bt)

is between the two dotted lines with 90% probability.

This is enough for many applications.

Suppose, however, that we were interested in the entire

curve. This curve is the set of two-dimensional points

of the form

{ (t, 8(t)) | !4 < t < 4 } .

For two distinct times t1 and t2, a pair (a, b) that puts

8(t1) between the lines may put 8(t2) outside the dotted

lines. Therefore, the entire curve does not fall between

the two dotted lines with 90% probability. A 90%

region for the entire curve would need to be wider than

the band shown in Figure 7.7. This subtle issue is

revisited for frequentist estimation in Section 7.2.4.5. 

7.2.4 Frequentist Estimation with 

Loglinear Model

The frequentist method has several variations, which

have been implemented in various software packages.

They are presented here, applied to the example of

Table 7.1, and the results are compared to each other

and to the Bayesian results.

Assume a loglinear model, ln8(t) = a + bt. Statistical

software packages present their products using some

technical terms, summarized here.

• General linear model: the mean of the observable

random variable is a linear function of unknown

parameters. This is NOT useful for the present

problem. It is mentioned only to point out the

possible confusion with the generalized linear

model below.

• Loglinear model: the logarithm of the mean of the

observable random variable is a linear function of

unknown parameters. This is exactly the model

considered in this section.

• Generalized linear model: a transformation of the

mean of the observable random variable is a linear

function of unknown parameters. This includes the

loglinear model as a special case, when the transformation is chosen to be the logarithm.

7.2.4.1 Point Estimation

Analysis of the loglinear model finds the maximum

likelihood estimates (MLEs) of a and b, based on the

Poisson counts. The discussion below will sometimes

call this the Poisson-maximum-likelihood method. 
Trends and Aging

7-7

Figure 7.8 Frequency of unplanned HPCI demands,

from Figure 6.22, with exponentially decreasing fitted

trend line overlaid.

This approach is applied here to Example 6.5. No

calculations are given for Example 7.1 because they

would be very similar, differing only by using age

instead of calendar year.

A loglinear model was fitted to the HPCI demand

data of Table 7.1 (Example 6.5). When the years

were coded as 87 through 93, the estimates of a and

b were 20.5305 and !0.2355. The second number

is the slope of ln8(t), and the first is the fitted value of

ln8(t) when t = 0, that is, in the year 1900. (Of

course, no HPCI systems existed in 1900, but the

model does not know that and fits a value.) When,

instead, the years are coded from 0 to 6, the slope is

the same, but the intercept parameter is different,

because now the value t = 0 is the year 1987. The

estimate of a, the intercept, is 0.0389, the fitted value

of ln8(t) for 1987.

The fitted value of 8(t) is the same, whichever coding

method is used. The fitted values are shown in Figure 7.8, overlaid on Figure 6.22. Each point and

vertical confidence interval is based on data from a

single year, but the fitted trend uses all the data.

7.2.4.2 Confidence Intervals for a and b

With the point estimates and , almost all software a$ $

b

will also report standard errors, estimates of the standard deviations of the estimators. The estimators are

assumed to be approximately normally distributed.

This is a good approximation if the number of observations in each bin is not too small. (One rule of thumb is

that the count in most bins be at least five. This is

based on the observation that each bin’s Poisson

distribution is approximately normal if the mean is five

or more. This rule of thumb is sufficient, but perhaps

unnecessarily conservative.)

A 100(1 ! ")% confidence interval for b is

(7.1) $ ( ) / b z se b a ± 1 2 −

where is the estimate, and z1 ! "/2 is the 1 ! "/2 $

b

quantile of the normal distribution. For example, for a

90% confidence interval, " equals 0.1 and the 0.95

quantile is 1.645. The term se(b) is the corresponding

standard error of b, the estimated standard deviation of

. $

b

A confidence interval for a is constructed in a similar

way, but is normally much less interesting. Who cares

what value a has? That parameter is just the intercept

at some arbitrarily coded time with t = 0. The parameter a is of interest only because it can be used to construct confidence intervals for ln8(t) = a + bt.

7.2.4.3 Test for Presence of Trend

Let two hypotheses be defined by:

H0: 8(t) is constant.

H1: 8(t) = exp(a + bt), b ... 0.

The loglinear model is used as an illustration for the

alternative hypothesis, but any other specific model

could be used, as long as it is not constant.

Note that 8(t) = exp(a + bt) is constant if and only if b

is zero. Therefore, the test of H0 is the same as a test

that b = 0.

As mentioned in Appendix B, tests for hypotheses

about b are intimately related to confidence intervals for

b. The hypothesis

H0: b = b0

is rejected in favor of the hypothesis 

H1: b ... b0

at significance level " if and only if the 100(1 ! ")%

confidence interval does not contain b0. In particular,

the hypothesis

H0: b = 0 ,

the hypothesis of no trend, is rejected at level 0.10 if the

90% confidence interval for b is entirely on one side of

0. The hypothesis is rejected at level 0.05 if the 95%

confidence interval is entirely on one side of 0, and so
Trends and Aging

7-8

forth. Most software packages print out a significance

level at which H0: b = 0 is rejected, the p-value for the

trend.

Similarly, software packages typically print a significance level at which the hypothesis a = 0 is rejected.

This should be ignored, because the value of a has no

inherent interest.

We now compare the above test to an earlier one.

Section 6.2.3.2.2 gave tests for the presence of a trend,

illustrated with Example 6.5. The only test it gave with

binned data was the chi-squared test of:

H0: 8(t) is constant.

H1: 8(t) is not constant.

Section 6.2.3.2.2 commented that the test is not very

powerful, because it considers such a broad class of

possibilities as the alternative hypothesis.

In Example 6.5, the chi-squared test rejected the

hypothesis of constant 8 with p-value 0.009. The

present test of b = 0 rejects this hypothesis with pvalue 0.0004. Although both tests reject H0, the test

based on the loglinear model finds stronger evidence

against constant 8 than the chi-squared test did. In

an example with a less clear trend, the test based on

b = 0 might find a statistically significant trend when

the chi-squared test did not.

Suppose that 8 were not constant, but went up and

down in an irregular way with no persistent increasing

or decreasing trend. The chi-squared test might discover this, but the test based on b would not discover

the nonconstancy of 8 — more precisely, the test of

b = 0 might “discover” the nonconstancy because the

random data might appear to indicate a trend in spite of

the true non-trending pattern of 8, but this would only

be an accident. The test to use depends on the alternatives that the analyst regards as credible. A test that

focuses on those alternatives will be more powerful

than a test that is designed for different alternatives.

7.2.4.4 Confidence Interval for 8(t) at Fixed t

Most software packages also can find approximate

confidence intervals for ln8(t) at particular values of t.

It is worthwhile understanding the approach, because

the software output may require modification to coincide with the analyst’s needs. The idea is that the

MLEs and are approximately normally distrib- a$ $

b

uted. The software finds an approximate 100(1 ! ")%

confidence interval for ln8(t) as

$ (7.2) $ [ ( )] / a bt z se a bt +± × + 1 2 −α

where z1 ! "/2 is as defined earlier, and se(a + bt) is the

standard error, the estimated standard deviation of

$ . The standard error depends on the value of t. $ a + bt

It is found by the software — it cannot be found in a

naive way from the standard errors of a and b, because

the MLEs and are correlated, not independent. a$ $

b

Expression 7.2 is a confidence interval for ln8(t). The

confidence interval for 8(t) itself is found by taking the

exponential of the two bounds in Expression 7.2.

Understanding this algebraic form may be useful. For

example, suppose that the software insists on giving

only a 95% confidence interval for 8(t), and the analyst

desires a 90% interval instead. The following modification can be made. The " corresponding to a 95%

confidence interval is 0.05. First, take logarithms of the

reported upper and lower confidence limits for 8(t).

Use these two values, and the form of Expression 7.2,

to find

z0.975 × se(a + bt) .

This follows from the fact that a 95% confidence

interval corresponds to 1 ! "/2 = 0.975. Using z0.975

and z0.95 from a table of the normal distribution, calculate the value of 

z0.95 × se(a + bt) .

From this, calculate the 90% confidence interval for

8(t),

exp[ $ . (7.3) $ ( )] . a bt z se a bt +± + 0 95

7.2.4.5 Simultaneous Confidence Band at All t

The above confidence interval is for a particular t. For

many values of t, many such confidence intervals could

be calculated. Each is a valid confidence interval, but

they are not simultaneously valid. This is a subtle

point. To appreciate it, recall the interpretation of the

90% confidence interval for 8(t) for some particular

time t1:

Pr[ confidence interval for 8(t1) contains true

occurrence rate at time t1 ] = 0.90. (7.4)

Here, the data set is thought of as random. If many data

sets could be generated from the same set of years, each

data set would allow the calculation of a confidence

interval for 8(t1), and 90% of these confidence intervals

would contain the true occurrence rate.
Trends and Aging

7-9

Events/reactor-critical-year 0.0

0.5

1.0

1.5

2.0

87 88 89

Year

90 91 92 93

GC99 0357 10

Observed frequency

Fitted frequency

Simultaneous 90% confidence 

 band for frequency

90% confidence interval 

 for frequency in 1993

Figure 7.10 Simultaneous 90% confidence band for

8(t), and 90% confidence interval for one frequency

of special interest, 8(93).

A similar confidence statement applies to each time.

The simultaneous statement would involve

Pr[ confidence interval for 8(t1) contains true

occurrence rate at time t1 AND

confidence interval for 8(t2) contains true

occurrence rate at time t2 AND

so forth ]. (7.5)

This probability is hard to quantify, because the intervals are all calculated from the same data set, and thus

are correlated. However, Expression 7.5 is certainly

smaller than 0.90, because the event in square brackets

in Expression 7.5 is more restrictive that the event in

brackets in Equation 7.4. 

This problem is familiar in the context of least squares

fitting. For example, Neter and Wasserman (1974)

discuss it, and attribute the solution to Working,

Hotelling, and Scheffé. A simple adaptation to the

present setting is sketched by Atwood (1995, App.

B-7). The simultaneous confidence band is obtained by

replacing z0.95 in Expression 7.3 by

[ ( )] , .

/

χ 0 90

2 12 r

where the expression in square brackets is the 90th

percentile of a chi-squared distribution with r degrees

of freedom. Here, r is the number of unknown parameters, 2 in this example.

The simultaneous 90% confidence band is about 30%

wider than the band of 90% confidence intervals,

because 

[ ( )] = 2.15, .

/

χ 0 90

2 12 r

which is about 30% larger than

z0.95 = 1.645 

when r = 2.

Figure 7.9 again uses the HPCI unplanned-demand

data of Example 6.5. The annual estimated event

frequencies are shown, along with the fitted frequency, the simultaneous 90% confidence band on

the frequency, and the band constructed from the

individual 90% confidence intervals.

Simultaneous confidence bands typically are not

calculated by software packages. They can be calculated by the user, however, based on the formulas above

and the information produced by the software package.

Figure 7.9 Simultaneous 90% confidence band and

band formed by individual 90% confidence intervals

for Poisson event rate, 8(t).

Which should be presented to users, the simultaneous

band or the band of individual confidence intervals? If

the user’s interest is in a single time, such as the most

recent time, then clearly the confidence interval at that

time is of greatest interest. If, on the other hand, the

user will look at the entire plot, for example to judge

the existence of a trend, then the simultaneous confidence band is a better indication of the uncertainty in

the estimated line. To satisfy both types of users, the

graph could show the simultaneous confidence band

and the confidence interval at the time of greatest

interest. Figure 7.10 shows such a plot, assuming that

a user would be most interested in the most recent time,

1993.

7.2.4.6 Alternative Using Least-Squares

Since the model assumes

ln8(t) = a + bt ,
Trends and Aging

7-10

one might decide simply to use least-squares software

as follows. First, estimate 8 based for each bin, based

on only the data for that bin;

$

λ / i ii = x s

Then fit ln to a + bti by least squares. In principle, $

λi

this works. In practice, the method has several twists in

the road, described next.

First, if the observed count is zero in any bin, the MLE $

λ

will be zero for that bin, and the logarithm will be

undefined. This is the case for the final bin of Example 7.1. The following ways around this have been

proposed.

• Instead of estimating 8i

 by xi/si, use (xi + 1⁄2)/si.

This is equivalent to replacing the MLE by the

posterior mean based on the Jeffreys noninformative prior.

• Estimate 8i

 by the posterior mean based on the

constrained noninformative prior. In this case, the

constraint could be that the prior mean equals the

observed overall mean Ei

xi/Eisi, or equals a modification to guarantee a positive number, (Ei

xi +

1⁄2)/Ei

si.

Such ways tend to reduce the trend slightly, because

they add a constant to all the failure counts, slightly

flattening out any original differences.

The second point that must be considered is that the

variance of Xi /si

 is not constant. Ordinary least-squares

fitting has some optimality properties if the variance is

constant. Otherwise, it is more efficient to use weighted

least squares, with weights inversely proportional to the

variances of the observations. Many statistical software

packages perform weighted least squares estimation.

For simplicity, this issue will be explained for the case

with no zero counts, and with ln8i

 estimated by ln( ) $

λi

= ln(xi/si). The variance of ln(Xi/si) is approximately the

relative variance of Xi /si , defined as

var(Xi /si

)/E2(Xi /si). This is 1/E(Xi) = 1/(8isi) if Xi has a

Poisson(8i

si) distribution. 

Unfortunately, the variances depend on the 8i

 values,

which are unknown. Therefore, the following

iteratively reweighted least-squares method can be

used. Begin by assuming that 8 is constant, and fit

ln(xi /si

) to a straight line with weighted least squares,

and weights si

. Calculate the resulting estimates of 8i,

. Then refit the data to a straight $ exp( $ $ λ ) i i = + a bt

line, using weighted least squares, and weights . $

λi is

Repeat this process until the estimates stabilize.

The final point is that least-squares fitting typically

assumes that the data are approximately normally

distributed around the straight line. In the present

context, this means that ln(Xi

/si) is assumed to be

approximately normally distributed. This assumption

is acceptable, unless the mean count is close to zero.

The variance of the normal distribution is then estimated from the scatter around the fitted line. This

differs from the typical treatment of Poisson data,

where the mean determines the variance.

A 90% confidence interval for 8(t) at a particular t is

given by 

exp{ $ (7.6) $ [ ( ) ( )]} . a bt t d se a bt +± × + 0 95

where t0.95(d) is the 95th percentile of Student’s t

distribution with d degrees of freedom. The software

will report the value of d. It is the number of bins

minus the number of estimated parameters, 7 ! 2 in

Example 6.5. The form of this equation is very similar

to the form of Equation 7.3, although the estimates and

standard deviation are calculated somewhat differently.

A simultaneous 90% confidence band has the same

form, but the multiplier t0.95(d) is replaced by

[2F0.90(r, d)]1⁄2 ,

where F0.90(r, d) is the 90th percentile of the F distribution with r and d degrees of freedom. This modification

of Equation 7.6 is analogous to that for Equation 7.3 to

get a simultaneous confidence band.

7.2.5 Comparison of Methods

The three frequentist methods are compared here.

Following that comparison, the Bayesian method is

compared to the frequentist methods, first for the

current example and then in general.

Figure 7.11, from Atwood (1995), shows results from

three frequentist analyses of the data of Table 7.1.

As can be seen in the figure, the fitted lines are

similar for all three analyses. The unweighted-leastsquares method gives an unnecessarily wide confidence band. This shows the inefficiency of unweighted least squares when the variance is not

constant. The Poisson-maximum-likelihood ap-
Trends and Aging

7-11

proach has a slightly narrower confidence band than

does the weighted-least-squares approach. The

reason is that the least-squares approach introduces

an additional parameter that must be estimated, the

variance around the line. The pure Poisson approach calculates the variance from the fitted mean.

The price of estimating an extra parameter is a larger

multiplier, [2F0.90(2, 5)]1⁄2 = 2.75 instead of

[ ( )] = 2.15. .

/

χ 0 90

2 12 2

One other reason for the difference between the two

plots on the right of the figure might be that they use

different estimators of the variance. In this example,

however, the scatter around the line agrees almost

perfectly with the scatter predicted by the Poisson

assumptions, so the two calculations of the variance

agree. This is coincidence, but it eliminates a possible

distraction in comparing the two methods.

In summary, all three calculations are valid, and yield

similar fitted lines. The method of unweighted least

squares uses the data in a somewhat inefficient way,

however, and therefore produces an unrealistically wide

confidence band. The method of iteratively reweighted

least squares provides a narrower confidence band, and

the loglinear model provides the narrowest band of all.

None of the calculations is exact: the least-squares

method treats ln(count/time) as approximately normal,

and the Poisson-maximum-likelihood method treats the

parameter estimators as approximately normal. However, the analysis based on the Poisson-maximumlikelihood method is preferable (if the Poisson assumption is correct), because it gives the tightest confidence

band, and reweighted least squares is second best. It

may be that extra sources of variation, or

“overdispersion,” have entered the data, variation

beyond that caused by the Poisson distribution. If so,

reweighted least squares would be best and the Poissonmaximum-likelihood method would produce an unrealistically narrow band.

Now the frequentist methods are compared with the

Bayesian method, first for the particular example, and

then in general.

The corresponding figure from the Bayesian analysis

is Figure 7.7. Careful examination of the figures

shows the following:

• The posterior median in Figure 7.7 is close to

the fitted line (the MLE) in Figure 7.9, which is

the middle panel of Figure 7.11.

• The 90% credible band in Figure 7.7 shows a

band that is valid for any one time, but not

simultaneously for all time. It is close to the

inner band in Figure 7.9, which also is valid for

any one time but not simultaneously. These

bands are somewhat narrower than the simultaneous bands of Figure 7.11. 

The following comments apply in general, not just to

the example:

• Frequentist estimation relies on approximate

normality of the estimators, and therefore does not

work well with data having few observed events.

Bayesian estimation obtains normal posteriors

Figure 7.11 Fitted Poisson event occurrence rate and simultaneous 90% confidence band, based on three ways

of fitting the HPCI unplanned-demands data.
Trends and Aging

7-12

when the data set is large, but does not fail entirely

when the data set is small – it merely obtains a

different, non-normal, posterior.

• Most frequentist software packages for analyzing

trends include calculations for investigating the

goodness of fit of the model, as will be seen in

Section 7.2.6. Current Bayesian software may

neglect this issue of model validation.

7.2.6 Model Validation

The three assumptions for a nonhomogeneous Poisson

process are given at the beginning of Section 7.2.2.

The first assumption is difficult to test from data. The

second, dealing with common cause failures, has been

addressed in Sections 2.2.4 and 6.2.3.3. The third

assumption is that event counts in disjoint intervals are

independent. This was addressed using a test of serial

correlation in Section 6.2.3.4 when 8 is constant, but

the analogue is too complicated to consider here. When

the data are collected into bins, as assumed here, serial

dependence can result in an unusually high or low event

count in a single bin. This can be discovered by

goodness-of-fit tests, considered below.

The final assumption made when fitting a trend model

is the form of the model. Methods to examine the

goodness of fit will be illustrated with the loglinear

model, following the precedent of the rest of this

section.

7.2.6.1 Graphical Check for Goodness of Fit

The natural graphical check is to compare the observed

values to the fitted line. This is illustrated by Figure 7.8. In that figure, the 90% confidence interval for

each year overlaps the fitted trend line. Because no

year deviates strongly from the overall trend, the data

appear consistent with the assumption of an exponential

trend. Even if one 90% interval had failed to overlap

the fitted trend line, one would not necessarily conclude

that the exponential-trend assumption is violated. The

reason for not being concerned about a single failure to

overlap is that some 90% confidence intervals are

expected to miss the true value. In the long run, as

many as 10% of the intervals may fail to contain the

true value. In the short run, one miss in seven is 14%,

so one miss in the graph is not alarming.

The above discussion is written from a statistical

viewpoint. An engineering viewpoint may reveal more.

For example, if the estimates for the individual bins

(plotted as dots in this section) drop very suddenly, it

may be that the mechanism has changed. If the time

bins correspond to plant age, the frequent early events

may correspond to a learning period. Such conjectured

causes should be investigated, and confirmed or rejected based on more detailed study of the events. As

is typical, a statistical analysis only puts up road signs,

pointing to interesting subjects for engineering investigations.

The statistical analysis is illustrated with an example

here. The examples given earlier in this chapter could

be used, but Example 2.1 is more interesting when

investigating lack of fit.

Example 2.1 stated that a particular plant had 34

unplanned reactor trips while at power in 1987-1995.

Table 7.3 gives the dates of those initiating events.

This data set is a portion of the database used by

Poloski et al. (1999a). This particular plant had its

initial criticality on 1/3/87 and its commercial start on

5/2/87.

Table 7.3 Dates of initiating events at one

plant, 1987-1995. (from Example 2.1)

01/21/87

01/22/87

02/27/87

03/11/87

03/13/87

03/31/87

04/03/87

04/12/87

04/14/87

04/21/87

04/22/87

05/24/87

06/17/87

06/21/87

06/22/87

07/09/87

08/04/87

11/07/87

11/08/87

03/09/88

10/14/88

10/30/88

01/16/89

02/06/89

02/07/89

02/22/89

03/14/89

10/09/89

06/03/91

07/12/92

07/15/92

07/17/92

10/12/95

11/05/95

These events were grouped by calendar year.

Because reactor trips occur only when the reactor is

at power, the relevant normalizing time is critical

time, given in Table 7.4 as critical years.

The now-familiar picture is given in Figure 7.12. This

figure shows that the first observed value, for 1987,

is well above the fitted line, and the second observed

value, for 1988, is well below the fitted line. In fact,

the assumed model seems to try to force data with

an L-shaped trend into a smooth exponentially

decreasing trend.

It appears that the plant had a learning period, during

which initiating events were quite frequent, followed

by a period with a much smaller frequency. Examination of Table 7.4 shows that the learning period

seems to have lasted until the summer of 1987 (three

events in June, one each in July and August, and

only infrequent events after that). It is not certain

that the explanation is “learning” in the usual sense,

but it is clear that the event frequency dropped

suddenly about six months after the initial criticality.
Trends and Aging

7-13

Figure 7.13 Standardized residuals, also called the

Pearson chi-squared residuals, for data of

Figure 7.12.

Figure 7.12 Annual frequency of initiating events,

with fitted exponentially decreasing 8(t) and

simultaneous 90% confidence band on 8(t).

Table 7.4 Initiating events and reactorcritical-years.

Calendar

year

Initiating

events

Reactor-criticalyears

1987 19 0.70936

1988 3 0.75172

1989 6 0.79482

1990 0 0.89596

1991 1 0.81529

1992 3 0.75123

1993 0 0.99696

1994 0 0.82735

1995 2 0.83760

Incidentally, a plot based on Bayesian calculations

would show the same general information as Figure

7.12. Replace the MLE fitted line by the posterior

median, and replace each confidence interval by a

credible interval for 8 based on a noninformative prior

and one year’s data. For an example of such a Bayesian

plot, see Figure 7.18 in Section 7.4.3.

In ordinary least-squares fitting, it is standard to plot

residuals, where each residual is defined as the observed value minus the fitted value. Under the assumed

model, the residuals do not all have the same variance,

so sometimes the standardized residuals are plotted,

where a standardized residual is the residual divided by

its theoretical standard deviation.

In the present context, the ith count, Xi

, is assumed to be

Poisson with mean si

8(ti). The ith residual, denoted ri,

is

r x st . i i ii = − $λ( )

The variance of a Poisson random variable equals the

mean. Therefore, the standardized residual is

r st . i i i / $λ( )

In the context of binned data, these are also sometimes

called the Pearson residuals or the chi-squared

residuals, because the sum of the squared Pearson

residuals is equal to the Pearson chi-squared statistic.

A plot of the these residuals against time may be

helpful. 

Figure 7.13 plots the standardized residuals against

calendar year, for the example of Tables 7.3 and 7.4.

This plot shows severe lack of fit. The standardized

residuals should be approximately normal (0,1), and

so should be mostly between !2 and 2. A value

greater than 3.5 is just too large. The plot also

shows something that may not be evident from

Figure 7.12. The largest value corresponds to 1995,

not 1987. This reflects the fact that in Figure 7.12,

the 1995 confidence interval is farthest from the fitted

line, in relative terms.

An informative plot for this example is the simple

cumulative plot, introduced in Chapter 6 (see Figure 6.23). This plot would normally be used to check

on whether the event occurrence rate is constant. The

slope is the event rate, and a nonconstant slope corresponds to a departure from a straight line.

The cumulative event count is plotted against event

date in Figure 7.14. In this example, the plot shows

a clear nonconstant slope, and moreover, the form of

the nonconstancy is shown: a very large rate (slope)

during the first year, followed by a somewhat smaller

rate, and then a very small rate in the last years.
Trends and Aging

7-14

0

5

10

15

20

25

30

35

1/1/87 1/1/89 1/1/91 1/1/93 1/1/95

Date

Cumulative number

GC99 0357 14

Figure 7.14 Cumulative count of initiating events, for

data of Table 7.3 (Example 2.1).

Two comments must be made. First, this figure requires the unbinned data, which may not be available in

every problem. Second, the use of calendar time

instead of critical time on the horizontal axis may

distort the figure somewhat.

If there is lack of fit to the exponential trend model, the

analyst should try to identify the causes of the lack of

fit. One frequent cause of lack of fit is systematic

variation — the assumed form of the model is incorrect.

In the examples of this section, systematic variation

means that ln8(t) is not of the form a + bt. This was

revealed by Figures 7.12 and 7.14. Another possible

cause is extra-Poisson variance, additional sources of

variation that are not accounted for in the Poisson

model. Figure 7.13 may show this. The residual for

1995 is surprisingly large. Table 7.3 and Figure 7.14

both show that two events occurred in relatively quick

succession in 1995, and that three events occurred in

quick succession in 1992. If any of these events were

dependent on each other, such dependence would

exaggerate the normal random variation in the counts

from year to year.

One must be careful about how to correct lack of fit. In

this example, it is reasonable to delete the early history

of the plant, the part corresponding to the learning

period. This would flatten the fitted line for the rest of

Figure 7.12, making it lower in 1988 and higher in

1995. Thus, dropping early data would make the late

data fit better. One would have to perform the analysis

to know whether this proposed solution will completely

remove the lack of fit in the late years or only reduce it.

The above approach does not consist of throwing away

data. Instead, it divides the data into two relatively

homogeneous sets, which never should have been

pooled. The set after the end of the learning period can

be analyzed as described above. The other set, during

the learning period, can also be analyzed. If it is thrown

away, that is only because no one chooses to analyze it.

If dividing the data into homogeneous pieces does not

correct the lack of fit, another option is to construct a

more complex model. For example, 8 could be modeled as a function of more variables than just t. Or one

could postulate a random count with larger variance

than the Poisson variance. Such topics are beyond the

scope of this handbook.

7.2.6.2 Statistical Test for Goodness of Fit

7.2.6.2.1 Test Based on Poisson Maximum 

Likelihood

When Poisson maximum likelihood is used to fit the

trend to the data, some software packages give two

measures of goodness of fit, the Pearson chi-squared

statistic and the deviance. The Pearson chi-squared

statistic, denoted X2

, is the sum of squares of the Pearson residuals. The deviance is based on the theory of

generalized linear models. It is defined as

D = 2Exi

{ln(xi) ! ln[si ]} . $

λ( ) ti

For more details, see Atwood (1995) or books on the

generalized linear model.

The assumed model is

H0: 8(t) = a + bt

for some (unknown) constants a and b. If this model is

correct, and if the number of observations per bin is

large, both X2

 and D have approximately a chi-squared

distribution, with degrees of freedom equal to the

number of bins minus the number of unknown parameters. In fact, the two statistics are asymptotically equal.

For small samples, on the other hand, the two are not

necessarily nearly equal to each other, nor is their

distribution approximately chi-squared. The distribution of X2

 typically approaches the chi-square distribution faster than the distribution of D does.
Trends and Aging

7-15

These two statistics can be used to test whether H0 is

true. Four situations can arise in practice.

• Both X2

 and D are in the upper tail of the chisquared distribution, larger than, say, the 95th

percentile. This is evidence of lack of fit to the

model, H0. Report the p-value based on X2

. For

example, if X2

 is at the 98th percentile of the chisquared distribution, report a p-value of 0.02.

Investigate the data to try to discover the reason for

the lack of fit.

• Both X2

 and D are in the middle of the chi-squared

distribution, say between the 5th and 95th percentiles. Then the model appears to fit adequately.

• Both X2

 and D are in the lower tail of the chisquared distribution. This is an indication of

overfit, with too complex a model to be justified by

the data. Although such a situation will probably

not arise with the two-parameter models of this

chapter, it can arise when the model contains many

factors, such as component age, manufacturer,

environment, etc.

• X2

 and D are so different from each other that they

give conflicting conclusions. That is, one statistic

is in the upper tail of the chi-squared distribution

and the other is in the lower tail, or one is in a tail

and the other is in the middle. This can indicate

one of two possibilities. (1) The data set may be

too small to allow an accurate assessment of the

goodness of fit. The problem often can be remedied by pooling the data to some extent. For

example, it is possible to fit a loglinear model

using nine one-year bins from Table 7.4. If X2

 and

D conflict, try pooling the data into two-year bins,

and so forth. (2) H0 may be false in a way that one

statistic detects and the other does not. X2

 and D

are asymptotically equal only if H0 is true, not if H0

is false. In this case, if it is really important to

decide whether H0 should be rejected, one could

try simulating data from the fitted model, and

seeing what fraction of the simulated data sets

produce a simulated X2

 or D as large as actually

observed. That fraction would approximate the

exact p-value, without relying on the asymptotic

chi-squared approximation.

For the HPCI unplanned demand data in Table 7.1,

the loglinear model seems to fit well. The values of

X2

 and D are 4.90 and 5.12, respectively. These are

both in the middle of a chi-squared distribution. The

degrees of freedom, 5, equals the number of bins, 7,

minus the number of unknown parameters, a and b.

For the initiating-event data of Table 7.4, X2

 and D

are 28.93 and 24.17. These are both far out in the

right tail of a chi-squared distribution with 7 degrees

of freedom. This is very strong evidence against the

loglinear model.

The results of these statistical tests are consistent with

the conclusions based on the graphs.

7.2.6.2.2 Test Based on Weighted Least-Squares

Fit

Consider fitting a function of the form

y = a + bt

based on observations yi

 at times ti. In the present

context, y equals ln(xi /si

). In Section 7.2.4.6, the

weighted sum of squares

E wi[yi ! (a + bti)]2

was minimized, with the weights equal to the inverses

of the estimated variances of Yi

 = ln(Xi /si). If the model

assumptions are correct, and if the Yi

s are approximately normally distributed, the weighted sum of

squares has approximately a chi-squared distribution.

The degrees of freedom d is the number of bins minus

the number of unknown parameters. The degrees of

freedom are 5 in the HPCI-demand example and 7 in

the initiating-events example. (Purists will note that the

chi-squared distribution applies if the weights are fixed

in advance, not derived from the random data. This

slight departure from theory is commonly ignored.)

If the weighted sum of squares is in the right tail of the

chi-squared distribution, such as beyond the 95th

percentile, this is evidence of lack of fit. As mentioned

above, one common cause of lack of fit is systematic

variation — the assumed form of the model is incorrect.

In the examples of this section, that means that ln8(t) is

not of the form a + bt. Another possible cause is extraPoisson variance, additional sources of variation that

are not accounted for in the Poisson model. To gain

insight as to which contributors to lack of fit seem to be

present, examine plots similar to Figures 7.12 through

7.14.

7.3 Unbinned Poisson Data

Example 6.6 and Table 7.3 are typical examples of the

type of data considered here. That is, the exact event

times are used. The corresponding summary tables of

counts, given in Tables 7.1 and 7.4, are not used.

In principle, the exact event times contain more information than the summaries of counts in bins. The

counts can be calculated from the exact event times, but
Trends and Aging

7-16

the exact event times cannot be retrieved from the count

totals. Therefore, in principle, better procedures can be

squeezed out of the exact event times. However,

software based on binned event counts is more widely

available. Also, little information is lost by grouping

the event times into bins, unless the bins are very large

and coarse. Therefore, use of binned data is usually the

most practical method for the data analyst.

7.3.1 Bayesian Analysis

When the exact event times are used, it is difficult to

write out the likelihood. In principle, one must write

the likelihood of each event time, conditional on the

previous times, and finally the probability of no events

after the final observed event until the end of the

observation period. [See Section 3.3 of Cox and Lewis

(1966).] Formulas are given for several cases by

Atwood (1992), but they are not intuitive.

Binning the data, as in Section 7.2, is much simpler.

Moreover, the bins may be made as numerous and small

as the analyst desires. Many of the bins would then

have event counts of zero, but that is allowed. This

approach would capture virtually all of the information

in the data. In practice there is little advantage in

constructing very fine bins, but the analyst who was

intent of squeezing every last bit of information from

the data could do it.

7.3.2 Frequentist Analysis

Frequentist analysis is also simpler when the data are

binned, although Atwood (1992) works out the formulas for the MLEs and approximate confidence intervals

for several cases that use the exact event times. The

unified notation in that article does not make the

expression immediately obvious for any particular

model. Typically, the MLEs must be found through

numerical iteration rather than directly from algebraic

formulas.

The simplest approach is to bin the data and use the

methods of Section 7.2. Remember that the bins must

not be too fine; a conservative rule of thumb says that

most of the bins should have expected event counts of

five or more.

The exception — the only relatively easy case with

unbinned data — is the power-law model when the

process is observed from time zero. Typically, the data

collection begins at some time in the middle of operation, but in those rare cases when the data collection

starts at time zero and ends at some time J, the MLEs of

the parameters in the two parameterizations of the

power-law model are:

, $

bn t / ln( / )i

i

n

=

=

∑ τ

1

$ / , and / $

c n

b = τ1

$ $ . / $

A nb b = τ

Here n is the number of events, ti is the time of the ith

event, and J is the final time in the data-observation

period. These formulas can be obtained by translation

into the notation of Section 7.2.2.1 of formulas in Bain

and Engelhardt (1991, Chap. 9, Eq. 13) or Atwood

(1992, Section 6.1). Those references also consider

confidence intervals and goodness-of-fit tests.

7.4 Binomial Data

This section parallels Section 7.2 closely. Only the

formulas are different, because the section deals with

failures on demand instead of events in time. Because

of the similarity to Section 7.2, some of the topics are

given a cursory treatment here. These topics are completely analogous to the material in Section 7.2, where

a fuller description can be found.

7.4.1 Examples

Example 6.10 can be used. This example consisted of

63 demands for the HPCI system during 7years, and 12

failures. It is convenient to combine the data into bins,

such as calendar months, calendar years, etc. Such

binning summarizes the data in a compact way. For

goodness-of-fit tests, discussed in Section 7.4.6, binning is not merely convenient – it is required. If the

bins are too fine (too few failures and successes expected in each bin) then the goodness-of-fit statistics X2

and D will be inconsistent with each other, and neither

will have an approximate chi-squared distribution under

H0. On the other hand, the bins must not be too coarse.

As in Section 7.2, denote the midpoint of the ith bin by

ti

. The bins must be small enough so that the expected

number of failures in the bin can be approximated by

the number of demands in the bin times p(ti

).

The data from Example 6.10 are summarized by

calendar year in Table 6.14, which is repeated here as

Table 7.5.

In this example the bins correspond to calendar years.

Other examples could be constructed in which the bins
Trends and Aging

7-17

correspond to ages, so that p would be modeled as a

function of age rather than calendar time.

Table 7.5 HPCI failures on demand, by year

(from Example 6.10).

Calendar year Failures Demands

1987 4 16

1988 2 10

1989 1 7

1990 3 13

1991 2 9

1992 0 6

1993 0 2

7.4.2 Model

7.4.2.1 General Model

The model is the same as that in Sections 2.3.1 and 6.3,

except now the probability p depends on time, t. Thus,

the model assumptions are:

1. The outcome of a demand at time t is a failure with

some probability p(t), and a success with probability 1 ! p(t).

2. Occurrences of failures on different demands are

statistically independent.

The number of demands and their times are assumed to

be fixed, and the outcome on each demand is assumed

to be random.

7.4.2.2 Logit Model

By far the most commonly used functional form for p(t)

is the logit model. In this model the logit transform of

p(t) is a function of unknown parameters. Fitting such

a model to data is sometimes called logistic regression.

The model that will be used in this section is

logit[p(t)] = a + bt ,

where the logit function is defined as

logit(p) = ln[ p/(1 ! p) ] .

This function was also encountered in Section 6.3.2.5.2

and in Appendix A.7.9, where the logistic-normal

distribution is introduced. Like the loglinear model for

8(t), the logit model for p(t) is a model in the class of

generalized linear models.

A frequently used relation is that

y = logit[p(t)] / ln{ p(t)/[1 ! p(t)] }

is equivalent to 

p(t) = logit!1

(y) / ey/(1 + ey) , (7.7)

denoting the inverse function of the logit by logit!1

.

Figure 7.15 shows logit[p(t)] as a function of t, and

Figure 7.16 shows p(t) itself as a function of t. Notice

that in Figure 7.16 the value of p(t) stays between 0.0

and 1.0, as it should.

Figure 7.15 Plot of ln{ p(t)/[1 ! p(t)] } = a + bt, with

a = !2 and three values of b.

Figure 7.16 Plot of p(t) as a function of t, corresponding to Figure 7.15.

The parameters have simple interpretations: a is the

value of the logit of p when t = 0, and b is the slope of

the logit of p.
Trends and Aging

7-18

model

{

 for (i in 1:m) {

 p[i] <- exp(a + i*b)/(1 + exp(a+i*b))

 x[i] ~ dbin(p[i], n[i])

 }

 a ~ dnorm(0.0, 0.0001)

 b ~ dnorm(0.0, 0.0001)

}

Figure 7.17 BUGS script for analyzing data of

Example 6.10.

7.4.2.3 Loglinear Model

If p is small, then logit(p) is close to ln(p), and the logit

model could be approximated by

ln[p(t)] = a + bt .

This is a loglinear model, just as in Section 7.2.2.2.

Software programs for analyzing a generalized linear

model always include the logit model as one special

case, and software programs for logistic regression are

based on the logit model. However, if for some reason

the analyst has software that only allows for the logtransformation, not the logit-transformation, that

software is probably adequate as long as p is small.

7.4.3 Bayesian Estimation with Logit

Model

The large-sample theory mentioned in Section 7.2.3

applies here as well. As the data set becomes large

(many demands and failures) the form of the likelihood

function approaches the form of a normal density for

the two variables a and b. Therefore, with large data

sets the conjugate prior is normal: if a and b are given

a normal prior distribution, the posterior will be approximately normal, with very good approximation as the

data set becomes large. The corresponding

noninformative prior for a and b is the limiting case as

the variance approaches infinity, which is a constant

density.

For the work here, it will be assumed that a computer

program produces a sample from the posterior distribution. The theory sketched above will then lead us to the

conclusions that the posterior distributions not only

appear normal, they really are normal, or very close to

normal.

A convenient, and free, software package is BUGS

(1995), Bayesian inference Using Gibbs Sampling.

This was also used in Section 7.2.3, where it is described in more detail. Example 6.10 is analyzed here

using the Windows version, WinBUGS, assuming a

logit model.

Using the data of Example 6.10 (Table 7.5), BUGS

was used to model logitp(i) = a + bi, for i from 1 to 7.

Then X(i) was modeled as having a binomial(n(i),

p(i)) distribution, where n(i) is the number of demands in year i. Finally, a and b were given very

diffuse prior normal distributions.

Figure 7.17 shows the BUGS script that was used to

analyze the data.

This uses the Equation 7.7 for expressing logit(p) in

terms of the normally distributed quantity a + ib.

Note the way BUGS happens to parameterize

distributions, putting p before n in the list of binomial

parameters, and parameterizing the normal distribution in terms of the precision J = 1/F2

. The reason for

using precision is explained in Section 6.6.1.2.1. A

precision of 0.0001 in the script corresponds to a

standard deviation of 100. That gives a very diffuse

distribution.

The script was executed with four separate starting

values of a and b, generating four chains of values,

each 10,000 elements long. The first 500 elements

of each chain were discarded, and the remaining

38,000 elements were used to estimate the posterior

distributions. Table 7.6 summarizes the posterior

distributions for a and b. When interpreting these

summaries, be aware that a and b are not independently distributed.

Table 7.6 Posterior statistics for a and b, for

loglinear model of Example 6.5.

a b

mean !0.8838 !0.2085

median !0.8654 !0.204

st. dev. 0.6477 0.1961

5th percentile !1.981 !0.5395

95th percentile 0.1471 0.1063

The table shows that the mean and median are

nearly equal in each case, and the 5th and 95th

percentiles are approximately 1.645 standard deviations from the mean. This approximation is poorest

for a, with !0.8838 + 1.645 × 0.6477 = 0.182, some-
Trends and Aging

7-19

87 88 89 90 91 92 93

t (year)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

p (failures/demand)

Mean and 90% posterior interval

 from constrained noninformative prior

Posterior median

5th and 95th percentiles

Figure 7.18 Posterior trend line for p with 90%

credible band, for data of Table 7.5. In addition,

annual estimates and 90% credible intervals are

shown, based on constrained noninformative prior.

what larger than the reported 95th percentile. That

is, the distributions appear to be approximately

normal, as anticipated by the theory mentioned

earlier, but the approximation is not perfect in the

tails.

The mean of b is negative, but not strongly so, only

1.06 standard deviations below 0. Treating b as

normally distributed, a table of the normal distribution

shows that b is negative with probability 0.86 and

positive with probability 0.14. Therefore, we are not

really sure that the trend is downward.

The posterior distribution of p is approximately

logistic-normal in any year. It is shown in Figure

7.18. The median is plotted as a solid line and the

5th and 95th percentiles are shown as dashed lines.

This is analogous to Figure 7.7 for 8.

Figure 7.7 also plotted the MLEs, based on each

year’s data, as dots. To illustrate the graphical

possibilities, Figure 7.18 is constructed somewhat

differently. The total data set has 12 failures in 63

demands. Therefore we constructed the constrained

noninformative prior with mean 12/63. Interpolation

of Table C.8 shows that this prior is approximately

beta(0.324, 1.376). For each year of data, this prior

was updated to obtain the posterior for that year; the

90% credible interval was plotted as a vertical line,

with a dot showing the posterior mean.

All the intervals overlap the fitted trend line. This is

graphical evidence that the logit model fits the data

well. (Many other models might also fit this sparse

data set well.)

7.4.4 Frequentist Estimation with 

Logit Model

7.4.4.1 Point Estimation

The model is analyzed by finding the MLEs of a and b,

based on binomial counts. The discussion below will

sometimes call this the binomial-maximum-likelihood

method. The software finds the estimates by numerical

iteration.

When this model is fitted to the data of Table 7.5, a

fitted trend is found, which can be overlaid on

Figure 6.38. It is not shown here, but will be displayed with a simultaneous confidence band in

Section 7.4.4.4.

7.4.4.2 Confidence Intervals for a and b

As in Section 7.2.4.2, almost all software for estimating

a and b reports standard errors, estimates of the standard deviations of the maximum likelihood estimators.

The estimators are assumed to be approximately

normally distributed, which is valid unless the sample

size is small.

Therefore, as in Section 7.2.4.2, a 100(1 ! ")% confidence interval for b is

$ ( ) / b z se b ± 1 2 −α

where is the estimate, and z1 ! "/2 is the 1 ! "/2 quan- $

b

tile of the normal distribution. The term se(b) is the

standard error of b, the estimated standard deviation of

the estimator.

The confidence interval for a is similar.

7.4.4.3 Test for Presence of Trend

Consider the two hypotheses defined by:

H0: p(t) is constant.

H1: p(t) = logit!1

(a + bt), b ... 0.

Note, the null hypothesis H0 is true if p(t) = logit!1

(a +

bt) and b is zero. Therefore, with this choice of an

alternative hypothesis, the test of H0 is the same as a

test that b = 0, the test given above based on a confidence interval for b.

As in Section 7.2.4.2, the hypothesis

H0: b = 0 ,
Trends and Aging

7-20

the hypothesis of no trend, is rejected at level 0.10 if the

90% confidence interval for b is entirely on one side of

0. The hypothesis is rejected at level 0.05 if the 95%

confidence interval is entirely on one side of 0, and so

forth. Most software packages print out a significance

level at which the hypothesis that b = 0 is rejected, the

p-value for the trend.

This is different from the tests of Chapter 6. Section

6.3.3.2.2 uses a different alternative hypothesis,

H1: p(t) is not constant .

It also uses a different test, the chi-squared test. Section

6.3.3.2.2 commented that the test is not very powerful

against the alternative of a trend in p, because it considers such a broad class of possibilities as the alternative

hypothesis.

Consider Example 6.10, with the HPCI failures

during unplanned demands, as summarized in

Table 7.5. In Section 6.3.3.2.2, the chi-squared test

rejected the hypothesis of constant p with p-value

0.77. That is, the test found no evidence of nonconstant p. The present test of b = 0 rejects this hypothesis with p-value 0.30. That is, this test still does

not reject the hypothesis of constant p. However, it

notices the slightly falling values of p in Figure 6.38,

and therefore sees somewhat stronger evidence

against constant p than the chi-squared test did.

Incidentally, the test based on and the Wilcoxon- $

b

Mann-Whitney test for trend (Section 6.3.3.2.2) reach

very similar conclusions.

7.4.4.4 Confidence Intervals and Confidence

Bands

Confidence intervals for p(t) at a particular t and

simultaneous confidence bands valid for all t both are

based on the approximate normality of the MLEs a$

and . The software finds an approximate $

b

100(1 ! ")% confidence interval for logit[p(t)] as

$ (7.8) $ [ ( )] / a bt z se a bt +± × + 1 2 −α

where, as before, z1 ! "/2 is the 100(1 ! "/2) percentile

of the standard normal distribution, and se(a + bt) is the

standard error, the estimated standard deviation of

$ . The standard error depends on the value of t, $ a + bt

and accounts for the fact that the MLEs and are a$ $

b

statistically correlated, not independent. The confidence interval for p(t) itself is found by inverting the

logit function. If L and U are the lower and upper

confidence bounds for logit[p(t)], that is, for a + bt,

then

logit!1

(L) = eL/(1 + eL) and

logit!1

(U) = eU/(1 + eU) (7.9)

are the corresponding confidence bounds for p(t).

Manipulation of Equation 7.8 allows the analyst to

convert from one degree of confidence to another, say

from 90% to 99%, by using different percentiles of the

normal distribution and the single standard error found

by the software.

As discussed in Section 7.2.4.5, a confidence interval is

valid at one t, and the band constructed from the

individual confidence intervals is not simultaneously

valid for all t. A simultaneous 100(1 ! ")% confidence

band for logit[p(t)] is found by replacing z1 ! "/2 in

Equation 7.8 by

[ ( )] , /χ 1 α

2 12

− r

with r equal to the number of estimated parameters, 2 in

Equation 7.8. This is exactly the value that was used in

Section 7.2.4.5. The only difference is that there the

confidence band was for ln8(t) and here it is for

logit[p(t)]. The confidence band for 8(t) was found by

inverting the logarithm function, that is, by taking an

exponential. The confidence band for p(t) is found by

inverting the logit function: if L and U are now used to

denote the lower and upper edges of the simultaneous

confidence band for logit[p(t)] at some t, the corresponding points on the confidence band for p(t) are

given by Equation 7.9.

Figure 7.19 shows the data from Table 7.5, plotted

as in Figure 6.38 but now with the fitted trend line

and the simultaneous 90% confidence band overlaid.

The confidence band can easily contain a horizontal

line; this is consistent with the fact that the hypothesis of constant p cannot be rejected.

7.4.4.5 Alternative Using Least-Squares Software

The model assumes that logit[p(t)] = a + bt . Therefore,

as in Section 7.2.4.6, one might decide simply to use

least-squares software as follows. First, estimate p for

each bin, based on only the data for that bin:

p xn $ / . i ii =

Then fit logit( ) to a + bti p$ by least squares. The same i

problems that were mentioned in Section 7.2.4.6 must

be addressed here.
Trends and Aging

7-21

Figure 7.19 Annual estimates of p, fitted trend line

and 90% confidence band for p(t), for data of Table

7.5.

First, if any observed failure count, xi, equals either 0 or

the demand count ni

, the MLE will be 0.0 or 1.0 for p$

that bin, and the logit will be undefined. In the data of

Table 7.5, this happens for the final two years. The

following ways around this have been proposed,

analogues of proposals in Section 7.2.4.6:

1. Instead of estimating pi

 by xi/ni, use (xi + 1⁄2)/

(ni

 + 1). This is equivalent to replacing the

MLE by the posterior mean based on the

Jeffreys noninformative prior.

2. Estimate pi

 by the posterior mean based on the

constrained noninformative prior. In this case,

the constraint could be that the prior mean

equals the observed overall mean Ei

xi /Eini, or

equals a modification to guarantee a positive

number, (Ei

xi + 1⁄2)/(Eini + 1).

Such ways tend to reduce the trend slightly, because

they add a constant to all the failure and demand counts,

slightly flattening out any original differences.

The second point that must be considered is that the

variance of the estimator of p is not constant. Therefore, the following iteratively reweighted least-squares

method can be used. Assume that p in the ith bin is

estimated by (Xi

 + ")/(ni + " + $). If the simple MLE

is used, then " and $ are both zero. If method 1 above

is used, then " = 1⁄2 and $ = 1⁄2. If method 2 above is

used, then " and $ must be found from Table C.8.

Neter and Wasserman (1974, Eq. 9.51) state that the

asymptotic variance of logit(MLE of pi

) is

1/[ni pi

(1 ! pi)] .

The method given here is a generalization when " and

$ are not both zero, setting the weight wi to the inverse

of the asymptotic variance of the estimator.

Begin by assuming that p is constant, and let be p$i

some simple estimate of p, the same for all i. Fit

logit[(xi

+")/(ni+*)] to a straight line with weighted least

squares, and weights

w . np n p

np p n

i

ii i i

ii i i

= + −+

− ++

( $ )[ ( $ ) ]

$ ( $ )( )

α β

α β

2 2

2

1

1

Calculate the resulting estimates of pi,

 = logit!1 p$ ( ) . i $ $ a bti +

Recalculate the weights with these estimates, and refit

the data to a straight line using weighted least squares.

Repeat this process until the estimates stabilize.

The third and final point is that least-squares fitting

typically assumes that the data are approximately

normally distributed around the straight line. In the

present context, this means that logit( ) is assumed to p$i

be approximately normally distributed. This assumption is acceptable if the number of failures in each bin

is not close to zero or to the number of demands. The

variance of the normal distribution is then estimated

from the scatter around the fitted line. This differs from

typical treatment of binomial data, where the mean

determines the variance.

A 90% confidence interval for p(t) at a particular t is

given by 

(7.10) logit { $ $ [ ( ) ( )]} .

− +± × + 1

0 95 a bt t d se a bt

where t0.95(d) is the 95th percentile of Student’s t

distribution with d degrees of freedom, just as with 8(t)

in Section 7.2.4.6. The software will report the value of

d. It is the number of bins minus the number of estimated parameters, 7 ! 2 in the example of Table 7.5.

The form of this equation is very similar to other

equations in Sections 7.2 and 7.4, although the estimates and standard deviation are calculated somewhat

differently.

A simultaneous 90% confidence band has the same

form, but the multiplier t0.95(d) is replaced by

[2F0.90(r, d)]1⁄2 ,

where F0.90(r, d) is the 90th percentile of the F distribution with r and d degrees of freedom.
Trends and Aging

7-22

7.4.5 Comparison of Bayesian and 

Frequentist Estimation with 

Logit Model

When the Bayesian analysis uses very diffuse priors, the

conclusions of the two analyses will be numerically

similar.

The posterior median in Figure 7.18 is very close to

the fitted line (the MLE) in Figure 7.19. The 90%

credible band in Figure 7.18 is narrower than the

simultaneous confidence band in Figure 7.19, because the simultaneous confidence band is based on

an inequality. It would be close to the frequentist

bounds that are valid at any one t, if such a graph

were calculated. The vertical lines in Figure 7.18,

representing credible intervals for p based on individual years of data, are generally close to the confidence intervals in Figure 7.19, except for the years

with little data, 1992 and 1993. For those two years,

the confidence intervals are quite wide, but the

credible intervals are shorter, under the influence of

the prior mean of 0.19.

Frequentist estimation relies on approximate normality

of the estimators, and therefore does not work well with

data having few observed events. Bayesian estimation

obtains normal posteriors when the data set is large, but

does not fail entirely when the data set is small – it

merely obtains a different, non-normal, posterior.

Most frequentist software packages for analyzing trends

include calculations for investigating the goodness of fit

of the model, as will be seen in Section 7.4.6. Current

Bayesian software may neglect this issue of model

validation.

7.4.6 Model Validation

The two assumptions for a time-dependent binomial

process are given at the beginning of Section 7.4.2.

The first assumption is difficult to test from data. The

other assumption is that outcomes on distinct demands

are independent. One kind of dependence is serial

dependence. Positive serial dependence means that

failures tend to be followed by more failures, for

example if a failure is misdiagnosed the first time, or if

a single cause results in a number of failures before it is

corrected. Negative serial dependence means that

failures tend to be followed by successes, for example

if the major cause of failure is wearout, at which time a

new component is installed (without any failures from

installation problems).

Positive serial dependence results in failures tending to

cluster together, with relatively long gaps between

failures. When the data are collected into bins, this can

translate into unusually high or low event counts in

individual bins. This can be discovered by goodnessof-fit tests, considered below. However, it is impossible to decide, from the failure counts alone, whether

the outcomes are serially correlated or whether p is

going up and down. The cause can be determined only

by an investigation to discover the failure mechanisms.

A negative serial dependence results in less-thanexpected variation in the event counts. A goodness-offit test will report a p-value near 1.0, indicating surprisingly good fit, too good to be believable.

The final assumption made when fitting a trend model

is the form of the model. Goodness-of-fit tests are

designed for testing this assumption. In fact, a

goodness-of-fit test is an all-purpose test for the various

assumptions, although it is not good at deciding which

assumption may be violated.

7.4.6.1 Graphical Check for Goodness of Fit

The natural graphical check is to compare the observed

values to the fitted line.

Figure 7.18 and 7.19 each show such a plot for the

data of Table 7.5 (Example 6.10). Either figure may

be used. In Figure 7.18, each credible interval

overlaps the fitted trend line, and in Figure 7.19, the

90% confidence interval for each year overlaps the

fitted trend line. Because no year deviates strongly

from the overall trend, the data appear consistent

with the assumption of the logit model.

The discussion at the end of Section 7.2.6.1 applies

here as well, concerning interpretation of a few intervals’ failure to overlap the fitted line, and concerning

the need for an engineering assessment of any strange

patterns.

As with the loglinear model for 8(t), the residuals and

standardized residuals can be plotted for p(t). Software

may report these as the “raw residuals” and the “Pearson chi-squared residuals,” respectively.

In the present context, the ith count, Xi

, is assumed to be

binomial with mean ni

p(ti). The ith residual, denoted ri,

is

r x npt . i i ii = − $( )
Trends and Aging

7-23

The variance of a binomial(n, p) random variable

equals np(1 ! p). Therefore, the standardized residual

is

r n . ip ti p ti i / $( )[1 − $( )]

These are also sometimes called the Pearson residuals

or chi-squared residuals. A plot of the standardized

residuals against time may be helpful, as in Section 7.2.6.1.

Also, a simple cumulative plot may be informative, as

it was in Section 7.2.6.1. In the present example, the

cumulative plot is given in Figure 6.39, which shows no

pattern of interest. In other data sets, such a plot might

not only show nonconstancy in p, but it might suggest

the form of the nonconstancy.

The cumulative plot shows failures per demand, revealing changes in p as a function of the demand count.

However, if the rate of demands is changing, as it is in

the present example, the plot can give a distorted

picture if p is regarded as a function of calendar time.

When p is constant, the issue of distortion is irrelevant

— a constant p is constant, whether it is regarded as a

function of demand count or of calendar time.

Just as in Section 7.2, lack of fit may be caused by

systematic variation or by extra-binomial variance,

additional sources of variation that are not accounted

for in the binomial model. See the discussion in Section 7.2.6.1.

7.4.6.2 Statistical Test for Goodness of Fit

7.4.6.2.1 Test Based on Binomial Maximum 

 Likelihood

Just as in Section 7.2, software packages that use

binomial maximum likelihood may give two measures

of goodness of fit, the Pearson chi-squared statistic,

denoted X2

, and the deviance, denoted D. The discussion in Section 7.2.6.2.1 applies here as well.

For the HPCI failure data in Table 7.5, the logit

model seems to fit well. The values of X2 and D are

2.12 and 2.91, respectively. These are both in the

middle of a chi-squared distribution with five degrees

of freedom. The degrees of freedom, five, equals the

number of bins, seven, minus the number of unknown parameters, a and b. The p-value for lack of

fit, based on X2

, is 0.84, indicating very good fit.

7.4.6.2.2 Test Based on Weighted 

Least-Squares Fit

Consider fitting a function of the form

y = a + bt

based on observations yi

 at times ti. In the present

context, y equals logit(xi

/ni). As in Section 7.2, if the

model assumptions are correct, and if the Yi

s are approximately normally distributed, the weighted sum of

squares has approximately a chi-squared distribution.

The degrees of freedom d is the number of bins minus

the number of unknown parameters. The number of

degrees of freedom is 5 in the HPCI-failure example.

The discussion in Section 7.2.6.2.2 applies here as well.

7.5 Discussion

This section ties together some specific methods given

in Chapters 6 and 7, showing the unifying formulations.

Readers who are happy simply using the earlier recipes

may skip this section.

7.5.1 Generalized Linear Models

Some software packages that implement the loglinear

and logit models do so in the framework of the generalized linear model. Such models are described in a

highly mathematical way by McCullagh and Nelder

(1989), and in an introductory way in Appendix B-2 of

Atwood (1995). This model has several elements, a

random component, a systematic component, and a

link, a function relating the random and the systematic

components.

• The random component consists of some independent observations Y1, ... , Ym. This is thought of as

an m-dimensional vector, Y. The examples of this

chapter have been the normalized Poisson event

count, Yi

 = Xi/si, and the fraction of binomial

failures on demand Yi

 = Xi/ni. 

• The systematic component is an m-dimensional

vector 0, with the ith element 0i

 related to explanatory variables, and to unknown parameters in a

linear way. The example of this chapter has been

0i = a + bti, where a and b are unknown parameters

and the explanatory variable ti

 is the calendar time

or age for Yi

.

• The link is a function g with

0i = g[E(Yi)] . (7.11)
Trends and Aging

7-24

In this chapter, the links have been the log function

for Poisson data and the logit function for binomial

data.

Thus the two examples of this chapter have been the

Poisson example, with

ln[E(Xi /si

)] = ln[8(ti)] = a + bti ,

and the binomial example, with

logit[E(Xi /ni

)] = logit[p(ti)] = a + bti .

This is the terminology used by many statistical software packages. The analyst must specify the distribution for the random component, the form of the systematic component in terms of the unknown parameters and

the explanatory variables, and the link function.

Other software packages use a slight variant of this

terminology. This is a generalized linear model with an

offset, replacing Equation 7.11 by

g[E(Yi)] = 0i + offseti . (7.12)

In the Poisson example, let Yi

 be the Poisson count

itself, Xi

, not the normalized count Xi/si. Then the

expected value of Yi

 is si8(ti), with ln8(ti) modeled as a

+ bti

. To satisfy Equation 7.12, let the offset term be

ln(si

). Then we have:

g[E(Yi

)] = ln[E(Xi)]

= ln[si

8(ti)]

= ln[8(ti

)] + ln(si)

= a + bti

 + ln(si)

= 0i

 + offseti .

In this version of the model, the software package

requires the analyst to specify the distribution of the

random component, the form of the systematic component, the link function, and the offset. The disadvantage

of this formulation is the extra term that must be specified, the offset. The advantage is that the distribution of

Xi

 is Poisson, whereas the distribution of Xi/si is hard to

specify because it does not have a standard name.

Much more elaborate models can be constructed in this

general framework, by adding more explanatory variables. For example, both calendar time and age of the

individual component could be treated together in one

model as explanatory variables. The explanatory

variables do not even have to be continuous. Manufacturer, system, and plant could be used as discrete

explanatory variables. The possibilities are limited only

by the availability of data. However, such models go

beyond the limited scope of this handbook.

7.5.2 The Many Appearances of the 

Chi-Squared Test

In Chapter 6, the Pearson chi-squared test was used to

test whether 8 or p was constant. In Chapter 7, 8 or p

is assumed to be nonconstant, yet the chi-squared test is

used anyway. Also, the chi-squared test was used in

Section 6.6.2.3.2 to test whether durations had an

assumed distributional form. 

To see the unity in this apparent diversity, note first that

in every case the Pearson chi-squared statistic has the

form

X2

 = Ei ( observedi ! expectedi )2 / expectedi .

To clarify possible confusion one must think about the

hypothesis being tested. The big general theorem states

that when H0 is true, X2

 has approximately a chi-squared

distribution, and the degrees of freedom is the number

of unknown parameters under H1 minus the number of

unknown parameters under H0. The approximation is

valid when the degrees of freedom stays constant and

the size of the data set becomes large. This is now

applied to the specific cases in this handbook.

Consider first Poisson data with event rate 8. In

Chapter 6, the null and alternative hypotheses were:

H0: 8 is constant.

H1: 8 is not constant.

The data were partitioned into c cells. These cells may

have corresponded to different sources of data, such as

different plants, or they may have resulted from binning

the data, for example, corresponding to c years. If H0

is true there is one unknown parameter, the single value

of 8. If, instead, H1 is true, there are c unknown parameters, the values of 8 in the different cells. Therefore,

by the big general theorem, when H0 is true X2

 is

approximately chi-squared with c ! 1 degrees of

freedom. This is the result stated in Section 6.2.3.1.2.

Consider now the corresponding case with binomial

data. The data fall into a 2×J contingency table. The

value J corresponds to the J sources of data or J bins,

and the two rows correspond to failures and successes.

The null and alternative hypotheses were:

H0: p is constant

H1: p is not constant .
Trends and Aging

7-25

If H0 is true, there is one unknown parameter, p. If,

instead, H1 is true there are J unknown parameters,

corresponding to the J bins or data sources. Therefore,

the big general theorem says that the degrees of freedom for X2

 is J ! 1, just as stated in Section 6.3.3.1.2.

Consider now the setting of Section 7.2, with 8(t)

modeled as a + bt. The time axis was divided into c

bins. To test the goodness of fit, the hypotheses were:

H0: 8(t) = a + bt,

H1: 8 for each bin is arbitrary.

Under H0 there are two unknown parameters, a and b.

Under H1, the number of unknown parameters is the

number of bins, since each can correspond to a different

8. Therefore, the big general theorem says that the

degrees of freedom, when testing H0, is c ! 2. This

agrees with Section 7.2.6.2. Recall that the chi-squared

distribution is approximate, for large data sets. The

deviance was used as a backup check, to help ensure

that the data set was large enough.

The treatment of p in Section 7.4.6.2 is exactly parallel

to that of 8.

Finally, the chi-square test was used in Section 6.6.2.3.2

to test whether durations follow a distribution of an

assumed form. A duration was denoted as T. To be

specific, consider a case in which the assumed distribution of T had two unknown parameters. Possibilities

include the lognormal(:, F) distribution and the

gamma(", $). The hypotheses in this setting were:

H0: T has a distribution of the assumed form.

H1: T has some other distribution.

The data were partitioned into c bins, so that every

observed value of T fell into one bin. Only the counts

in the bins were used, not the individual duration times.

If H0 is true, there are two unknown parameters (: and

F, or " and $, or whatever the two parameters of the

assumed distribution are).

If H1 is true, there are c ! 1 parameters. These parameters are Pr(T falls in bin 1), Pr(T falls in bin 2), etc.

There are c bins, but only c ! 1 parameters, because the

probabilities must sum to 1.0. Thus, any c ! 1 parameters determine the final one.

The big general theorem should say that the degrees of

freedom are (c ! 1) ! 2. However, a subtle complication arises. An assumption of the big theorem is that

the two unknown parameters of the distribution are

estimated using the maximum likelihood estimated

based on the bin counts. In this setting, however, it is

far easier to estimate those parameters from the raw

data. Therefore, as stated in Section 6.6.2.3.2, the

degrees of freedom fall somewhere between c ! 3 and

c ! 1.

To summarize this section, the Pearson chi-squared test

has many applications. To avoid confusion, the analyst

must clearly specify the null hypothesis being tested and

the alternative hypothesis.

7.5.3 Nonparametric Estimators of 8(t)

The nonparametric estimators of a density in Section 6.6.3 can also be used to estimate a time-dependent

event-occurrence rate 8(t). In each case, the data

consist of a number of times, durations in Section 6.6.3

and event times in the present case. The only difference

is the scale: a density integrates to 1.0, and an occurrence rate does not. To use an estimator from Section

6.6.3 in the occurrence-rate setting, multiply the estimate by the total number of observed events. This is

the correct scale correction.

The estimators in Section 6.6.3.1.1 showed a problem

at 0, estimating a positive density to the left of zero

even though a duration time cannot be negative. This

problem was corrected by reflecting the data around 0,

and initially using a data set that contained both the true

data and the mirror images (a value at !t for every

value of t). Such a problem also occurs with estimation

of 8(t). If data are collected in a time period from J0 to

J1, simple kernel estimates will have this problem at

both ends. To correct this problem, reflect the data at

each end, so that artificial data have been constructed

beyond J0 on the left and beyond J1 on the right.

Construct the density estimate based on this augmented

data set. Then truncate the density — set it to zero

outside the observation interval, and multiply it by 3 so

that it again integrates to 1.0. Finally, convert it from

a density estimate to an estimate of the Poisson intensity

by multiplying it by the number of observed events.

When interpreting the resulting graphed function, be

aware that the estimate will be flat at the two ends, by

the way the estimate was constructed. The slope of the

line at the two ends cannot be used to draw inferences

about whether the Poisson intensity is changing.

This method was applied to the data of Table 7.3

(Example 2.1), unplanned scrams at a new reactor.

The normal kernel was used. The 34 event dates

were converted to consecutive days, and the standard deviation was 874 days. The formula 

h = 1.06 F n!1/5
Trends and Aging

7-26

0

5

10

15

20

1987 1988 1989 1990 1991 1992 1993 1994 1995

Year

Events/reactor-critical year

GC00 0462 1

Figure 7.20 Estimate of 8(t) using standard normal

kernel, with bandwidth h = 400.

0

5

10

15

20

25

30

1987 1988 1988 1990 1991 1992 1993 1994 1995

Year GC00 0462 2

Events/reactor-critical year Figure 7.21 Estimate of 8(t) using standard normal

kernel, with bandwidth h = 200.

resulted in h = 457 days. This is only a very preliminary suggested value for two reasons: F was

estimated, not known, and the true shape of the

intensity function is not close to normal. Because it

is wise to undersmooth rather than oversmooth, a

value h = 400 days was used.

The estimated Poisson intensity function, 8(t), is

shown in Figure 7.20. The calculations were performed in terms of calendar days, and converted to

reactor-critical-years by assuming 0.82 critical years

per calendar year. This is the average of the values

shown in the right column of Table 7.4.

The curve in Figure 7.20 can be compared with the

exponential curve in Figure 7.12. The curve using

the kernel estimator follows the ups and downs of the

data more closely than does the simple exponential

curve. However, the kernel-estimator curve at the

beginning of 1987 is only 17.4, substantially below

the simple maximum likelihood estimate of 26.8,

based on the 1987 data only. Two factors contribute

to this. First, the bandwidth of h = 400 days is

apparently too wide. The learning period at the plant

lasted less than one year, so a smaller bandwidth

would better reflect the rapid learning that was taking

place. Second, the conversion from calendar time to

reactor-critical-years used the average value for nine

years, 0.82. In fact, the first calendar year had only

0.71 critical years. Therefore, the estimate during

the first year should be about 15% larger than

shown.

Figure 7.21 shows the kernel estimator with a smaller

bandwidth, h = 200 days. It follows the rapid drop in

the scram frequency at the beginning of the plant’s

history much more closely. It also is more sensitive

to small, perhaps random, clusters of events later in

the plant’s history. The constant conversion rate of

0.82 reactor-critical-years per calendar year has

been used, with the same effect as in the previous

figure.

This example has illustrated both some advantages and

some difficulties of nonparametric estimation of a

Poisson intensity function. The same advantages and

difficulties were seen for nonparametric density estimation. See Section 6.6.3 for more discussion of these

issues.
8-1

Figure 8.1 Hierarchical model for Poisson data.

8. PARAMETER ESTIMATION USING DATA 

FROM DIFFERENT SOURCES

8.1 The Hierarchical Model

This chapter contains the most complex mathematics of

the handbook, although most of the formulas can be

skipped by those who use preprogrammed software.

The formulas are of interest primarily to those who

must write the programs. Comments throughout the

chapter state which sections can be skipped by most

readers.

In this chapter, data come from a number of similar, but

not identical, sources. For simplicity, the discussion is

in terms of data from a number of nuclear power plants.

However, the ideas can be applied much more widely.

The situation is described by a hierarchical model,

with two levels. The first level models the plants as a

family, with the members resembling each other. The

second level models the data that are generated at each

plant.

To be more specific, suppose that initiating events are

to be modeled, so the parameter of interest is 8.

Level 1 of the model says that 8 varies among the m

plants, but only to a limited degree. Thus, the plants are

not identical, but they resemble each other. This is

modeled by a distribution g that describes the population variability. Before any data are generated, the

distribution g is invoked m times, producing values 81

through 8m. These values of 8i are independently

generated, but they all come from the same distribution,

g. For each i, 8i

 is assigned to plant i. That is Level 1

of the hierarchical model. It is shown on the left side of

Figure 8.1.

Level 2 of the model says that, conditional on the 8i

values, the plants independently produce data. Thus,

for each i, plant i is observed for time ti

, and it experiences a random number of initiating events, Xi

, with Xi

having a Poisson(8iti) distribution. This is shown on the

right side of Figure 8.1.

The population-variability distribution g could be a

gamma(", $) distribution, which has the computational

advantage of being conjugate to the Poisson distribution. But that is not the only allowed distribution. It

could also be a lognormal(:, F2

) distribution, or some

other distribution.

The data consist of the observation times, t1 through tm,

which are known and treated as fixed, and the event

counts, x1 through xm, which are treated as randomly

generated. The unknown parameters consist of 81

through 8m, as well as any unknown parameters of g.

These parameters of g could be " and $, or : and F2

, or

some other parameters, depending on the assumed form

of g. To emphasize the contrast between the two levels,

the parameters of g, such as " and $ or : and F2

, are

sometimes called hyperparameters.

When the data instead are failures on demand, the

situation is very similar. The population-variability

distribution generates parameters p, one for each plant.

The distribution g might be a beta distribution, or it

might be some nonconjugate distribution, such as

(truncated) lognormal or logistic-normal. The unknown

parameters consist of the parameters of g, and the

parameters p1 through pm. The data consist of the

counts of failures and demands at each plant, (x1, n1)

through (xm, nm). Examples 8.1 and 8.2 illustrate the

two types of data.
Parameter Estimation Using Data From Different Sources 

8-2

Example 8.1 Initiating events from many plants.

The number of unplanned scrams at power, x, and the number of 1000 critical hours, t, are listed below for one

calendar year (1984) at 66 plants. The data are from Martz et al. (1999).

Plant x t Plant x t Plant x t Plant x t

Arkansas 1

Arkansas 2

Beaver Valley 1

Big Rock Point

Brunswick 2

Callaway

Calvert Cliffs 1

Cook 1

Cook 2

Cooper Station

Crystal River 3

Davis-Besse

Diablo Canyon 1

Dresden 2

Dresden 3

Duane Arnold

Farley 1

3

12

4

2

3

12

5

3

7

3

2

4

5

3

8

6

2

6.2500

7.6433

6.4516

6.8966

2.6549

1.5038

7.5758

8.1081

5.3030

6.0000

8.3333

5.5556

1.0846

6.5217

3.8835

6.5934

6.8966

Farley 2

Fort Calhoun

Ginna

Grand Gulf

Haddam Neck

Hatch 1

Hatch 2

Indian Point 2

Indian Point 3

Kewaunee

LaSalle 1

LaSalle 2

Maine Yankee

McGuire 1

McGuire 2

Millstone 1

Millstone 2

6

1

1

7

3

7

7

4

7

4

9

11

7

4

16

0

3

8.3333

5.2632

6.6667

2.0896

6.5217

5.6452

3.1111

4.7059

6.9307

7.5472

6.2937

5.4726

6.6667

6.0606

6.9869

6.9902

8.5714

Monticello

North Anna 1

North Anna 2

Oconee 1

Oconee 2

Oconee 3

Oyster Creek

Palisades

Pt. Beach 1

Pt. Beach 2

Prairie Island 1

Prairie Island 2

Quad Cities 1

Quad Cities 2

Robinson 2

Salem 1

Salem 2

0

8

4

3

0

4

2

1

0

0

4

0

3

2

0

10

10

0.8106

4.7619

6.1538

7.5000

8.7840

6.5574

1.6949

1.5625

6.4201

7.5442

8.3333

7.8440

4.7619

6.8966

 0.6161

2.6738

3.3898

San Onofre 2

San Onofre 3

St. Lucie 1

St. Lucie 2

Summer

Surry 1

Surry 2

Susquehanna 1

Susquehanna 2

Turkey Point 3

Turkey Point 4

Vermont Yank.

Wash. Nucl. 2

Zion 1

Zion 2

5

7

6

9

11

8

14

7

7

8

9

2

23

6

7

5.2632

5.0725

5.5556

7.3770

5.5556

5.2980

7.4468

6.5421

2.1472

7.3394

5.0847

7.1429

4.3643

6.3158

6.3063

Example 8.2 Failure to start of AFW motor-driven segments at many plants.

The number of failures to start on unplanned demands for motor-driven segments of the auxiliary feedwater

(AFW) system are tabulated for 68 plants, for 1987-1995. Here, x is the number of failures and n is the number

of demands. Common-cause failures are excluded. The data are from Poloski et al. (1998, Table E-4).

Plant x n Plant x n Plant x n Plant x n

Arkansas 1

Arkansas 2

Beaver Valley 1

Beaver Valley 2

Braidwood 1

Braidwood 2

Byron 1

Byron 2

Callaway

Calvert Cliffs 1

Calvert Cliffs 2

Catawba 1

Catawba 2

Comanche Pk 1

Comanche Pk 2

Cook 1

Cook 2

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 14

 9

 24

 43

 13

 24

 11

 26

 57

 12

 15

 41

 89

 66

 14

 18

 36

Crystal River 3

Diablo Canyon 1

Diablo Canyon 2

Farley 1

Farley 2

Fort Calhoun

Ginna

Harris

Indian Point 2

Indian Point 3

Kewaunee

Maine Yankee

McGuire 1

McGuire 2

Millstone 2

Millstone 3

North Anna 1

 1

 0

 0

 0

 0

 0

 0

 0

 1

 2

 0

 0

 0

 0

 1

 0

 0

 16

 46

 30

 34

 54

 5

 28

 98

 24

 32

 26

 23

 45

 44

 11

 54

 20

North Anna 2

Oconee 1

Oconee 2

Oconee 3

Palisades

Palo Verde 1

Palo Verde 2

Palo Verde 3

Point Beach 1

Point Beach 2

Prairie Island 1

Prairie Island 2

Robinson 2

Salem 1

Salem 2

San Onofre 2

San Onofre 3

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 1

 0

 0

 0

 0

 18

 18

 18

 12

 13

 7

 12

 9

 8

 16

 3

 7

 28

 24

 32

 13

 17

Seabrook

Sequoyah 1

Sequoyah 2

South Texas 1

South Texas 2

St. Lucie 1

St. Lucie 2

Summer

Surry 1

Surry 2

Three Mile Island 1

Vogtle 1

Vogtle 2

Waterford 3

Wolf Creek

Zion 1

Zion 2

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 0

 17

 30

 41

 69

 87

 35

 21

 24

 26

 32

 6

103

 45

 38

 51

 13

 8

In Example 8.1, most of the plants experience at least

one initiating event, and the total number of events is

361. Thus, the data set is large, and the methods given

below perform well. Example 8.2, on the other hand, is

a small data set. That is, most of the plants experienced

no failures, and the total number of failures is only 6.

This example was deliberately chosen for this handbook

to illustrate problems that can occur with sparse data.

Two methods are given in Sections 8.2 and 8.3 for

analyzing data by means of a hierarchical model. The

results of each analysis include both an estimate of the

population-variability distribution, g, and estimates of

all the plant-specific parameters, 8i

 or pi.
Parameter Estimation Using Data From Different Sources 

8-3

8.2 The Parametric Empirical

Bayes Method

8.2.1 General Approach

In spite of the name, this is not a truly Bayesian method. Instead, it is a kind of hybrid, involving a nonBayesian step followed by a Bayesian step.

Step 1. Look at the variability in the data from the

plants, and estimate g. That is, based on the

data from all the plants, estimate the parameters of g by maximum likelihood, and obtain

the resulting estimate of the distribution. Call

the estimate . g$

Step 2. Now treat as a prior distribution. Perform g$

the usual Bayesian update, with the prior

distribution and data from a single plant, g$

to get the posterior distribution for the plantspecific parameter, 8i

 or pi.

Thus, the method yields both an estimate of the population variability and plant-specific estimates at each

plant.

The method as just explained underestimates the

uncertainty in the answers, because it treats as if it g$

were equal to the true distribution g. Therefore, the

best implementations of the empirical Bayes method

add a final adjustment to Step 2, which makes the

plant-specific posterior distributions somewhat more

diffuse. This largely accounts for the inaccuracy in

equating to g$ g.

The name “empirical Bayes” follows from the two

steps. The plant-specific estimates are found in a

Bayesian way, by updating a prior distribution with

plant-specific data. However, the prior is not based on

prior data or on prior belief, but instead is based on the

existing data — the prior is determined empirically.

Step 1 can be carried out in a simple way only if the

distributions have convenient forms. Thus, parametric

empirical Bayes estimation assumes that g is conjugate

to the distribution of the data at any plant. That is, g is

a gamma distribution when the data are initiating

events or other data from a Poisson process, and g is a

beta distribution when the data are failures on demand.

This is a limitation to the method. One reason for

introducing the hierarchical Bayes method, in Section

8.3, is to overcome this limitation.

Some people might object that the method double

counts the data. It uses the data to decide on the prior

distribution, and then it uses the same data again to

update the prior to obtain the plant-specific estimates.

There are two responses to this: (1) The objection is

not important in practice, unless the number of plants in

the study is very small, or if a small number of plants

dominate the data. If no single plant contributes much

to the estimate of g, then there is very little double

counting that influences the final estimate for that plant.

(2) The hierarchical Bayes method, given in Section

8.3, will avoid this difficulty entirely.

For failures on demand, Martz et al. (1996) give a

tutorial on the empirical Bayes method, illustrated with

nuclear power plant (NPP) data. Siu and Kelly (1998)

also explain the method as part of their tutorial article.

Carlin and Louis (2000) give a full treatment, including

worked-out examples.

8.2.2 MLE Equations for the GammaPoisson Model

Readers who do not need to know the equations can

skip directly to Section 8.2.5.

The gamma-Poisson model is used for initiating events.

The data at plant i consist of a count of events, xi

, in

time ti

. Conditional on the plant-specific parameter 8i,

it is assumed that xi

 was generated from a Poisson(8iti)

distribution. However 8i

 was generated from the distribution g, which is assumed to be gamma(", $). The

equations for the maximum likelihood estimates

(MLEs) of " and $ are now given.

The conditional distribution of X, conditional on 8, is

Poisson. However, the unconditional distribution of

X, when 8 might be any value generated by the

population-variability distribution g, is more complicated. It can be shown that the unconditional distribution equals the distribution conditional on 8, averaged

over the possible values of 8. In equation form, this is

Pr( | , ) Pr( | ) ( | , ) Xx Xx g d === . ∫ α β λ λα β λ

Substituting the formulas for the Poisson and gamma

distributions, it can be shown that this equals

(8.1)

Pr( | , )

( )

!() ( / )( / ) ( )

X x

x

n

t t x x

=

= ++ − +

α β

α

α

β β α . Γ

Γ

1
Parameter Estimation Using Data From Different Sources 

8-4

As mentioned in Section 6.2.3.5, Equation 6.8, this is

the gamma-Poisson distribution, also often called the

negative binomial distribution. This distribution is not

conditional on 8. Therefore, 8 does not appear in this

expression. Instead, Equation 8.1 gives the probability

of seeing x events in time t at a plant with a randomly

assigned 8. 

To find the MLEs of " and $, write the product of

terms of the form of Expression 8.1, using values (xi

, ti)

for i from 1 to m. That product is the joint unconditional likelihood of the data. Take the logarithm of that

expression and maximize it. There are several ways to

accomplish this.

One approach is to maximize it numerically as a

function of two variables, using some version of Newton’s method. This is the approach of Siu and Kelly

(1998). The derivatives of the log-likelihood, used in

performing the maximization, are given below, as

stated by Engelhardt (1994). Here ln L denotes the

logarithm of the likelihood.

∂

∂αln [ ( ) ( ) ln( / )] Lx t ψα ψα β i i

i

m

= +− − + =

∑ 1 1

 . ∂

∂β β

α

β

ln L x x

t

t i

i

i

i

i

m

=− − +

+

⎡

⎣

⎢ ⎤

⎦

⎥ =

∑

1

1

The function R is the digamma function, R(u) =

(d/du)ln'(u). It is built into many computer packages.

Because xi

 is an integer, the expression involving R can

be rewritten as

ψα ψα . (8.2)

α

( ) () +− = = + − x ∑ j i

j

xi 1

1 1

A second approach reduces the problem to solving one

equation, as follows. At the maximum of the loglikelihood, the two derivatives are equal to zero.

Therefore, do the following:

1. Set the two derivatives equal to zero. The solutions, to be found, are and . α$ $

β

2. Solve the second equation for , as a function of α$

. $

β

3. Substitute this expression into the first equation.

4. Solve the resulting equation numerically for . $

β

5. Calculate from the numerical value of . α$ $

β

The necessary equations to carry out these steps are the

following. The equation for , as a function of , is α$ $

β

$ . (8.3) $

$ $ α β

β β = +

⎛

⎝

⎜ ⎞

⎠

⎟

+

⎛

⎝

⎜ ⎞

⎠

⎟ = =

∑ ∑ x

t

t

t

i

i i

m

i

i i

m

1 1

Substitute Equation 8.3 into 

[ ( $ ) ( $) ln( / $ ψα ψα β +− − + = )] =

∑ x t i i

i

m

1 0 1

and solve that equation numerically for . Having $

β

obtained the numerical value of , find from $

β α$

Equation 8.3.

Sometimes the equations do not have a solution. If the

plants do not appear to differ much — for example, the

naive plant-specific estimates xi /ti

 are all similar — the

maximum likelihood estimate of g may be degenerate,

concentrated at a single point. That says that the plants

appear to have a single common 8. Engelhardt (1994)

recommends aborting the estimation process, not trying

to fit a model, if the estimate of $ becomes greater than

Eti during the iterations. The population-variability

distribution g would be gamma(", $), with the second

parameter greater than Eti

. But simply pooling the data

(and using a Jeffreys prior) would result in a gamma(Exi

 + 1⁄2, Eti) posterior distribution. Thus the

empirical Bayes distribution would produce a betweenplant distribution that is more concentrated (larger

second parameter) than the distribution when the plant

data are pooled. This is not the intent of the hierarchical model.

8.2.3 MLE Equations for the 

Beta-Binomial Model

The beta-binomial model is used for failures on demand. The data at plant i consist of a count of failures,

xi

, and demands, ni. Conditional on the plant-specific

parameter pi

, it is assumed that xi was generated from a

binomial(ni

, pi) distribution. However, pi was generated

from the distribution g, which is assumed to be beta(",

$). The equations for the MLEs of " and $ are now

given.

The conditional distribution of X, conditional on p, is

binomial. However, the unconditional distribution of

X, when p might be any value generated by the

population-variability distribution g, equals the distribution conditional on p, averaged over the possible values

of p. That is,
Parameter Estimation Using Data From Different Sources 

8-5

Pr( | , ) Pr( | ) ( | , ) X x X x p g p dp === . ∫ αβ αβ

Substituting the formulas for the binomial and beta

distributions, and using some standard identities

relating the beta function and the gamma function, it

can be shown that this equals

Pr( )

!

!( )!

( )

( )

( )

( )

( )

( )

X x

n

xn x

x nx

n

=

= −

+ +− +

+ +

Γ

Γ

Γ

Γ

Γ

Γ

α

α

β

β

α β

α β .

This is called the beta-binomial distribution. If both x

and n are integers, this can be rewritten without the

gamma function, as follows:

Pr( )

!

!( )! ( )( ) ( )

X x

n

xn x jj j

j

n x

j

x

j

n

=

= − + + ++ =

− −

=

−

=

−

∏ ∏ α β αβ ∏0

1

0

1

0

1

 .

As just stated, this is valid if x and n are integers. Are

they not always integers? No, they are not, if the data

set only gives an estimate of the number of demands,

which is not necessarily an integer. In that case, the

expression with the gamma function is the only one that

can be used.

The likelihood is the product of terms of one of these

forms, containing values (xi

, ni) for i = 1 to m. To find

the MLE, take the logarithm of the likelihood and

maximize it.

The maximization can be done in a variety of ways.

One approach, following Atwood (1994), does not deal

with " and $ directly. Instead, it reparameterizes,

working with

* = " + $ and

: = "/* .

The intuitive reason for this reparameterization is that

: is the mean of the binomial distribution, and in most

models the mean is one of the easiest things to estimate. The letter * was chosen as a mnemonic for

“dispersion,” because the variance of the binomial

distribution is :(1 ! :)/(* + 1). Thus, * is related

directly to the variance. Recall that in Section

6.3.2.2.2, the prior and posterior values of " + $, which

we are calling * here, were interpreted as the prior and

posterior number of demands.

After working with : and *, and finding the MLEs of

these parameters, we will translate back to find the

MLEs of " and $, using the equations:

" = :*

$ = (1 ! :)* . 

The MLE is found by setting the derivatives with

respect to : and * to zero. After some manipulation,

the equations can be expressed as

(8.4a)

{ }

{ }

ψ μδ ψ μδ

ψ μδ ψ μδ

( ) ()

(( ) ) (( ) )

+ −

= − +− − −

=

=

∑

∑

x

n x

i

j

m

i i

i

m

1

1

1 1

(8.4b)

{ }

{ }

ψ μδ ψ μδ

ψδ ψδ

(( ) ) (( ) )

( ) ()

1 1 1

1

− +− − −

= +−

=

=

∑

∑

n x

n

i i

j

m

i

j

m

 .

Here R is the digamma function, the derivative of ln',

just as in Section 8.2.2. If xi

 and ni are integers for all

i, Equation 8.2 can be used to rewrite Equation 8.4 as

(8.5a)

1

1

1

0

1

1

0

1

1

μδ

μ δ

+

∑

⎡

⎣

⎢ ⎤

⎦ ∑ ⎥

= − +

∑ ⎡

⎣

⎢ ⎤

⎦ ∑ ⎥

=

−

=

=

− −

=

j

j

j

x

i

m

j

n x

i

m

i

i i

( )

and

(8.5b)

1

1

1

0

1

1

0

1

1

( ) − +

∑ ⎡

⎣

⎢ ⎤

⎦ ∑ ⎥

= +

∑

⎡

⎣

⎢ ⎤

⎦ ∑ ⎥

=

− −

=

=

−

=

μ δ

δ

j

j

j

n x

i

m

j

n

i

m

i i

i

 .

The Equations 8.4 or 8.5 must be solved for : and *.

One method, suggested by Atwood (1994) is to begin

with a trial value of *. Solve Equation 8.4a or 8.5a

numerically for :. This typically needs only a few

iterations. Substitute this value into Equation 8.4b or

8.5b, and solve the resulting equation for *. Continue

alternating between the two equations until the estimates converge.

The estimates do not always converge. If the plants

have very similar data, the maximum likelihood estimate of g may be concentrated at a single point, degen-
Parameter Estimation Using Data From Different Sources 

8-6

erate. This would say that the plants all have the same

p. Atwood (1994) recommends aborting the iterations

if the value of * becomes greater than Eni

. Allowing *

to be greater than Eni would produce a populationvariability distribution that is more concentrated than

the distribution corresponding to simply pooling the

data.

8.2.4 Adjustment for Uncertainty in the 

Estimate of g

As mentioned above, the method as presented so far

underestimates the uncertainty in the final answers,

because it does not account for the uncertainty in . g$

Kass and Steffey (1989) present a correction to the

final estimates, to approximately account for this

uncertainty. The plant-specific posterior means are

unchanged, but the posterior variances are increased

somewhat. Kass and Steffey (1998) state that the

adjustment is very important if there are few data

subsets (plants, in the present discussion) and many

observations (initiating events or demands). Conversely, the adjustment is unimportant when there are

many data subsets and few observations. No harm is

done by automatically applying the adjustment in every

case. The formulas are given here.

8.2.4.1 Equations for the Gamma-Poisson Case

With the gamma-Poisson distribution, it is computationally advantageous to reparameterize in terms of :

= "/$ and ". Denote the maximum likelihood estimators for the hyperparameters : and " by and . It μ$ α$

turns out that these estimators are asymptotically

uncorrelated, causing certain terms in the formulas to

be zero.

The method as given in Section 8.2.2 finds the estimates and , which can be reparameterized α$ $

β

as and . These are the estimated parame- $ $ / $ μ αβ = α$

ters of the gamma prior distribution g. The method

then updates the estimated prior by plant-specific data.

The posterior distribution of 8i

 is also a gamma distribution, with posterior mean

E (8.6) x

t i

i

i

post( ) $

$ / $ λ α

α μ = ++

and posterior variance 

 . ( ) var ( ) $

$ / $ post λ α

α μ i

i

i

x

t = +

+ 2

The Kass-Steffey adjustment increases the variance to

(8.7)

var ( )

( )

$ var( $) ( )$ var( $)

post

post post .

λ

∂ λ

∂μ μ

∂ λ

∂α α

i

E E i i +

⎡

⎣

⎢ ⎤

⎦

⎥ +

⎡

⎣

⎢ ⎤

⎦

⎥

2 2

A covariance term would normally also be present, but

this term is zero when the parameterization is in terms

of : and ".

We now develop the formulas that must be substituted

into Equation 8.7. From Equation 8.6, the derivatives

of are Epost i ( ) λ

 and

∂ λ

∂μ

α α

α μ

E x

t

i i

i

post( )

$

$( $ )

( $ $ ) = − +

+ 2

 . ∂ λ

∂α

μ μ

α μ

E x t

x

i i i

i

post( )

$

$( $ )

( $ $) = − −

+ 2

From the asymptotic theory of maximum likelihood

estimation, the variances are found as follows. The

information matrix, J, is the negative of the matrix of

second derivatives of the log-likelihood:

J (8.8)

J J

J J

E

L L

L L = ⎡

⎣

⎢ ⎤

⎦

⎥ = −

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

11 12

21 22

2

2

2

2 2

2

∂

∂μ

∂

∂μ∂α

∂

∂μ∂α

∂

∂α

evaluated at and . The inverse of this matrix is μ$ α$

asymptotically equal to the variance-covariance matrix:

 . var( $) cov( $, $)

cov( $, $) var( $)

μ μα

μα α

⎡

⎣

⎢ ⎤

⎦

⎥

When this is carried out, we have

J t

t

i

i i

m

11

1

= ∑= +

$

$ $$

α

μ αμ

J x [ ] t

t i

i

m

i

i i

m

22

1 1

= ′ − ′ + − ∑ ∑ = = +ψα ψα μ

α αμ

( $) ( $ ) $

$ $$

 . J J 12 21 = = 0
Parameter Estimation Using Data From Different Sources 

8-7

If all the xi

 values are integers, the difference of ψ

terms can be rewritten using Equation 8.2, and the

difference of derivatives can be written algebra- ψ′

ically, avoiding use of a special function.

Because the off-diagonal elements are zero, the inverse

consists of the inverses of the diagonal terms, and it

follows that

var(μ$) / = 1 and 11 J

var(α$) / = 1 . 22 J

The final step of the empirical Bayes method is to

substitute the expressions just found into the KassSteffey adjustment for the posterior variance, Equation 8.7. Then approximate the posterior distribution

by a gamma distribution having the original posterior

mean and the adjusted posterior variance. An example

will be given below.

8.2.4.2 Equations for the Beta-Binomial Case

As in Section 8.2.3, we parameterize in terms of : =

"/(" + $) and * = " + $. Denote the maximum likelihood estimators by and . Altho μ$ ugh these estima- $

δ

tors are asymptotically not exactly uncorrelated, as was

the case for the gamma-Poisson model, they are nearly

uncorrelated. The equations are given by Atwood

(1995).

The method as given in Section 8.2.3 finds the estimates and , the estimated parameters of the beta μ$ $

δ

prior distribution g. The method then updates the

estimated prior by plant-specific data. The posterior

distribution of pi

 is also a beta distribution, with

posterior mean

E p (8.9) x

n i

i

i

post( ) $ $

$ = +

+

μδ

δ

and posterior variance

var ( ) ( )[ ( )] / ( . $ ) post post post pEp Ep n i i ii = − ++ 1 1 δ

The Kass-Steffey adjustment increases the variance to

(8.10)

var ( )

( )

$ var( $) ( )

$ var( $)

( )

$

( )

$ cov( $, $)

post

post post

post post .

p

Ep Ep

Ep Ep

i

i i

i i

+

⎡

⎣

⎢ ⎤

⎦

⎥ +

⎡

⎣

⎢ ⎤

⎦

⎥

+

⎡

⎣

⎢ ⎤

⎦

⎥

⎡

⎣

⎢ ⎤

⎦

⎥

∂

∂μ μ

∂

∂δ

δ

∂

∂μ

∂

∂δ μ δ

2 2

From Equation 8.9, the two derivatives of are E p post i ( )

∂

∂μ

δ

δ

E p

n

i

i

post( )

$

$

$ = +

 . ∂

∂δ

μ

δ

E p x n

n

i i i

i

post( )

$

$

( $ ) = − −

+ 2

The variances and covariance are found from inverting

the matrix in Equation 8.8, with * used now instead of

". The terms can be found as follows. Define

[ ] S xi

i

m

1

1

= ′ − ′ + =

∑ ψ μδ ψ μδ ( $ $) ( $ $ )

[ ] S nx i i

i

m

2

1

= ′ 1 1 − − ′ − +− =

∑ ψ μδ ψ μδ (( $) $) (( $) $ )

 . [ ] S ni

i

m

3

1

= ′ − ′ + =

∑ ψδ ψδ ( $) ( $ )

Then the information matrix is given by

J SS 11

2 = +1 2 $

δ ( )

J S SS 22

2

1

2 = +− − 1 2 3 μ μ $ ( $)

JJ S S . 12 21 1 2 = = −− $ 1 $ ( $) $ μδ μ δ 

The variances and covariance follow from standard

formulas for inverting a 2×2 matrix. Define the determinant

D = J11J22 ! (J12)

2

 .

Then we have

var(μ$) / = J D 22

var( $δ ) / = J D 11

cov( $, . $ μ δ ) / = −J D 12

To complete the Kass-Steffey adjustment, substitute the

above expressions into the equation for the adjusted

variance, Equation 8.10. Then find the beta distribution

having the posterior mean and the adjusted posterior

variance. Use this as the approximate posterior distribution for pi

.

8.2.5 Application to Examples

The parametric empirical Bayes method is now illustrated with Examples 8.1 and 8.2. First, a chi-squared

test will be performed, to test whether the plants can be
Parameter Estimation Using Data From Different Sources 

8-8

1.E-03 1.E-02 1.E-01 1.E-00 1.E+01 1.E+02

Industry

Callaway

Wash. Nuclear 2

Diablo Canyon 1

Salem 1

Grand Gulf

Susquehanna 2

Salem 2

McGuire 2

Hatch 2

Dresden 3

46 other plants

Crystal River 3

Fort Calhoun

Ginna

Robinson 2

Monticello

Point Beach 1

Millstone 1

Point Beach 2

Prairie Island 2

Oconee 2

Events/1000-Critical-Hours GC00 0419 1

Figure 8.2 Plant-specific MLEs and 90% confidence

intervals for 8.

Industry

Callaway

Wash. Nuclear 2

Salem 1

Diablo Canyon 1

Grand Gulf

Susquehanna 2

Salem 2

McGuire 2

Hatch 2

LaSalle 2

46 other plants

Quad Cities 2

Vermont Yankee

Fort Calhoun

Crystal River 3

Ginna

Point Beach 1

Millstone 1

Point Beach 2

Prairie Island 2

Oconee 2

Events/1000-Critical-Hours GC00 0419 2

1.E-03 1.E-02 1.E-01 1.E-00 1.E+01 1.E+02

Figure 8.3 Plant-specific posterior means and 90%

credible intervals for 8.

pooled. In each example, the difference between plants

will be found to be either statistically significant or

very close to statistically significant. Then plantspecific confidence intervals will be found, each based

only on the data for a single plant. Then the empirical

Bayes method will be used, and the resulting 90%

credible intervals will be shown, based on the plantspecific posterior distributions, using the Kass-Steffey

adjustment. The plant-specific intervals resulting from

the empirical Bayes analysis will be compared to the

(less sophisticated) plant-specific confidence intervals.

8.2.5.1 Example 8.1, Initiating Events

To test poolability of the plants in Example 8.1, the

Pearson chi-squared test was performed, as presented in Section 6.2.3.1.2. The test statistic X2

 was

equal to 378.5. Because there were 66 plants, the

value of X2

 should be compared to a chi-squared

distribution with 65 degrees of freedom. The value

of 378.5 is very far out in the tail of the chi-squared

distribution, off the table. Thus, the evidence is

extremely strong, beyond question, that the plants

do not all have the same 8.

To show this graphically, 90% confidence intervals

for 8 were plotted, with each confidence interval

based on the data from a single plant. These are

shown in Figure 8.2. Because t has been written in

terms of 1,000 critical hours, the units of 8 are events

per thousand critical hours. The order of the plants

is not alphabetical, but instead is by decreasing

estimate of 8. Because the example has so many

plants, only the plants with the 10 highest and 10

lowest estimates are individually identified in the

figure.

A 90% confidence interval is plotted at the top of the

plot for the pooled industry data. Of course the

interval is too short to be realistic, because pooling

of the data is completely unjustified in this example.

In fact, the interval is too short even to be visible.

Nevertheless, the overall pooled mean is a useful

reference value for comparison with the individual

plant results. For this reason, a vertical dashed line

is drawn through the pooled mean.

Because the plant-specific estimates differ so greatly, the figure uses a logarithmic scale. This means

that some of the point estimates, those with zero

values, cannot be plotted.

Figure 8.3 is based on the empirical Bayes method.

For each plant, the mean and 90% credible interval

are shown, based on the posterior distribution and

the Kass-Steffey adjustment. The mean and 90%

interval for the industry are also plotted, and a

vertical dashed line is drawn through the mean.
Parameter Estimation Using Data From Different Sources 

8-9

The most striking feature seen by comparing the two

figures is that the empirical Bayes estimates vary

less from each other than do the MLEs. Of course,

if a plant has no events, the lower confidence limit is

zero, and any Bayesian method will give a non-zero

lower limit. Such a difference appears enormous

when plotted on a logarithmic scale. However, the

effect is seen not only at the bottom of Figures 8.2

and 8.3 but also at the top: the largest plant-specific

posterior means are closer to the industry average

than are the corresponding MLEs. Indeed, just as

was seen for Bayes methods in general, the empirical Bayes method gives posterior means that are

between the MLEs and the industry (i.e., the prior)

mean.

Those who wish to make some detailed comparisons can find a few numerical values listed in Tables

8.1 through 8.3.

Table 8.1 Portion of frequentist analysis 

results for Example 8.1.

Plant x, t MLE and 90% 

confidence intervala

Industry

Callaway

Wash. Nuc. 2

Diablo Can. 1

Salem 1

Grand Gulf

Pt. Beach 2

Prairie Isl. 2

Oconee 2

361, 374.229

 12, 1.5038

 23, 4.3643

 5, 1.0846

 10, 2.6738

 7, 2.0896

 0, 7.5442

 0, 7.8440

 0, 8.7840

(0.883, 0.965, 1.05)

 (4.60, 7.98, 12.9)

 (3.60, 5.27, 7.47)

 (1.82, 4.61, 9.69)

 (2.03, 3.74, 6.34)

 (1.57, 3.35, 6.29)

 (0.0, 0.0, 0.397)

 (0.0, 0.0, 0.382)

 (0.0, 0.0, 0.341)

a. Format is (lower bound, MLE, upper bound).

The order of the plants is not exactly the same in

Figures 8.2 and 8.3. The reason is that estimates

for different plants are pulled toward the industry

mean by different amounts. This can cause some

rearrangement of the ranking of the plants. For

example, Salem 1 and Diablo Canyon 1 appear in

reverse order in the two figures (and in Tables 8.1

and 8.3). The reason is that Diablo Canyon 1 has

about half as much data (5 events in 1,085 hours) as

Salem 1 (10 events in 2,674 hours). Therefore

Diablo Canyon 1 is pulled more toward the industry

mean.

We notice also, by comparing Tables 8.2 and 8.3,

that the Kass-Steffey adjustment is very small in this

example. The data set is so large that g can be

estimated quite well. Any error in equating the

estimate to the true distribution is minor, as reflected

in the small effect of the Kass-Steffey adjustment.

Table 8.2 Portion of empirical Bayes 

analysis, without Kass-Steffey

adjustment.

Plant Gamma 

parameters,

", $

Posterior mean and

90% credible interval a

Industry

Callaway

Wash. Nuc. 2

Salem 1

Diablo Can. 1

Grand Gulf

Pt. Beach 2

Prairie Isl. 2

Oconee 2

 1.39, 1.211

13.39, 2.715

24.39, 5.575

11.39, 3.885

 6.39, 2.296

 8.39, 3.301

 1.39, 8.755

 1.39, 9.055

 1.39, 9.995

(0.118, 1.15, 3.07)

 (2.94, 4.93, 7.34)

 (3.03, 4.37, 5.93)

 (1.66, 2.93, 4.49)

 (1.25, 2.78, 4.81)

 (1.29, 2.54, 4.14)

(0.0164, 0.159, 0.424)

(0.0158, 0.154, 0.410)

(0.0143, 0.139, 0.372)

a.

 Format is (lower bound, mean, upper bound).

Table 8.3 Portion of empirical Bayes analysis,

with Kass-Steffey adjustment.

Plant Gamma 

parameters,

", $

Posterior mean and

90% credible interval a

Industry

Callaway

Wash. Nuc. 2

Salem 1

Diablo Can. 1

Grand Gulf

Pt. Beach 2

Prairie Isl. 2

Oconee 2

 1.39, 1.211

12.13, 2.460

23.40, 5.348

11.03, 3.762

 6.08, 2.185

 8.15, 3.204

 1.33, 8.382

 1.33, 8.665

 1.33, 9.554

(0.118, 1.15, 3.07)

 (2.86, 4.93, 7.47)

 (3.00, 4.37, 5.96)

 (1.65, 2.93, 4.52)

 (1.22, 2.78, 4.86)

 (1.27, 2.54, 4.16)

(0.0151, 0.159, 0.431)

(0.0146, 0.154, 0.417)

(0.0132, 0.139, 0.378)

a.

 Format is (lower bound, mean, upper bound).

An empirical Bayes estimator is sometimes called a

shrinkage estimator, or a shrinker, because the

method pulls all the MLEs in towards the industry

mean. The intuitive justification for such shrinkage is

the recognition that extreme data are produced by a

combination of extreme parameters and luck. Thus, the

plant with the highest observed frequency appears so

extreme because of a combination of large 8 and some

bad luck. Likewise, the plant with the best performance, Oconee 2, which ran for 366 days straight

without a single scram, can attribute its perfect performance to a combination of low 8 and good luck. The

empirical Bayes method tries to remove the effect of

luck when estimating the 8 values.
Parameter Estimation Using Data From Different Sources 

8-10

1.E-03 1.E-02 1.E-01 1.E+00

Industry

Millstone 2

Crystal River 3

Indian Point 3

Indian Point 2

Robinson 2

(63 plants with

no failures)

p GC00 0419 3

1.E-04

Figure 8.4 Plant-specific MLEs and 90% confidence

intervals for p.

As always when performing a statistical analysis, one

should try to combine statistical calculations with

engineering understanding. It is known that newly

licensed plants sometimes experience more initiating

events than they do after acquiring more experience.

This was mentioned in the discussion of Example 2.1,

and it is seen again here.

Of the 66 plants, 9 did not have their commercial

starts until 1984 or later. These 9 young plants

are all among the 19 with the highest event frequencies. For example, consider the two plants

with the highest estimated frequencies, based on

the 1984 data. Both of these plants had their

commercial starts in December 1984.

The hierarchical model is intended for plants that are

nominally identical. The variability among the plants

is unexplained, and modeled as random. An important

assumption is that each plant is assigned a 8 from the

same distribution, g. As a result, each plant is as likely

as any other to have a large 8 or a small 8. The

parameters 8i

 are called exchangeable if any 8i is as

likely as any other to correspond to a particular plant.

As discussed by Gelman et al. (1995, Section 5.2),

when we know nothing about the plants, exchangeability is a reasonable assumption. When we know the ages

of the plants, however, exchangeability is no longer

reasonable. The most immature plants are expected to

have larger values of 8.

Thus, the analysis of Example 8.1 really should be

modified. One way would be to separate the plants into

two groups, mature and immature, and perform an

empirical Bayes analysis on each group. A more

sophisticated way would be to try to model the age of

the plant as a continuous explanatory variable. Then

the otherwise random 8i

 would be multiplied by some

function of the age of plant i, a large factor for immature plants and a smaller factor for mature plants. Such

models are beyond the scope of this handbook, however.

8.2.5.2 Example 8.2, AFW Segment Failures to

Start

This example has 68 plants, with 6 failures in 1993

demands.

Poloski et al. (1998) perform a chi-squared test to

see if p is the same at all plants. This test is explained in Section 6.3.3.1.2. The test statistic X2

equals 113.1. Because there are 68 plants, the

degrees of freedom is 67. The reported p-value is

0.0004, meaning that 113.1 is the 99.96th percentile

of the chi-squared distribution with 67 degrees of

freedom. However, the data set has so few failures

that the chi-squared distribution is not a good

approximation for the distribution of X2

. The expected number of failures at a plant with 30 demands

(a typical number of demands) is 6×30/1993 = 0.09.

This is much less than the recommended minimum

of 0.5. Therefore, the calculated p-value is quite

suspect.

Poloski et al. (1998) chose to model between-plant

differences with a hierarchical model, partly because

of the above calculated p-value, and partly on the

grounds that modeling possible differences between

plants is more conservative (reflecting more variability) than simply pooling the data.

The empirical Bayes estimate of the populationvariability distribution, g, is a beta (0.137, 36.34)

distribution. The mean of this distribution is

3.77E!3. The 5th and 95th percentiles are 5.99E!12

and 2.12E!2. Note, the first parameter of the beta

distribution is very small, well below 0.5. As a result,

the 5th percentile is unrealistically small, implying

less than one failure in one hundred billion demands.

This unrealistic lower bound carries over to the

posterior distribution of all plants that have zero

failures. Figures 8.4 and 8.5 are the analogues of

Figures 8.2 and 8.3.
Parameter Estimation Using Data From Different Sources 

8-11

1.E-03 1.E-02 1.E-01 1.E+00

p GC00 0419 4

 Industry

 Indian Point 3

 

Millstone 2

 Crystal River 3

 

 Indian Point 2

 

Robinson 2

(63 plants with

no failures)

1.E-04

Figure 8.5 Plant-specific posterior means and 90%

credible intervals for p.

The first figure shows plant-specific MLEs and 90%

confidence intervals, while the second shows the

results of the empirical Bayes analysis, posterior

means and 90% credible intervals. Only the five

plants that had failures are individually identified in

the figures.

Some numerical details are given in Tables 8.4

through 8.6.

Table 8.4 Portion of frequentist analysis 

results for Example 8.2.

Plant x, d MLE and 90% conf. int. a

Industry

Millstone 2

Crystal River 3

Indian Point 3

Indian Point 2

Robinson 2

Prairie Island 1

Vogtle 1

6, 1993

1, 11

1, 16

2, 32

1, 24

1, 28

0, 3

0, 103

(1.31E!3, 3.01E!3, 5.93E!3)

(4.65E!3, 9.09E!2, 3.64E!1)

(3.20E!3, 6.25E!2, 2.64E!1)

(1.12E!2. 6.25E!2, 1.84E!1)

(2.13E!3, 4.17E!2, 1.83E!1)

(1.83E!3, 3.57E!2, 1.59E!1)

(0.0, 0.0, 3.32E!1)

(0.0, 0.0, 2.87E!2)

a

 Format is (lower bound, MLE, upper bound).

Just as with Example 8.1, the empirical Bayes

method pulls the plant-specific MLEs toward the

industry mean. This is seen in both the figures and

the tables. Also, the Kass-Steffey adjustment increases the width of the plant-specific intervals by

a noticeable amount, for example, by about 30%

for Indian Point 3. This is best seen by comparing

Tables 8.5 and 8.6. This comparison shows that

the estimates of the parameters have noticeable

uncertainty, even if the assumption of a beta distribution is accepted.

Table 8.5 Portion of empirical Bayes analysis,

without Kass-Steffey adjustment.

Plant Beta parameters,

", $

Posterior mean and 90%

credible interval a

Industry

Indian Point 3

Millstone 2

Crystal River 3

Indian Point 2

Robinson 2

Prairie Isl. 1

Vogtle 1

0.137, 36.34

2.137, 68.34

1.137, 47.34

1.137, 52.34

1.137, 60.34

1.137, 64.34

0.137, 39.34

0.137, 139.3

(6.0E!11, 3.77E!3, 2.12E!2)

(6.14E!3, 3.12E!2, 7.15E!2)

(1.70E!3, 2.40E!2, 6.78E!2)

(1.53E!3, 2.17E!2, 6.14E!2)

(1.33E!3, 1.88E!2, 5.33E!2)

(1.24E!3, 1.76E!2, 5.01E!2)

(5.5E!12, 3.48E!3, 1.96E!2)

(1.6E!12, 9.86E!4, 5.52E!3)

a. Format is (lower bound, mean, upper bound).

Table 8.6 Portion of empirical Bayes analysis,

with Kass-Steffey adjustment.

Plant Beta parameters,

", $

Posterior mean and 90%

credible interval a

Industry

Indian Point 3

Millstone 2

Crystal R. 3

Indian Point 2

Robinson 2

Prairie Isl. 1

Vogtle 1

0.137, 36.34

1.149, 35.65

0.596, 24.29

0.663, 29.94

0.754, 39.34

0.793, 44.14

0.133, 37.98

0.127, 128.9

(6.0E!11, 3.77E!3, 2.12E!2)

(2.27E!3, 3.12E!2, 8.77E!2)

(2.26E!4, 2.40E!2, 8.54E!2)

(3.16E!4, 2.17E!2, 7.44E!2)

(4.34E!4, 1.88E!2, 6.17E!2)

(4.78E!4, 1.76E!2, 5.69E!2)

(2.6E!12, 3.48E!3, 1.97E!2)

(2.8E!13, 9.86E!4, 5.59E!3)

a.

 Format is (lower bound, mean, upper bound).

Poloski et al. (1998) carry out the above empirical

Bayes analysis, but they do not report the calculated lower bounds for plants that experience no

failures. They recognize that those lower bounds

are unrealistically small, and that such calculated

values are an artifact of using a beta distribution. 

Therefore, they simply report that the lower bound

is <1E-8. The next section gives a way to avoid

entirely the assumption of a beta distribution.
Parameter Estimation Using Data From Different Sources 

8-12

Figure 8.6 Hierarchical Bayes model for Poisson data.

8.3 The Hierarchical Bayes

Method

8.3.1 General Approach

In the preceding discussion of parametric empirical

Bayes, the unknown hyperparameters of the population-variability (or prior) distribution were estimated by

maximum likelihood. The empirical Bayes estimate of

the population-variability distribution is the corresponding distribution in which these maximum likelihood estimates have been inserted.

The hierarchical Bayes method is entirely different. It

embodies a complete (or full) Bayesian approach to the

problem of estimating the unknown population-variability distribution based on the available data. The

hierarchical Bayes approach expresses the initial

uncertainty (that is, uncertainty before the data are

considered) about the unknown hyperparameters using

yet another prior, a so-called second-order or hyperprior distribution. For example, in Example 8.1, the

population-variability distribution can be a gamma

distribution, with parameters (called hyperparameters

in this context) " and $ . The distribution g could also

be lognormal with parameters : and F2

. Any desired

distribution can be used, with any parameterization.

Figure 8.6 denotes the parameters of g generically as "

and $. The uncertainty in the state-of-knowledge about

the values of " and $ is expressed by a suitably specified joint hyperprior distribution on " and $. This

expands Figure 8.1 to be Figure 8.6. We almost always

desire such hyperprior distributions to be diffuse

because we almost never have very precise (or informative) information at the hyperprior level of such a

model.

Figure 8.6 is drawn showing " and $ with separate

distributions. In general, the hyperparameters together

have a joint distribution, which does not have to be the

product of independent distributions.

In the full Bayesian model all the unknown parameters,

including prior-distribution hyperparameters, are

assigned prior distributions that express the analyst’s

initial uncertainty about these parameters. This is

known as a hierarchical Bayes model. Berger (1985)

and Gelman et al. (1995) discuss the basic notions of

hierarchical Bayes modeling. In Example 8.1, the

parameters of interest to be estimated at the overall

population-variability level of the analysis are " and $,

while the plant-specific parameters to be estimated are

the 66 8i

 values. Each of these 68 parameters is

assigned an appropriate prior distribution in a hierarchical Bayes analysis.

The solution to the hierarchical Bayes method requires

conditioning on the data and obtaining the required

posterior distributions of all the parameters of interest.

This is done using Markov chain Monte Carlo (MCMC)

simulation (see Section 8.3.3). The desired point and

interval estimates of the parameters are then directly

(and easily) obtained from these posterior distributions.

This process will be illustrated for Examples 8.1 and

8.2 in Sections 8.3.4 and 8.3.5, respectively.

It is well known (Berger 1985 and Gelman et al. 1995)

that parametric empirical Bayes can be viewed as an

approximation to a full hierarchical Bayes analysis.

However, there are several important advantages of

hierarchical Bayes over parametric empirical Bayes.

First, parametric empirical Bayes essentially requires

the use of a conjugate population-variability distribution in order to obtain the required unconditional

distribution of X in closed form. Because hierarchical

Bayes analysis is implemented in practice using Monte

Carlo simulation, non-conjugate population-variability
Parameter Estimation Using Data From Different Sources 

8-13

t

i

μ i

α β

λ i

xi

Plant i

Figure 8.7 Directed graph for the hierarchical Bayes

analysis of Example 8.1.

distributions can be used as easily as conjugate distributions. For the Poisson example, a lognormal(:, F2

)

distribution on 8 is as easy as a gamma(", $) distribution.

Second, when using hierarchical Bayes, there is no need

to worry about double counting of the data. The

hierarchical model and associated Bayesian analysis

ensures that this cannot occur.

Finally, as mentioned above, the hierarchical Bayes

method is conveniently and easily implemented in

practice by means of Markov chain Monte Carlo

simulation using existing software, which is presently

available for free download from the Web (see Section

8.3.3.3).

8.3.2 Directed Graphs

Those who do not need to know the formulas can skim

this section and then jump directly to Section 8.3.4.

The first conceptual step in a hierarchical Bayes analysis should be to construct a directed graph representing the hierarchical Bayes model. Briefly, such a graph

represents all quantities as nodes in a directed graph, in

which arrows between nodes represent directed influences. A directed graph for Example 8.1 is shown in

Figure 8.7, where we have defined :i

 = 8iti.

Note that a solid arrow indicates a stochastic dependency, while a dashed arrow indicates a logical function.

The hierarchical Bayes approach for the gamma-Poisson case proceeds as follows. First specify hyperprior

distributions for the two hyperparameters outside the

“plant i” box in Figure 8.7. Inference on the

hyperparameters ", $ and the scram rate vector 8 = (81,

82, ..., 866) requires that we obtain Monte Carlo samples from the joint posterior g(", $, 8 | x), where the

data vector x is defined as x = (x1, x2, ..., x66). The

letter g is used here to denote both prior and posterior

densities. Generate these samples, and then use summary statistics from these samples to obtain the desired

estimates, such as point and interval estimates of these

parameters.

In order to calculate samples from this joint posterior

we must successively sample from the full conditional

distributions. That is, we must successively sample

from the conditional distribution of each stochastic

node given all the other stochastic nodes in the graph.

However, conditional independence is expeditiously

exploited in directed graphs in order to simplify these

full conditional distributions. For example, given $

and 8i, " in Figure 8.7 is conditionally independent

of xi

.

8.3.3 Markov Chain Monte Carlo

(MCMC) Simulation

Readers who do not need the programming details can

skip directly to Section 8.3.4.

Markov chain Monte Carlo (MCMC) sampling

techniques give the required samples from the joint

posterior distribution of all the unknown parameters.

The desired hierarchical Bayes point and interval

estimates can thus be directly computed from the

corresponding simulated sample observations without

the need for tedious analytical or numerical calculations. MCMC is a Monte Carlo integration technique

that is implemented using Markov chains. MCMC

draws these samples by running a cleverly constructed

Markov chain for a long period of time. Good introductions to MCMC are provided by Gilks et al. (1996)

and Gelman et al. (1995).

In the Poisson example, the required hierarchical Bayes

estimates can be obtained by means of Gibbs sampling,

a basic MCMC technique that is described next. The

equations are sketched here first. Then a publicly

available software package, BUGS, is described. The

package implements the equations without requiring the

users to understand the details.
Parameter Estimation Using Data From Different Sources 

8-14

8.3.3.1 Gibbs Sampling

Gibbs sampling is a technique that can be used to

generate a random sample from the joint posterior

distribution indirectly, provided that we can directly

sample each of the full conditional distributions (which

are described below).

The Gibbs sampling method, sometimes also called

simply the Gibbs sampler, is briefly described here.

Suppose that we have a set of p parameters 21, 22, ...,

2p whose joint posterior distribution g(21, 22, ..., 2p | x)

is unknown but is of interest to be estimated. This is

the usual case when using the hierarchical Bayes

method. In Example 8.1, the 2i

s consist of the two

hyperparameters plus the 66 8i

s, and the number of

parameters, p, is 68.

However, suppose that the full conditional distributions

g( 2i | 2j, x, j ... i) i = 1, 2, ..., p, are known in the sense

that sample values of 2i

, conditional on values of 2j, j ...

i, may be generated from these by some appropriate

method. Under mild conditions, these conditional

distributions uniquely determine the required joint

posterior distribution g(21, 22, ..., 2p | x); hence, they

determine all the unconditional marginal distributions

g( 2i | x), i = 1, 2, ..., p, as well.

The Gibbs sampler generates samples from the required

joint distribution as follows:

(1) Select an arbitrary starting set of values . θ θ 1

0 0

,..., p

Set j = 0.

(2) Draw from g( 21 θ | , ..., , x), then 1

j+1 θ2

j θ p

j

 from g(22 θ | , , ..., , x), and so on 2

j+1 θ1

j θ3

j θ p

j

up to from g(2p θ p | , ..., , x) to com- j+1 θ1

j θ p

j

−1

plete one iteration of the sampler.

(3) Increment j and repeat Step (2) until j+1 = n. After

n such iterations of Step (2), we have obtained the

sample ( , ..., ). Under mild conditions, as θ n 1

n θ p

n

6 4 this p-tuple converges in distribution to the

unknown joint posterior distribution g(21, 22, ...,

2p | x).

Typical implementation of the Gibbs sampling algorithm generates an initial “large” number of iterations

(called the burn-in) until the samples have converged.

The burn-in samples are discarded, and the samples

generated thereafter are used as sample observations

from the joint posterior distribution g(21, 22, ..., 2p | x).

Nonparametric density estimators, such as those given

in Section 6.7.3, can then be used to approximate the

posterior distribution using the post burn-in samples.

Examples 8.1 and 8.2 are analyzed in this way in

Sections 8.3.4 and 8.3.5.

In Example 8.1, 68 full conditional distributions are

required in order to use the Gibbs sampler:

(1)

g xg

e i

i

(|,,) (|,)

( )

αβλ αβλ

β

α

λ

α α

α

≡

∝ ⎡

⎣

⎢ ⎤

⎦

⎥ ⎛

⎝

⎜ ⎞

⎠

⎟ =

−

Γ ∏

66

1

66

(2)

g($ | ", 8, x) / g($ | ", 8 )

= gamma 66 0 0625 0 0625 1

66

α λ + +

⎛

⎝

⎜ ⎞

⎠

⎟ =

., . ∑ i

i

(3)

g(8i | ", $, x) = gamma("+xi, $+ti), i = 1, ..., 66 .

It is easy to sample directly from the gamma distributions. The first distribution, however, the distribution

for ", is not of a familiar form, and in fact is not

known fully. It is only known to be proportional to the

stated expression, with the normalizing constant

unknown. The Metropolis-Hastings method is

described next, as a way to sample from a distribution such as the distribution for ".

8.3.3.2 Metropolis-Hastings Step

It is sometimes the case that one or more of the full

conditional distributions g( 2i

 | 2j, x, j ... i) i = 1, 2, ...,

p, required in Step (2) of the Gibbs sampler may not be

available in closed form. This may happen as follows.

These full conditional distributions are usually obtained

using Bayes’ Theorem and, while the two terms (the

likelihood and the prior distribution) in the numerator

of Bayes’ Theorem are usually known, the integration

required to obtain the normalizing factor (the denominator) in Bayes’ Theorem cannot be performed in

closed form. That is, the required full conditional

distribution is known only up to a multiplicative constant, the normalizing factor. The corresponding full

conditional distribution is thus unavailable in closed

form, and sample values from this distribution cannot

be directly obtained as required in Step (2) of the Gibbs

sampler. 

Denote the full conditional distribution, which is known

only up to a normalizing constant, as g(2i

 | 2j, x, j ... i).

For convenience, we suppress the conditioning terms in

the notation below. In this situation, sample observations may be obtained in Step (2) of the Gibbs sampler
Parameter Estimation Using Data From Different Sources 

8-15

by using a so-called Metropolis-Hastings step (Hastings 1970) as follows:

(1) Initialize 2i

0

 and set j = 0.

(2) Generate an observation 2i

*

 from a candidate

distribution q(2i

*

| 2i

j

), where q(y|x) is a probability

density in y with parameter (for example mean) x.

(3) Generate a uniform(0, 1) observation u .

(4) Let

 θ θ αθ θ

θ i

j+ i i

j

i

i

j

1 u = ⎧ ≤ ⎨

⎩

* * , (,)

,

 if 

 otherwise 

where

α(,) . ()(|)

()(|) x y

g yqxy

g xq yx =

(5) Increment j and go to (2).

Because " uses a ratio, g(y)/g(x), it can be calculated

even though the normalizing constant for g is unknown.

The candidate distribution in Step (2) can be almost any

distribution (Gilks et al. 1996), although a symmetric

distribution such as a normal distribution results in a

simplification of the algorithm, and is called simply a

Metropolis step (as opposed to a Metropolis-Hastings

step). A common choice for q(y|x) is a normal distribution with mean x and a variance that allows the

random deviates to be a representative sample from the

entire complete conditional distribution. A preliminary

rule of thumb suggested by Gelman et al. (1995, Sec.

11.5) is that the variance be such that the new value,

2i*, is picked in Step (4) about 30% of the time, and the

old value, 2i

j

, is picked about 70% of the time. The

new value should be picked more often in problems

with few parameters and less often in problems with

many parameters.

Actually, BUGS favors a method called adaptive

rejection sampling (Gilks and Wild 1992) instead of

the Metropolis-Hastings algorithm. This method uses

more storage space but fewer iterations. It requires that

the conditional distributions in the Gibbs sampler be

log-concave (George et al. 1993). This requirement is

satisfied for the commonly used prior distributions. If

the user happens to select a prior that leads to a problem, BUGS will give a diagnostic message.

8.3.3.3 BUGS (Bayesian Inference Using Gibbs

Sampling)

Fortunately, for a wide range of common problems,

there is little need to actually program the Gibbs

sampler in practice. Gibbs sampling has been conveniently implemented through the BUGS software

project (BUGS 1995, Spiegelhalter et al. 1995, and

Gilks et al. 1994). It is currently available for free

download at WWW URL 

http://www.mrc-bsu.cam.ac.uk/bugs/ .

The classic BUGS program uses text-based model

description and a command-line interface, and versions

are available for major computing platforms.

A Windows version, WinBUGS, has an optional

graphical user interface (called DoodleBUGS) as well

as on-line monitoring and convergence diagnostics.

BUGS is reasonably easy to use and, along with a user

manual, includes two volumes of examples. Section

8.3.4 illustrates how WinBUGS was used in Example

8.1 to obtain the hierarchical Bayes estimates of the 66

plant-specific scram rates and of the corresponding

population-variability distribution. Section 8.3.5 uses

WinBUGS in Example 8.2, to obtain the hierarchical

Bayes estimates of p at each of the 68 plants and of the

corresponding population-variability distribution.

8.3.4 Application to Example 8.1, 

Initiating Events

8.3.4.1 Development of Hyperprior Distributions

It will be necessary at the start to develop hyperprior

distributions. Therefore, this topic is discussed here in

general terms. One must choose hyperpriors that are

diffuse enough to contain the true values of the

hyperparameters. That means that the analyst must

have some vague idea of the true values of the hyperparameters. In principle, this should come from experience with other, similar data sets. Often, however, the

current data set is used to provide guidance on the

plausible hyperparameter values. The diffuseness of the

hyperpriors can be assessed by using the rough rule of

thumb that a quantity can plausibly vary from its mean

by up to two standard deviations.

In the examples below, the plausible values will be

obtained from the previous empirical Bayes analyses.

However, it is possible to develop hyperprior distributions without such a sophisticated analysis, just by

examining the data carefully. For example, one can

easily get an estimate of the mean of 8, and one can

also find an upper bound that 8 rarely exceeds. Such

considerations can provide rough guidance on plausible

bounds for the hyperparameters " and $. Then one

constructs hyperprior distributions that achieve these
Parameter Estimation Using Data From Different Sources 

8-16

model

{

 for (i in 1:M) {

 lambda[i] ~ dgamma(alpha,beta)

 mu[i] <- lambda[i]*t[i]

 x[i] ~ dpois(mu[i])

 }

 alpha ~ dexp(1.0)

 beta ~ dgamma(0.0625, 0.0625)

}

Figure 8.8 WinBUGS model specification for

Example 8.1.

bounds without going far into the tails of the distributions. The process is not always easy, but some analysis of this type is necessary.

It might seem tempting to just use distributions with

enormous standard deviations, say mean 1 and standard

deviation 1,000. In principle this is possible, but in

practice it may challenge the numerical accuracy of the

software. Therefore, it is recommended that the data be

examined and that hyperpriors be constructed that are

diffuse enough to include anything consistent with the

data, but that are not unrealistically diffuse.

8.3.4.2 Example Analysis

Let us now illustrate the hierarchical Bayes method

in Example 8.1. We use the Gibbs sampler in BUGS

to calculate all the required population and plantspecific scram rate estimates. 

To begin, we assume that the population-variability

distribution is gamma(", $) just as in Section 8.2.

First the hyperprior distributions must be constructed

for " and $, based on the considerations in Section

8.3.4.1. The empirical Bayes analysis of Section

8.2.5.1 gave " = 1.4 and $ = 1.2. Therefore, we

choose both hyperpriors to have mean 1, agreeing

(to one significant digit) with the empirical Bayes

results. The hyperparameter " is given an exponential hyperprior distribution with mean and variance of

1, while the hyperparameter $ is given an independent gamma(0.0625, 0.0625) hyperprior distribution.

Thus, $ is assumed to have a hyperprior mean and

standard deviation of 1 and 4, respectively. The

hyperpriors are diffuse (large variances), and have

plausible means, so they will probably not bias the

final answers much.

Figure 8.8 contains the WinBUGS model used here

for this Poisson example. The initial values considered are: alpha = 1, beta = 1, and lambda[i] = 1, i =

1, ..., 66.

After 1,000 burn-in iterations (to remove the effect of

the initial starting values and to achieve convergence

of the Markov chain), 10,000 additional simulated

posterior sample values of ", $, and 8 = (81, 8 2, ...,

866) were recorded. These 10,000 sample values

were then used to calculate the required posterior

point and credible interval estimates of ", $, and

each 8i. For example, the hierarchical Bayes estimated posterior mean of the Callaway scram rate is

calculated to be 4.97 per 1,000 critical hours. The

corresponding 90% credible interval on 81 is [2.87,

7.51].

In addition, the marginal posterior mean and standard deviation of " are calculated to be 1.38 and

0.30, respectively, whereas those for $ are computed

to be 1.21 and 0.32. A hierarchical Bayes 90%

credible interval for " is [0.94, 1.93], while the corresponding interval for $ is [0.76, 1.78]. The marginal

posterior correlation between " and $ is also easily

calculated from the 10,000 pairs of corresponding

posterior (", $) values to be 0.89. From Table 8.3

we see that the empirical Bayes point estimates of "

and $ are 1.39 and 1.21, respectively, which are in

near perfect agreement with the hierarchical Bayes

estimates.

Figure 8.9 illustrates the hierarchical Bayes results.

For each plant, the posterior mean and 90% credible

interval are shown. The mean and 90% credible

interval for the population-variability distribution are

also shown, and a vertical dashed line is drawn

through the mean. Actually, this population-variability distribution is the gamma distribution evaluated

when " and $ are set equal to their posterior means.

It does not reflect the uncertainty in these two hyperparameters. Figure 8.9 agrees very closely with

Figure 8.3.

Table 8.7 contains the same portion of the numerical

hierarchical Bayes analysis results for Example 8.1

as are displayed in Table 8.3.

The point and interval estimates in Table 8.7 are all

in good agreement with the empirical Bayes estimates in Table 8.3.

8.3.5 Application to Example 8.2, AFW 

Segment Failures to Start

Recall that this example has 68 plants with sparse failure

data consisting of only 6 failures in 1,993 demands.

Because the data are so sparse, the form of the prior, the
Parameter Estimation Using Data From Different Sources 

8-17

Industry

Callaway

Wash. Nuclear 2

Salem 1

Diablo Canyon 1

Grand Gulf

Susquehanna 2

Salem 2

McGuire 2

Hatch 2

LaSalle 2

46 other plants

Quad Cities 2

Vermont Yankee

Fort Calhoun

Crystal River 3

Ginna

Point Beach 1

Millstone 1

Point Beach 2

Prairie Island 2

Oconee 2

Events/1000-Critical-Hours GC00 0419 5

1.E-03 1.E-02 1.E-01 1.E-00 1.E+01 1.E+02

Figure 8.9 Plant-specific posterior means and 90%

credible intervals for 8, from hierarchical Bayes

analysis.

model 

{ 

 for (i in 1:M) { 

 x[i] ~ dbin(p[i],n[i]) 

 p[i] ~ dbeta(alpha, beta) 

 } 

 alpha ~ dexp(1.0) 

 beta ~ dgamma(1.0,0.035) 

}

Figure 8.10 WinBUGS model specification for a

beta prior in Example 8.2.

population-variability distribution, can strongly influence

the answers. Therefore, the example is analyzed using two

population-variability distributions, first a beta distribution, as in Section 8.2, and then a logistic-normal distribution. In each case, diffuse hyperpriors with plausible

means are used. Therefore, the exact choices made for the

hyperpriors have little influence on the answer.

Table 8.7 Portion of hierarchical Bayes analysis results for Example 8.1.

Plant Posterior mean and 90%

credible interval a

Industry

Callaway

Wash. Nuc. 2

Salem 1

Diablo Can. 1 

Grand Gulf

Pt. Beach 2

Prairie Isl. 2

Oconee 2

(0.116, 1.14, 3.06)

(2.87, 4.97, 7.51)

(3.01, 4.39, 5.99)

(1.65, 2.94, 4.57)

(1.24, 2.83, 4.99)

(1.27, 2.57, 4.21)

(0.013, 0.156, 0.429)

(0.012, 0.152, 0.410)

(0.011, 0.137, 0.374)

a.

 Format is (lower bound, mean, upper bound).

The least significant digit may be inaccurate by 2

or more, because of Monte Carlo sampling error.

8.3.5.1 Analysis with Beta Prior

We assume that the population-variability distribution

is a beta(", $) distribution. In this case, the empirical

Bayes analysis found " = 0.1 and $ = 36. We will

construct diffuse hyperpriors that contain these

values.

The hyper-parameter " is assigned an exponential

(1) hyperprior distribution with a hyperprior mean and

standard deviation of 1, while the hyperparameter $

is assigned an independent gamma(1.0, 0.035)

hyperprior distribution. Thus, we assume that $ has

a hyperprior mean and standard deviation both equal

to approximately 30. The forms of these hyperprior

distributions ensure that the joint posterior distribution will be log-concave, and the diffuseness of the

hyperpriors ensures that they will not influence the

final answers greatly.

The chosen hyperpriors include the desired values of

0.1 and 36, and much more. In particular, the exponential distribution for " allows any value below the

mean of 1, because the exponential density decreases monotonically. Likewise, the gamma density

of $ also is monotonically decreasing, and so allows

any value below the mean. As for values larger than

the mean, we apply the rule of thumb that says that

most random variables can easily deviate from the

mean by up to two standard deviations. This says

that " could be as large as 3 and $ as large as about

90. Because of the knowledge gained from the

empirical Bayes analysis, we believe that the

hyperpriors are diffuse enough. If the prior belief

about the parameters were more rough, based on a

cruder analysis of the data, we would want to compensate by choosing more diffuse hyperpriors. See

Section 8.3.4.1 for more details.

Fig. 8.10 contains the WinBUGS model used here

for this binomial example. The initial values were:

alpha = 1, beta = 1, and p[i] = 0.01, i = 1, ..., 68.
Parameter Estimation Using Data From Different Sources 

8-18

1.E-03 1.E-02 1.E-01 1.E+00

p GC00 0419 4

1.E-04

 Industry

 Indian Point 3

 

Millstone 2

 Crystal River 3

 

 Indian Point 2

 

Robinson 2

(63 plants with

no failures)

Figure 8.11 Plant-specific posterior means and 90%

credible intervals for p, from hierarchical Bayes

analysis with beta population-variability distribution.

After 10,000 burn-in iterations (to remove the effect

of the initial starting values and to achieve convergence of the Markov chain), 90,000 additional

simulated posterior sample values of ", $, and p =

(p1, p2, ..., p68) were recorded. These 90,000 sample

values were then used to calculate the required

posterior point and credible interval estimates of ",

$, and each pi.

In addition, the marginal posterior mean and standard deviation of " are calculated to be 0.188 and

0.136, respectively, whereas those for $ are computed to be 46.4 and 32.4. A hierarchical Bayes

90% credible interval for " is [0.046, 0.442], while

the corresponding interval for $ is [9.95, 109.5].

Note the large uncertainties associated with the

hierarchical Bayes estimates of " and $ because of

the sparseness of the data. Table 8.6 shows that the

empirical Bayes point estimates of " and $ are 0.137

and 36.34, respectively, which are well within the

90% credible intervals of the corresponding hierarchical Bayes estimates.

Figure 8.11 illustrates the hierarchical Bayes results.

For each plant, the posterior mean and 90% credible

interval are shown. The mean and 90% credible

interval for the population-variability distribution are

also shown, and a vertical dashed line is drawn

through the mean.

Table 8.8 contains the same portion of the numerical

hierarchical Bayes analysis results for Example 8.2

as given in Table 8.6. The results are presented to

only two significant digits, because the Monte Carlo

errors reported by BUGS show that the third significant digit is not meaningful.

Table 8.8 Portion of hierarchical Bayes 

analysis results using beta prior for

Example 8.2.

Plant Posterior mean and 90% 

credible interval a

Industry

Indian Point 3

Millstone 2

Crystal R. 3

Indian Point 2

Robinson 2

Prairie Isl. 1

Vogtle 1

(1.6E!9, 4.0E!3, 2.1E!2)

(5.5E!2, 3.2E!2, 7.9E!2)

(1.6E!2, 2.6E!2, 8.3E!2)

(1.5E!2, 2.3E!2, 7.1E!2)

(1.3E!2, 1.9E!2, 5.9E!2)

(1.2E!2, 1.8E!2, 5.4E!2)

(1.5E!15, 4.1E!3, 2.1E!2)

(2.5E!16, 1.2E!3, 6.4E!3)

a. Format is (lower bound, mean, upper bound).

The point and interval estimates in Table 8.8 are all

in reasonably close agreement with the empirical

Bayes estimates in Table 8.6.

8.3.5.2 Analysis with Logistic-Normal Prior

One of the primary advantages in using the hierarchical

Bayes method is the ability to consider non-conjugate

population-variability (or prior) distributions. We now

illustrate this for Example 8.2.

The previous analysis considered a conjugate beta prior in

this example. Table 8.8 shows that, for x = 0, the use of

a beta prior produces lower 5% credible limits on the

order of 10-15 or 10-16, which are unrealistically small.

This result is a consequence of the fitted L-shaped beta

prior distribution, with high density close to p = 0.

We can avoid such unrealistic results by using a nonconjugate logistic-normal prior distribution (see Section

6.3.2.5.2 or A.7.9) in the hierarchical Bayes approach.

Recall that, while such a prior is extremely difficult to

consider in an empirical Bayes approach, it is extremely

easy to do in hierarchical Bayes.

Figure 8.12 contains the WinBUGS model specification for using a logistic-normal prior in Example 8.2.

Observe that this is no more difficult than using the
Parameter Estimation Using Data From Different Sources 

8-19

model

{

 for (i in 1:M) {

 x[i] ~ dbin(p[i],n[i])

 y[i] ~ dnorm(mu,tau)

 p[i] <- exp(y[i])/(1 + exp(y[i]))

 }

 mu ~ dnorm(-5,0.0001)

 tau ~ dgamma(1,7)

}

Figure 8.12 WinBUGS model specification for a

logistic-normal prior in Example 8.2.

0.005 0.01 0.015 0.02

50

100

150

200

250

300

p

Density

Logistic-Normal

Beta

Figure 8.13 Fitted population-variability distributions

in Example 8.2.

1.E-03 1.E-02 1.E-01 1.E+00

p GC00 0419 4

1.E-04

 Industry

 Indian Point 3

 

Millstone 2

 Crystal River 3

 

 Indian Point 2

 

Robinson 2

(63 plants with

no failures)

Figure 8.14 Plant-specific posterior means and 90%

credible intervals for p, from hierarchical Bayes

analysis with logistic-normal population-variability

distribution.

hierarchical Bayes model based on a conjugate beta

prior shown in Figure 8.10. Note that BUGS parameterizes the normal distribution in terms of J = 1/F2

. A

commonly used prior distribution for J is gamma, and

that choice is used here. Thus, assigning : a prior

precision of 0.0001 is equivalent to assigning it a

prior variance of 10,000, or a prior standard deviation

of 100.

Again using 10,000 burn-in iterations and 90,000

replications of the Gibbs sampler for the model in

Figure 8.12, WinBUGS likewise calculated posterior

means and 90% credibility intervals for :, F, and

each pi.

The marginal posterior mean and standard deviation

of : are calculated to be –5.097 and 0.09517, respectively, whereas those for F are computed to be

0.640 and 0.238. A hierarchical Bayes 90% credible

interval for : is [-5.253, -4.939], while the corresponding interval for F is [0.322, 1.08].

Figure 8.13 shows the two estimated populationvariability distributions, when the form is assumed to

be beta (the conjugate distribution) or logistic-normal.

The mean of the beta prior is 0.004 and the mean of

the logistic-normal prior is 0.007, nearly twice as

large. Note that, unlike the beta prior, the logisticnormal prior avoids the high probability density close

to p = 0.

Figure 8.14 illustrates the hierarchical Bayes results

using the logistic-normal prior. As in Figure 8.11, the

posterior mean and 90% credible interval are shown

for each plant. The mean and 90% credible interval

for the population-variability distribution are also

shown, and a vertical dashed line is drawn through

the mean. This plot may be directly compared with

Figure 8.11.

Table 8.9 contains the same portion of the numerical

hierarchical Bayes analysis results for the logisticnormal as given in Table 8.8 for the beta prior.

Note that the posterior means and 90% credible

intervals in Table 8.9 are all larger than those in

Table 8.8. As stated above, this is a direct consequence of using the logistic-normal prior in Figure

8.13. Observe also that the lower 90% credibility

limits for plants with no failures, such as Prairie

Island 1 and Vogtle 1, are now much more realistic

than the corresponding limits in Table 8.8.
Parameter Estimation Using Data From Different Sources 

8-20

Table 8.9 Portion of hierarchical Bayes analysis results using logistic-normal

prior for Example 8.2.

Plant Posterior mean and 90% credible interval a

Industry

Indian Point 3

Millstone 2

Crystal R. 3

Indian Point 2

Robinson 2

Prairie Isl. 1

Vogtle 1

(2.1E!3, 7.4E!3, 1.7E!2)

(6.2E!3, 3.4E!2, 8.7E!2)

(3.0E!3, 3.1E!2, 9.7E!2)

(2.8E!3, 2.6E!2, 8.0E!2)

(2.5E!3, 2.1E!2, 6.2E!2)

(2.4E!3, 1.9E!2, 5.7E!2)

(6.6E!4, 1.2E!2, 4.4E!2)

(4.3E!4, 4.E!3, 1.3E!2)

a.

 Format is (lower bound, mean, upper bound).

8.4 Discussion

This chapter concludes with several important observations.

8.4.1 Hierarchical Bayes Is Still Bayes

The hierarchical Bayes model is a special case of the

familiar Bayesian model. It is not some new kind of

construction. To see this, consider Levels 0 and 1

together in Figure 8.6. The prior parameter is a vector

2, consisting of the hyperparameters and the 8is. Thus,

the prior 2 is a vector with dimension 2+ m. The prior

distribution on 2 is specified: the joint distribution of

the hyperparameters " and $ is given by the hyperprior

in Level 0, and the conditional distributions of the 8i

s

are independent, and specified by g, conditional on "

and $. Thus, Levels 0 and 1 together specify the prior

parameter vector and its prior distribution. The posterior distribution is therefore given by Bayes’ Theorem:

gpost(2 | data) % Pr(data | 2) × gprior(2) .

This differs from the applications of Bayes’ Theorem

elsewhere in this handbook in only two ways: the

parameter is a high-dimensional vector, and the prior

distribution has a lot of structure, as shown in Figure 8.6.

A practical consequence of the high dimension of 2 is

that the tools of Chapter 6, numerical integration and

simple random sampling methods, do not work well.

More recently developed methods, versions of Markov

chain Monte Carlo sampling, must be used. Conceptually, however, hierarchical Bayes modeling fits perfectly within the framework of Bayes’ Theorem. In

particular, everything is legal, with no double counting

of data.

8.4.2 The “Two-Stage” Bayesian

Method

Kaplan (1983) introduced a “two-stage” Bayesian

method, which has sometimes been used in PRA work.

It is described here in terms of Figure 8.6. The method

singles out the plant of special interest, say Plant 1. It

then estimates the hyperparameters, " and $, in a

Bayesian way, using the data from all the plants except

Plant 1. It then uses the estimated g(8 | ", $) as a prior,

combining it with the data from Plant 1 to estimate 81

in the usual Bayesian way.

The originally intended reason for not using Plant 1 in

the first stage was to avoid double counting. As mentioned in Section 8.4.1, the hierarchical Bayes method

is based directly on Bayes’ Theorem, and therefore

does not involve double counting. Therefore, the twostage Bayesian method should no longer be used, but

should be replaced by the conceptually cleaner hierarchical Bayes method. Now that numerical algorithms

have been developed to sample from the posterior

distributions, this is completely feasible.

8.4.3 Lower Bounds on Parameters

Example 8.2 illustrated that different populationvariability distributions (prior distributions) can lead to

radically different lower percentiles of the parameters’

posterior distributions. This occurred in that example

for those plants that had experienced no failures. A

beta prior led to 5th percentiles for p on the order of

1E!15, whereas a logistic-normal prior led to 5th

percentiles on the order of 5E!4. No one believes the

first answers, but many people could believe the second

answers. Does that mean that the answers from the

logistic-normal prior are “right”?

In fact, the lower bounds are an artifact of the model, in

both cases. The plants that experienced no failures

reveal only that p is “small” at those plants. They do

not give information about how small p might be. If

many plants have no failures, as in Example 8.2, then

we have very little information about the lower end of

the population-variability distribution. In contrast to

this, the plants that experienced one or more failures

convey much more information, revealing both how

large and how small p could plausibly be at those
Parameter Estimation Using Data From Different Sources 

8-21

plants. Therefore, the 95th percentile of p at any plant

is somewhat dependent on the assumed form of the

population-variability distribution (beta, logisticnormal, or whatever). But when many plants have no

observed failures, the 5th percentile of p at any of those

plants is extremely dependent on this assumed form.

And why was a particular form assumed for the population-variability distribution? For convenience only!

Thus, even if the answers from a logistic-normal prior

look credible, we do not “know” that they are correct.

We may choose to discard the results from using a beta

prior, because we do not want to publish 5th percentiles

that could be ridiculed. We might also choose to

publish the results from using a logistic-normal prior,

knowing that the 5th percentiles appear credible. But it

is a delusion to think that we “know” lower bounds on

p at the plants with no observed failures. The calculated lower bounds remain an artifact of the assumed

model.

Fortunately, lower bounds are not a concern for risk.

Means and upper bounds are the important quantities,

and they can be estimated with much less dependence

on the model.

8.4.4 Empirical Bayes as an 

Approximation to Hierarchical

Bayes

As remarked elsewhere, Figures 8.3 and 8.9 are very

similar to each other, and Figures 8.5 and 8.11 are

similar to each other. That is, in both Examples 8.1 and

8.2, the empirical Bayes results are numerically close to

the hierarchical Bayes results, when (a) the empirical

Bayes method includes the Kass-Steffey adjustment,

and (b) both methods use the conjugate populationvariability distribution, a gamma distribution for

Poisson data and a beta distribution for binomial data.

This agreement between the methods is more than

coincidence. Kass and Steffey (1989) developed their

method specifically with this intent: to make the

empirical Bayes approach give a first-order approximation to the hierarchical Bayes approach with a diffuse

hyperprior. The method does this well in the two

examples. Of course, when the hierarchical Bayes

method does not use a conjugate population-variability

distribution, as in Section 8.3.5.2, there is no corresponding empirical Bayes method.
Parameter Estimation Using Data From Different Sources 

8-22
9-1

9. REFERENCES

ANS and IEEE, 1983, PRA Procedures Guide A Guide to the Performance of Probabilistic Risk Assessments for

Nuclear Power Plants, NUREG/CR-2300, 1983, American Nuclear Society, LaGrange, IL, and The Institute of

Electrical and Electronics Engineers, New York, NY.

ASME, 2002, Standard for Probabilistic Risk Assessment for Nuclear Power Plant Applications, ASME RA-S2002. American Society of Mechanical Engineers, New York, NY.

Ascher, H., and H. Feingold, 1984, Repairable Systems Reliability - Modeling, Inference, Misconceptions and Their

Causes. Marcel Dekker, Inc., NY.

ATV, 1992, T-Book, 3rd ed., ATV Office and Studsvik, Vattenfall AB, S-16287 Vällingby, Sweden.

Atwood, C. L., 1992, Parametric Estimation of Time-Dependent Failure Rates for Probabilistic Risk Assessment in

Reliability Engineering and System Safety, Vol. 37, No. 3, pp. 181-194.

Atwood, C. L., 1994, Hits per Trial: Basic Analysis of Binomial Data, EGG-RAAM-11041, Idaho National

Engineering and Environmental Laboratory, Idaho Falls, ID.

Atwood, C. L., 1995, Modeling Patterns in Count Data Using Loglinear and Related Models, INEL-95/012, Idaho

National Engineering and Environmental Laboratory, Idaho Falls, ID.

Atwood, C. L., 1996, Constrained Noninformative Priors in Risk Assessment, in Reliability Engineering and System

Safety, Vol. 53, Issue 1, pp. 37-46.

Atwood, C. L., and C. D. Gentillon, 1996, Bayesian Treatment of Uncertainty in Classifying Data: Two Case

Studies, in Probabilistic Safety Assessment and Management 96 (ESREL’96 -- PSAM-III), P. C. Cacciabue and I. A.

Papazoglou, ed., pp. 1283-1288. Springer, London.

Atwood, C.L., D. L. Kelly, F. M. Marshall, D. A. Prawdzik, and J. W. Stetkar, 1998, Evaluation of Loss of Offsite

Power Events at Nuclear Power Plants: 1980-1996, NUREG/CR-5496, INEEL/EXT-97-00887, Idaho National

Engineering and Environmental Laboratory, Idaho Falls, ID. 

Atwood, C. L., and M. Engelhardt, 2003, Bayesian Estimation of Unavailability, STATWOOD 2003-2. Submitted

to Reliability Engineering and System Safety.

Bain, L. J., and M. Engelhardt, 1991, Statistical Analysis of Reliability and Life-Testing Models, Theory and

Methods, 2nd ed. Marcel Dekker, Inc., NY.

Bain, L. J., and M. Engelhardt, 1992, Introduction to Probability and Mathematical Statistics, 2nd ed. PWS-Kent

Pub., Boston, MA.

Battle, R. E., and D. J. Campbell, 1983, Reliability of Emergency AC Power Systems at Nuclear Power Plants,

NUREG/CR-2989, Oak Ridge National Laboratory, Oak Ridge, TN.

Berger, J. O., 1985, Statistical Decision Theory and Bayesian Analysis. 2nd ed., Springer-Verlag, NY.

Bernardo, J. M., and A. F. M. Smith, 1994, Bayesian Theory, John Wiley and Sons, New York.

Bertucio, R. C., and J. A. Julius, 1990, Analysis of Core Damage Frequency: Surry, Unit 1 Internal Events,

NUREG/CR-4550,Vol. 3, Rev. 1, Part 1, SAND86-2084, Sandia National Laboratories, Albuquerque, NM.
References

9-2

Blanton, C. H., and S. A. Eide, 1993, Savannah River Site Generic Data Base Development, WSRC-TR-93-262,

Westinghouse Savannah River Company, Aiken, SC.

Box, G. E. P., 1980, Sampling and Bayes’ Inference in Scientific Modelling and Robustness (with discussion) in

Journal of the Royal Statistical Society, Series A, Vol. 143, Part 4, pp. 383-430.

Box, G. E. P., and G. C. Tiao, 1973, Bayesian Inference in Statistical Analysis. Addison-Wesley, Reading, MA.

Breeding, Roger J., Timothy J. Leahy, and Jonathan Young, 1985, Probabilistic Risk Assessment Course

Documentation Volume 1: PRA Fundamentals, NUREG/CR-4350 Volume 1 of 7, SAND85-1495 Volume 1 of 7,

Sandia National Laboratories, Albuquerque NM, August 1985.

Budnitz, R.J., et al. 1997. Recommendations for Probabilistic Seismic Hazard Analysis: Guidance on the Use of

Experts, NUREG/CR-6372, Lawrence Livermore National Laboratory, Livermore, CA.

BUGS: Bayesian Inference Using Gibbs Sampling, Version 0.50, 1995, MRC Biostatistics Unit, Cambridge, UK. 

Available at www.mrc-bsu.cam.ac.uk/bugs/.

Carlin, B. P., and T. A. Louis, 2000, Bayes and Empirical Bayes Methods for Data Analysis, Second Edition,

Chapman & Hall/CRC, Boca Raton, FL.

Carlson, D. D., D. R. Gallop, A. M. Kolaczkowski, G. J. Kolb, and D. W. Stack, 1983, Interim Reliability

Evaluation Program Procedures Guide, NUREG/CR-2728, SAND82-1100, Sandia National Laboratories,

Albuquerque, NM.

Chu, T.L., et al., 1993, Evaluation of Potential Severe Accidents During Low Power and Shutdown Operations at

Surry Unit 1, NUREG/CR-6144, BNL-NUREG-52399, Brookhaven National Laboratory, Upton, NY.

Cleveland, W. S., 1985, The Elements of Graphing Data. Wadsworth Advanced Books and Software, Monterey,

CA.

Com Ed, 1981, Zion Probabilistic Safety Study, Commonwealth Edison, Chicago, IL.

Conover, W. J. 1999. Practical Nonparametric Statistics, Third Edition. John Wiley & Sons, NY.

Cox, D. R., and D. V. Hinkley, 1974, Theoretical Statistics, Chapman and Hall, London, England.

Cox, D. R., and P. A. W. Lewis, 1966, The Statistical Analysis of Series of Events. Chapman and Hall, London,

England.

CPC, 1981, Big Rock Point Nuclear Power Plant Probabilistic Risk Assessment, Consumers Power Company,

Jackson, MI.

Denson, G., G. Chandler, W. Crowell, A. Clark, and P. Jaworski, 1995, Nonelectronic Parts Reliability Data - 1995,

NPRD-95. Reliability Analysis Center, Griffiss AFB, Rome, NY.

Denson, W., W. H. Crowell, D. Mahar, and P. Jaworski, 1996, Electronics Parts Reliability Data - 1997, A

Compendum of Commerical and Military Device Field Failure Rates: Volumes 1 and 2, EPRD-97, Reliability

Analysis Center, Rome, NY. 

Drago, J. P., R. J. Borkowski, D. H. Pike, and F. F. Goldberg, 1982, The In-Plant Reliability Data Base for NuclearPower-Plant Components: Data Collection and Methodology Report, NUREG/CR-2641, Oak Ridge National

Laboratory, Oak Ridge, TN.
References

9-3

Droguett, E., 1999, A Methodology for the Treatment of Model Uncertainty, PhD Dissertation, Dept. of Materials

and Nuclear Engineering, University of Maryland, College Park, MD.

Drouin, M. T., F. T. Harper, and A. L. Camp, 1990, Analysis of Core Damage Frequency From Internal Events,

Methodology Guidelines: Volume 1, NUREG/CR-4550, SAND-86-2084. Sandia National Laboratories,

Albuquerque, NM.

Duke, 1984, Oconee Unit III Probabilistic Risk Assessment. Duke Power Company, Charlotte, NC.

Eide, S. A., S. T. Beck, M. B. Calley, W. J. Galyean, C. D. Gentillon, S.T. Khericha, S.D. Novack, and T.E.

Wierman, 1999a, Reliability Study: Westinghouse Reactor Protector System, 1984-1995, NUREG/CR-5500,Vol. 2,

INEEL/EXT-97-00740, Idaho National Engineering and Environmental Laboratory, Idaho Falls, ID.

Eide, S. A., S. T. Beck, M. B. Calley, W. J. Galyean, C. D. Gentillon, S.T. Khericha, and T.E. Wierman, 1999b,

Reliability Study: General Electric Reactor Protector System, 1984-1995, NUREG/CR-5500,Vol. 3, INEEL/EXT97-00740, Idaho National Engineering and Environmental Laboratory, Idaho Falls, ID.

Eide, S. A., S. V. Chmielewski, and T. D. Swantz, 1990, Generic Component Failure Data Base for Light Water

and Liquid Sodium Reactor PRAs (Probabilistic Risk Assessments), EGG-SSRE-8875, Idaho National Engineering

and Environmental Laboratory, Idaho Falls, ID.

Eide, S. A., S. T. Khericha, M. B. Calley, D. A. Johnson, and M. L. Marteeny, 1991, Component External Leakage

and Rupture Frequency Estimates, EGG-SSRE-9639, Idaho National Engineering and Environmental Laboratory,

Idaho Falls, ID.

Engelhardt, M. E., 1994, Events in Time: Basic Analysis of Poisson Data, EGG-RAAM-11088, Idaho National

Engineering and Environmental Laboratory, Idaho Falls, ID.

Engelhardt, M. E., 1995, Reliability Estimation and Applications, chap. 6 in The Exponential Distribution: Theory,

Methods, and Applications, N. Balakrishnan and A. P. Basu, ed., pp. 77-91. Gordon and Breach Publishers,

Amsterdam, Holland. 

Engelhardt, M. E., 1996, Statistical Analysis of Random Duration Times, INEL-95/0206, Idaho National

Engineering and Environmental Laboratory, Idaho Falls, ID. 

EPRI, 1982, ATWS: A Reappraisal, Part 3: Frequency of Anticipated Transients (Interim Report), EPRI NP-2230. 

Electric Power Research Institute, Palo Alto, CA. 

EPRI, 1989, Advanced Light Water Reactor Requirements Document, Electric Power Research Institute, Palo Alto,

CA.

EPRI, 1992, Nuclear Plant Reliability: Data Collection and Usage Guide, EPRI TR-100381, Halliburton NUS

Environmental Corporation, Walnut Creek, CA.

Everitt, B. S., 1992, The Analysis of Contingency Tables, 2nd ed. Chapman and Hall, London, England. 

Federal Register Notice for revisions to 10 CFR 50.72 and 10 CFR 50.73, October 2000.

Fleming, K.N., and A. Mosleh, 1985, Classification and Analysis of Reactor Operating Experience Involving

Dependent Events, EPRI NP-3967, Electric Power Research Institute, Palo Alto, CA.

Gallaher, R.B., et al., 1984, Sequence Coding and Search System for Licensee Event Reports, NUREG/CR-3905,

ORNL/NSIC-223, Oak Ridge National Laboratory, Oak Ridge, Tennessee.
References

9-4

Gelman, A., J. B. Carlin, H. S. Stern, and D. B. Rubin. 1995. Bayesian Data Analysis. Chapman & Hall, London.

George, E. I., U. E. Makov, and A. F. M. Smith, 1993, Conjugate Likelihood Distributions, in Scandinavian Journal

of Statistics, Vol. 20, pp. 147-156.

Gertman, D. I., W. E. Gilmore, W. J. Galyean, M. R. Groh, and C. D. Gentillon, 1990, Nuclear Computerized

Library for Assessing Reactor Reliability (NUCLARR). Volume 1, Revision 1. Summary Description, NUREG/CR4639, EFF-2458-VOL-1-REV-1, Idaho National Engineering and Environmental Laboratory, Idaho Falls, ID.

Gilks, W. R., S. Richardson, and D. J. Spiegelhalter, 1996, Markov Chain Monte Carlo in Practice. Chapman &

Hall, London.

Gilks, W. R., A. Thomas, and D. J. Spiegelhalter, 1994, A Language and Program for Complex Bayesian Modeling,

in The Statistician, Vol. 43, pp. 169-178.

Gilks, W. R., and P. Wild, 1992, Adaptive Rejection Sampling for Gibbs Sampling, in Applied Statistics, Vol. 41,

pp. 337-348.

Grant, G.M., W. S. Roesener, D. G. Hall, C. L. Atwood, C. D. Gentillon, and T. R. Wolf, 1999a, Reliability Study:

High-Pressure Coolant Injection (HPCI) System, 1987-1993, NUREG/CR-5500, Vol. 4, INEL-94/0158, Idaho

National Engineering and Environmental Laboratory, Idaho Falls, ID.

Grant, G.M., J. P. Poloski, A. J. Luptak, C. D. Gentillon, and W. J. Galyean, 1999b, Reliability Study: Emergency

Diesel Generator Power System 1987-1993, NUREG/CR-5500, Vol. 5, INEL-95/0035, Idaho National Engineering

and Environmental Laboratory, Idaho Falls, ID.

Grant, G.M., J. P. Poloski, C. D. Gentillon, and W. J. Galyean, 1999c, Reliability Study: Isolation Condenser System

1987-1993, NUREG/CR-5500, Vol. 6, INEL-95/0478, Idaho National Engineering and Environmental Laboratory,

Idaho Falls, ID.

Hagler-Bailly, 1972, Nuclear Power Experience Database. Hagler-Bailly Inc., Boulder, CO. 

Hart, Jeffrey D., 1997, Nonparametric Smoothing and Lack-of-Fit Tests, Springer, New York.

Hastings, W. K., 1970, Monte Carlo Sampling Methods Using Markov Chains and Their Applications, in

Biometrika, Vol. 57, pp. 97-109.

Hatch, S. W., P. Cybulskis, and R. O. Wooton, 1981, Reactor Safety Study Methodology Applications Program: 

Grand Gulf #1 BWR Power Plant, NUREG/CR-1659/4, Sandia National Laboratories, Albuquerque, NM.

Hoeting, Jennifer A., David Madigan, Adrian E. Raftery, and Chris T. Volinsky, 1999, Bayesian Model Averaging:

A Tutorial, in Statistical Science, Vol. 14, No. 4, pp. 382-417. 

Hollander, M., and D. A. Wolfe, 1999, Nonparametric Statistical Methods, 2nd ed. John Wiley & Sons, NY.

Houghton J. R., 2001a, Component performance study – air-operated valves, 1987 – 1998, NUREG-1715, Vol. 3,

Nuclear Regulatory Commission, Washington, DC.

Houghton J. R., 2001b, Component performance study – motor-operated valves, 1987 – 1998, NUREG-1715, Vol.

4, Nuclear Regulatory Commission, Washington, DC.

Houghton J. R. and H. G. Hamzehee, 2000a, Component performance study – turbine-driven pumps, 1987 – 1998,

NUREG-1715, Vol. 1, Nuclear Regulatory Commission, Washington, DC.
References

9-5

Houghton J. R., and H. G. Hamzehee, 2000b, Component performance study – motor-driven pumps, 1987 – 1998,

NUREG-1715, Vol. 2, Nuclear Regulatory Commission, Washington, DC.

Høyland, A., and M. Rausand. 1994. System Reliability Theory. Wiley-Interscience, John Wiley & Sons, NY.

Hubble, W. H., and C. Miller, 1980, Data Summaries of Licensee Event Reports of Valves at U. S. Commercial

Nuclear Power Plants, NUREG/CR-1363, Vol. 1, EGG-EA-5125, Idaho National Engineering and Environmental

Laboratory, Idaho Falls, ID.

IEEE, 1983, IEEE Guide to the Collection and Presentation of Electrical, Electronic, Sensing Component, and

Mechanical Equipment Reliability Data for Nuclear Power Generating Stations, IEEE-Std 500-1984. Institute of

Electrical and Electronics Engineers, NY.

INPO, 1998, Equipment Performance and Information Exchange System (EPIX), Volume1, Instructions for Data

Entry Maintenance Rule and Reliability Information Module, Vol. 1, INPO 98-001, Institute of Nuclear Power

Operations, Atlanta, GA.

Johnson, N. L., S. Kotz, and A. W. Kemp, 1992, Univariate Discrete Distributions, 2nd ed. John Wiley & Sons, NY. 

Johnson, N. L., S. Kotz, and N. Balakrishnan, 1994, Continuous Univariate Distributions, Volume 1, 2nd ed. John

Wiley & Sons, NY.

Johnson, N. L., S. Kotz and N. Balakrishnan, 1995, Continuous Univariate Distributions, Volume 2, 2nd ed. John

Wiley & Sons, NY.

Kaplan, S., 1983, On a “Two-Stage” Bayesian Procedure for Determining Failure Rates from Experiential Data, in

IEEE Transactions on Power Apparatus and Systems, Vol. PAS-102, No. 1, pp. 195-202.

Kass, R. E., and D. Steffey. 1989. Approximate Bayesian Inference in Conditionally Independent Hierarchical

Models (Parametric Empirical Bayes Models), in Journal of the American Statistical Association, Vol. 84, pp 717-

726.

Koehler, K. J., and K. Larntz, 1980, An Empirical Investigation of Goodness-of-Fit Statistics for Sparse

Multinomials, in Journal of the American Statistical Association, Vol. 75, Issue 370, pp. 336-344. 

Kolaczkowski, A. M., and A. C. Payne, 1983, Station Blackout Accident Analysis, NUREG/CR-3226, SAND-82-

2450, Sandia National Laboratories, Albuquerque, NM.

Kotra, J.P., M.P. Lee, N.A. Eisenberg, and A.R. DeWispelare. 1996. Branch Technical Position on the Use of

Expert Elicitation in the High-Level Radioactive Waste Program, NUREG/CR-1563, U.S. Nuclear Regulatory

Commission, Washington, DC.

Kvarfordt, K.J., M.J. Cebull, S.T. Wood, and A. Mosleh, 1998, Common-Cause Failure Database and Analysis

System: Overview, NUREG/CR-6268, Idaho National Engineering and Environmental Laboratory, Idaho Falls, ID. 

Lee, Peter M., 1997, Bayesian Statistics: An Introduction, Second Edition, Arnold, a member of the Hodder

Headline Group, London.

Lehtinen, E., U. Pulkkinen, and K. Pörn, 1997, Statistical Trend Analysis Methods for Temporal Phenomena, SKI

Report 97:10, Swedish Nuclear Power Inspectorate, Stockholm, Sweden.

Lewontin, R. C., and J. Felsenstein, 1965, The Robustness of Homogeneity Tests in 2×N Tables, in Biometrics, No.

21, pp. 19-33. 
References

9-6

Lofgren, E. V., and M. Thaggard, 1992, Analysis of Standby and Demand Stress Failures Modes: Methodology and

Applications to EDGs and MOVs, NUREG/CR-5823, SAIC 87-029, Science Applications International Corporation.

Lwin, T., and N. Singh, 1974, Bayesian Analysis of the Gamma Distribution Model in Reliability Engineering, in

IEEE Transactions on Reliability, Vol. R-23, pp. 314-319.

Mackowiak, D. P., C. D. Gentillon, and K. L. Smith, 1985, Development of Transient Initiating Event Frequencies

for Use in Probabilistic Risk Assessments, NUREG/CR-3862, EGG-2323, EG&G Idaho, Inc., Idaho Falls, ID.

Manning, J. J., et al., 1986, A Guide for Completing Input Data Forms for CREDO, A Centralized Reliability,

Availability and Maintainability Analysis Center for Liquid-Metal-Cooled Reactor and Test Facility Components,

ORNL/TM-9892, Oak Ridge National Laboratory, Oak Ridge, TN.

Marshall, F.M., D.M. Rasmuson, and A. Mosleh, 1998, Common-Cause Parameter Estimations, NUREG/CR-5497,

Idaho National Engineering and Environmental Laboratory, Idaho Falls, ID.

Martz, H.F., and R.G. Almond, 1997, Using Higher-Level Failure Data in Fault Tree Quantification, in Reliability

Engineering and System Safety, Vol. 56, pp. 29-42.

Martz, H. F., and M. C. Bryson, 1983, On Combining Data for Estimating the Frequency of Low-Probability Events

with Application to Sodium Valve Failure Rates, in Nuclear Science and Engineering, Vol. 83, No. 2, pp. 267-280.

Martz, H. F., and M. S. Hamada, 2003, Uncertainty in Counts and Operating Time in Estimating Poisson Occurrence

Rates, in Reliability Engineering and System Safety, Vol. 80, pp. 75-79.

Martz, H. F., R. L. Parker, and D. M. Rasmuson, 1999, Estimation of Trends in the Scram Rate at Nuclear Power

Plants, in Technometrics, Vol. 41, pp. 352-364.

Martz, H. F., and R. R. Picard, 1995. Uncertainty in Poisson Event Counts and Exposure Time in Rate Estimation,

in Reliability Engineering and System Safety, Vol. 48, pp. 181-190.

Martz, H. F., P. H. Kvam, and L. R. Abramson, 1996. Empirical Bayes Estimation of the Reliability of NuclearPower-Plant Emergency Diesel Generators, in Technometrics, Vol. 38, pp. 11-24.

Martz, H. F., and R. A. Waller, 1991, Bayesian Reliability Analysis. R. E. Krieger Publishing Co., Malabar, FL. 

McClymont, A. S., and B. W. Poehlman, 1982a, ATWS: A Reappraisal, Part 3: Frequency of Anticipated

Transients, Interim Report, EPRI-NP-2230-PT.3, Electric Power Research Institute, Palo Alto, CA.

McClymont, A. S., and B. W. Poehlman, 1982b, Loss of Off-site Power at Nuclear Power Plants: Data and

Analysis, EPRI NP-2301, Electric Power Research Institute, Palo Alto, CA.

McCullagh, P., and J. A. Nelder, 1989, Generalized Linear Models, 2nd ed. Chapman & Hall, NY.

Microsoft Excel, 2001, Microsoft Corporation, Seattle, WA.

Moore, D. S., 1986, Tests of Chi-Squared Type, chap. 3 in Goodness-of-Fit Techniques, Ralph B. D'Agostino and

Michael A. Stephens, ed., pp. 63-95. Marcel Dekker, Inc., NY.

Mosleh, A., 1986, Hidden Sources of Uncertainty: Judgment in the Collection and Analysis of Data, in Nuclear

Engineering and Design, Vol. 93, pp. 187-198.

Mosleh, A., and G. Apostolakis, 1985, The Development of a Generic Data Base for Failure Rates, in Proceedings

of the ANS/ENS International Topical Meeting on Probabilistic Safety Methods and Applications, held in San

Francisco, CA on February 24 - March 1, EPRI NP-3912-SR.
References

9-7

Mosleh, A., K. N. Fleming, G. W. Parry, H. M. Paula, D. H. Worledge, and D. M. Rasmuson, 1988, Procedures for

Treating Common Cause Failures in Safety and Reliability Studies, NUREG/CR-4780, EPRI NP-5613, PLG-0547,

Vol. 1.

Mosleh, A., K. N. Fleming, G. W. Parry, H. M. Paula, and D. H. Worledge, 1989, Procedures for Treating Common

Cause Failures in Safety and Reliability Studies: Analytical Background and Techniques, NUREG/CR-4780V2,

PLG-0547-VOL-2, EPRINP-5613-VOL-2, Pickard, Lowe, and Garrick, Inc., Newport Beach, CA.

Mosleh, A., D.M. Rasmuson, and F.M. Marshall, 1998, Guidelines on Modeling Common-Cause Failures in

Probabilistic Risk Assessments, NUREG/CR-5485, Idaho National Engineering and Environmental Laboratory,

Idaho Falls, ID.

Mosleh, A., N. Siu, C. Smidts, and C. Lui, 1994, Proceedings of Workshop I in Advanced Topics in Risk and

Reliability Analysis. Model Uncertainty: Its Characterization and Quantification, NUREG/CP-1038.

Nelson, W., 1982, Applied Life Data Analysis. John Wiley & Sons, NY. 

Neter, J., and W. Wasserman, 1974, Applied Linear Statistical Models: Regression, Analysis of Variance, and

Experimental Designs. Richard D. Irwin, Inc., Homewood, IL.

Nickolaus, J.R., N. E. Moffitt, B. F. Gore, and T. V. Vo, 1992, Auxiliary Feedwater System Risk-Based Inspection

Guide for the North Anna Nuclear Power Plants, NUREG/CR-5837, PNL-7923, Battelle Pacific Northwest

Laboratory, Richland, WA. 

Nowlen, S. P., A. Boone, A. Kolaczkowski, D. Kelly, N. Siu, and H. W. R. Woods, 2002, Fire Detection and

Suppression: PRA Modeling and Data Analysis, in Proceedings of the 6th International Conference on Probabilistic

Safety Assessment and Management. Elseveir, 2002, Volume I, pgs 243-249.

NRC, 1975a, Reporting of Operating Information - Appendix A Technical Specifications, Draft Regulatory Guide

1.16, Nuclear Regulatory Commission, Washington, D.C.

NRC, 1975b, Reactor Safety Study - An Assessment of Accident Risks in U.S. Commercial Nuclear Power Plants,

WASH-1400, NUREG-75/014, Nuclear Regulatory Commission, Washington, D.C.

NRC, 1990, Severe Accident Risks: An Assessment for Five U.S. Nuclear Power Plants: Final Summary Report,

NUREG-1150-V1, Nuclear Regulatory Commission, Washington, D.C.

NRC, 1995a, Use of Probabilistic Risk Assessment Methods in Nuclear Activities: Final Policy Statement, Federal

Register, Vol. 60, pp. 42622. Washington, D. C.

NRC, 1995b, Licensed Operating Reactors – Status Summary Report, NURG-0020, Nuclear Regulatory

Commission, Washington, D.C.

NRC, 1997, Revised Contents of the Monthly Operating Report, Generic Letter 97-02, Nuclear Regulatory

Commission, Washington, D.C.

NRC, 1998, Event Reporting Guidelines 10 CFR 50.72 and 50.73, NUREG-1022, Rev. 1, Nuclear Regulatory

Commission, Washington, D.C.

NRC, 2000a, Event Reporting Guidelines 10 CFR 50.72 and 50.73, NUREG-1022, Rev. 2, Nuclear Regulatory

Commission, Washington, D.C.

NRC, 2000b, Technical Basis and Implementation Guidelines for A Technique for Human Event Analysis, NUREG1624, Rev. 1, Nuclear Regulatory Commission, Washington, D.C.
References

9-8

OREDA, 1997, Offshore Reliability Data Handbook, OREDA-97, 3rd edition, SINTEF Industrial Management, Det

Norske Veritas, Veritasveien 1, P.O. Box 300, –1322 Høvik, Norway.

Oswald, A. J., C. D. Gentillon, S. D. Matthews, and T. R. Meachum, 1982, Generic Data Base for Data and Models

Chapter of the National Reliability Evaluation Program (NREP) Guide, EGG-EA-5887, Idaho National Engineering

and Environmental Laboratory, Idaho Falls, ID.

Papazoglou, I. A., R. A. Bari, A. J. Buslik, R. E. Hall, and D. Ilberg, 1984, Probabilistic Safety Analysis Procedures

Guide, NUREG/CR-2815, Brookhaven National Laboratory, Upton, NY.

Parry, G.W., B.O.Y. Lydell, A.J. Spurgin, P. Moieni, and A. Beare, 1992, An Approach to the Analysis of Operator

Actions in Probabilistic Risk Assessment, EPRI TR-100259, Electric Power Research Institute, Palo Alto, CA. 

PECO, 1982, Probabilistic Risk Assessment: Limerick Generation Station, Revision 4, Philadelphia Electric

Company, Philadelphia, PA.

Poloski, J. P., and W. H. Sullivan, 1980, Data Summaries of Licensee Event Reports of Diesel Generators at U.S.

Commercial Nuclear Power Plants, January 1, 1976 to December 31, 1978, NUREG/CR-1362, EGG-EA-5092,

Idaho National Engineering and Environmental Laboratory, Idaho Falls, ID.

Poloski, J. P., G. M. Grant, C. D. Gentillon, W. J. Galyean, and J. K. Knudsen, 1998, Reliability Study:

Auxiliary/Emergency Feedwater System, 1987-1995, NUREG/CR-5500, Vol. 1, INEEL/EXT-97/00740, Idaho

National Engineering and Environmental Laboratory, Idaho Falls, ID.

Poloski, J.P., D. G. Marksberry, C. L. Atwood, and W. J. Galyean, 1999a, Rates of Initiating Events at U.S. Nuclear

Power Plants: 1987-1995, NUREG/CR-5750, INEEL/EXT-98-00401, Idaho National Engineering and

Environmental Laboratory, Idaho Falls, ID.

Poloski, J. P., G. M. Grant, C. D. Gentillon, W. J. Galyean, and W. S. Roesener, 1999b, Reliability Study: Reactor

Core Isolation Cooling System, 1987-1993, NUREG/CR-5500, Vol. 7, INEL-95/0196, Idaho National Engineering

and Environmental Laboratory, Idaho Falls, ID.

Poloski, J. P., G. M. Grant, C. D. Gentillon, and W. J. Galyean, 1999c, Reliability Study: High Pressure Core Spray

System, 1987-1993, NUREG/CR-5500, Vol. 8, INEL-95/00133, Idaho National Engineering and Environmental

Laboratory, Idaho Falls, ID.

Poloski, J. P., J. K. Knudsen, C.L. Atwood, and W. J. Galyean, 2000, Reliability Study: High Pressure Safety

Injection System, 1987-1997, NUREG/CR-5500, Vol. 9, INEEL/EXT-99-00373. Idaho National Engineering and

Environmental Laboratory, Idaho Falls, ID.

Pörn, K., 1990, On Empirical Bayesian Inference Applied to Poisson Probability Models, Linköping Studies in

Science and Technology, Dissertation No. 234. Division of Quality Technology, Department of Mechanical

Engineering, Linköping University, Linköping, Sweden, ISBN 91-7890-696-3.

Pörn, K., 1996, “The two-stage Bayesian Method used for the T-Book Application,” in Reliability Engineering and

System Safety, Vol. 51, Issue 2, pp. 169-179.

Pörn, K., R. Blomquist, and K. Shen, 1993, Initiating Events in Swedish Nuclear Power Plants – Classification and

Trends, in Proceedings of the International Topical Meeting on Probabilistic Safety Assessment, PSA '93, clearwater

Beach, FL. Published by the American Nuclear Society, ISBN 0-89448-180-0, pp. 1190-1194.

Pörn, K., K. Shen, and R. Nyman, 1994, I-Book, Version 2. Initiating Events at the Nordic Nuclear Power Plants

(in Swedish), SKI Report 94:12, Swedish Nuclear Power Inspectorate, Stockholm, Sweden. ISSN 1104-1374, ISRN

SKI-R-94-12-SE.
References

9-9

Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992, Numerical Recipes in Fortran 77:

Volume 1 of Fortran Numerical Recipes, Second Edition, Cambridge University Press, New York. (Also available

for C++ and Pascal.)

Quattro Pro, 2001, Corel Corporation, Ottawa, Canada.

Raiffa, H., and R Schlaifer, 1961, Applied Statistical Decision Theory, The M.I.T. Press, Cambridge, MA.

Roscoe, J. T., and J. A. Byars, 1971, An Investigation of the Restraints with Respect to Sample Size Commonly

Imposed on the Use of the Chi-square Statistic, in Journal of the American Statistical Association, Vol. 66, Issue

336, pp. 755-759.

Ross, S., 1983, Stochastic Processes, John Wiley and Sons, NY. 

Rubin, D. B., 1996, Multiple Imputation after 18+ Years (with Discussion), in Journal of the American Statistical

Association, Vol. 91, pp. 444-489.

Samanta, P. K., I. S. Kim, T. Mankamo, and W. E. Vesely, 1994, Handbook of Methods for Risk-Based Analyses of

Technical Specifications, NUREG/CR-6141, BNL-NUREG-52398, Brookhaven National Laboratory, Upton, NY.

Sandtorv, H. A., P. Hokstad, and D. W. Thompson, 1996, Practical Experiences with a Data Collection Project: The

OREDA project, in Reliability Engineering and System Safety, Vol. 51, Issue 2, pp. 159-167.

SAS, Version 8. 2000. SAS Institute, Cary, NC.

SAS/INSIGHT Users Guide, Version 6, Third Edition, 1995, SAS Institute Inc., SAS Campus Drive, Cary, NC.

Shah, V. N., M. B. Sattison, C. L. Atwood, A. G. Ware, and G. M. Grant, 1998, Assessment of Pressurized Water

Reactor Primary System Leaks, NUREG/CR-6582, INEEL/EXT-97-01068, Idaho National Engineering and

Environmental Laboratory, Idaho Falls, ID. 

Silverman, B. W., 1986, Density Estimation for Statistics and Data Analysis. Chapman and Hall, London, England.

Simard, R. L., 1983, Recent Initiatives in Information Exchange Through the International Nuclear Community, in

Progress in Nuclear Energy, Vol. 12, No. 3, pp. 221-225.

Simard, R. L., 1985, Application of NPRDS Data to Enhance Nuclear Plant Design and Operations, in Proceedings:

International Topical Meeting on Probabilistic Safety Methods and Applications Volume: Sessions 1-8, conference

held in Palo Alto, CA, on Feb. 24 - March 1, EPRI NP-3912-SR, Volume 1, Electric Power Research Institute, Palo

Alto, CA.

Siu, N., and G. Apostolakis, 1984, Models for the Detection and Suppression of Fires in Nuclear Power Plants and

Some Related Statistical Problems, UCLA-ENG-8440.

Siu, N., and G. Apostolakis, 1986, A Methodology for Analyzing the Detection and Suppression of Fires in Nuclear

Power Plants, in Nuclear Science and Engineering, Vol. 94, pp. 213-226.

Siu, N. O., and D. L. Kelly, 1998, Bayesian Parameter Estimation in Probabilistic Risk Assessment, in Reliability

Engineering and System Safety, Volume 62, Issue 1-2, pp. 89-116.

Smith, A. F. M., and A. E. Gelfand, 1992, Bayesian Statistics without Tears: A Sampling-Resampling Perspective, in

The American Statistician, Vol. 46, No. 2, pp. 84-88. 

Spiegelhalter, D. J., A. Thomas, N. G. Best, and W. R. Gilks, 1995, BUGS: Bayesian Inference Using Gibbs

Sampling, Version 0.50, MRC Biostatistics Unit, Cambridge, UK. Available at www.mrc-bsu.cam.ac.uk/bugs/
References

9-10

Staple, B.D., T.D. Brown, J.J. Gregory, and J.L. LaChance, 1999, Risk Comparisons of Scheduling Preventative

Maintenance for Boiling Water Reactors During Shutdown and Power Operation, NUREG/Cr-5593, SAND99-

0676, Sandia National Laboratories, Albuquerque, NM.

STATISTICA for Windows, Vols. I and III, 1995, StatSoft, Inc., 23255 East 13th Street, Tulsa, OK.

StatXact, 1999, CYTEL Software Corp., Cambridge, MA.

Swain, A.D, 1987, Accident Sequence Evaluation Program Human Reliability Analysis Procedure, NUREG/CR4772, Sandia National Laboratories, Albuquerque, NM.

Swain, A.D., and H.E. Guttman, 1983, Handbook of Human Reliability Analysis with Emphasis on Nuclear Power

Plant Applications, NUREG/CR-1278, Sandia National Laboratories, Albuquerque, NM.

Tashjian, B. M., 1982, Nuclear Plant Reliability Data System Data Usage in Availability Engineering, in IEEE

Transactions on Power Apparatus and Systems, Vol. PAS-101, No. 8, pp. 2582-2590.

Thompson, W. A., Jr., 1981, On the Foundations of Reliability, in Technometrics, Vol. 23, No. 1, pp. 1-13. 

Trojovsky, M., 1982, Data Summaries of Licensee Event Reports Of Pumps at U. S. Commercial Nuclear Power

Plant, NUREG/CR-1205, Revision 1, EGG-EA-5524, Idaho National Engineering and Environmental Laboratory,

Idaho Falls, ID. 

TUD Office and Pörn Consulting, 2000, T-Book: Reliability Data of Components in Nordic Nuclear Power Plants,

5th Edition, Battenfall Inhouse-Grafiska, Vällingby, Sweden.

Tukey, J. W., 1977, Exploratory Data Analysis. Addison-Wesley Publishing Co., Reading, MA. 

Vesely, W.E., F. F. Goldberg, N. H. Roberts, and D. F. Haasl, 1981, Fault Tree Handbook, NUREG-0492, U. S.

Nuclear Regulatory Commission, January 1981.

Walpole, R. E., and R. H. Myers, 1985, Probability and Statistics for Engineers and Scientists, 2nd edition. 

Macmillan, NY.

Wheeler, T.A., S. C. Hora, W. R. Crammand, and S. D. Unwin, 1989, Analysis of Core Damage Frequency from

Internal Events: Expert Judgment Elicitation, NUREG/CR-4550, Vol. 2, SAND86-2084, Sandia National

Laboratories, Albuquerque, NM.

Whitehead, D.W., H.M. Paula, G.W. Perry, and D.M. Rasmuson, 1993, Enhancements to Data Collection and

Reporting of Single and Multiple Failure Events, NUREG/CR-5471, SAND89-2562, Sandia National Laboratories,

Albuquerque, NM.

Wierman, T.E., S.T. Beck, M.B. Calley, S.A. Eide, C.D. Gentillon, and W.E. Kohn, 2002a, Reliability Study:

Combustion Engineering Reactor Protection System, 1984-1998, NUREG/CR-5500, Vol. 10, INEL/EXT-97-00740, 

Idaho National Engineering and Environmental Laboratory, Idaho Falls, ID.

Wierman, T.E., S.T. Beck, M.B. Calley, S.A. Eide, C.D. Gentillon, and W.E. Kohn, 2002b, Reliability Study:

Babcock & Wilcox Reactor Protection System, 1984-1998, NUREG/CR-5500, Vol. 11, INEL/EXT-97-00740, 

Idaho National Engineering and Environmental Laboratory, Idaho Falls, ID.

Wilk, M. B., and R. Gnanadesikan, 1968, Probability Plotting Methods for the Analysis of Data, in Biometrika, Vol.

55, pp. 1-17.

Zacks, S., 1992, Introduction to Reliability Analysis: Probability Models and Statistical Methods, Springer-Verlag,

New York.
APPENDICES

A-1

E1

E2

E3

GC00 0432 1

Figure A.1 Venn diagram, showing ten outcomes

and three events.

A. BASICS OF PROBABILITY

A.1 Events

Any repeatable process for which the result is uncertain

can be considered an experiment, such as counting

failures over time or measuring time to failure of a

specific item of interest. The result of one execution of

the experiment is referred to as an outcome. Due to

uncertainty associated with the process, repetitions or

trials of a defined experiment would not be expected to

produce the same outcomes. The set of all possible

outcomes of an experiment is defined as the sample

space.

Sample spaces can contain discrete points (such as pass,

fail) or points in a continuum (such as measurement of

time to failure). An event E is a specified set of possible outcomes in a sample space S (denoted E d S, where

d denotes subset).

Most events of interest in practical situations are

compound events, formed by some composition of two

or more events. Composition of events can occur

through the union, intersection, or complement of

events, or through some combination of these.

For two events, E1 and E2, in a sample space S, the

union of E1 and E2 is defined to be the event containing

all sample points in E1 or E2 or both, and is denoted by

the symbol (E1 c E2). Thus, a union is simply the event

that either E1 or E2 occurs or both E1 and E2 occur.

For two events, E1 and E2, in a sample space S, the

intersection of E1 and E2 is defined to be the event

containing all sample points that are in both E1 and E2,

denoted by the symbol (E1 1 E2). The intersection is

the event that both E1 and E2 occur.

Figure A.1 shows a symbolic picture, called a Venn

diagram, of some outcomes and events. In this example, the event E1 contains three outcomes, event E2

contains five outcomes, the union contains seven

outcomes, and the intersection contains one outcome.

The complement of an event E is the collection of all

sample points in S and not in E. The complement of E

is denoted by the symbol &E and is the outcomes in S

that are not in E occur. In Figure A.1, the complement

of E1 is an event containing seven outcomes.

It is sometimes useful to speak of the empty or null set,

a set containing no outcomes. In Figure A.1, the event

E3 is empty. It cannot occur.

Two events, E1 and E2, are said to be mutually exclusive if the event (E1 1 E2) contains no outcomes in the

sample space S. That is, the intersection of the two

events is the null set. Mutually exclusive events are

also referred to as disjoint events. Three or more

events are called mutually exclusive, or disjoint, if each

pair of events is mutually exclusive. In other words, no

two events can happen together.

A.2 Basic Probability Concepts

Each of the outcomes in a sample space has a probability associated with it. Probabilities of outcomes are

seldom known; they are usually estimated from relative

frequencies with which the outcomes occur when the

experiment is repeated many times. Once determined,

the probabilities must satisfy two requirements:

1. The probability of each outcome must be a number

$ 0 and # 1.

2. The probabilities of all outcomes in a given sample

space must sum to 1.

Associated with any event E of a sample space S is the

probability of the event, Pr(E). Since an event represents a particular set of outcomes of an experiment, the

values of Pr(E) are built from the probabilities of the

outcomes in E.

Probabilities are associated with each outcome in the

sample space through a probability model. Probability
Basics of Probability

A-2

models are often developed on the basis of information

derived from outcomes obtained from an experiment.

Probability models are also formulated in the context of

mathematical functions. 

 

The values of Pr(E) estimated from the experimental

outcomes are often defined as being representative of

the long-run relative frequency for event E. That is, the

relative frequency of an outcome will tend toward some

number between 0 and 1 (inclusive) as the number of

repetitions of the experiment increases. Thus, the

probability of the outcome is the number about which

the long-term relative frequency tends to stabilize.

This interpretation forms the basis of the relative

frequency definition of probability, also referred to as

the frequentist view of probability. In the frequentist

view, a mathematical theory of probability is developed

by deriving theorems based on the axioms of probability given in the next subsection. The probability of an

event is considered to be a fixed quantity, either known

or unknown, that is a property of the physical object

involved and that can be estimated from data. A

theorem derived from the three axioms describes the

frequentist view:

If an experiment is repeated a large number of times, n,

the observed relative frequency of occurrence, nE /n, of

the event E (where nE = the number of repetitions when

event E occurred) will tend to stabilize at a constant,

Pr(E), referred to as the probability of E.

Another interpretation of probability leads to the socalled classical definition of probability, which can be

stated as follows:

If an experiment can result in n equally likely and

mutually exclusive outcomes and if nE of these outcomes contain attribute E, then the probability of E is

the ratio nE / n.

For example, if each of the outcomes in Figure A.1 had

equal probability, 0.1, then Pr(E1) = 0.3, Pr(E2) = 0.5,

Pr(E11E2) = 0.1, Pr(E1cE2) = 0.7, and Pr(E3) = 0.

The classical definition is limited, because it assumes

equally likely outcomes. However, it helps motivate

the frequentist axioms mentioned above. These axioms

provide a mathematical framework for probability, an

overview of which is addressed in Section A.3. Some

texts, including parts of this handbook, use the terms

classical and frequentist interchangeably.

Another interpretation of probability is as a subjective

probability. Probabilities obtained from the opinions

of people are examples of subjective probabilities. In

this concept, probability can be thought of as a rational

measure of belief. Any past information about the

problem being considered can be used to help assign

the various probabilities. In particular, information

about the relative frequency of occurrence of an event

could influence the assignment of probabilities.

The notion of subjective probability is the basis for

Bayesian inference. In contrast to the relative frequency definition of probability that is based on properties of events, subjective probability can be extended to

situations that cannot be repeated under identical

conditions. However, the assignment of subjective

probabilities can be done according to certain principles

so that the frequency definition requirements of probability are satisfied. All the mathematical axioms and

theorems developed for frequentist probability apply to

subjective probability, but their interpretation is different.

Martz and Waller (1991) present subjective probability

as dealing not only with events but with propositions.

A proposition is considered to be a collection of events

that cannot be conceived as a series of repetitions, for

example, a nuclear power plant meltdown. The degree

of belief in proposition A, Pr(A), represents how

strongly A is believed to be true. Thus, subjective

probability refers to the degree of belief in a proposition. At the extremes, if A is believed to be true, Pr(A)

= 1; if A is believed to be false, Pr(A) = 0. Points

between 0 and 1 represent intermediate beliefs between

false and true.

A.3 Basic Rules and Principles of

Probability

The relative frequency, classical, and subjective

probability definitions of probability satisfy the following axiomatic requirements of probability:

If Pr(E) is defined for a type of subset of the sample

space S, and if

1. Pr(E) $ 0, for every event E,

2. Pr(E1 c E2 c @ @ @) = Pr(E1) + Pr(E2) + @ @ @ , where

the events E1, E2, . . . , are such that no two have a

point in common, and

3. Pr(S) = 1,

then Pr(E) is called a probability function.

A probability function specifies how the probability is

distributed over various subsets E of a sample space S.
Basics of Probability

A-3

From this definition, several rules of probability follow

that provide additional properties of a probability

function.

The probability of an impossible event (the empty or

null set) is zero, written as:

Pr(i) = 0,

where i is the null set. The probability of the complement of E is given by:

Pr(&E) = 1 ! Pr(E).

In general, the probability of the union of any two

events is given by:

Pr(E1 c E2) = Pr(E1) + Pr(E2) ! Pr(E1 1 E2).

If E1 and E2 are mutually exclusive, then Pr(E1 1 E2) =

Pr(i) = 0, and

Pr(E1 c E2) = Pr(E1) + Pr(E2),

which is a special case of the second axiom of probability stated above and is sometimes referred to as the

addition rule for probabilities.

For three events,

Pr(E1 c E2 c E3) = Pr(E1) + Pr(E2) + Pr(E3)

! Pr(E1 1 E2) ! Pr(E1 1 E3)

! Pr(E2 1 E3) + Pr(E1 1 E2 1 E3 ).

This rule is also referred to as the inclusion-exclusion

principle and can be generalized to n events. It is

widely used in PRA to calculate the probability of an

“or” gate (a union of events) in a fault tree (NRC 1994).

The inclusion-exclusion principle also provides useful

upper and lower bounds on the probability of the union

of n events that are not mutually exclusive. One such

upper bound, referred to as the rare event approximation, is:

Pr(E1 c E2 c ... c En) # Pr(E1) + Pr(E2) + ... + Pr(En).

The rare event approximation should only be used when

the probabilities of the n events are all very small (NRC

1994). If the n events are mutually exclusive, the error

is zero. A bound on the error is

 max [Pr(Ei

)], n

2

⎛

⎝

⎜ ⎞

⎠

⎟

which is valid regardless of the independence of events

(NRC 1994, though printed with a misprint there). The

error in the rare-event approximation arises from the

remaining terms in the full expansion of the left-hand

side of the inequality. This approximation is frequently

used in accident sequence quantification.

Many experimental situations arise in which outcomes

are classified by two or more events occurring simultaneously. The simultaneous occurrence of two or more

events (the intersection of events) is called a joint event,

and its probability is called a joint probability. Thus,

the joint probability of both events E1 and E2 occurring

simultaneously is denoted by Pr(E1 1 E2).

The probability associated with one event, irrespective

of the outcomes for the other events, can be obtained by

summing all the joint probabilities associated with all

the outcomes for the other events, and is referred to as

a marginal probability. A marginal probability is

therefore the unconditional probability of an event,

unconditioned on the occurrence of any other event.

Two events E1 and E2 are often related in such a way

that the probability of occurrence of one depends on

whether the other has or has not occurred. The conditional probability of one event, given that the other has

occurred, is equal to the joint probability of the two

events divided by the marginal probability of the given

event. Thus, the conditional probability of event E2,

given event E1 has occurred, denoted Pr(E2*E1), is

defined as:

Pr(E2*E1) = Pr(E1 1 E2) / Pr(E1), (A.1)

for Pr(E1) > 0. If Pr(E1) = 0, Pr(E2*E1) is undefined.

Rearranging this equation yields:

Pr(E1 1 E2) = Pr(E1) Pr(E2*E1)

(A.2)

= Pr(E2) Pr(E1*E2).

Calculation of joint probability requires the concept of

statistical independence. An event E2 is statistically

independent of E1 if the probability of E2 does not

change whenever E1 occurs or does not occur. Thus, E2

is independent of E1 if

Pr(E2*E1) = Pr(E2).

It follows from Equation A.1 that E2 is independent of

E1 if their joint probability is equal to the product of the

unconditional, or marginal, probabilities of the events:

Pr(E1 1 E2) = Pr(E1) Pr(E2).
Basics of Probability

A-4

This is sometimes referred to as the multiplication rule

for probabilities. In this formulation, it is clear that E2

is independent of E1 if E1 is independent of E2, and we

say simply that E1 and E2 are statistically independent.

If Pr(E2) varies depending on whether or not event E1

has occurred, then events E1 and E2 are said to be

statistically dependent.

If E1, E2, ... are mutually exclusive, and if the union of

E1, E2, ... equals the entire sample space, then the events

E1, E2, ... are said to form a partition of the sample

space. Exactly one of the events must occur, not more

than one but exactly one. In this case, the law of total

probability says

Pr(A) = 3Pr(A |Ei

) Pr(Ei) .

A special case can be written when there are only two

sets. In this case, write E1 simply as E and E2 as &E . 

Then the law of total probability simplifies to

Pr(A) = Pr(A*E)Pr(E) + Pr(A*&E )Pr(&E ) 

for any event A. This formula is the basis for event

trees, which are frequently used to diagram the possibilities in an accident sequence. 

The concepts of mutually exclusive events and statistically independent events are often confused. If E1 and

E2 are mutually exclusive events and Pr(E1) and Pr(E2)

are nonzero, Pr(E1 1 E2) = Pr(i) = 0. From Equation

A.1, Pr(E2*E1) = 0, which does not equal Pr(E2). Thus,

the two events are not independent. Mutually exclusive

events cannot be independent and independent events

cannot be mutually exclusive.

Equation A.2 can be used to calculate the probability of

the intersection of a set of events (the probability that

all the events occur simultaneously). For two events E1

and E2, the probability of simultaneous occurrence of

the events is equal to the probability of E1 times the

probability of E2 given that E1 has already occurred. In

general, the probability of the simultaneous occurrence

of n events can be written as:

Pr(E1 1 E2 1 ... 1 En) =

Pr(E1) Pr(E2*E1) Pr(E3*E2 1 E1) ... Pr(En*En-1 1 ... 1 E1),

which is referred to as the chain rule. This rule can be

used to calculate the probability that a given accident

sequence occurs, with E1 denoting the initiating event

and the remaining events corresponding to the failure or

success of the systems that must function in order to

mitigate such an accident.

The probability of occurrence of at least one of a set of

statistically independent events yields a result that is

important to PRA and fault tree applications. If E1, E2,

..., En are statistically independent events, the probability that at least one of the n events occurs is:

Pr(E1 c E2 c ... c En) = (A.3)

1 ! [1 ! (Pr(E1)][1 ! (Pr(E2)] ... [1 ! (Pr(En)],

which is equivalent (with expansion) to using the

inclusion-exclusion rule. For the simple case where

Pr(E1) = Pr(E2) = ... = Pr(En) = p, the right-hand side of

this expression reduces to 1 ! (1 ! p)

n

.

The general result in Equation A.3 has application in

PRA and fault tree analysis. For example, for a system

in which system failure occurs if any one of n independent events occurs, the probability of system failure is

given by Equation A.3. These events could be failures

of critical system components. In general, the events

represent the modes by which system failure (the top

event of the fault tree) can occur. These modes are

referred to as the minimal cut sets of the fault tree and,

if independent of each other (no minimal cut sets have

common component failures), Equation A.3 applies.

[See Vesely et al. (1981) for further discussion of fault

trees and minimal cut sets.]

If the n events are not independent, the right side of

Equation A.3 may be greater than or less than the left

side. However, for an important situation that frequently arises in PRA, the right side of Equation A.3

forms an upper bound for the left side.

If the n events are cut sets that are positively associated

[see Esary and Proschan (1970, 1963)], then the right

side is an upper bound for Pr(E1 c E2 c ... c En) and is

known as the min cut upper bound (NRC 1994). This

name arises from common PRA applications where Ei

is the i

th minimal cut set of a system or accident sequence of interest. In this case, the min cut upper

bound is superior to the rare event approximation and

can never exceed unity (as can happen with the rare

event approximation). If the n events satisfy conditions

similar to those of the rare event approximation, the min

cut set upper bound is a useful approximation to the left

side of Equation A.3. Note that the min cut upper

bound is not applicable for mutually exclusive events.
Basics of Probability

A-5

A.4 Random Variables and 

Probability Distributions

A.4.1 Random Variables

A random variable is any rule that associates real

numbers with the outcomes of an experiment. For

example, the number of initiating events in one year, the

number of failures to start in 12 demands, and the time

to complete a repair of a pump are all random variables.

If the numbers associated with the outcomes of an

experiment are all distinct and countable, the corresponding random variable is called a discrete random

variable. The number of initiating events and the

number of failures to start are examples of discrete

random variables.

If the sample space contains an infinite number of

outcomes (like those contained in any interval), the

random variable is continuous. Time T is a common

continuous random variable, for example, time to

failure, time between failures, or time to repair, where

the random variable T can assume any value over the

range 0 to 4.

A.4.2 Probability Distributions

A probability function (introduced at the beginning of

Section A.3) associates a probability with each possible

value of a random variable and, thus, describes the

distribution of probability for the random variable. For

a discrete random variable, this function is referred to

as a discrete probability distribution function

(p.d.f.). A discrete p.d.f., commonly denoted by f, is

also referred to as a discrete distribution, or discrete

probability mass function.

If x denotes a value that the discrete random variable X

can assume, the probability distribution function is

often denoted Pr(x). The notation used here is that a

random variable is denoted by an upper-case letter and

an observed or observable value of the random variable

(a number) is denoted by a lower-case letter. The sum

of the probabilities over all the possible values of x

must be 1. Thus, we write f(x) = Pr(X = x), and require

Eif(xi) = 1.

Certain discrete random variables have wide application

and have therefore been defined and given specific

names. The two most commonly used discrete random

variables in PRA applications are the binomial and

Poisson random variables, which are presented in

Section A.6.

A continuously distributed random variable has a

density function, a nonnegative integrable function,

with the area between the graph of the function and the

horizontal axis equal to 1. This density function is also

referred to as the continuous probability distribution

function (p.d.f.). If x denotes a value that the continuous random variable X can assume, the p.d.f. is often

denoted as f(x). The probability that X takes a value in

a region A is the integral of f(x) over A. In particular,

Pr( ) ( ) a X b f x dx a

b ≤≤= ∫

and

Pr(x # X # x + )x) . f(x))x (A.4)

for small )x. Also,

f x dx ( ) = −∞

∞

∫ 1

The most commonly used continuous distributions in

PRA are the lognormal, exponential, gamma, and

beta distributions. Section A.7 summarizes the essential facts about these distributions, and also about less

common but occasionally required distributions:

uniform, normal, Weibull, chi-squared, inverted

gamma, logistic-normal, Student’s t, F, and Dirichlet.

A.4.3 Cumulative Distribution Functions

Discrete probability distributions provide point probabilities for discrete random variables and continuous

p.d.f.s provide point densities for continuous random

variables. A related function useful in probability and

PRA is the cumulative distribution function (c.d.f.).

This function is defined as the probability that the

random variable assumes values less than or equal to

the specific value x, and is denoted F(x).

For a discrete random variable X, with outcomes xi

, and

the corresponding probabilities Pr(xi

), F(x) is the sum

of the probabilities of all xi # x. That is,

Fx X x x . i

x x i

( ) = Pr( ≤ ) =

≤

∑ Pr( )

For a continuous random variable X, F(x) is the area

beneath the p.d.f. f(x) up to x. That is, F(x) is the

integral of f(x):

F(x) = Pr(X # x) = . f y dy

x

( )

−∞

∫
Basics of Probability

A-6

If X takes on only positive values, the lower limit of

integration can be set to 0. The upper limit is x, and f(x)

is the derivative of F(x). Note that, because F(x) is a

probability, 0 # F(x) # 1. If X ranges from -4 to +4,

then

F(-4) = 0 and F(+4) = 1.

If X has a restricted range, with a and b being the lower

and upper limits of X respectively, a < X < b, then

F(a) = 0 and F(b) = 1.

Also, F(x) is a nondecreasing function of x, that is,

if x2 > x1, F(x2) $ F(x1).

Another important property of F(x) is that

Pr(x1 < X # x2) = F(x2) ! F(x1)

for discrete random variables and

Pr(x1 # X # x2) = F(x2) ! F(x1)

for continuous random variables.

An example of a p.d.f. and the associated c.d.f. for a

continuous distribution is shown in Figure A.2.

Figure A.2 Probability density function (p.d.f.) and

cumulative distribution function (c.d.f.).

A.4.4 Reliability and Hazard Functions

A.4.4.1 Definitions

There are also characterizations that have special

interpretations for time-to-failure distributions. Let T

denote the random time to failure of a system. The

reliability function of a system is defined as

R(t) = Pr(T > t) .

Hence, R(t), called the reliability at time t, is the

probability that the system does not fail in the time

interval (0, t) or equivalently, the probability that the

system is still operating at time t. (This discussion uses

the notation (a, b) to mean the set of times > a and # b,

but the distinction between < and # is a mathematical

fine point, not important in practice.) The reliability

function is also sometimes called the survival function.

It is equal to 1 ! F(t).

When used as a reliability criterion, it is common to

state a time, say t0, called the mission time, and require

for a system that the reliability at mission time t0 be at

least some prescribed level, say R0. For example, a

pump might be required to operate successfully for at

least 12 hours with probability at least 0.95. The

requirement in this case is R0 = 0.95 and t0 = 12. In

terms of the reliability function, this would mean R(12)

$ 0.95. One interpretation would be that such a pump

would perform for the required mission time for 95% of

the situations when it is called on to do so. Another

interpretation is that 95% of all such pumps would

perform as required.

Consider a system that operates for a particular mission

time, unless it fails. If it fails, no immediate repairs are

attempted, so some authors call the system

nonrepairable. A common way to characterize this

system’s reliability is in terms of the hazard function.

Suppose that the system is still operating at time t, and

consider the probability that it will fail in a small

interval of time (t, t + )t). This is the conditional

probability Pr(t < T # t + )t * T > t). The hazard

function, h, is defined so that when )t is small,

h(t))t . Pr(t < T # t + )t * T > t) . (A.5)

This function is also encountered, under the name of 8,

in some treatments of Poisson processes. Equation A.5

gives, approximately,

ht t

tTt t

T t

ft t

R t

( ) Pr( )

Pr( )

( )

( )

∆

∆

∆

≈ < ≤+

>

≈

This is the basis for the formal definition of h:

h t

f t

R t

( )

( )

( ) =
Basics of Probability

A-7

Figure A.3 The reliability function, hazard function

and cumulative hazard function.

For details, see Bain and Engelhardt (1992, p. 541).

Equation A.5 is analogous to Equation A.4, except that

the probability in Equation A.5 is conditional on the

system having survived until t, whereas Equation A.4

refers to all systems in the original population, either

still surviving or not. Suppose a large number, say N,

of identical systems are put into operation at time t = 0,

and n is the number which fail in the interval (t, t + )t).

It follows that f(t))t . n/N, the observed relative

frequency of systems failed in the interval (t, t + )t).

On the other hand, if Nt

 denotes the number of the

original N systems which are still in operation at time t,

then h(t))t . n/Nt

, the observed relative frequency of

surviving systems which fail in this same interval.

Thus, f(t) is a measure of the risk of failing at time t for

any system in the original set, whereas h(t) is a measure

of the risk of failing at time t, but only for systems that

have survived this long.

The hazard function is used as a measure of “aging” for

systems in the population. If h(t) is an increasing

function, then systems are aging or wearing out with

time. Of course, in general the hazard function can

exhibit many types of behavior other than increasing

with time. In actuarial science the hazard function is

called the force of mortality, and it is used as a measure of aging for individuals in a population. More

generally, the hazard function gives an indication of

“proneness to failure” of a system after time t has

elapsed. Other terms which are also used instead of

hazard function are hazard rate and failure rate. The

term failure rate is often used in other ways in the

literature of reliability [see Ascher and Feingold (1984),

p. 19]. 

A.4.4.2 Relations among p.d.f., Reliability, and

Hazard

Any one of the functions F, f, R, and h completely

characterizes the distribution, and uniquely determines

the other three functions. The definition

h t f t

R t ( ) ( )( ) =

was given above. The right side can be written as the

derivative of !ln[R(t)], leading to

R t h u du H t ( ) ( ) t( ) exp ( ) exp ( ) = =− − ∫0

where the function H(t) is called the cumulative

hazard function. The reliability function, R(t), and the

c.d.f., F(t) = 1 ! R(t), are therefore uniquely determined

by the hazard function, h(t), and the p.d.f. can be

expressed as

f t h t h u du ( ) . t( ) ( ) exp ( ) = − ∫0

Figure A.3 shows the reliability, hazard and the cumulative hazard function for the example of Figure A.2.

The hazard function in Figure A.3 is an increasing

function of time. Therefore, it would be consistent with

systems with a dominant wear-out effect for the entire

life of the system. The lifetime of a system may be

divided into three typical intervals: the burn-in or

infant period, the random or chance failure period,

and the wear-out period. During the useful period, the

dominant cause of failures is “random” failures. For

example, systems might fail due to external causes such

as power surges or other environmental factors rather

than problems attributable to the defects or wear-out in

the systems. This example is somewhat idealized

because for many types of systems the hazard function

will tend to increase slowly during the later stages of

the chance failure period. This is particularly true of

mechanical systems. On the other hand, for many

electrical components such as transistors and other

solid-state devices, the hazard function remains fairly

flat once the burn-in failure period is over.

A.4.5 Joint, Marginal, and Conditional

Distributions

Many statistical methods are based on selecting a

sample of size n from a probability distribution f(x).

Such a sample is denoted by

(X1 = x1, X2 = x2, ..., Xn = xn) = (x1, x2, ..., xn),
Basics of Probability

A-8

where x1, x2, ..., xn are the actual values of the random

variable X which has the distribution f(x).

The concepts of simultaneous events and joint, marginal, and conditional probability, discussed in Section

A.3, also pertain to random variables and probability

distributions. Two random variables X1 and X2 (both

continuous, both discrete, or one of each) can have a

joint distribution, or joint p.d.f., denoted f(x1, x2). The

point (x1, x2) can be thought of as a point in two-dimensional Euclidean space. Similarly, n random variables

have joint distribution f(x1, x2, ..., xn). Also, the n

random variables have joint cumulative distribution

F(x1, x2, ..., xn).

The marginal distribution of Xi

 is defined as the joint

p.d.f. integrated (for continuous random variables) or

summed (for discrete random variables) over the n!1

other corresponding dimensions, resulting in a function

of xi

 alone. Thus, the marginal distribution of Xi is the

unconditional p.d.f. of Xi

, fi(xi).

The conditional distribution of X1 given X2, denoted

f(x1 | x2), is defined by

f(x x ) , f(x , x )

f (x ) 1 2

1 2

2 2

 = 

where f2(x2) ... 0. This conditional distribution can be

shown to satisfy the requirements of a probability

function. Sampling from a conditional p.d.f. would

produce only those values of X1 that could occur for a

given value of X2 = x2. The concept of a conditional

distribution also extends to n random variables.

Two random variables X1 and X2 are independent if

their joint p.d.f. is equal to the product of the two

individual p.d.f.s. That is,

f(x1, x2) = f(x1) f(x2).

In general, X1, X2, ..., Xn are independent random

variables if

f(x1, x2, ..., xn) = f(x1) @ f(x2) @ . . . @ f(xn).

A.4.6 Characterizing Random Variables

and Their Distributions

A.4.6.1 Distribution Characteristics

Probability distributions have many characteristics of

interest, some of which are described by distribution

parameters. The term parameter is used to refer to a

fixed characteristic. In contrast to a statistic, which

changes from sample to sample, a parameter for a

particular distribution is a constant and does not

change. However, when a parameter’s value is not

known, sample statistics can be used to estimate the

parameter value. Parameter estimation is discussed in

Appendix B.

A very useful distribution characteristic is the parameter

that serves as a measure of central tendency, which can

be viewed as a measure of the middle of a distribution.

When a change in the parameter slides the distribution

sideways, as with the mean of a normal distribution, the

parameter is referred to as the location parameter. It

serves to locate the distribution along the horizontal

axis. Sometimes, however, a change in the parameter

squeezes or stretches the distribution toward or away

from zero, as with the mean of the exponential distribution. In that case, the parameter is a scale parameter.

In any case, the most common measure of central

tendency is the mean, μ, of the distribution, which is a

weighted average of the outcomes, with the weights

being probabilities of outcomes. For a discrete random

variable X,

μ . X i i

i

=∑ x x Pr( )

For a continuous random variable X,

μ . X = x f x dx

−∞

∞

∫ ( )

(In Section A.4.6.2 below, the mean of X will be

denoted E(X), the “expected value” of X.)

Another distribution characteristic commonly used as a

measure of central tendency, or location, is the median.

For a continuous distribution, the median is the point

along the horizontal axis for which 50% of the area

under the p.d.f. lies to its left and the other 50% to its

right. The median of a random variable, X, is commonly designated med(X) or x.50 and, for a continuous

distribution, is the value for which Pr(X # x.50) = .50

and Pr(X $ x.50) = .50. In terms of the cumulative

distribution, F(x.50) = .50. The median is a specific case

of the general 100"th percentile, x", for which F(x") =

". When the factor of 100 is dropped, x" is called the

" quantile. Along with the median as the 50th percentile (or equivalently, the 0.5 quantile), the 25th and 75th

percentiles are referred to as quartiles of a distribution.
Basics of Probability

A-9

0.25

0.50

0.75

1.00

x0.25 x0.50 x0.75 GC00 0432 3

Mean

Figure A.5 Cumulative distribution function (c.d.f.)

showing quartiles, median, and mean.

x0.25 x0.50 x0.75 GC00 0432 2

Mean

Each area = 0.25

Figure A.4 Density, showing quartiles, median, and

mean.

Figure A.4 shows the quartiles, x0.25 and x0.75, the

median, x0.50, and the mean. The quartiles and the

median divide the area under the density curve into four

pieces, each with the same area. Note that the mean is

greater than the median in this example, which is the

usual relation when the density has a long right tail, as

this one does.

Figure A.5 shows the same quantities plotted with the

c.d.f. By definition, the q quantile, xq, satisfies F(xq)

= q.

The mean and the median are used to measure the

center or location of a distribution. Since the median is

less affected by tail-area probabilities, it can be viewed

as a better measure of location than the mean for

highly-skewed distributions. For symmetric distributions, the mean and median are equivalent.

A different measure of center or location of a distribution is the mode, which indicates the most probable

outcome of a distribution. The mode is the point along

the horizontal axis where the “peak” or maximum of the

p.d.f. is located. Note that the mode does not necessarily have to be near the middle of the distribution. It

simply indicates the most likely value of a distribution.

Note also that a peak does not have to exist and, in

some cases, more than one peak can exist.

Another important characteristic of a distribution is its

variance, denoted by F2

. The variance is the average

of the squared deviations from the mean. The standard

deviation, F, of a distribution is the square root of its

variance. Both the variance and standard deviation are

measures of a distribution’s spread or dispersion. For

a discrete random variable X,

σ μ . X i i

i

x x 2 2 = − ∑ ( Pr( ) )

For a continuous random variable X,

σ μ . X x f x dx 2 2 = −

−∞

∞

∫ ( ) ()

Though less used than the mean and variance, the

skewness is defined as

E(X ! :)

3

/F3 .

It measures asymmetry. It is usually positive if the

density has a longer right tail than left tail, and negative

if the density has a longer left tail than right tail. For

example, the density in Figure A.4 has positive skewness.

A.4.6.2 Mathematical Expectation

The definitions of distribution means and variances

arise from mathematical expectation and moments of

a distribution, which form an important method for

calculating the parameters of a known p.d.f. In general,

the expectation (expected value or mathematical

expectation) of a function g(X), denoted E[g(X)], is

EgX gx x , i i

i

[ ( )] ( ) Pr( ) = ∑

when X is discrete, and

E g X g x f x dx [ ( )] ( ) ( ) = ,

−∞

∞

∫

when X is continuous.
Basics of Probability

A-10

Because of their wide use, several expectations have

special names. For g(X) = X, the expectation E(X)

becomes the mean of X. Thus, the mean is also commonly referred to as the expected value (or expectation)

of the random variable X. In addition, for g(X) = X, the

expectation E(X) is known as the first moment about

the origin.

The variance, FX

2

, also denoted by Var(X), of a distribution is defined by mathematical expectation with g(X)

= (X ! :X)

2

. Thus,

Var(X) = FX

2

 = E[(X ! :X)

2

] = E(X 2) ! [E(X)]2,

which is known as the second moment about the

mean.

Ordinary moments (moments about the origin) of a

random variable X are defined as

Mr = E(Xr

),

for r = 1, 2, ... . Thus,

Var(X) = FX

2

 = E(X 2) ! [E(X)]2 = M2 ! M1

2

.

Central moments (moments about the mean) of a

random variable X are defined as being equal to E[(X !

:)

r

] for r = 2, 3, ... . The ordinary and central moments

can be seen to define characteristics of distributions of

random variables.

An important rule of expectation commonly used in

PRA is that the expected value of a product of independent random variables is the product of their respective expected values. That is, E(X1AX2A ... AXn) =

E(X1)AE(X2)A ... AE(Xn) when all Xi

 are independent. This

rule also applies to conditionally independent random

variables. If the random variables X2, X3, ..., Xn are all

conditionally independent given X1 = x1, then

f(x2, x3, ..., xn*x1) = f(x2*x1)Af(x3*x1)A ... Af(xn*x1).

It follows that 

E(X2AX3A ... AXn*x1) = E(X2*x1)AE(X3*x1)A ... AE(Xn*x1).

Thus,

E(X1AX2A ... AXn) = E[X1AE(X2*x1)AE(X3*x1)A ... AE(Xn*x1)].

The following facts are also often useful:

• E(Ei Xi

) = Ei E(Xi), whether or not the Xis are

independent.

• Var(Ei Xi

) = Ei Var(Xi), if the Xis are independent.

• E(aX + b) = aE(X) + b .

• Var(aX + b) = a2

Var(X) .

• The last two give useful special cases when a = 1

or b = 0.

A.4.6.3 Moment-Generating Functions

Another special mathematical expectation is the

moment-generating function of a random variable.

For a random variable X with p.d.f. f(x), the momentgenerating function of X (or of the distribution) is

defined by M(t) = E(etX), if M exists for some interval

!h < t < h. Therefore, if X is a continuous random

variable,

M t e f x dx . tx () ( ) =

−∞

∞

∫

If X is a discrete random variable,

M t ef x . tx

i

i

i () ( ) =∑

Note that not every distribution has a moment-generating function.

The importance of the moment-generating function is

that, when it does exist, it is unique and completely

specifies the distribution of the random variable. If two

random variables have the same moment-generating

function, they have the same distribution.

It can be shown that the moments of a distribution can

be found from the series expansion of M(t). The

moments of the distribution can also be determined

from the moment-generating function by differentiating

the moment-generating function with respect to t and

setting t = 0. See Martz and Waller (1991) and any of

several mathematical statistics texts, such as Hogg and

Craig (1995), for further details on moment-generating

functions.

A.4.6.4 Covariance and Correlation

For two random variables, X and Y, with means :x and

:y, the expected value E[(X ! :x)(Y ! :y)] is called the

covariance of X and Y, denoted Cov(X, Y). The

covariance of X and Y divided by the product of the

standard deviations of X and Y is called the correlation

coefficient (or correlation) between X and Y, denoted

Cor(X, Y). That is,
Basics of Probability

A-11

Cor

Cov

Var Var ( ,) ( ,)

( ) ()

X Y

X Y

X Y =

 . = − −

− −

E X EY

E X EY

X Y

X Y

( )( )

[( ) ] [( ) ]

μ μ

μ μ 2 2

The correlation coefficient measures the degree of

association between X and Y, that is, the strength of a

linear relationship between X and Y. It is always

between !1 and 1. Positive correlation (correlation >

0) means that X and Y tend to be large together and

small together in a linear way. Negative correlation

means that X tends to be large when Y is small and vice

versa, in a linear way. If X and Y are independent, then

their correlation is zero. The converse is not true;

examples can be constructed where X and Y are dependent (in a nonlinear way) yet have zero correlation.

A.4.7 Distribution of a Transformed 

Random Variable

This section considers the distribution of Y = h(X),

when X has a known distribution and h is a known

function. The problem is straightforward when X has a

discrete distribution. When X is continuous and h is

monotone, either increasing or decreasing, the c.d.f.s

are also related in the natural way, as follows. Let F be

the c.d.f. of X and let G be the c.d.f. of Y. Then we

have

G(y) = Pr(Y # y) = Pr[h(X) # y] .

If h is monotone increasing, this equals

Pr[X # h-1(y)] = F(x),

where x and y are related by y = h(x), x = h-1(y). In

summary, G(y) = F(x).

If, instead, h is monotone decreasing, then a similar

argument gives

G(y) = 1 ! F(x) .

The surprise comes with the densities. Differentiate

both sides of either of the above equations with respect

to y, to obtain the density of y. This involves using the

chain rule for differentiation. The result is

gy f x . dx

dy () () =

That is, the density of Y is not simply equal to the

density of X with a different argument. There is also a

multiplier, the absolute value of the derivative.

Two important special cases are given here. If Y =

exp(X), then

g(y) = f[ln(y)](1/y) .

If Y = 1/X, then

g(y) = f(1/y)(1/y2) .

These formulas form the basis for the densities of the

lognormal distribution and the inverted gamma distribution.

A.5 Bayes’ Theorem

It is frequently desired to calculate the probability of an

event A given that another event B has occurred at some

prior point in time. It can also be of interest to calculate

the probability that a state of nature exists given that a

certain sample is observed, for example, belonging to a

certain population based on a sample measurement or

observation. Conditional probability leads directly to

Bayes’ Theorem, which, along with subjective probability, forms the basis for Bayesian inference commonly

used in PRA.

Recall the definition of a partition from Section A.3:

A1, A2, ..., An are a partition of the sample space if the

are mutually exclusive and their union equals the entire

sample space. Bayes’ Theorem states that if A1, A2, ...,

An are a partition of the sample space and if B is any

other event such that Pr(B) > 0, then

Pr( , (A.6) Pr( Pr(

Pr(

A B BA A

B i

i i ) = ) )

)

where

Pr( Pr( ( B BA A j j .

j

n

) =

=

∑ ) Pr )

1

This last equation is the law of total probability (Section A.3). Equation A.6 follows from the definition of

conditional probability in Equation A.1:

Pr( . Pr(

Pr(

Pr( Pr(

Pr(

A B

B A

B

BA A

B i

i ii ) = ∩ ) = )

)

)

)
Basics of Probability

A-12

The Pr(Ai

*B) is the posterior (or a posteriori) probability for the event Ai

, meaning the probability of Ai once

B is known. The Pr(Ai

) is the prior (or a priori) probability of the event Ai

 before experimentation or observation. The event B is the observation. The Pr(B*Ai

) is

the probability of the observation given that Ai

 is true.

The denominator serves as a normalizing constant.

Calculating the posterior probabilities Pr(Ai

*B) requires

knowledge of the probabilities Pr(Ai

) and Pr(B*Ai), i =

1, 2, ..., n. The probability of an event can often be

determined if the population is known, thus, the

Pr(B*Ai

) can be determined. However, the Pr(Ai), i = 1,

2, ..., n, are the probabilities that certain states of nature

exist and are either unknown or difficult to ascertain.

These probabilities, Pr(Ai

), are called prior probabilities

for the events Ai

 because they specify the distribution of

the states of nature prior to conducting the experiment.

Application of Bayes’ Theorem utilizes the fact that

Pr(B*Ai

) is easier to calculate than Pr(Ai*B). If probability is viewed as degree of belief, then the prior belief

is changed, by the test evidence, to a posterior degree of

belief. In many situations, some knowledge of the prior

probabilities for the events A1, A2, ..., An exists. Using

this prior information, inferring which of the sets A1, A2,

..., An is the true population can be achieved by calculating the Pr(Ai

*B) and selecting the population that

produces the highest probability.

Equation A.6 pertains to disjoint discrete events and

discrete probability distributions. Bayes’ Theorem has

analogous results for continuous p.d.f.’s. The continuous version is given here. Suppose X is a discrete or

continuous random variable, with p.d.f. depending on

parameter 2, and with conditional p.d.f. of X, given 2,

specified by f(x*2). Suppose that 2 has a continuous

probability distribution with p.d.f. g(2). This can

happen in two ways: either 2 is a possible value of the

random variable 1 (using the convention of denoting

random variables with uppercase letters), or else 2 is an

uncertain parameter with a subjective uncertainty

distribution. The second case is the more common one.

Call g(2) the prior p.d.f. Then for every x such that

f(x) > 0 exists, the posterior p.d.f. of 2, given X = x, is

g x , (A.7) fx g

f x ( ) ( )()( )

θ

θ θ =

where

fx fx g d () ( )() = ∫ θ θθ

is the marginal p.d.f. of X. Again, the prior and posterior p.d.f.’s can be used to represent the probability of

various values 2 prior to and posterior to observing a

value of another random variable X. This is valid

whether “probability of 2” has a frequentist or subjective interpretation.

A.6 Discrete Random Variables

A.6.1 The Binomial Distribution

The binomial distribution describes the number of

failures X in n independent trials. The random variable

X has a binomial distribution if:

1. The number of random trials is one or more and is

known in advance.

2. Each trial results in one of two outcomes, usually

called success and failure (although they could be

pass-fail, hit-miss, defective-nondefective, etc.).

3. The outcomes for different trials are statistically

independent.

4. The probability of failure, p, is constant across

trials.

Equal to the number of failures in the n trials, a binomial random variable X can take on any integer value

from 0 to n. The probability associated with each of

these possible outcomes, x, is defined by the binomial(n, p) p.d.f. as

Pr( ) ( )

..., .

X x , n

x p p

x n

x nx == −

=

⎛

⎝

⎜ ⎞

⎠

⎟ − 1

0,

Here

n

x

n

xn x

⎛

⎝

⎜ ⎞

⎠

⎟ = −

!

!( )!

is the binomial coefficient. The symbol

n! = n(n !1)(n !2) ... (2)(1)

denotes n factorial, with 0! defined to be equal to 1.

This binomial coefficient provides the number of ways

that exactly x failures can occur in n trials (number of

combinations of n trials selected x at a time).

The binomial distribution has two parameters, n and p,

of which n is known. (Although n may not always be

known exactly, it is treated as known in this handbook.)

The mean and variance of a binomial(n, p) random

variable X are
Basics of Probability

A-13

Pr( ) !

( )

! ,

X x e

x

e t

x

x

t x

= =

=

−

−

μ

λ

μ

λ

0

0.1

0.2

0.3

0.4

0.5

01234

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 2 4 6 8 10 12

0.00

0.04

0.08

0.12

0.16

0 10 20 30 40

Probability Probability Probability

GC00 0432 4

n p = 4, = 0.25

n p = 12, = 0.25

n p = 40, = 0.25

Figure A.6 Three binomial probability distribution

functions.

E(X) = np

and

Var(X) = np(1 ! p).

Figure A.6 shows three binomial probability distribution functions, with parameter p = 0.25, and n = 4, 12,

and 40. In each case, the mean is np. The means have

been aligned in the three plots.

A.6.2 The Poisson Distribution

The Poisson distribution provides a discrete probability

model that is appropriate for many random phenomena

that involve counts. Examples are counts per fixed time

interval of the number of items that fail, the number of

customers arriving for service, and the number of

telephone calls occurring. A common use of the

Poisson distribution is to describe the behavior of many

rare event occurrences. The Poisson distribution is also

frequently used in applications to describe the occurrence of system or component failures under steadystate conditions.

The count phenomena that occur as Poisson random

variables are not necessarily restricted to occurring over

a time interval. They could also be counts of things

occurring in some region, such as defects on a surface

or within a certain material. A process that leads to a

Poisson random variable is said to be a Poisson process.

The Poisson distribution describes the total number of

events occurring in some interval of time t (or space).

The p.d.f. of a Poisson random variable X, with parameter : = 8t, is

(A.8)

for x = 0, 1, 2, ..., and x! = x(x ! 1)(x ! 2) ... (2)(1), as

defined previously.

The Poisson distribution has a single parameter :,

denoted Poisson(:). If X denotes the number of events

that occur during some time period of length t, then X

is often assumed to have a Poisson distribution with

parameter : = 8t. In this case, X is considered to be a

Poisson process with intensity 8 > 0 (Martz and Waller

1991). The variable 8 is also referred to as the event

rate (or failure rate when the events are failures).

Note that 8 has units 1/time; thus, 8t = : is

dimensionless.

If only the total number of occurrences for a single time

period t is of interest, the form of the p.d.f. in Equation

A.8 using : is simpler. If the event rate, 8, or various

time periods, t, are of interest, the form of the p.d.f. in

Equation A.8 using 8t is more useful.

The expected number of events occurring in the interval

0 to t is : = 8t. Thus, the mean of the Poisson distribution is equal to the parameter of the distribution, which

is why : is often used to represent the parameter. The

variance of the Poisson distribution is also equal to the

parameter of the distribution. Therefore, for a Poisson(:) random variable X,

E(X) = Var(X) = : = 8t.
Basics of Probability

A-14

Figure A.7 Three Poisson probability distribution

functions.

Figure A.7 shows three Poisson probability distribution

functions, with means : = 1.0, 3.0, and 10.0, respectively. The three means have been aligned in the

graphs. Note the similarity between the Poisson distribution and the binomial distribution when : = np and n

is not too small.

Several conditions are assumed to hold for a Poisson

process that produces a Poisson random variable:

1. For small intervals, the probability of exactly one

occurrence is approximately proportional to the

length of the interval (where 8, the event rate or

intensity, is the constant of proportionality).

2. For small intervals, the probability of more than

one occurrence is essentially equal to zero (see

below).

3. The numbers of occurrences in two non-overlapping intervals are statistically independent.

More precise versions of condition 2 are: (1) the

probability of more than one event occurring in a very

short time interval is negligible in comparison to the

probability that only one event occurs (Meyer 1970),

(2) the probability of more than one event occurring in

a very short time interval goes to zero faster than the

length of the interval (Pfeiffer and Schum 1973), and

(3) simultaneous events occur only with probability

zero (Çinlar 1975). All of these versions have the

practical interpretation that common cause events do

not occur. The phrase “do not occur” is used in this

handbook, as it is in Thompson (1981).

The Poisson distribution also can serve as an approximation to the binomial distribution. Poisson random

variables can be viewed as resulting from an experiment

involving a large number of trials, n, that each have a

small probability of occurrence, p, of an event. However, the rare occurrence is offset by the large number

of trials. As stated above, the binomial distribution

gives the probability that an occurrence will take place

exactly x times in n trials. If p = :/n (so that p is small

for large n), and n is large, the binomial probability that

the rare occurrence will take place exactly x times is

closely approximated by the Poisson distribution with

: = np. In general, the approximation is good for large

n, small p, and moderate : (say : # 20) [see Derman et

al. (1973)].

The Poisson distribution is important because it describes the behavior of many rare event occurrences,

regardless of their underlying physical process. It also

has many applications to describing the occurrences of

system and component failures under steady-state

conditions. These applications utilize the relationship

between the Poisson and exponential (continuous

random variable, see Section A.7.4) distributions: the

times between successive events follow an exponential

distribution.

A.7 Continuous Random Variables

A.7.1 The Uniform Distribution

A uniform distribution, also referred to as a rectangular

distribution, represents the situation where any value in

a specified interval, say [a, b], is equally likely. For a

uniform random variable, X, because the outcomes are

equally likely, f(x) is equal to a constant. The p.d.f. of

a uniform distribution with parameters a and b, denoted

uniform(a, b), is

f x b a ( ) = −

1
Basics of Probability

A-15

a GC00 0432 6

Area = 1

0

1/( - ) b a

b

Figure A.8 Density of uniform(a, b) distribution.

Figure A.9 Two normal densities.

for a # x # b.

Figure A.8 shows the density of the uniform(a, b)

distribution.

The mean and variance of a uniform(a, b) distribution

are

E X b a ( ) = +

2

and

Var( ) ( ) X . b a = − 2

12

A.7.2 The Normal Distribution

One of the most widely encountered continuous probability distributions is the normal distribution, which has

the familiar bell shape and is symmetrical about its

mean value. The importance of the normal distribution

is due to: (1) its applicability in describing a very large

number of random variables that occur in nature and (2)

the fact that certain useful functions of nonnormal

random variables are approximately normal. Details on

the derivation of the normal distribution can be found

in many basic mathematical statistics textbooks [e.g.,

Hogg and Craig (1995)].

The normal distribution is characterized by two parameters, : and F. For a random variable, X, that is normally distributed with parameters : and F, the p.d.f. of

X is

f x (A.9)

x ( ) exp = − ⎛ −

⎝

⎜ ⎞

⎠

⎟ ⎡

⎣

⎢ ⎤

⎦

⎥ 1

2

1

2

2

σ π

μ

σ

for !4 < x < 4, !4 < : < 4, and F > 0. Increasing :

moves the density curve to the right and increasing F

spreads the density curve out to the right and left while

lowering the peak of the curve. The units of : and F

are the same as for X.

The mean and variance of a normal distribution with

parameters : and F are

E(X) = :

and

Var(X) = F2

.

The normal distribution is denoted normal(:, F2

).

Figure A.9 shows two normal(:, F2

) densities. The

distribution is largest at : and is more concentrated

around : when F is small than when F is large.

Note the similarity of the normal density to a binomial

p.d.f. with large np or a Poisson p.d.f. with large :.

This illustrates the fact that a normal distribution can

sometimes be used to approximate those distributions.

The normal(0, 1) distribution is called the standard

normal distribution, which, from Equation A.9, has

p.d.f.

φ (A.10)

π

( ) exp x

x = −⎛

⎝

⎜ ⎞

⎠

⎟ 1

2 2

2

for !4 < x < 4. The cumulative distribution of the

standard normal distribution is denoted by M. Tables

for the standard normal distribution are presented in

Appendix C and in almost all books on statistics.

It can be shown that the transformed random variable Z

= (X ! :)/F is normal(0, 1). Thus, to calculate probabilities for a normal(:, F2

) random variable, X, when :

... 0 and/or F2 ... 1, the tables for the standard normal can

be used. Specifically, for any number a,
Basics of Probability

A-16

Pr[ X # a ] = Pr[ (X ! :)/F # (a !:)/F ]

= Pr[ Z # (a ! :)/F ]

= M[ (a ! :)/F ] .

Part of the importance of the normal distribution is that

it is the distribution that sample sums and sample means

tend to possess as n becomes sufficiently large. This

result is known as the central limit theorem, which

states that, if X1, X2, ..., Xn, are independent random

variables, each with mean : and variance F2

, the sum of

these n random variables, 'i

Xi, tends toward a normal(n:, nF2

) distribution for large enough n. Since the

sample mean is a linear combination of this sum, the

central limit theorem also applies. Thus, = 'i

Xi X /n

tends to a normal(:, F2/n) distribution. The importance

of the central limit theorem is it can be used to provide

approximate probability information for the sample

sums and sample means of random variables whose

distributions are unknown. Further, because many

natural phenomena consist of a sum of several random

contributors, the normal distribution is used in many

broad applications.

Because a binomial random variable is a sum, it tends

to the normal distribution as n gets large. Thus, the

normal distribution can be used as an approximation

to the binomial distribution. One rule of thumb is that

the approximation is adequate for np $ 5.

A Poisson random variable also represents a sum and,

as presented previously, can also be used as an approximation to the binomial distribution. It follows that the

normal distribution can serve as an approximation to

the Poisson distribution when : = 8t is large. One

rule of thumb is that the approximation is adequate for

: $ 5.

A.7.3 The Lognormal Distribution

Use of the lognormal distribution has become increasingly widespread. It is commonly used as a distribution

for failure time and in maintainability analysis (Martz

and Waller 1991). It has also been widely used as a

prior distribution for unknown positive parameters.

The lognormal distribution arises from the product of

many independent random variables. If Y = Y1@Y2@ ... @Yn

=(iYi is the product of n independent positive random

variables that are (nearly) identically distributed, then

ln(Y) = ln((i

Yi) = 'iln(Yi) is a sum that tends toward a

normal distribution.

The distribution of Y is defined to be lognormal when

the distribution of ln(Y) is normal. That is, when Y is

lognormal, ln(Y) is normal(:, F2

). The parameters of

the lognormal distribution are : and F, the parameters

from the underlying normal distribution. For a random

variable, Y, that is lognormally distributed with parameters : and F, denoted lognormal(:, F2

), the p.d.f. of Y

is

f y ( )

y

( ) exp ln = −−y

⎡

⎣

⎢ ⎤

⎦

⎥

1

2

1

2 2

2

σ π σ

μ

for 0 < y < 4, !4 < : < 4, and F > 0. Note the y in the

denominator, for reasons explained in Section A.4.7.

The mean and variance of a lognormal(:, F2

) distribution are

E(Y) = exp(: + F2

/2)

and

Var(Y) = exp(2: + F2

)[exp(F2) ! 1].

In addition, the median of a lognormal distribution is

exp(:) and the mode is exp(:!F2

). See Martz and

Waller (1991) for more information on the lognormal

distribution.

Sometimes the median of Y = exp(:) is used as a

parameter. In addition, a parameter commonly used in

PRA is the error factor (EF), where EF = exp(1.645F).

This definition causes EF to satisfy

Pr[med(Y)/EF # Y # med(Y)*EF] = 0.90.

Figure A.10 shows three lognormal densities. The

value : = !7 corresponds to a median of about 1.E!3.

[More exactly, it corresponds to exp(!7) = 9.E!4.]

The value : = !6.5 corresponds to a median of about

1.5E!3. The value F = 0.67 corresponds to an error

factor EF = 3, and F = 1.4 corresponds to an error

factor EF = 10.

The two distributions with F = 0.67 and different values

of : have essentially the same shape, but with different

scales. The larger : corresponds to spreading the

distribution out more from zero. The distribution with

F = 1.4, and therefore EF = 10, has a very skewed

distribution.

To calculate probabilities for a lognormal(:, F2

)

random variable, Y, the tables for the standard normal

can be used. Specifically, for any number b,
Basics of Probability

A-17

0E0 1E-3 2E-3 3E-3

μ = −7, σ=1.4

μ = −7, σ = 0.67

μ = −6.5, σ = 0.67

Figure A.10 Three lognormal densities.

0 GC00 0433 1

Figure A.11 Two exponential densities.

Pr[ Y # b ] = Pr[ ln(Y) # ln(b) ]

= Pr[ X # ln(b) ]

= M[ (ln(b) ! :)/F ] ,

where X = ln(Y) is normal(:, F2

).

A.7.4 The Exponential Distribution

The exponential distribution is widely used for

modeling time to failure and is inherently associated

with the Poisson process [see Martz and Waller

(1991)]. For a Poisson random variable X defining the

number of failures in a time interval t and for a random

variable T defining the time to failure, it can be shown

that T has the exponential p.d.f.

f(t) = 8e!8t

,

for t > 0. Thus, the time to first failure and the times

between successive failures follow an exponential

distribution and the number of failures in a fixed time

interval follows a Poisson distribution.

Figure A.11 shows two exponential densities, for two

values of 8 . The intercept (height of the curve when t

= 0) equals 8 . Thus, the figure shows that the distribution is more concentrated near zero if 8 is large. This

agrees with the interpretation of 8 as a frequency of

failures and t as time to first failure.

The exponential distribution parameter, 8, corresponds

to the 8t parameterization of the Poisson p.d.f. in

Equation A.8 and is referred to as the failure rate if the

component or system is repaired and restarted immediately after each failure. It is called the hazard rate if

the component or system can only fail once and cannot

be repaired. Section A.4.4.2 discusses modeling

duration times with different distributions and defines

the hazard rate as h(t) = f(t)/[1 ! F(t)]. For the exponential distribution, the hazard rate is constant, 8. It

can be shown that the exponential distribution is the

only distribution with a constant hazard rate.

The c.d.f. of the exponential distribution is

F(t) = 1 ! e!8t

.

The exponential distribution with parameter 8 is

denoted exponential(8). The mean and variance of an

exponential(8) distribution are

E(T) = 1/8

and

Var(T) = 1/82

.

The relationship of the exponential distribution to the

Poisson process can be seen by observing that the

probability of no failures before time t can be viewed in

two ways. First, the number of failures, X, can be

counted. The probability that the count is equal to 0 is

given by Equation A.8 as 

 . Pr( ) ( )

!

X e te t t == = − − 0

0

0

λ λ λ

Alternatively, the probability that first failure time, T, is

greater than t is

Pr(T > t) = 1 ! Pr(T # t)

 = 1 ! F(t)

 = 1 ! [1 ! e!8t

]

 = e!8t

.
Basics of Probability

A-18

Thus, the two approaches give the same expression for

the probability of no failures before time t.

The assumptions of a Poisson process require a constant

failure rate, 8 , which can be interpreted to mean that

the failure process has no memory (Martz and Waller

1991). Thus, if a device is still functioning at time t, it

remains as good as new and its remaining life has the

same exponential(8) distribution. This constant failure

rate corresponds to the flat part of the common bathtub

curve (frequency of failures plotted against time) and

does not pertain to initial (burn-in) failures and wearout failures.

A different, sometimes useful, parameterization uses :

= 1/8 = E(T). For example, if T represents a time to

failure, : is called the mean time to failure. If T is the

time to repair, or to fire suppression, or to some other

event, the name for : is the mean time to repair, or

other appropriate name. The exponential(:) distribution for T has density

f(t) = (1/:)exp(!t/:), for t $ 0

and c.d.f.

F(t) = 1 ! exp(!t/:), for t $ 0 .

The units of : are the same as the units of t, minutes or

hours or whatever the data have. The mean and variance are

E(T) = :

var(T) = :2 .

A.7.5 The Weibull Distribution

The Weibull distribution is widely used in reliability

and PRA and generalizes the exponential distribution to

include nonconstant failure or hazard rates (Martz and

Waller 1991). Different Weibull distributions have

been successfully used to describe initial failures and

wear-out failures. The Weibull distribution is appropriate when a system is composed of a number of components, and system failure is due to any one of the

components failing. It, therefore, is commonly referred

to as a distribution corresponding to failure of the

weakest link.

For a random variable, T, that has a Weibull distribution, the p.d.f. is

f t , t t ( ) exp = ⎛ −

⎝

⎜ ⎞

⎠

⎟ − − − ⎛

⎝

⎜

⎜

⎜

⎞

⎠

⎟

⎟

⎟

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

β

α

θ

α

θ

α

β 1 β

for t $ 2 $ 0 and parameters " > 0 and $ > 0. The

parameter 2 is a location parameter and corresponds to

a period of guaranteed life that is not present in many

applications (Martz and Waller 1991). Thus, 2 is

usually set to zero. The c.d.f. for T is

F t , t ( ) exp =− − − ⎛

⎝

⎜

⎜

⎜

⎞

⎠

⎟

⎟

⎟

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

1 θ

α

β

for t $2 and " > 0 and $ > 0.

The " parameter is a scale parameter that expands or

contracts the density along the horizontal axis. The $

parameter is a shape parameter that allows for a wide

variety of distribution shapes. [See Martz and Waller

(1991) for further discussion and examples.] When $

= 1, the distribution reduces to the exponential distribution. Therefore, the Weibull family of distributions

includes the exponential family of distributions as a

special case.

A Weibull distribution with parameters ", $, and 2 is

referred to as Weibull(", $, 2) and, when 2 = 0,

Weibull(", $). The mean and variance of the Weibull

distribution are given by Martz and Waller (1991) as

2 + "'(1 + 1/$)

and

"2['(1 + 2/$) ! '2(1 + 1/$) .

Here, ' is the gamma function, defined in Section

A.7.6.

Figure A.12 shows four Weibull densities, all with the

same scale parameter, ", and all with location parameter 2 = 0. The shape parameter, $, varies. When $ < 1,

the density becomes infinite at the origin. When $ = 1,

the distribution is identical to the exponential distribution. Surprisingly, the distribution is not asymptotically

normal as $ becomes large, although it is approximately

normal when $ is near 3.

A.7.6 The Gamma and Chi-Squared

Distributions

The gamma distribution is an extension of the exponential distribution and is sometimes used as a failure

time model (Martz and Waller 1991). It is also often

used as a prior distribution in Bayesian estimation (see

Appendix B) of the failure rate parameter 8 from

Poisson(8t) or exponential(8) data. The chi-squared

distribution is a re-expression of a special case of the

gamma distribution.
Basics of Probability

A-19

Figure A.12 Four Weibull densities, all having 2 = 0

and all having the same ".

The gamma distribution arises in many ways. The

distribution of the sum of independent exponential(8)

random variables is gamma, which forms the basis for

a confidence interval for 8 from exponential(8) data.

Because the sum of n independent exponentially

distributed random variables has a gamma distribution,

the gamma distribution is often used as the distribution

of the time, or waiting time, to the nth event in a Poisson process. In addition, the chi-squared distribution is

the distribution for a sum of squares of independent,

identically distributed normal random variables, which

forms the basis for a confidence interval for the variance of a normal distribution. The gamma distribution

is also often used as a distribution for a positive random

variable, similar to the lognormal and Weibull distributions. In PRA work, it is often used as a Bayesian

distribution for an uncertain positive parameter.

Two parameterizations of the gamma distribution are

common, with various letters used for the parameters.

The parameterization given here is most useful for

Bayesian updates, the primary use of the gamma

distribution in this handbook. For a random variable, T,

that has a gamma distribution, the p.d.f. is

, ft t t

a

( ) ( ) = − exp( ) − β

α β α

Γ

1

for t, ", and $ > 0. 

Here

Γ ( ) α α = − − ∞

∫ x e dx 1 x

0

is the gamma function evaluated at ". If " is a positive integer, '(") = (" ! 1)!.

A gamma distribution with parameters " and $ is

referred to as gamma(", $). The mean and variance of

the gamma(", $) random variable, T, are:

E(T) = "/$

and

Var(T) = "/$ 2.

The parameters " and $ are referred to as the shape and

scale parameters. The shape parameter " allows the

density to have many forms. If " is near zero, the

distribution is highly skewed. For " = 1, the gamma

distribution reduces to an exponential($!1

) distribution.

Also, the gamma(" = n/2, $ = 1⁄2) distribution is known

as the chi-squared distribution with n degrees of

freedom, denoted P2

(n). The p.d.f. for the P2(n) distribution is found by substituting these values into the

above formula for the gamma p.d.f. It also can be

found in many statistics texts [e.g., Hogg and Craig

(1995, Chapter 4)].

In addition, if T has a gamma(", $) distribution, then

2$T has a P2

(2") distribution, which forms the defining

relationship between the two distributions. The gamma

and chi-squared distributions can, therefore, be viewed

as two ways of expressing one distribution. Since the

chi-squared distribution usually is only allowed to have

integer degrees of freedom, the gamma distribution can

be thought of as an interpolation of the chi-squared

distribution.

Percentiles of the chi-squared distribution are tabulated

in Appendix C. These tables can be used as follows to

find the percentiles of any gamma distribution. The

100×p percentile of a gamma(", $) distribution is

P2

p(2")/(2$), where P2p(2") denotes the 100×p percentile of the chi-squared distribution with 2" degrees of

freedom.

Figure A.13 shows gamma densities with four shape

parameters, ". When " < 1, the density becomes

infinite at 0. When " = 1, the density is identical to an

exponential density. When " is large, the distribution

is approximately a normal distribution.

As stated previously, the sum of exponential lifetimes

or waiting times has a gamma distribution, with the

shape parameter " equal to the number of exponential

lifetimes. Also, it has been stated that in general the

sum of independent, identically distributed random

variables is approximately normal. This is the reason

why the gamma distribution is approximately normal

when " is large.
Basics of Probability

A-20

Figure A.13 Gamma densities with four shape parameters.

Figure A.14 Four inverted gamma densities, having the

same scale parameter, $, and various shape parameters,

".

An alternative parameterization of the gamma distribution uses the scale parameter, say J = $!1

. If T has a

gamma(", J) distribution, its p.d.f. is

ft t t ( ) ( ) = − exp( / ) − 1 1

τ α

τ α

α

Γ

for t, ", and J > 0. The mean and variance of the

gamma(", J) random variable, T, are:

E(T) ="J

and

Var(T) = "J2

.

This alternative parameterization is useful in a very

small portion of this handbook.

A.7.7 The Inverted Gamma and Inverted

Chi-Squared Distributions

The inverted gamma distribution is often used as a

prior distribution for Bayesian estimation of the mean

of an exponential distribution (Martz and Waller 1991).

It is also used as a prior and posterior distribution for F2

when the data have a normal distribution with variance

F2 (Box and Tiao 1973, Lee 1997).

For a gamma(", $ ) random variable, T, W = 1/T has an

inverted gamma distribution with p.d.f.

f w , w w( ) ( ) = exp

⎛

⎝

⎜ ⎞

⎠

⎟ −⎛

⎝

⎜ ⎞

⎠

⎟

+ β

α

β α α

Γ

1 1

for w, ", and $ > 0. The parameters here are the same

as for the gamma distribution. For example, if T has

units of time then w and $ both have units 1/time. A

comparison of this density with the gamma density

shows that this density has an extra w2

 in the denominator, for reasons explained in Section A.4.7.

The parameters of the inverted gamma distribution are

" and $ and this distribution is denoted inverted

gamma(", $). Just as with the gamma(", $) distribution, " is the shape parameter and $ is the scale parameter. The distribution can also be parameterized in terms

of J = $!1

.

The mean and variance of an inverted gamma(", $)

random variable, W, are

E W , " > 1, ( ) = −

β

α 1

and

Var( ) , " > 2.

( )( ) W = − −

β

α α

2

2 1 2

Note that for " # 1 the mean and higher moments do

not exist. For 1 < " # 2 the mean exists but the variance does not exist (Martz and Waller 1991).

Figure A.14 shows four inverted gamma distributions,

all having the same scale parameter, $, and having

various shape parameters, ".
Basics of Probability

A-21

0 0.1 0.2 0.3 0.4 0.5

α β

α β

α β

α β

 = 0.5, = 4.5

 = 1, = 9

 = 2, = 18

 = 3, = 27

GC00 0433 5

Figure A.16 Four beta distributions with mean 0.1.

Figure A.15 Beta distributions with mean = 0.5.

In the special case with " = n/2 and $ = 1⁄2, the distribution is called the inverted chi-squared distribution

with n degrees of freedom. Values from this distribution are sometimes denoted P!2

(n). This form of the

distribution is often used in connection with a prior for

F2 when the data are normally distributed.

A.7.8 The Beta Distribution

Many continuous quantitative phenomena take on

values that are bounded by known numbers a and b.

Examples are percentages, proportions, ratios, and

distance to failure points on items under stress. The

beta distribution is a versatile family of distributions

that is useful for modeling phenomena that can range

from 0 to 1 and, through a transformation, from a to b.

The beta distribution family includes the uniform

distribution as well as density shapes that range from

decreasing to uni-modal right-skewed to symmetric to

U-shaped to uni-modal left-skewed to increasing (Martz

and Waller 1991). It can serve as a model for a reliability variable that represents the probability that a system

or component lasts at least t units of time. The beta

distribution is also widely used in Bayesian estimation

and reliability analysis as a prior distribution for the

binomial distribution parameter p that represents a

reliability or failure probability.

The p.d.f. of a beta random variable, Y, is

fy y y ( ) , ( )

()() = ( ) + − Γ − −

Γ Γ

α β

α β

α β 1 1 1

for 0 # y # 1, with the parameters ", $ > 0. The distribution is denoted beta(", $). The gamma functions at

the front of the p.d.f. form a normalizing constant so

that the density integrates to 1.

The mean and variance of the beta(", $) random

variable, Y, are

E Y( ) = +

α

α β

and

Var( ) . ( )( )Y = + ++

αβ

αβαβ 2 1

Various beta distributions are shown in Figures A.15

and A.16. Figure A.15 shows beta densities with " = $,

and therefore with mean 0.5. When " < 1, the density

becomes infinite at 0.0, and when $ < 1, the density

becomes infinite at 1.0. When " = $ = 1, the density is

uniform. When " and $ are large, the density is approximately normal.

Figure A.16 shows densities with mean 0.1. Again,

when " < 1, the density becomes infinite at 0.0, and

when " > 1, the density is zero at 0.0. As the parameters " and $ become large, the density approaches a

normal distribution.

Another parameterization of the beta distribution uses

the parameters x0 = " and n0 = " + $. This parameterization is used by Martz and Waller (1991) because it

simplifies Bayes formulas and Bayesian estimation.

The p.d.f. of a beta(x0, n0) is

f y , n

x nx

y y

x nx ( ) ( )

( )( ) = − ( ) −

Γ − −−

Γ Γ

0

0 00

1 1 0 00 1

for 0 # y # 1, with the parameters x0 and n0 satisfying 

n0 > x0 > 0.
Basics of Probability

A-22

0.0 0.2 0.4 0.6 0.8 1.0

σ = 0.5

σ = 1

σ = 2

Figure A.17 Three logistic-normal densities with

median = 0.5.

The mean and variance of the beta(x0, n0) random

variable, Y, are

E Y x

n

( ) = 0

0

and

Var( ) . )

( )

(

Y xn x

n n = −+

00 0

0

2

0 1

Percentiles of the beta distribution occur in the formula

for a confidence interval for p, and in the formula for a

Bayes credible interval for p when a conjugate prior is

used. Some percentiles are tabulated in Appendix C.

In addition, many software packages, including some

commonly used spreadsheets, can calculate these

percentiles. If none of these work, Martz and Waller

(1991) give a method for finding the beta percentiles

from the corresponding percentiles of an F distribution,

discussed in Section A.7.11. The F distribution is

tabulated in most statistics books, and can be interpolated if necessary with good accuracy. The relation is

betaq(", $) = " / [" + $F1 ! q(2$, 2")]

for small q, and 

betaq(", $) = "Fq(2", 2$) / [$ + " Fq(2", 2$)]

for large q. Here betaq(", $) denotes the q quantile, or

the 100×q percentile, of the beta(", $) distribution, and

Fq(d1, d2) denotes the q quantile of an F distribution

with d1 and d2 degrees of freedom. So if all else fails,

and a statistics book with F tables is nearby, the first

formula can be used to find the lower percentile of the

beta distribution and the second formula can be used to

find the upper percentile. This method is not discussed

further here, because it is not expected to be needed

often. 

A.7.9 The Logistic-Normal Distribution

While not widely used in PRA, this distribution is

commonly used for Bayesian inference in other fields of

application, especially as a prior for the binomial

parameter p when p could plausibly be fairly large. X

has a logistic-normal distribution if ln[X/(1 ! X)] is

normally distributed with some mean : and variance F2

.

The function ln[X/(1 ! X)] may appear strange, but it is

common enough in some areas of application to have a

name, the logit function. Therefore, the above statements could be rewritten to say that X has a logisticnormal distribution if logit(X) is normally distributed.

Properties of the logistic-normal distribution are

summarized here.

• Let y = ln[x/(1 ! x)]. Then x = ey / (1 + ey). This

implies that x must be between 0 and 1.

• As x increases from 0 to 1, y = ln[x/(1 ! x)] increases monotonically from !4 to +4. Thus, y can

be generated from a normal distribution with no

problem of forcing x outside its possible range.

• The monotonic relation between x and y means that

the percentiles match. For example, the 95th

percentile of Y is : + 1.645F. Denote this by y0.95.

Therefore, the 95th percentile of X is 

x0.95 = exp( y0.95) / [1 + exp( y0.95)].

Alternatively, this can be written as

y0.95 = ln[x0.95 / (1 ! x0.95 )] .

• If X is close to 0 with high probability, so that

X/(1 ! X) is close to X with high probability, then

the logistic-normal and lognormal distributions are

nearly the same.

The third bullet shows how to find the percentiles of a

logistic-normal distribution. Unfortunately there is no

equally easy way to find the moments, such as the mean

or variance. Moments must be found using numerical

integration.

Figure A.17 shows several logistic normal distributions

that all have median 0.5. These correspond to a normally distributed y with mean : = 0 and with various

values of F. Figure A.18 shows several logistic normal

distributions that all have median 0.1. These correspond to a normally distributed y with mean : = !2.2 =

ln[0.1/(1 ! 0.1)].
Basics of Probability

A-23

0.0 0.1 0.2 0.3 0.4 0.5

σ = 0.5

σ = 1

σ = 2

Figure A.18 Three logistic-normal densities with

median = 0.1.

Note the general similarities to the beta distributions in

Figures A.15 and A.16. Note also the differences:

Logistic-normal distributions are characterized most

easily by percentiles, whereas beta distributions are

characterized most easily by moments. Also, the beta

densities can be J-shaped or U-shaped, but the logisticnormal densities always drop to zero at the ends of the

range.

A.7.10 Student’s t Distribution

The Student’s t distribution is not used in a central way

in PRA. However, it appears in a peripheral way in

places in this handbook, when dealing with the parameters of a normal or lognormal distribution, or in largesample situations when a distribution is approximated

as normal or lognormal. Therefore, the basic facts are

summarized here.

If (1) Z has a standard normal distribution, (2) X has a

chi-squared distribution with d degrees of freedom, and

(3) Z and X are statistically independent, then

T

Z

X d = /

has a Student’s t distribution with d degrees of freedom.

Therefore, T has a distribution that is symmetrical about

0, and it can take values in the entire real line. If d is

large, the denominator is close to 1 with high probability, and T has approximately a standard normal distribution. If d is smaller, the denominator adds extra variability, and the extreme percentiles of T are farther out

than are the corresponding normal percentiles. Tables

of the distribution are given in Appendix C.

Although not needed for ordinary work, the p.d.f. and

first two moments of T are given here. [See many

standard texts, such DeGroot (1975) or Bain and Engelhartd (1992).] The p.d.f. is

f t [ ] . d

d d

t

d

( ) [( ) / ]

( ) (/) ( /) /

( )/

= ++

Γ − +

Γ

1 2

2 1 2 1 2 2 1 2

π

If d > 1 the mean is 0. If d > 2 the variance is d/(d!2).

If d # 2 the variance does not exist. If d = 1, even the

mean does not exist; in this case the distribution is

called a Cauchy distribution.

A.7.11 F Distribution

The F distribution, also called Snedecor’s F distribution, arises as follows. If Y and Z are independent chisquared random variables with m and n degrees of

freedom, respectively, then 

X

Y m

Z n = /

/

has an F distribution with m and n degrees of freedom.

This is sometimes written as an F(m, n) distribution.

This can be re-expressed in terms of a ratio of gammadistributed variables, because the chi-squared distribution is a special case of a gamma distribution.

The density of an F distribution is almost never needed,

although it is given in mathematical statistics books as

f x

mn m n

m n

x

mx n

mn m

m n ( ) [( ) / ]

( /)(/) ( )

/ / ( /)

= ( )/

+

+

−

+

Γ

Γ Γ

2

2 2

2 2 21

2

for x $ 0. Bain and Engelhardt (1992) give the moments:

E(X) = n/(n ! 2)

Var(X) = .

2 2

2 4

2

2

nmn

mn n

( )

( )( )

+ −

− −

The mean is defined only if n > 2, and the variance only

if n > 4.

It follows from the definition in terms of a ratio of chisquared variables that the percentiles are related to each

other as follows. If Fq(m, n) is the q quantile (that is,

the 100q percentile) of an F(m, n) distribution, then

Fq(m, n) = 1/F1 ! q(n, m) . (A.11)
Basics of Probability

A-24

The F distribution is also related to the beta distribution, and Equation (A.11) forms the basis for the two

different forms of the relation given near the end of

Section A.7.8.

The distribution is not tabulated in Appendix C for two

reasons: the distribution is used only minimally for the

applications in this handbook, and the percentiles and

probabilities are given by many commonly used software packages.

A.7.12 Dirichlet Distribution

The Dirichlet distribution is a multivariate generalization of the beta distribution. Let m variables Y1, ..., Ym

be such that EiYi = 1. Their distribution can be described in terms of any m ! 1 of them, such as Y1, ...,

Ym!1 with

Y Y . m i

i

m

= − =

−

1 ∑

1

1

The m variables have a Dirichlet distribution with

parameters "1, ..., "m if the joint density of the first m !

1 variables is

fy y

yy y

m

m

i m

m i

i

m

m

m

( ,..., ) ( ... )

() ( ) 1 1

1

1

1

1

1

1

1 1

1 1 1

−

−

−

−

=

− −

= + +

⋅⋅⋅

× ⋅⋅⋅ −

⎛

⎝

⎜ ⎞

⎠

⎟ −

∑

Γ

Γ Γ

α α

α α

α α

α

Observe that when m = 2 this reduces to a beta distribution for Y1 with parameters "1 and "2. (Some authors

say that Y1, ..., Ym have the Dirichlet distribution, while

others say that Y1, ..., Ym!1 have this distribution. The

distribution is the same whichever way it is described.)

Many of the properties of the distribution are described

most easily in terms of an additional parameter ",

defined as " = "1 + ... + "m. Some of these properties

are the following.

Individually, each Yi

 has a beta("i, " ! "i) distribution.

Therefore, we have

E(Yi

) = "i /", and

Var(Yi

) ="i(" ! "i)/["2(" + 1)].

It can also be shown that the covariance terms are given

by

Cov(Yi

, Yj) = !"i"j /["2(" + 1)].

Thus, the ratio of each "i

 to " determines the corresponding mean. Once the means are fixed, the magnitude of " determines the variances and covariances,

with large " corresponding to small variances. The

covariances are negative, meaning that if one variable

is larger than its mean, each other variable tends to be

smaller than its mean; this is not surprising for variables

that must sum to 1.

One application of the Dirichlet distribution in PRA is

to multiple-branch nodes in event trees. If an event tree

has a node with m branches, m > 2, the probability of

the ith branch (also called the ith “split fraction”) can be

denoted pi

. The probabilities must satisfy p1 + ... + pm

= 1. They are not known exactly, and therefore are

assigned a joint distribution that describes their uncertainty in a Bayesian way. The Dirichlet distribution is

a natural distribution to use.

For further information about this distribution, see the

article in the Encyclopedia of Statistical Sciences, or 

Kotz, Balakrishnan, and Johnson (2000).
Basics of Probability

A-25

REFERENCES FOR APPENDIX A

Ascher, H., and H. Feingold, 1984, Repairable Systems Reliability – Modeling, Inference, Misconceptions and Their

Causes, Marcel Dekker, Inc., New York.

Bain, L. J., and M. Engelhardt, 1992, Introduction to Probability and Mathematical Statistics, 2nd Ed., PWS-Kent

Pub., Boston.

Box, G. E. P., and G. C. Tiao, 1973, Bayesian Inference in Statistical Analysis. Addison-Wesley, Reading, MA.

Çinlar, E., 1975, Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.

DeGroot, Morris H., 1975, Probability and Statistics, Addison-Wesley, Reading, MA.

Derman, C., L. J. Gleser, and I. Olkin, 1973, A Guide to Probability Theory and Application. Holt, Rinehart and

Winston, NY.

Encyclopedia of Statistical Sciences, Vol. 2, 1982, S. Kotz and N. L. Johnson, eds., John Wiley and Sons, NY, pp.

386-387.

Esary, J. D., and F. Proschan, 1963, Coherent Structures of Non-Identical Components, in Technometrics, Vol. 6,

pp. 191-209.

Esary, J. D., and F. Proschan, 1970, A Reliability Bound for Systems of Maintained, Interdependent Components in

Journal of the American Statistical Association, Vol. 65, pp. 329-338.

Hogg, R. V., and A. T. Craig, 1995, Introduction to Mathematical Statistics, 5th Edition. Macmillan, NY.

Kotz, S., N. Balakrishnan, and N. L. Johnson, 2000, Continuous Multivariate Distributions, Volume 1, Models and

Applications, 2nd Edition, Chapter 49. Wiley-Interscience, NY.

Lee, Peter M., 1997, Bayesian Statistics: An Introduction, Second Edition, Arnold, a member of the Hodder

Headline Group, London.

Martz, H. F., and R. A. Waller, 1991, Bayesian Reliability Analysis. R. E. Krieger Publishing Co., Malabar, FL. 

Meyer, P. L., 1970, Introductory Probability and Statistical Applications. Addison-Wesley Pub. Co., Reading, MA. 

NRC, 1994, A Review of NRC Staff Uses of Probabilistic Risk Assessment, NUREG-1489, U. S. Nuclear Regulatory

Commission, Washington, D.C.

Pfeiffer, P. E., and D. A. Schum, 1973, Introduction to Applied Probability. Academic Press, NY.

Snedecor, G. W., and W. G. Cochran, 1989, Statistical Methods. Iowa State University Press, Ames, IA.

Thompson, W. A., Jr., 1981, On the Foundations of Reliability, in Technometrics, Vol. 23, No. 1, pp. 1-13. 

Vesely, W. E., F. F. Goldberg, N. H. Roberts, and D. F. Haasl, 1981, Fault Tree Handbook. NUREG-0492. U.S.

Nuclear Regulatory Commission, Washington, D.C.
Basics of Probability

A-26
B-1

B. BASICS OF STATISTICS

B.1 Random Samples

When sampling from a distribution (or population), it is

usually assumed that the n observations are taken at

random, in the following sense. It is assumed that the

n random variables X1, X2, ..., Xn are independent. That

is, the sample X1, X2, ..., Xn, taken from a distribution

f(x), has the joint p.d.f. h satisfying

h(x1, x2, ..., xn) = f(x1) A f(x2) A . . . A f(xn).

This follows the definition of independent random

variables given in Section A.4.5. A sample taken in this

way is called a random sample. (As elsewhere in this

handbook, upper case letters denote random variables

and lower case letters denote particular values, number.)

The random variables X1, X2, ..., Xn forming such a

random sample are referred to as being independent and

identically distributed. If n is large enough, the sampled values will represent the distribution well enough

to permit inference about the true distribution.

B.2 Sample Moments

Mathematical expectation and moments provide characteristics of distributions of random variables. These

ideas can also be used with observations from a random

sample from a distribution to provide estimates of the

parameters that characterize that distribution.

A statistic is a function of one or more random variables that does not depend on any unknown parameters.

A function of random variables that can be computed

from the collected data sample is thus a statistic. Note

that a function of random variables is also a random

variable that has its own probability distribution and

associated characteristics.

If X1, X2, ..., Xn denote a random sample of size n from

a distribution f(x), the statistic

X

X

n

i

i

n

= =

∑1

is the mean of the random sample, or the sample

mean and the statistic

(B.1)

S

X X

n

i

i

n

2

2

1

1 =

−

−

=

∑ ( )

is the variance of the random sample. Note that n !

1 is used as the denominator in the S2 statistic to make

the statistic an unbiased estimator of the population

variance, F2

 (unbiased estimators are discussed in

Section B.4.1). Some authors use n in the denominator

instead of n ! 1, with corresponding adjustment of

formulas that involve S, but this handbook uses Equation B.1 consistently. In applications with computer

packages, note which definition is used and make any

necessary adjustments to formulas in this handbook.

Although not used as much as the sample mean and

sample variance, the sample skewness is occasionally

of interest. The definition can vary in detail, but one,

used by SAS (1988) is

 . n

n n XX S i

i

n

( )( ) ( ) − − − =

∑ 1 2

3

1

3

Similarly, the statistics defined by

m , X

n r

i

r

i

n

=

=

∑1

for r = 1, 2, ..., are called the sample moments.

One of the common uses of statistics is estimating the

unknown parameters of the distribution from which the

sample was generated. The sample mean, or average,

X, is used to estimate the distribution mean, or population mean, :, the sample variance, S2

, is used to

estimate the population variance, F2

, and so forth.

B.3 Statistical Inference

Since values of the parameters of a distribution are

rarely known, the distribution of a random variable is

rarely completely known. However, with some assumptions and information based on a random sample of

observations from the distribution or population, values

of the unknown parameters can often be estimated.

Probabilities can then be calculated from the corresponding distribution using these parameter estimates.
Basics of Statistics

B-2

Statistical inference is the area of statistics concerned

with using sample data to answer questions and make

statements about the distribution of a random variable

from which the sample data were obtained. Parameter

estimators are functions of sample data that are used to

estimate the distribution parameters. Statements about

parameter values are inferred from the specific sample

to the general distribution of the random variable or

population. This inference cannot be perfect; all

inference techniques involve uncertainty. Understanding the performance properties of various estimators has

received much attention in the statistics field.

For the purposes of this handbook, statistical inference

procedures can be classified as follows:

• parameter estimation

- estimation by a point value

- estimation by an interval

• hypothesis testing

- tests concerning parameter values

- goodness-of-fit tests and other model-validation

tests.

Parametric statistical inference assumes that the

sample data come from a particular, specified family of

distributions, with only the parameter values unknown.

However, not all statistical inference is based on

parametric families. In many cases, in addition to not

knowing the distribution parameter values, the form of

the parametric family of distributions is unknown.

Distribution-free, also called nonparametric, techniques are applicable no matter what form the distribution may have. Goodness-of-fit tests are an important

type of nonparametric tests that can be used to test

whether a data set follows a hypothesized distribution.

For statistical inference, two major approaches exist,

the frequentist approach and the Bayesian approach.

The two resulting sets of inference tools are summarized in Sections B.4 and B.5. In PRA work, Bayesian

estimators are normally used for parameter estimation.

See, for example, NUREG-1489 (NRC 1994). However, frequentist hypothesis tests are often used for

model validation, especially when the hypothesis to be

tested does not involve a simple parameter. This use of

Bayesian techniques for estimation and frequentist

techniques for model validation is also recommended

by Box (1980).

NUREG-1489 (NRC 1994) lists a number of “advantages” and “disadvantages” for each of the Bayesian

and frequentist approaches. An “advantage” is often in

the eye of the beholder. For example, is it an advantage

or disadvantage that frequentist methods use only the

data at hand, not external or prior information? Therefore, the lists from that report are presented in modified

and augmented form in Table B.1, where the points are

not called advantages or disadvantages, but simply

“features.”

B.4 Frequentist Inference

Frequentist estimation of distribution parameters uses

only the information contained in the data sample and

assumptions about a model for the sample data. In

contrast to Bayesian estimation (discussed in Section

B.5), degree of belief is not incorporated into the

estimation process of frequentist estimation.

In the frequentist approach to estimation, a distribution

parameter is treated as an unknown constant and the

data to be used for estimation are assumed to have

resulted from a random sample. Information outside

that contained in the sample data is used minimally.

The random variability in the sample data is assumed to

be due directly to the process under study. Thus, the

frequentist approach addresses variation in parameter

estimates and how far estimates are from the true

parameter values.

Frequentist testing of a hypothesis follows the same

spirit. The hypothesis is assumed, and the data are

compared to what would have been expected or predicted by the hypothesis. The frequentist analyst asks whether the observed values come from the likely part of

the distribution or from the extreme tails, and decides in

this way whether the data are consistent with the

hypothesis.

B.4.1 Point Estimation

Many situations arise in statistics where a random

variable X has a p.d.f. that is of known functional form

but depends on an unknown parameter 2 that can take

on any value in a set. The different values for 2

produce a family of distributions. One member of the

family corresponds to each possible value of 2.

Estimators of the distribution parameter are functions

of sample data that are used to estimate the distribution

parameters. Thus, estimators are themselves random

variables. The specific value of the estimator computed

from a random sample provides an estimate of the

distribution parameter. Note the distinction between

estimator, a random variable, and estimate, a particular

value. An estimate of a distribution parameter in the

form of a single number is called a point estimate of

that parameter. The sample mean is a point estimate of
Basics of Statistics

B-3

Table B.1 Features of Bayesian and frequentist approaches.

Bayesian Approach Frequentist Approach

Bayesian methods allow the formal introduction of prior

information and knowledge into the analysis, which can

be especially useful when sample data are scarce, such as

for rare events. For the nuclear industry, this knowledge

often exists in the form of industry-wide generic data.

Thus, Bayesian estimation allows the use of various types

of relevant generic data in PRA.

Results depend only on the data sample. Including

relevant information about a parameter that is external to

the random sample is complicated.

If the prior distribution accurately reflects the uncertainty

about a parameter, Bayesian parameter estimates are

better than classical estimates.

Bayesian estimation can be sensitive to the choice of a

prior distribution. Therefore:

Identifying suitable prior distributions and justifying and

gaining acceptance for their use can be difficult.

The choice of a prior distribution is open to criticism that

the choice is self-serving and may reflect inappropriate,

biased, or incorrect views.

Because Bayesian probability intervals can be interpreted

as probability statements about a parameter, they are

easily combined with other sources of uncertainty in a

PRA using the laws of probability.

A confidence interval cannot be directly interpreted as a

probability that the parameter lies in the interval.

Bayesian distributions can be propagated through fault

trees, event trees, and other logic models.

It is difficult or impossible to propagate frequentist

confidence intervals through fault and event tree models

common in PRA to produce corresponding interval

estimates on output quantities of interest.

Using Bayes’ Theorem, Bayesian estimation provides a

method to update the state of knowledge about a parameter as additional data become available.

Frequentist methods can update an earlier analysis if the

original data are still available or can be reconstructed.

In complicated settings, Bayesian methods require

software to produce samples from the distributions.

In complicated settings, frequentist methods must use

approximations. In some cases they may be unable to

analyze the data at all.

Bayesian hypothesis tests are commonly used only with

hypotheses about a parameter value.

A well-developed body of hypothesis tests exists, useful

for model validation. These are appropriate for investigating goodness of fit, poolability of data sources, and

similar questions that do not involve a simple parameter.

Both Approaches

When the quantity of data is large, both approaches produce good estimates.

Both types of computation are straightforward when estimating a parameter in a simple setting.
Basics of Statistics

B-4

the mean of the distribution and the sample variance is

a point estimate of the variance of the distribution. For

another sample drawn from the same population, a

different sample mean and variance would be calculated. In fact, these sample statistics are specific values

of random variables. Thus, viewed as random variables

the sample statistics have their own sampling distributions. For example, it can be shown that has mean X

: and variance F2/n, regardless of the distribution from

which the samples are drawn.

Different techniques exist for obtaining point estimates

for unknown distribution characteristics or parameters.

Two of the most common methods are presented here

[see Hogg and Craig (1995) for more information]:

maximum likelihood estimation and the method of

moments.

A distribution of a random variable X that depends on

an unknown parameter 2 will be denoted f(x; 2). If X1,

X2, ..., Xn is a random sample from f(x; 2), the joint

p.d.f. of X1, X2, ..., Xn is f(x1; 2)Af(x2; 2)A ... Af(xn; 2).

This joint p.d.f. may be viewed as a function of the

unknown parameter 2 and, when so viewed, is called

the likelihood function, L, of the random sample.

Thus, the likelihood function is the joint p.d.f. of X1, X2,

..., Xn, denoted

L xx x fx , n i

i

n

( ; , ,..., ) ( ; ) θ θ 1 2

1

= =

∏

viewed as a function of 2. The maximum likelihood

estimate of 2 is defined as the value such that L( ; $θ $

θ

x1, x2, ..., xn) $ L(2; x1, x2, ..., xn) for every value of 2.

That is, the maximum likelihood estimate of 2 is the

value that maximizes the likelihood function. In $θ

many cases, this maximum will be unique and can often

be obtained through differentiation. Note that solving

the derivative set to zero for 2 may be easier using

ln(L), which is equivalent since a function and its

natural logarithm are maximized at the same value of

2.

The maximum likelihood estimate is a function of the

observed random sample x1, x2, ..., xn. When is $θ

considered to be a function of the random sample X1,

X2, ..., Xn, then is a random variable and is called the $

θ

maximum likelihood estimator of 2.

Another method of point estimation is the method of

moments, which involves setting the distribution

moments equal to the sample moments:

Mr = E(Xr) = mr = 'ixi

r

/n,

for r = 1, 2, ..., k, if the p.d.f. f(x; 21, 22, ..., 2k) has k

parameters. The k equations can be solved for the k

unknowns 21, 22, ..., 2k and the solutions $ , $ ,..., $ θθ θ 1 2 k

are the method-of-moments estimators.

How “well” a point estimator estimates a parameter has

received a large amount of attention. Numerous desirable properties of point estimators exist. One desirable

property of estimators, alluded to previously in Section

B.2, is unbiasedness. An unbiased estimator is one

whose mean value is equal to the parameter being

estimated. That is, an estimator is unbiased for a $

θ

parameter 2 if E( ) = θ

$ 2. For a random sample from

a normal distribution, the sample mean, , and the X

sample variance, S2, are unbiased estimators of : and

F2, respectively. (Recall that S2 is defined by Equation

B.1, with n ! 1 in the denominator.) However, the

method of moments estimator of the variance is biased.

The bias of an estimator is defined as . $

θ E( $

θ θ ) −

Minimum variance is another desirable property of an

estimator. An unbiased estimator is said to have

minimum variance if its variance is less than or equal to

the variance of every other unbiased statistic for 2.

Such an estimator is referred to as an unbiased, minimum variance estimator.

Another desirable property of estimators is sufficiency.

For a random sample X1, X2, ..., Xn from f(x; 21, 22, ...,

2m), and functions (statistics) of the Xis, $ , $ , ..., $ θθ θ 1 2 m

the statistics are jointly sufficient statis- $ , $ , ..., $ θθ θ 1 2 m

tics if the conditional p.d.f. of the Xis given the statistics

, g(x1, x2, ..., xn* ), is independent of the $

θjs $ , $ , ..., $ θθ θ 1 2 m

parameters (Martz and Waller, 1991).

Sufficiency can be thought of as exhausting all the

possible information about a parameter that is contained

in the random sample. When a sufficient statistic exists,

it may serve as the basis for a minimum variance or

“best” estimator of the parameter. Sufficiency is also

important because it simplifies Bayesian estimation

methods.

Under certain commonly occurring conditions, as the

sample size gets large, the maximum likelihood

estimator is approximately normally distributed, approximately unbiased, and has approximately the

minimum variance. It is, therefore, a very good

estimator for large data sets. The maximum likelihood

estimator is not necessarily good for small data sets.
Basics of Statistics

B-5

Several other methods of estimation and desirable

properties for estimators exist. Further information can

be found in Hogg and Craig (1995) or Kendall and

Stuart (1973).

B.4.2 Interval Estimation

Another way of estimating a parameter is to identify

that it falls in some interval (lcl, ucl) with a specified

degree of certainty, or confidence, where lcl denotes the

lower confidence limit and ucl denotes the upper

confidence limit. The interval (lcl, ucl) is referred to as

an interval estimate of the parameter. The lcl and ucl

values are calculated from the random sample from the

given distribution. Associating a level of desired

confidence with an interval estimate produces a confidence interval. The level of desired confidence is also

referred to as the confidence coefficient.

Confidence intervals are based on estimators associated

with a random sample (functions of the data), LCL for

the lower confidence limit and UCL for the upper

confidence limit, such that, prior to observing the

random sample, the probability that the unknown

parameter, 2, is contained in the interval [LCL, UCL] is

known. That is,

Pr[LCL # 2 # UCL] = 1 ! "

for 0 < " < 1.

Once the random sample has been generated, the

functions LCL and UCL produce two values, lcl and

ucl. The interval (lcl, ucl) is called a two-sided confidence interval with confidence level 1 ! ", or equivalently, a 100(1 ! ")% two-sided confidence interval.

Similarly, upper one-sided confidence intervals or

lower one-sided confidence intervals can be defined

that produce only an upper or lower limit, respectively.

Since the true parameter value, although unknown, is

some constant, the interval estimate either contains the

true parameter value or it does not. A 95% confidence

interval is interpreted to mean that, for a large number

of random samples from the same distribution, 95% of

the resulting intervals (one interval estimate of the same

population parameter constructed the same way for

each sample) would contain the true population parameter value, and 5% of the intervals would not. The " =

.05 risk of obtaining an interval that does not contain

the parameter can be increased or decreased. Values

for 1 ! " should be decided upon prior to obtaining the

random sample, with .99, .95, and .90 being typical.

Note that higher confidence levels result in wider

interval estimates.

Confidence intervals cannot be interpreted as probability statements about the parameter being estimated,

because the parameter is assumed to be an unknown

constant and not a random variable. The level of

confidence pertains to the percentage of intervals, each

calculated from a different random sample from the

same distribution, that are expected to contain the true

parameter value. The confidence does not pertain to

the specific calculated interval (it could be from the

unlucky 5% of intervals that do not contain the true

parameter value).

As an example, a confidence interval for the parameter

: can be produced from a random sample drawn from

a normal(:, F2

) population by calculating the appropriate functions of the data. Recall that, if each sample

value is drawn from a normal distribution, the sample

mean has a normal(:, F2 X /n) distribution, where n is

the sample size. Even if the sample values are drawn

from a distribution that is not normal, by the central

limit theorem, will be approximately normal( X :,

F2/n) for sufficiently large n. Assuming that F2 is

known (from previous data and experience), the standardized normal random variable

Z

X

n = − μ

σ /

is normal(0, 1), and tabulated in Appendix C. From

these tables, values of w can be found for which

Pr[!w # Z # w] = 1 ! ". (B.2)

For example, for " = .05, w = 1.96. In this case, w is

the 97.5th percentile of the standard normal distribution, commonly denoted z0.975, or z1!"/2 for " = .05.

Substituting for Z in Equation B.2 above, along with

some algebraic manipulation, produces

Pr[ ] X w , nX w

n

− ≤ ≤ + =−

σ

μ

σ

1 α

which defines a 100(1 ! ")% confidence interval for

the population mean :, where

LCL X w (B.3)

n = − σ

and

UCL X w , (B.4)

n

= +

σ
Basics of Statistics

B-6

with w = z1!"/2.

A random sample will yield a specific sample mean.

The numbers w and n are known, and F was assumed to

be known. Therefore, for a preassigned confidence

level, values for LCL and UCL can be calculated to

produce a specific 100(1 ! ")% confidence interval for

:. Each of the random variables LCL and UCL is a

statistic, and the interval (LCL, UCL) is a random

interval formed from these statistics.

Usually the value of F is not known. In this case, the

unbiased estimator of the population variance, S2

, can

be used to produce S, which can be used in the above

equations in place of F. Thus, the following standardized random variable, T, can be formed:

T

X

S n = − μ

/

(This formula requires the definition of S based on

Equation B.1.) For sufficiently large n (say 25 or 30),

T follows a normal(0, 1) distribution. If n is not sufficiently large, T follows a Student’s t distribution, for

which tabulated probabilities exist in many statistics

books, and in Appendix C. The Student’s t distribution

depends on a parameter called the degrees of freedom.

In the present example, this parameter equals n ! 1.

Confidence intervals for the population mean can then

be calculated similarly to the case where F is known,

using either the Student’s t distribution or, when n is

large, the normal distribution.

Confidence intervals can also be constructed for differences of means and many other population parameters,

such as variances, probabilities, quantiles, and distribution characteristics (see, for example, Hogg and Craig

1978).

B.4.3 Hypothesis Testing

Testing a statistical hypothesis is another major area of

statistics. A hypothesis is a statement about the distribution of the observable random variable. Often this

statement is expressed as a statement about one or more

parameters of the distribution. As discussed previously,

estimation uses information in the data from a random

sample to infer something about the magnitude of a

parameter value. Similar to estimation, hypothesis

testing also uses information from the random sample.

However, the objective of hypothesis testing is to

determine whether the specific statement about the

distribution is true.

The hypothesis to be tested is referred to as the null

hypothesis, denoted by H0. The alternative to the null

hypothesis is referred to as the alternative hypothesis,

denoted H1 or Ha. A test of a hypothesis is a rule or

procedure for deciding whether to reject or accept the

null hypothesis. This rule or procedure is based upon

information contained in the random sample and

produces a single number, called a test statistic, which

leads to a decision of whether the sample values do not

support H0. The entire set of values that the test statistic may assume is divided into two regions, one corresponding to the rejection region and the other to the

acceptance region.

If the test statistic computed from a particular sample

has a value in the rejection region, H0 is rejected. If the

test statistic falls in the acceptance region, H0 is said to

be accepted, due to lack of evidence to reject. For each

of the two possible cases for H0, true or false, the test

either rejects or does not reject H0, producing four

distinct possibilities. These possibilities (using conditional probability notation), along with some concepts

and terms associated with hypothesis testing, are

summarized in Table B.2 (Martz and Waller, 1991).

Table B.2 Possible hypothesis test outcomes.

H0 True H0 False

Accept H0 Pr(accept H0 | H0 is true) = 1!"

= Level of confidence

Pr(accept H0 | H0 is false) = $

= Pr(Type II Error)

Reject H0 Pr(reject H0 | H0 is true) = "

= Level of significance

= Pr(Type I Error)

Pr(reject H0 | H0 is false) = 1!$

= Power
Basics of Statistics

B-7

A stated null hypothesis is either true or false. One of

two errors can occur in hypothesis testing:

1. rejection of the null hypothesis when it is true,

referred to as the Type I error; and

2. acceptance of the null hypothesis when it is false,

referred to as the Type II error.

The probability of making a Type I error, denoted by

", is referred to as the significance level of the test.

Thus, 1 ! " is the probability of making a correct

decision when H0 is true. The probability of making a

correct decision when H0 is false, denoted 1 ! $, is

referred to as the power of the test. The probability of

making a Type II error is equal to one minus the power

of the test, or $.

The goodness of a statistical hypothesis test is measured

by the probabilities of making a Type I or a Type II

error. Since " is the probability that the test statistic

will fall in the rejection region, assuming H0 to be true,

increasing the size of the rejection region will increase

" and simultaneously decrease $ for a fixed sample

size. Reducing the size of the rejection region will

decrease " and increase $. If the sample size, n, is

increased, more information will be available for use in

making the decision, and both " and $ will decrease.

The probability of making a Type II error, $, varies

depending on the true value of the population parameter. If the true population parameter is very close to the

hypothesized value, a very large sample would be

needed to detect such a difference. That is, the probability of accepting H0 when H0 is false, $, varies depending on the difference between the true value and

the hypothesized value. For hypothesis tests, " is

specified prior to conducting the random sample. This

fixed " specifies the rejection region. For a deviation

from the hypothesized value that is considered practical

and that is wished to be detectable by the hypothesis

test, a sample size can be selected that will produce an

acceptable value of $.

Different alternative hypotheses will result in different

rejection regions for the same H0. This is seen most

easily for a hypothesis that is expressed in terms of a

parameter, for example, H0: : = :0 for some given

value :0. In this case, there is an exact correspondence

between one-sided and two-sided confidence intervals

and rejection regions for one-sided and two-sided

alternative hypotheses. If the hypothesized value falls

outside a 100(1 ! ")% confidence interval for the

corresponding population parameter, the null hypothesis would be rejected with level of confidence equal to

1 ! ".

For the example presented in the previous section,

Section B.4.2, the 100(1 ! ")% two-sided confidence

interval for a population mean is defined by the LCL

and UCL in Equations B.3 and B.4. For the hypothesized value of the mean, say :0, if :0 < lcl or :0 > ucl,

H0 would be rejected. Equivalently, the test statistic in

Equation B.2 can be computed using : = :0 and, for

" = .05, if it is greater than 1.96 or less than -1.96, H0

would be rejected with 95% level of confidence.

To further illustrate these concepts, a more detailed

example is presented. Atwood et al. (1998) assert that

for non-momentary losses of offsite power with plantcentered causes, the recovery times are lognormally

distributed with median 29.6 minutes and error factor

10.6. This is equivalent to X being normally distributed

with : = ln(29.6) = 3.388 and F = ln(10.6)/1.645 =

1.435, where X = ln(recovery time in minutes). Suppose that a plant of interest has experienced five such

losses of offsite power in recent history. It is desired to

test whether the plant’s recovery times follow the

claimed distribution.

To simplify the situation, the question is formulated in

terms of : only, assuming that F = 1.435. The null

hypothesis is

H0: : = 3.388 .

Because only long recovery times are of concern from

a risk standpoint, the alternative hypothesis is defined

as

H1: : > 3.388 .

That is, values <3.388 are possible, but are not of

concern. The test statistic, based on n = 5 recovery

times, is to reject H0 if

Z . X = w − >

3388

1435 5

.

. /

To make ", the probability of Type I error, equal to

0.05, w is chosen to be the 95th percentile of the

standard normal distribution, 1.645. Then the test can

be re-expressed as rejecting H0 if 

X > 4.44 .

The upper part of Figure B.1 shows the density of X

when : = 3.388. The area to the right of 4.44 is 

Pr( . | X H > 4 44 , 0 is true)

which equals 0.05.
Basics of Statistics

B-8

Figure B.2 Power curves when n = 5 and n = 10. The

graph shows the probability of rejecting H0, as a

function of the true :.

Figure B.1 Probability of rejecting H0: : = 3.388, if in

fact H0 is true (upper distribution), and if H0 is false

with : = 4.09 (lower distribution).

What if H0 is false? For example, a median 60-minute

recovery time corresponds to : = ln(60) = 4.09. The

lower part of Figure B.1 shows the density of when X

: = 4.09. The area to the right of 4.44 is

Pr( . | . ) X > = 4 44 4 09 μ ,

which is equal to 0.29. This value represents the power

of the hypothesis test when : = 4.09 and is the probability of (correctly) rejecting H0. The probability of a

Type II error when : = 4.09 is 1 ! 0.29 = 0.71.

It can be useful to plot the power as a function of :.

The plot is called a power curve. Figure B.2 shows

two power curves, corresponding to n = 5 and n = 10.

The probability of Type I error, that is, the probability

of rejecting H0 when H0 is true, is shown as ". The

probability of Type II error, that is, the probability of

accepting H0 when H0 is false, is shown as $ for one

value of :, and equals 1 minus the power. The two

tests, with n = 5 and n = 10, have both been calibrated

so that " = 0.05. The power, for any value of : in H1,

is larger when n = 10 than when n = 5; equivalently, the

probability of Type II error is smaller.

The interpretation of confidence in hypothesis testing is

also the same as with confidence intervals. That is, the

confidence is not in one specific test statistic. The

confidence arises from the viewpoint that if the random

sample was collected a number of times in the same

way and if H0 was true, 100(1 ! ")% of the tests would

result in not rejecting H0.

As can be seen, interval estimation and hypothesis

testing are closely related. Some experimenters prefer

expressing inference as estimators. Others prefer to test

a particular hypothesized value for the parameter of

interest. 

B.4.4 Goodness-of-Fit Tests

The methods presented above are concerned with

estimating the parameters of a distribution, with the

actual form of the distribution assumed to be known (or

the central limit theorem applies with large n). Other

hypothesis tests do not assume that only a parameter is

unknown. In particular, goodness-of-fit tests are

special hypothesis tests that can be used to check on the

assumed distribution itself. Based on a random sample

from some distribution, goodness-of-fit tests test the

hypothesis that the data are distributed according to a

specific distribution. In general, these tests are based

on a comparison of how well the sample data agree with

an expected set of data from the assumed distribution.

Perhaps the most familiar goodness-of-fit test is the chisquare test. The test statistic used for this statistical

test has an approximate P2

 distribution, leading to the

name of the test. A random sample of n observations,

X1, X2, ..., Xn, can be divided or binned into k groups or

intervals, referred to as bins, producing an empirical
Basics of Statistics

B-9

distribution. The assumed distribution under the null

hypothesis, f0(x), is used to calculate the probability that

an observation would fall in each bin, with the probabilities denoted by p1, p2, ..., pk.

These probabilities are frequently referred to as cell

probabilities. The k bins are also called cells. The k

bin intervals do not overlap and they completely cover

the range of values of f0(x). It follows that 'i

k

=1pi

 = 1.

The expected frequency of the ith bin, denoted ei

, is ei

= npi, for i = 1, 2, ..., k. The ei are commonly referred

to as the expected cell counts. The observed frequencies for each of the k bins, denoted Oi

, are referred to as

observed cell counts.

The chi-square goodness-of-fit test compares the

observed frequencies to the corresponding expected

frequencies for each of the k groups by calculating the

test statistic:

X . O e

e

i i

i i

k

2

2

1

= −

=

∑ ( )

If the observations come from some distribution other

than that specified in the null hypothesis, the observed

frequencies tend to agree poorly with the expected

frequencies, and the computed test statistic, X2

, becomes large.

The distribution of the quantity X2

 can be approximated

by a chi-square distribution. The parameter that specifies the chi-square distribution is called the degrees of

freedom. Its value depends on the number of unknown

parameters and how they are estimated. When the nullhypothesis distribution is completely specified, such as

normal with both : and F known, the degrees of freedom are k ! 1. If, instead, H0 specifies the form of the

distribution but not the parameters, the degrees of

freedom must be adjusted. In the example, if and S2 X

from the sample are used to estimate : and F2 when

testing the distribution, the degrees of freedom are

between k ! 1 and k ! 1 ! m, where m is the number of

estimated parameters, 2. If the quantity X2

 is greater

that the 1 ! " quantile of the P2

(k ! 1) distribution, the

hypothesized probability distribution is rejected. If X2

is less than the 1 ! " quantile of the P2(k ! 1 ! m)

distribution, the data are concluded to be adequately

modeled by f0(x).

When the sample size is small, the P2

 distribution still

applies as long as the expected frequencies are not too

small. Larger expected cell counts make the chi-square

distribution approximation better. The problem with

small expected frequencies is that a single random

observation falling in a group with a small expected

frequency would result in that single value having a

major contribution to the value of the test statistic, and

thus, the test itself. In addition, small expected frequencies are likely to occur only in extreme cases. One rule

of thumb is that no expected frequency should be less

than 1 (see Snedecor and Cochran, 1989). Two expected frequencies can be near 1 if most of the other

expected frequencies are greater than 5. Groups with

expected frequencies below 1 should be combined or

the groups should be redefined to comply with this rule.

Note that k is the number of groups after such combination or redefinition.

Comparing how well sample data agree with an expected set of data leads to another common use of the

chi-square test: testing whether two or more classification criteria, used to group subjects or objects, are

independent of one another. Although not a goodnessof-fit test, the chi-square test for independence is

similar to the chi-square goodness-of-fit test.

For two grouping criteria, the rows of a two-way

contingency table can represent the classes of one of

the criteria and the columns can represent the classes of

the other criterion. To test the hypothesis that the rows

and columns represent independent classifications, the

expected number, eij, that would fall into each cell of

the two-way table is calculated and used to compute the

following chi-square test statistic:

X , O E

E

ij ij

i j ij

2

2

= − ∑ ( )

,

where i = 1, 2, ..., r (the number of rows); j = 1, 2, ..., c

(the number of columns); and Oij is the number observed to belong to the ith row and jth column. The eij

are calculated by

e , R C

n ij

i j =

where Ri and Cj are the total observed in the ith row and

jth column, respectively, and n is the total sample size

(n = 'Ri

 = 'Cj).

For this test, the P2

 test statistic follows a chi-square

distribution with (r ! 1)(c ! 1) degrees of freedom. If

the calculated X2

 exceeds the 1 ! " quantile of the P2

distribution with (r ! 1)(c ! 1) degrees of freedom, the

null hypothesis of independence is rejected and the

rows and columns are concluded to not represent

independent classifications.
Basics of Statistics

B-10

Figure B.3 The hypothesized distribution, the empirical distribution, and the Kolmogorov test statistic, D.

The Kolmogorov goodness-of-fit test tests the hypothesis that the observed random variable has c.d.f. F0(x),

versus the alternative hypothesis that the observed

random variable does not have c.d.f. F0(x). It does this

by comparing the sample c.d.f. (the empirical distribution function) to the hypothesized c.d.f. For a random

sample of n observations, X1, X2, ..., Xn, the test statistic

is defined as the maximum vertical distance between the

empirical c.d.f., and F0(x). The actual procedure $

F x( )

for calculating the test statistic can be found in many

statistics texts, including Martz and Waller (1991) and

Conover (1999). The test statistic is then compared to

the 1 ! " quantile of tabled values for the Kolmogorov

test, e.g. in Table C. If the calculated test statistic

exceeds the 1 ! " quantile, the hypothesized c.d.f. is

rejected. Otherwise, F0(x) is concluded to describe the

data. The Kolmogorov goodness-of-fit test is based on

each individual data point and therefore is equally

effective for small or large samples.

As an example, consider the previous example of lossof-offsite-power recovery times. Suppose that five

recovery times have been observed at the plant: 7, 22,

94, 185, and 220 minutes. The corresponding values of

x = ln(recovery time in minutes) are 1.95, 3.09, 4.54,

5.22, and 5.39. The null hypothesis and alternative

hypothesis are:

H0: X is normal with : = 3.388, F = 1.435 

H1: X has some other distribution .

Note, all possible alternative distributions are considered, not just normal distributions, or distributions with

F = 1.435.

Figure B.3 shows the distribution function specified by

H0 (the smooth curve) and the empirical distribution

function specified by the data (the step function). The

maximum distance between the two distributions is D,

the Kolmogorov test statistic. If D is large, the test

rejects H0 in favor of H1.

If the sample size is small, the Kolmogorov test may be

preferred over the chi-square test. The Kolmogorov

test is exact, even for small samples, while the chisquare test is an approximation that is better for larger

sample sizes. The Kolmogorov statistic can also be

used to construct a confidence region for the unknown

distribution function.

The Kolmogorov goodness-of-fit test is sometimes

called the Kolmogorov-Smirnov one-sample test.

Statistics that are functions of the maximum vertical

distance between and F0(x) are considered to be $

F x( )

Kolmogorov-type statistics. Statistics that are functions

of the maximum vertical distance between two empirical distribution functions are considered to be Smirnovtype statistics. A test of whether two samples have the

same distribution function is the Smirnov test, which is

a two-sample version of the Kolmogorov test presented

above. This two-sample test is also called the

Kolmogorov-Smirnov two-sample test. Conover

(1999) presents additional information and tests.

Another useful goodness-of-fit test is the AndersonDarling goodness-of-fit test and test for normality. The

Anderson-Darling test measures the squared difference

between the empirical distribution function (EDF) of a

sample and the theoretical distribution to be tested. It

averages this squared difference over the entire range of

the random variable, weighting the tails more heavily

than the center. This statistic is recommended to guard

against wayward observations in the tail and has generally good power. 

Because many statistical methods require the assumption of normality, some assessment of whether data

come from a normal population is helpful when considering appropriate analysis techniques. The AndersonDarling statistic provides a measure of how much

normal probability scores for the data (normal probability plot values) deviate from a straight line that would

arise under normality. A computer package is often

used to calculate this statistic and compare it to tabled

values for the statistic. If the calculated statistic is too

high, the deviations from the straight line are too large

to be attributed to the variation due to sampling observations from a normal distribution. Thus, the hypothesis of normality is rejected. See Stephens (1982) for

more information on the Anderson-Darling goodnessof-fit test.
Basics of Statistics

B-11

Certain patterns of deviations from linearity in normal

probability plots indicate common types of nonnormal

characteristics, such as skewness or kurtosis (presence

of long or short tails of the p.d.f.). Test for skewness or

kurtosis are also available. See Snedecor and Cochran

(1989) for more information on these tests.

B.5 Bayesian Estimation

B.5.1 Purpose and Use

Bayesian estimation is the other major class of statistical inference methods. Similar to frequentist estimation, both point and interval estimates can be obtained.

However, Bayesian estimation is different from classical estimation in both practical and philosophical

perspectives (NRC, 1994). Bayesian estimation incorporates degree of belief and information beyond that

contained in the data sample, forming the practical

difference from classical estimation. The subjective

interpretation of probability forms the philosophical

difference from frequentist methods.

The prior belief about a parameter’s value is contained

in what is referred to as the prior distribution, which

describes the state of knowledge (or subjective probability) about the parameter, prior to obtaining the data

sample. Therefore, in the Bayesian approach, the

parameters of the sampling distribution have probability

distributions. These probabilities do not model random

variability of the parameters, but the analyst’s degree of

belief about the true values of the unknown parameters.

The distributions are sometimes called “uncertainty

distributions.” A Bayesian uncertainty distribution

satisfies all the rules of a probability distribution.

Bayesian estimation consists of two main areas, both of

which use the notion of subjective probability. The first

area involves using available data to assign a subjective,

prior distribution to a parameter, such as a failure rate.

The degree of belief about the uncertainty in a parameter value is expressed in the prior distribution. This

assignment of a prior distribution does not involve the

use of Bayes’ Theorem. The second area of Bayesian

estimation involves using additional or new data to

update an existing prior distribution. This is called

Bayesian updating, and directly uses Bayes’ Theorem.

Bayes’ Theorem, presented in Section A.5, can be seen

to transform the prior distribution by the effect of the

sample data distribution to produce a posterior distribution. The sample data distribution, f(x*2), can be

viewed as a function of the unknown parameter, instead

of the observed data, xi

, producing a likelihood function, as discussed in Section B.4.1. Using the likelihood function, the fundamental relationship expressed

by Bayes’ Theorem is

Posterior Distribution . Prior Distribution Likelihood

Marginal Distribution = ×

The marginal distribution serves as a normalizing

constant.

In Bayesian updating, the sampling distribution of the

data provides new information about the parameter

value. Bayes’ Theorem provides a mathematical

framework for processing new sample data as they

become sequentially available over time. With the new

information, the uncertainty of the parameter value has

been reduced, but not eliminated. Bayes’ Theorem is

used to combine the prior and sampling distributions to

form the posterior distribution, which then describes the

updated state of knowledge (still in terms of subjective

probability) about the parameter. Point and interval

estimates of the parameter can then be obtained directly from the posterior distribution, which is viewed

as containing the current knowledge about the parameter. This posterior distribution can then be used as the

prior distribution when the next set of data becomes

available. Thus, Bayesian updating is successively

implemented using additional data in conjunction with

Bayes’ Theorem to obtain successively better posterior

distributions that model plant-specific parameters.

Bayesian point and interval estimates are obtained from

both the prior and posterior distributions. The interval

estimates are subjective probability intervals, or

credible intervals. The terminology is not yet universally standard. Berger (1985) and Bernardo and Smith

(2000) both use the term credible interval, but Box

and Tiao (1973) use Bayes probability interval,

Lindley (1965) uses Bayesian confidence interval, and

other authors have used other terms. A credible interval

can be interpreted as a subjective probability statement

about the parameter value, unlike classical interval

estimates. That is, the interpretation of a 95% Bayesian

posterior probability interval (a, b) is that, with 95%

subjective probability, the parameter is contained in the

interval (a, b), given the prior and sampling distributions.

B.5.2 Point and Interval Estimates

Bayesian parameter estimation involves four steps. The

first step is identification of the parameter(s) to be

estimated, which involves consideration of the assumed

distribution of the data that will be collected. The

second step is development of a prior distribution that
Basics of Statistics

B-12

appropriately quantifies the state of knowledge concerning the unknown parameter(s). The third step is collection of the data sample. The fourth and final step is

combining the prior distribution with the data sample

using Bayes’ Theorem, to produce the desired posterior

distribution.

For PRA applications, determining the prior distribution is usually based on generic data and the data

sample usually involves site-specific or plant-specific

operating data. The resulting posterior distribution

would then be the site-specific or plant-specific distribution of the parameter.

The plant-specific data collected are assumed to be a

random sample from an assumed sampling distribution.

The data are used to update the prior, producing the

posterior distribution. Point estimates, such as the most

likely value (the mode), the median, or (most commonly) the mean value, and probability interval estimates of the parameter can then be obtained. Other

bounds and other point values can also be obtained with

the Bayesian approach because the posterior parameter

distribution is entirely known and represents the available knowledge about the parameter.

Bayesian interval estimation is more direct than classical interval estimation and is based solely on the

posterior p.d.f.. A symmetric 100(1 ! ")% two-sided

Bayes probability interval estimate of the parameter

is easily obtained from the "/2 and 1 ! "/2 quantiles of

the posterior distribution. Lower and upper one-sided

Bayes probability interval estimates can similarly be

calculated. Again, note that the Bayes interval estimates are explicit probability statements of the true

parameter being contained in the interval.

In some applications, such as a planned facility, plantspecific data do not exist. In these cases, Bayes’

Theorem is not used. Only the generic data are used

and parameter estimates are based solely on the assumed prior distribution. Investigation of the sensitivity

of the results to the choice of the prior distribution is

important for these cases.

B.5.3 Prior Distributions

The prior distribution is fundamental to any Bayesian

analysis and represents all that is known or assumed

about the parameter 2 prior to collecting any data. The

information summarized by the prior distribution can be

objective, subjective, or both. Operational data and

data from a previous but comparable experiment could

be used as objective data. Subjective information could

involve personal experience and opinions, expert

judgement, assessments of degree of belief, and design

information.

The selection of prior distributions can be seen to be

somewhat subjective. A particular prior must be

evaluated to determine the sensitivity of the choice of

that prior on the parameter estimates. Consistency of

the prior information and data with the prior distribution must be tested.

Choices for the initial prior distribution and techniques

for handling various kinds of data are described in

detail in several references, such as Martz and Waller

(1991), Raiffa and Schlaifer (1961), and Siu and Kelly

(1998) . 

B.5.3.1 Noninformative Prior Distributions

One class of prior distributions that is widely used is

termed noninformative priors, also referred to as

priors of ignorance, or reference priors (Bernardo and

Smith 1994). These names refer to the situation where

very little a priori information about a parameter is

available in comparison to the information expected to

be provided by the data sample, or there is indifference

about the range of values the parameter could assume.

One might think that this indifference could be expressed by a prior distribution that is uniformly distributed over the interval of interest. Every value in the

interval is equally likely and no knowledge about any

specific value over another value is imposed.

However, uniform distributions do not necessarily best

reflect true noninformativeness (Box and Tiao 1973),

because models can be parameterized in various ways.

For example, if the time to failure, T, is exponentially

distributed, it is common to write the density of T as

ft e t ( ) = − λ λ

or alternatively as

ft e . t ( ) / = 1 −

μ

μ

The two parameters are related by 8 = 1/:.

A uniform distribution cannot be said to automatically

reflect ignorance and be used as a standard

noninformative prior distribution. For the exponential

example here, ignorance of 8 implies ignorance of :,

but 8 and : cannot both have a uniform distribution. In

fact, suppose that 8 has the uniform distribution in
Basics of Statistics

B-13

some finite range, say from a to b. Then : has a

density proportional to 1/:2

 in the range from 1/b to

1/a, as stated in Appendix A.4.7. The distribution of :

is not uniform.

Jeffreys’ rule (Jeffreys 1961) guides the choice of

noninformative prior distributions and provides the

Jeffreys prior distribution (Box and Tiao, 1973). The

Jeffreys prior distribution is commonly used in PRA

and involves using a specific parameterization of the

model (distribution). Jeffreys’ method is to transform

the model into a parameterization that is in terms of a

location parameter, a parameter that slides the distribution sideways without changing its shape. (See Box

and Tiao 1978, Secs. 1.2.3 and 1.3.4). This method

then uses the uniform distribution as the noninformative

prior for the location parameter. It is reasonable to

regard a uniform distribution as noninformative for a

location parameter. The distribution for any other

parameterization is then determined, and is called

noninformative.

In the exponential example, working with log(time), let

2 = log(:), S = log(T), and s = log(t). Using algebraic

formulas given in Section A.4.7 of Appendix A, it can

be shown that the density in this parameterization is

fs s e . s ( ) exp( ) exp( ) = − − − θ θ

Because 2 only appears in the expression s ! 2, a

change in 2 simply slides the distribution sideways

along the s axis. Therefore, 2 is a location parameter.

The Jeffreys noninformative prior is a uniform distribution for 2. This distribution translates to a density for

8 which is proportional to 1/8, and a density for :

which is proportional to 1/:. These are the Jeffreys

noninformative prior distributions for 8 and :.

A further argument for Jeffreys prior distributions is

that the resulting Bayes intervals are numerically equal

to confidence intervals (Lindley 1958), and the confidence intervals are based on the data alone, not on prior

belief. Unfortunately, the above approach cannot be

followed exactly when the data come from a discrete

distribution, such as binomial or Poisson. The original

parameter can only approximately be converted to a

location parameter. The resulting distribution is still

called the Jeffreys prior, however, even though it only

approximates the Jeffreys method.

To avoid the appearance of pulling prior distributions

out of the air, the general formula for the Jeffreys prior

is stated here, as explained by Box and Tiao (1973) and

many others. All the particular cases given in this

handbook can be found by working out the formula in

those cases. Let 2 denote the unknown parameter to be

estimated. Let L(2; x) denote the likelihood corresponding to a single observation. It is a function of 2,

but it also depends on the data, x. For example, x is the

number of Poisson events in a single unit of time, or the

number of failures on a single demand, or the length of

a single duration. Calculate

− . d

d

L x

2

2 θ ln[ ( ; )] θ

Now replace the number x by the random variable X,

and evaluate the expectation of the calculated derivative:

E . d

d − L X

⎛

⎝

⎜ ⎞

⎠

⎟

2

2 θ ln[ ( ; )] θ

The Jeffreys noninformative prior is a function of 2

proportional to the square root of this expectation.

B.5.3.2 Conjugate Prior Distributions

A conjugate prior distribution is a distribution that

results in a posterior distribution that is a member of the

same family of distributions as the prior. Therefore,

conjugate prior distributions are computationally convenient. The methodology for obtaining conjugate

priors is based on sufficient statistics and likelihood

functions (see Martz and Waller, 1991).

The beta family of distributions is the conjugate family

of prior distributions for the probability of failure of a

component in a binomial sampling situation. The

resulting posterior beta distribution can then be used to

provide point and interval estimates of the failure

probability.

A time-to-failure random variable is often assumed to

follow an exponential distribution, with the failure

events arising from a Poisson process. For this model,

with either exponential or Poisson data, the gamma

family of distributions is the conjugate family of prior

distributions for use in Bayesian reliability and failure

rate analyses.

Figure B.4 is a schematic diagram showing the relation

of the kinds of priors that have been mentioned so far.

There are many nonconjugate priors. A relatively small

family of priors is conjugate, typically a single type

such as the gamma distributions or beta distributions.

Finally, the Jeffreys noninformative prior is a single
Basics of Statistics

B-14

All priors

Conjugate

priors

Jeffreys prior

Figure B.4 Schematic diagram of types of priors.

distribution, shown in the diagram by a dot. In many

cases, the Jeffreys prior is also conjugate, as indicated

in the figure.

A popular nonconjugate prior is the lognormal distribution. It can be used as a prior distribution for both the

binomial sampling and Poisson process models above,

although it is not conjugate.

Conjugate prior distributions provide convenience, but

accurate modeling of prior degree of belief should not

be sacrificed for mathematical convenience. However,

when one expression of prior belief is viewed to be as

correct as another, the more convenient expression is

usually selected for use.

B.5.3.3 Other Prior Distribution Approaches

The prior distribution is the distribution of degree of

belief before data that provide new information are

obtained. Usually, the prior probabilities do not have

a direct frequency interpretation and cannot be experimentally confirmed.

When the prior distribution does have a frequency

interpretation, the observed data can be used to estimate

the prior distribution. This situation represents another

class of methods of statistical inference called empirical Bayes methods. The empirical Bayes prior distribution is empirically determined, for example, using

observed plant-specific data for a given set of plants.

Bayes’ Theorem can then be applied to combine this

prior with observed data from a specific plant to produce a posterior distribution. Thus, empirical Bayes

methods are useful when data from similar, but not

identical, sources exist. This situation also gives rise to

the use of so-called hierarchical Bayes methods (see

Gelman, et al., 1995, and Carlin and Louis, 1996).

Attempts have been made to remove some of the

subjectivity present in selecting prior distributions, with

the goal being to obtain one distribution for the same

given information. That is, different analysts using the

same information would decide upon the same prior

distribution. The result has been development of the

method of maximum entropy. If 2 is a parameter

with uncertainty distribution g, the entropy is defined as

− . ∫ g gd ( ) ln[ ( )] θ θθ

The distribution g that maximizes this expression is

called the maximum entropy distribution. For finite

ranges, the p.d.f. with the largest entropy is the uniform,

or flat, distribution. Thus, entropy can be viewed as a

measure of the variability in the height of a p.d.f., and

a maximum entropy prior would be the one with the

required mean that is as flat as possible. Siu and Kelly

(1998, Table 2) give the maximum entropy distributions

for a number of possible constraints.

Maximum entropy methods may see more use in the

future, but still do not produce a systematic approach to

selecting only one prior from a set of possible priors.

In fact, the same problem that the Jeffreys’ method

attempts to address (Section B.5.3.1) is present with the

maximum entropy approach: if a model is parameterized in two different ways, the maximum entropy

priors for the two parameters are inconsistent with each

other.

To address this lack of invariance, constrained

noninformative priors are obtained. They are based

on the maximum entropy approach in conjunction with

Jeffreys’ method. That parameterization is used for

which the parameter is a location parameter. Giving

maximum entropy to this parameter produces a distribution called the constrained noninformative prior

distribution. Atwood (1996) presents constrained noninformative priors and their application to PRA.

Constrained noninformative prior distributions are

seeing use in PRA, although not as much as Jeffreys’

priors.

Other ways of defining noninformative prior distributions exist. See Martz and Waller (1991) and Berger

(1985) for more information. 
Basics of Statistics

B-15

REFERENCES FOR APPENDIX B 

Atwood, C. L., 1996, Constrained Noninformative Priors in Risk Assessment, in Reliability Engineering and System

Safety, vol. 53, pp. 37-46.

Atwood, C. L., D. L. Kelly, F. M. Marshall, D. A. Prawdzik and J. W. Stetkar, 1998, Evaluation of Loss of Offsite Power

Events at Nuclear Power Plants: 1980-1996, NUREG/CR-5496, INEEL/EXT-97-00887, Idaho National Engineering

and Environmental Laboratory, Idaho Falls, ID. 

Berger, J. O., 1985, Statistical Decision Theory and Bayesian Analysis, Second Edition, Springer-Verlag, New York.

Bernardo, J. M., and A. F. M. Smith, 1994, Bayesian Theory, John Wiley & Sons, New York.

Box, G. E. P., 1980, Sampling and Bayes’ Inference in Scientific Modelling and Robustness (with discussion) in Journal

of the Royal Statistical Society, Series A, Vol. 143, Part 4, pp. 383-430.

Box, G. E. P., and G. C. Tiao, 1973, Bayesian Inference in Statistical Analysis. Addison-Wesley, Reading, MA. 

Carlin, B. P., and T. A. Louis, 1996, Bayes and Empirical Bayes Methods for Data Analysis. Chapman & Hall, London.

Conover, W. J., 1999, Practical Nonparametric Statistics, 3rd ed. John Wiley & Sons, NY.

Gelman, A., J. B. Carlin, H. S. Stern, and D. B. Rubin, 1995, Bayesian Data Analysis, Chapman & Hall, London.

Hogg, R. V., and A. T. Craig, 1995, Introduction to Mathematical Statistics, Macmillan, NY.

Jeffreys, H., 1961, Theory of Probability, Third Ed., Clarendon Press, Oxford, England.

Kendall, M. G., and A. Stuart, 1973, The Advanced Theory of Statistics, Vol. 2, Hafner, NY.

Lindley, D. V., 1958, Fiducial Distributions and Bayes’ Theorem, in Journal of the Royal Statistical Society, Series B,

Vol. 20, pp. 102-107. 

Lindley, D. V., 1965, Introduction to Probability and Statistics from a Bayesian Viewpoint, Cambridge University Press,

Cambridge, UK.

Martz, H. F., and R. A. Waller, 1991, Bayesian Reliability Analysis, R. E. Krieger Publishing Co., Malabar, FL. 

NRC, 1994, A Review of NRC Staff uses of Probabilistic Risk Assessment, NUREG-1489, U. S. Nuclear Regulatory

Commission, Washington, D.C.

Raiffa, H., and R Schlaifer, 1961, Applied Statistical Decision Theory, The M.I.T. Press, Cambridge, MA.

SAS, 1988, SAS Procedures Guide, Release 6.03 Edition, SAS Institute, Inc., Cary, NC.

Siu, N., and D. Kelly, 1998, Bayesian Parameter Estimation in Probabilistic Risk Assessment, in Reliability Engineering

and System Safety, Vol. 62, pp. 89-116.

Snedecor, G. W., and W. G. Cochran, 1989, Statistical Methods, Iowa State University Press, Ames, IA.

Stephens, M. A., 1982, Anderson-Darling Test for Goodness of Fit, in Encyclopedia of Statistical Sciences, Vol. 1., pp.

81-85, S. Kotz and N. L. Johnson, eds., John Wiley & Sons, NY.
Basics of Statistics

B-16
C-1

C. STATISTICAL TABLES

Table C.1. Standard normal cumulative distribution function, M. 

Φ(z) z

z M(z) z M(z) z M(z) z M(z)

-4.00

-3.50

-3.00

-2.99

-2.98

-2.97

-2.96

-2.95

-2.94

-2.93

-2.92

-2.91

-2.90

-2.89

-2.88

-2.87

-2.86

-2.85

-2.84

-2.83

-2.82

-2.81

-2.80

-2.79

-2.78

-2.77

-2.76

-2.75

-2.74

-2.73

-2.72

-2.71

-2.70

-2.69

-2.68

-2.67

-2.66

-2.65

-2.64

-2.63

-2.62

-2.61

-2.60

-2.59

-2.58

-2.57

-2.56

-2.55

-2.54

-2.53

3.2E-5

2.3E-4

0.0013

0.0014

0.0014

0.0015

0.0015

0.0016

0.0016

0.0017

0.0018

0.0018

0.0019

0.0019

0.0020

0.0021

0.0021

0.0022

0.0023

0.0023

0.0024

0.0025

0.0026

0.0026

0.0027

0.0028

0.0029

0.0030

0.0031

0.0032

0.0033

0.0034

0.0035

0.0036

0.0037

0.0038

0.0039

0.0040

0.0041

0.0043

0.0044

0.0045

0.0047

0.0048

0.0049

0.0051

0.0052

0.0054

0.0055

0.0057

-2.52

-2.51

-2.50

-2.49

-2.48

-2.47

-2.46

-2.45

-2.44

-2.43

-2.42

-2.41

-2.40

-2.39

-2.38

-2.37

-2.36

-2.35

-2.34

-2.33

-2.32

-2.31

-2.30

-2.29

-2.28

-2.27

-2.26

-2.25

-2.24

-2.23

-2.22

-2.21

-2.20

-2.19

-2.18

-2.17

-2.16

-2.15

-2.14

-2.13

-2.12

-2.11

-2.10

-2.09

-2.08

-2.07

-2.06

-2.05

-2.04

-2.03

0.0059

0.0060

0.0062

0.0064

0.0066

0.0068

0.0069

0.0071

0.0073

0.0075

0.0078

0.0080

0.0082

0.0084

0.0087

0.0089

0.0091

0.0094

0.0096

0.0099

0.0102

0.0104

0.0107

0.0110

0.0113

0.0116

0.0119

0.0122

0.0125

0.0129

0.0132

0.0136

0.0139

0.0143

0.0146

0.0150

0.0154

0.0158

0.0162

0.0166

0.0170

0.0174

0.0179

0.0183

0.0188

0.0192

0.0197

0.0202

0.0207

0.0212

-2.02

-2.01

-2.00

-1.99

-1.98

-1.97

-1.96

-1.95

-1.94

-1.93

-1.92

-1.91

-1.90

-1.89

-1.88

-1.87

-1.86

-1.85

-1.84

-1.83

-1.82

-1.81

-1.80

-1.79

-1.78

-1.77

-1.76

-1.75

-1.74

-1.73

-1.72

-1.71

-1.70

-1.69

-1.68

-1.67

-1.66

-1.65

-1.64

-1.63

-1.62

-1.61

-1.60

-1.59

-1.58

-1.57

-1.56

-1.55

-1.54

-1.53

0.0217

0.0222

0.0228

0.0233

0.0239

0.0244

0.0250

0.0256

0.0262

0.0268

0.0274

0.0281

0.0287

0.0294

0.0301

0.0307

0.0314

0.0322

0.0329

0.0336

0.0344

0.0351

0.0359

0.0367

0.0375

0.0384

0.0392

0.0401

0.0409

0.0418

0.0427

0.0436

0.0446

0.0455

0.0465

0.0475

0.0485

0.0495

0.0505

0.0516

0.0526

0.0537

0.0548

0.0559

0.0571

0.0582

0.0594

0.0606

0.0618

0.0630

-1.52

-1.51

-1.50

-1.49

-1.48

-1.47

-1.46

-1.45

-1.44

-1.43

-1.42

-1.41

-1.40

-1.39

-1.38

-1.37

-1.36

-1.35

-1.34

-1.33

-1.32

-1.31

-1.30

-1.29

-1.28

-1.27

-1.26

-1.25

-1.24

-1.23

-1.22

-1.21

-1.20

-1.19

-1.18

-1.17

-1.16

-1.15

-1.14

-1.13

-1.12

-1.11

-1.10

-1.09

-1.08

-1.07

-1.06

-1.05

-1.04

-1.03

0.0643

0.0655

0.0668

0.0681

0.0694

0.0708

0.0721

0.0735

0.0749

0.0764

0.0778

0.0793

0.0808

0.0823

0.0838

0.0853

0.0869

0.0885

0.0901

0.0918

0.0934

0.0951

0.0968

0.0985

0.1003

0.1020

0.1038

0.1056

0.1075

0.1093

0.1112

0.1131

0.1151

0.1170

0.1190

0.1210

0.1230

0.1251

0.1271

0.1292

0.1314

0.1335

0.1357

0.1379

0.1401

0.1423

0.1446

0.1469

0.1492

0.1515
Statistical Tables

C-2

Table C.1 Standard normal cumulative distribution function, M (continued).

z M(z) z M(z) z M(z) z M(z)

-1.02

-1.01

-1.00

-0.99

-0.98

-0.97

-0.96

-0.95

-0.94

-0.93

-0.92

-0.91

-0.90

-0.89

-0.88

-0.87

-0.86

-0.85

-0.84

-0.83

-0.82

-0.81

-0.80

-0.79

-0.78

-0.77

-0.76

-0.75

-0.74

-0.73

-0.72

-0.71

-0.70

-0.69

-0.68

-0.67

-0.66

-0.65

-0.64

-0.63

-0.62

-0.61

-0.60

-0.59

-0.58

-0.57

-0.56

-0.55

-0.54

-0.53

-0.52

0.1539

0.1562

0.1587

0.1611

0.1635

0.1660

0.1685

0.1711

0.1736

0.1762

0.1788

0.1814

0.1841

0.1867

0.1894

0.1922

0.1949

0.1977

0.2005

0.2033

0.2061

0.2090

0.2119

0.2148

0.2177

0.2206

0.2236

0.2266

0.2296

0.2327

0.2358

0.2389

0.2420

0.2451

0.2483

0.2514

0.2546

0.2578

0.2611

0.2643

0.2676

0.2709

0.2743

0.2776

0.2810

0.2843

0.2877

0.2912

0.2946

0.2981

0.3015

-0.51

-0.50

-0.49

-0.48

-0.47

-0.46

-0.45

-0.44

-0.43

-0.42

-0.41

-0.40

-0.39

-0.38

-0.37

-0.36

-0.35

-0.34

-0.33

-0.32

-0.31

-0.30

-0.29

-0.28

-0.27

-0.26

-0.25

-0.24

-0.23

-0.22

-0.21

-0.20

-0.19

-0.18

-0.17

-0.16

-0.15

-0.14

-0.13

-0.12

-0.11

-0.10

-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.3050

0.3085

0.3121

0.3156

0.3192

0.3228

0.3264

0.3300

0.3336

0.3372

0.3409

0.3446

0.3483

0.3520

0.3557

0.3594

0.3632

0.3669

0.3707

0.3745

0.3783

0.3821

0.3859

0.3897

0.3936

0.3974

0.4013

0.4052

0.4090

0.4129

0.4168

0.4207

0.4247

0.4286

0.4325

0.4364

0.4404

0.4443

0.4483

0.4522

0.4562

0.4602

0.4641

0.4681

0.4721

0.4761

0.4801

0.4840

0.4880

0.4920

0.4960

 0.00

 0.01

 0.02

 0.03

 0.04

 0.05

 0.06

 0.07

 0.08

 0.09

 0.10

 0.11

 0.12

 0.13

 0.14

 0.15

 0.16

 0.17

 0.18

 0.19

 0.20

 0.21

 0.22

 0.23

 0.24

 0.25

 0.26

 0.27

 0.28

 0.29

 0.30

 0.31

 0.32

 0.33

 0.34

 0.35

 0.36

 0.37

 0.38

 0.39

 0.40

 0.41

 0.42

 0.43

 0.44

 0.45

 0.46

 0.47

 0.48

 0.49

 0.50

0.5000

0.5040

0.5080

0.5120

0.5160

0.5199

0.5239

0.5279

0.5319

0.5359

0.5398

0.5438

0.5478

0.5517

0.5557

0.5596

0.5636

0.5675

0.5714

0.5753

0.5793

0.5832

0.5871

0.5910

0.5948

0.5987

0.6026

0.6064

0.6103

0.6141

0.6179

0.6217

0.6255

0.6293

0.6331

0.6368

0.6406

0.6443

0.6480

0.6517

0.6554

0.6591

0.6628

0.6664

0.6700

0.6736

0.6772

0.6808

0.6844

0.6879

0.6915

 0.51

 0.52

 0.53

 0.54

 0.55

 0.56

 0.57

 0.58

 0.59

 0.60

 0.61

 0.62

 0.63

 0.64

 0.65

 0.66

 0.67

 0.68

 0.69

0.70

 0.71

 0.72

 0.73

 0.74

 0.75

 0.76

 0.77

 0.78

 0.79

 0.80

 0.81

 0.82

 0.83

 0.84

 0.85

 0.86

 0.87

 0.88

 0.89

 0.90

 0.91

 0.92

 0.93

 0.94

 0.95

 0.96

 0.97

 0.98

 0.99

 1.00

 1.01

0.6950

0.6985

0.7019

0.7054

0.7088

0.7123

0.7157

0.7190

0.7224

0.7257

0.7291

0.7324

0.7357

0.7389

0.7422

0.7454

0.7486

0.7517

0.7549

0.7580

0.7611

0.7642

0.7673

0.7703

0.7734

0.7764

0.7794

0.7823

0.7852

0.7881

0.7910

0.7939

0.7967

0.7995

0.8023

0.8051

0.8078

0.8106

0.8133

0.8159

0.8186

0.8212

0.8238

0.8264

0.8289

0.8315

0.8340

0.8365

0.8389

0.8413

0.8438
Statistical Tables

C-3

Table C.1 Standard normal cumulative distribution function, M (continued).

z M(z) z M(z) z M(z) z M(z)

 1.02

 1.03

 1.04

 1.05

 1.06

 1.07

 1.08

 1.09

 1.10

 1.11

 1.12

 1.13

 1.14

 1.15

 1.16

 1.17

 1.18

 1.19

 1.20

 1.21

 1.22

 1.23

 1.24

 1.25

 1.26

 1.27

 1.28

 1.29

 1.30

 1.31

 1.32

 1.33

 1.34

 1.35

 1.36

 1.37

 1.38

 1.39

 1.40

 1.41

 1.42

 1.43

 1.44

 1.45

 1.46

 1.47

 1.48

 1.49

1.50

1.51

0.8461

0.8485

0.8508

0.8531

0.8554

0.8577

0.8599

0.8621

0.8643

0.8665

0.8686

0.8708

0.8729

0.8749

0.8770

0.8790

0.8810

0.8830

0.8849

0.8869

0.8888

0.8907

0.8925

0.8944

0.8962

0.8980

0.8997

0.9015

0.9032

0.9049

0.9066

0.9082

0.9099

0.9115

0.9131

0.9147

0.9162

0.9177

0.9192

0.9207

0.9222

0.9236

0.9251

0.9265

0.9279

0.9292

0.9306

0.9319

0.9332

0.9345

 1.52

 1.53

 1.54

 1.55

 1.56

 1.57

 1.58

 1.59

 1.60

 1.61

 1.62

 1.63

 1.64

 1.65

 1.66

 1.67

 1.68

 1.69

 1.70

 1.71

 1.72

 1.73

 1.74

 1.75

 1.76

 1.77

 1.78

 1.79

 1.80

 1.81

 1.82

 1.83

 1.84

 1.85

 1.86

 1.87

 1.88

 1.89

 1.90

 1.91

 1.92

 1.93

 1.94

 1.95

 1.96

 1.97

 1.98

 1.99

2.00

2.01

0.9357

0.9370

0.9382

0.9394

0.9406

0.9418

0.9429

0.9441

0.9452

0.9463

0.9474

0.9484

0.9495

0.9505

0.9515

0.9525

0.9535

0.9545

0.9554

0.9564

0.9573

0.9582

0.9591

0.9599

0.9608

0.9616

0.9625

0.9633

0.9641

0.9649

0.9656

0.9664

0.9671

0.9678

0.9686

0.9693

0.9699

0.9706

0.9713

0.9719

0.9726

0.9732

0.9738

0.9744

0.9750

0.9756

0.9761

0.9767

0.9772

0.9778

 2.02

 2.03

 2.04

 2.05

 2.06

 2.07

 2.08

 2.09

 2.10

 2.11

 2.12

 2.13

 2.14

 2.15

 2.16

 2.17

 2.18

 2.19

 2.20

 2.21

 2.22

 2.23

 2.24

 2.25

 2.26

 2.27

 2.28

 2.29

 2.30

 2.31

 2.32

 2.33

 2.34

 2.35

 2.36

 2.37

 2.38

 2.39

 2.40

 2.41

 2.42

 2.43

 2.44

 2.45

 2.46

 2.47

 2.48

 2.49

2.50

2.51

0.9783

0.9788

0.9793

0.9798

0.9803

0.9808

0.9812

0.9817

0.9821

0.9826

0.9830

0.9834

0.9838

0.9842

0.9846

0.9850

0.9854

0.9857

0.9861

0.9864

0.9868

0.9871

0.9875

0.9878

0.9881

0.9884

0.9887

0.9890

0.9893

0.9896

0.9898

0.9901

0.9904

0.9906

0.9909

0.9911

0.9913

0.9916

0.9918

0.9920

0.9922

0.9925

0.9927

0.9929

0.9931

0.9932

0.9934

0.9936

0.9938

0.9940

2.52

2.53

2.54

2.55

2.56

2.57

2.58

2.59

2.60

2.61

2.62

2.63

2.64

2.65

2.66

2.67

2.68

2.69

2.70

2.71

 2.72

 2.73

 2.74

 2.75

 2.76

 2.77

 2.78

 2.79

 2.80

 2.81

 2.82

 2.83

 2.84

 2.85

 2.86

 2.87

 2.88

 2.89

 2.90

 2.91

 2.92

 2.93

 2.94

 2.95

 2.96

 2.97

 2.98

 2.99

 3.00

 3.50

 4.00

0.9941

0.9943

0.9945

0.9946

0.9948

0.9949

0.9951

0.9952

0.9953

0.9955

0.9956

0.9957

0.9959

0.9960

0.9961

0.9962

0.9963

0.9964

0.9965

0.9966

0.9967

0.9968

0.9969

0.9970

0.9971

0.9972

0.9973

0.9974

0.9974

0.9975

0.9976

0.9977

0.9977

0.9978

0.9979

0.9979

0.9980

0.9981

0.9981

0.9982

0.9982

0.9983

0.9984

0.9984

0.9985

0.9985

0.9986

0.9986

0.9987

0.99977

0.999968
Statistical Tables

C-4

Table C.2. Percentiles of the chi-squared distribution.

χ

2

q

q

Deg. of

freedom P2

0.005 P20.01 P20.025 P20.05 P20.10 P20.20 P20.30 P20.40

0.25

0.5

1

2

3

4

5

6

7

8

9

4.8E-19

8.4E-10

3.93E-5

0.0100

0.0717

0.207

0.412

0.676

0.989

1.344

1.735

1.2E-16

1.35E-8

1.57E-4

0.0201

0.115

0.297

0.554

0.872

1.239

1.646

2.088

1.9E-13

5.27E-7

9.82E-4

0.0506

0.216

0.484

0.831

1.237

1.690

2.180

2.700

4.8E-11

8.44E-6

3.93E-3

0.103

0.352

0.711

1.145

1.635

2.167

2.733

3.325

1.24E-8

1.35E-4

0.0158

0.211

0.584

1.064

1.610

2.204

2.833

3.490

4.168

3.17E-6

2.16E-3

0.0642

0.446

1.005

1.649

2.343

3.070

3.822

4.594

5.380

8.12E-5

0.0110

0.148

0.713

1.424

2.195

3.000

3.828

4.671

5.527

6.393

8.11E-4

0.0350

0.275

1.022

1.869

2.753

3.655

4.570

5.493

6.423

7.357

10

11

12

13

14

15

16

17

18

19

2.156

2.603

3.074

3.565

4.075

4.601

5.142

5.697

6.265

6.844

2.558

3.053

3.571

4.107

4.660

5.229

5.812

6.408

7.015

7.633

3.247

3.816

4.404

5.009

5.629

6.262

6.908

7.564

8.231

8.907

3.940

4.575

5.226

5.892

6.571

7.261

7.962

8.672

9.390

10.12

4.865

5.578

6.304

7.042

7.790

8.547

9.312

10.09

10.86

11.65

6.179

6.989

7.807

8.634

9.467

10.31

11.15

12.00

12.86

13.72

7.267

8.148

9.034

9.926

10.82

11.72

12.62

13.53

14.44

15.35

8.295

9.237

10.18

11.13

12.08

13.03

13.98

14.94

15.89

16.85

20

21

22

23

24

25

26

27

28

29

7.434

8.034

8.643

9.260

9.886

10.52

11.16

11.81

12.46

13.12

8.260

8.897

9.542

10.20

10.86

11.52

12.20

12.88

13.56

14.26

9.591

10.28

10.98

11.69

12.40

13.12

13.84

14.57

15.31

16.05

10.85

11.59

12.34

13.09

13.85

14.61

15.38

16.15

16.93

17.71

12.44

13.24

14.04

14.85

15.66

16.47

17.29

18.11

18.94

19.77

14.58

15.44

16.31

17.19

18.06

18.94

19.82

20.70

21.59

22.48

16.27

17.18

18.10

19.02

19.94

20.87

21.79

22.72

23.65

24.58

17.81

18.77

19.73

20.69

21.65

22.62

23.58

24.54

25.51

26.48

30

35

40

45

50

55

60

70

13.79

17.19

20.71

24.31

27.99

31.73

35.53

43.25

14.95

18.51

22.16

25.90

29.71

33.57

37.48

45.42

16.79

20.57

24.43

28.37

32.36

36.40

40.48

48.75

18.49

22.47

26.51

30.61

34.76

38.96

43.19

51.74

20.60

24.80

29.05

33.35

37.69

42.06

46.46

55.33

23.36

27.84

32.34

36.88

41.45

46.04

50.64

59.90

25.51

30.18

34.87

39.58

44.31

49.06

53.81

63.35

27.44

32.28

37.13

42.00

46.86

51.74

56.62

66.40

80

90

100

125

150

51.14

59.17

67.30

88.01

109.1

53.52

61.74

70.05

91.17

112.7

57.15

65.64

74.22

95.94

118.0

60.39

69.13

77.93

100.2

122.7

64.28

73.29

82.36

105.2

128.3

69.21

78.56

87.95

111.5

135.3

72.92

82.52

92.13

116.3

140.5

76.19

85.60

95.81

120.4

145.0

For large degrees of freedom d, use , where zp χ ( ) is the corresponding percentile of a standard normal p p z d 2 2

≈+ − 212/

distribution.
Statistical Tables

C-5

Table C.2 Percentiles of the chi-squared distribution (continued).

P2

0.50 P20.60 P20.70 P20.80 P20.90 P20.95 P20.975 P20.99 P20.995

Deg. of

freedom

4.84E-3

0.0873

0.455

1.386

2.366

3.357

4.351

5.348

6.346

7.344

8.343

0.0210

0.188

0.708

1.833

2.946

4.045

5.132

6.211

7.283

8.351

9.414

0.0737

0.375

1.074

2.408

3.665

4.878

6.064

7.231

8.383

9.524

10.66

0.229

0.726

1.642

3.219

4.642

5.989

7.289

8.558

9.803

11.03

12.24

0.716

1.501

2.706

4.605

6.251

7.779

9.236

10.64

12.02

13.36

14.68

1.419

2.420

3.841

5.991

7.815

9.488

11.07

12.59

14.07

15.51

16.92

2.269

3.433

5.024

7.378

9.348

11.14

12.83

14.45

16.01

17.53

19.02

3.543

4.868

6.635

9.210

11.34

13.28

15.09

16.81

18.48

20.09

21.67

4.585

6.004

7.879

10.56

12.84

14.86

16.75

18.55

20.28

21.95

23.59

0.25

0.5

1

2

3

4

5

6

7

8

9

9.342

10.34

11.34

12.34

13.34

14.34

15.34

16.34

17.34

18.34

10.47

11.53

12.58

13.64

14.69

15.73

16.78

17.82

18.87

19.91

11.78

12.90

14.01

15.12

16.22

17.32

18.42

19.51

20.60

21.69

13.44

14.63

15.81

16.98

18.15

19.31

20.47

21.61

22.76

23.90

15.99

17.28

18.55

19.81

21.06

22.31

23.54

24.77

25.99

27.20

18.31

19.68

21.03

22.36

23.68

25.00

26.30

27.59

28.87

30.14

20.48

21.92

23.34

24.74

26.12

27.49

28.85

30.19

31.53

32.85

23.21

24.72

26.22

27.69

29.14

30.58

32.00

33.41

34.81

36.19

25.19

26.76

28.30

29.82

31.32

32.80

34.27

35.72

37.16

38.58

10

11

12

13

14

15

16

17

18

19

19.34

20.34

21.34

22.34

23.34

24.34

25.34

26.34

27.34

28.34

20.95

21.99

23.03

24.07

25.11

26.14

27.18

28.21

29.25

30.28

22.77

23.86

24.94

26.02

27.10

28.17

29.25

30.32

31.39

32.46

25.04

26.17

27.30

28.43

29.55

30.68

31.79

32.91

34.03

35.14

28.41

29.62

30.81

32.01

33.20

34.38

35.56

36.74

37.92

39.09

31.41

32.67

33.92

35.17

36.42

37.65

38.89

40.11

41.34

42.56

34.17

35.48

36.78

38.08

39.36

40.65

41.92

43.19

44.46

45.72

37.57

38.93

40.29

41.64

42.98

44.31

45.64

46.96

48.28

49.59

40.00

41.40

42.80

44.18

45.56

46.93

48.29

49.64

50.99

52.34

20

21

22

23

24

25

26

27

28

29

29.34

34.34

39.34

44.34

49.33

54.33

59.33

69.34

31.32

36.47

41.62

46.76

51.89

57.02

62.13

72.36

33.53

38.86

44.16

49.45

54.72

59.98

65.23

75.69

36.25

41.78

47.27

52.73

58.16

63.58

68.97

79.71

40.26

46.06

51.81

57.51

63.17

68.80

74.40

85.52

43.77

49.80

55.76

61.66

67.50

 73.31

79.08

90.53

46.98

53.20

59.34

65.41

71.42

77.38

83.30

95.03

50.89

57.34

63.69

69.96

76.15

 82.29

88.38

100.4

53.67

60.27

66.77

73.17

79.49

85.75

91.95

104.2

30

35

40

45

50

55

60

70

79.34

89.33

99.33

124.3

149.3

82.56

92.76

102.9

128.4

153.8

86.12

96.52

106.9

132.8

158.6

90.40

101.0

111.7

138.1

164.3

96.57

107.6

118.5

145.6

172.6

101.9

113.1

124.3

152.1

179.6

106.6

118.1

129.6

157.8

185.8

112.3

124.1

135.8

164.7

193.2

116.3

128.3

140.2

169.5

198.4

80

90

100

125

150
Statistical Tables

C-6

Table C.3. Percentiles of Student's t distribution. t

q

q

Deg. of

freedom

t0.6 t0.7 t0.8 t0.90 t0.95 t0.975 t0.99 t0.995

 1

 2

 3

 4

 5

 6

 7

 8

 9

 10

0.325

0.289

0.277

0.271

0.267

0.265

0.263

0.262

0.261

0.260

0.727

0.617

0.584

0.569

0.559

0.553

0.549

0.546

0.543

0.542

1.376

1.061

0.978

0.941

0.920

0.906

0.896

0.889

0.883

0.879

3.078

1.886

1.638

1.533

1.476

1.440

1.415

1.397

1.383

1.372

6.314

2.920

2.353

2.132

2.015

1.943

1.895

1.860

1.833

1.812

12.71

4.303

3.182

2.776

2.571

2.447

2.365

2.306

2.262

2.228

31.82

6.965

4.541

3.747

3.365

3.143

2.998

2.896

2.821

2.764

63.66

9.925

5.841

4.604

4.032

3.707

3.499

3.355

3.250

3.169

 11

 12

 13

 14

 15

 16

 17

 18

 19

 20

0.260

0.259

0.259

0.258

0.258

0.258

0.257

0.257

0.257

0.257

0.540

0.539

0.538

0.537

0.536

0.535

0.534

0.534

0.533

0.533

0.876

0.873

0.870

0.868

0.866

0.865

0.863

0.862

0.861

0.860

1.363

1.356

1.350

1.345

1.341

1.337

1.333

1.330

1.328

1.325

1.796

1.782

1.771

1.761

1.753

1.746

1.740

1.734

1.729

1.725

2.201

2.179

2.160

2.145

2.131

2.120

2.110

2.101

2.093

2.086

2.718

2.681

2.650

2.624

2.602

2.583

2.567

2.552

2.539

2.528

3.106

3.055

3.012

2.977

2.947

2.921

2.898

2.878

2.861

2.845

 21

 22

 23

 24

 25

 26

 27

 28

 29

 30

0.257

0.256

0.256

0.256

0.256

0.256

0.256

0.256

0.256

0.256

0.532

0.532

0.532

0.531

0.531

0.531

0.531

0.530

0.530

0.530

0.859

0.858

0.858

0.857

0.856

0.856

0.855

0.855

0.854

0.854

1.323

1.321

1.319

1.318

1.316

1.315

1.314

1.313

1.311

1.310

1.721

1.717

1.714

1.711

1.708

1.706

1.703

1.701

1.699

1.697

2.080

2.074

2.069

2.064

2.060

2.056

2.052

2.048

2.045

2.042

2.518

2.508

2.500

2.492

2.485

2.479

2.473

2.467

2.462

2.457

2.831

2.819

2.807

2.797

2.787

2.779

2.771

2.763

2.756

2.750

 40

 50

 60

 70

 80

 90

100

120

150

4

0.255

0.255

0.254

0.254

0.254

0.254

0.254

0.254

0.254

0.253

0.529

0.528

0.527

0.527

0.526

0.526

0.526

0.526

0.526

0.524

0.851

0.849

0.848

0.847

0.846

0.846

0.845

0.845

0.844

0.842

1.303

1.299

1.296

1.294

1.292

1.291

1.290

1.289

1.287

1.282

1.684

1.676

1.671

1.667

1.664

1.662

1.660

1.658

1.655

1.645

2.021

2.009

2.000

1.994

1.990

1.987

1.984

1.980

1.976

1.960

2.423

2.403

2.390

2.381

2.374

2.368

2.364

2.358

2.351

2.326

2.704

2.678

2.660

2.648

2.639

2.632

2.626

2.617

2.609

2.576

For percentiles below the 50th, use the fact that the t distribution is symmetrical about zero, so t 1!q = !t q .
Statistical Tables

C-7

Tables of the beta(", $) distribution are given on the

next pages. Because of the limited size of the paper,

each table has been split into two pieces and printed on

two facing pages. Each table contains a diagonal line,

shown by blackened cells. The table entries below this

line and to the left are the lower percentiles, such as the

10th or the 5th. The table entries above the diagonal

and to the right are the upper percentiles, such as the

90th or the 95th. In this way, both sets of percentiles

appear in a single table.

Only distributions with " < $ are tabulated. These

distributions have probability concentrated near zero,

and are the distributions usually encountered in PRA

work. For distributions with " > $, use the fact that if

X has a beta(", $) distribution then 1 ! X has a beta($,

") distribution. An example is given as a footnote to

each table.

The size of the page limits the number of parameter

pairs (", $) that can be tabulated. Therefore, interpolation is often necessary, which may give only rough

accuracy. If greater accuracy is required, the user can

find the beta distribution calculated by many commonly

used computer packages and spreadsheets. Similarly,

extrapolation beyond the table may sometimes be

necessary. A footnote to each table gives an approximate extrapolation formula when $ >> ", and an

example of its application. If greater accuracy is

needed, use a commercially available computer

program.
C-8

Table C.4. 90th and 10th percentiles of beta (", $) distribution.

$ for 90th percentiles

0.1 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5

1.00E+0 7.33E-1 3.49E-1 2.16E-1 1.55E-1 1.20E-1 9.84E-2 8.31E-2 7.20E-2 6.34E-2 5

0.00 E+0 9.76E-1 8.10E-1 6.49E-1 5.32E-1 4.48E-1 3.86E-1 3.39E-1 3.02E-1 2.72E-1 2

$for10th%iles

0.1 8.87E-8 0.00 + 9.00E-1 7.85E-1 6.84E-1 6.02E-1 5.36E-1 4.82E-1 4.38E-1 4.01E-1 3

0.5 3.46E-10 2.45E-2 0.0 0 E 8.44E-1 7.59E-1 6.85E-1 6.22E-1 5.68E-1 5.23E-1 4.84E-1 4

1.0 1.00E-10 1.00E-2 1.00E-1 0.00 +0 8.04E-1 7.38E-1 6.80E-1 6.28E-1 5.84E-1 5.45E-1 5

1.5 5.60E-11 6.18E-3 6.78E-2 1.56E-1 0. 0 00 7.75E-1 7.21E-1 6.73E-1 6.30E-1 5.92E-1 5

2.0 3.86E-11 4.46E-3 5.13E-2 1.23E-1 1.96E-1 0.0 0+0 7.53E-1 7.08E-1 6.67E-1 6.30E-1 5

2.5 2.93E-11 3.48E-3 4.13E-2 1.02E-1 1.65E-1 2.25E-1 0.0 0+0 7.36E-1 6.97E-1 6.61E-1 6

3.0 2.37E-11 2.86E-3 3.45E-2 8.64E-2 1.43E-1 1.97E-1 2.47E-1 0.0 0+0 7.21E-1 6.87E-1 6

3.5 1.98E-11 2.42E-3 2.97E-2 7.53E-2 1.26E-1 1.75E-1 2.21E-1 2.64E-1 0.00 E0 7.09E-1 6

4.0 1.71E-11 2.10E-3 2.60E-2 6.66E-2 1.12E-1 1.58E-1 2.01E-1 2.41E-1 2.79E-1 0. 0E0 6

4.5 1.50E-11 1.85E-3 2.31E-2 5.98E-2 1.01E-1 1.43E-1 1.84E-1 2.22E-1 2.58E-1 2.91E-1 0

5. 1.33E-11 1.66E-3 2.09E-2 5.42E-2 9.26E-2 1.32E-1 1.70E-1 2.06E-1 2.40E-1 2.71E-1 36. 1.09E-11 1.37E-3 1.74E-2 4.57E-2 7.88E-2 1.13E-1 1.47E-1 1.79E-1 2.10E-1 2.40E-1 27. 9.26E-12 1.17E-3 1.49E-2 3.95E-2 6.86E-2 9.91E-2 1.29E-1 1.59E-1 1.88E-1 2.15E-1 28. 8.04E-12 1.02E-3 1.31E-2 3.48E-2 6.08E-2 8.82E-2 1.16E-1 1.43E-1 1.69E-1 1.95E-1 29. 7.10E-12 9.02E-4 1.16E-2 3.11E-2 5.45E-2 7.95E-2 1.05E-1 1.30E-1 1.54E-1 1.78E-1 210. 6.36E-12 8.09E-4 1.05E-2 2.81E-2 4.95E-2 7.23E-2 9.57E-2 1.19E-1 1.42E-1 1.64E-1 112.5 5.04E-12 6.44E-4 8.39E-3 2.27E-2 4.01E-2 5.90E-2 7.86E-2 9.82E-2 1.18E-1 1.37E-1 1

15. 4.17E-12 5.35E-4 7.00E-3 1.90E-2 3.37E-2 4.99E-2 6.67E-2 8.37E-2 1.01E-1 1.17E-1 120. 3.11E-12 4.00E-4 5.25E-3 1.43E-2 2.56E-2 3.81E-2 5.12E-2 6.46E-2 7.81E-2 9.16E-2 130. 2.06E-12 2.65E-4 3.51E-3 9.61E-3 1.73E-2 2.59E-2 3.49E-2 4.43E-2 5.39E-2 6.36E-2 750. 1.23E-12 1.59E-4 2.10E-3 5.80E-3 1.05E-2 1.57E-2 2.14E-2 2.73E-2 3.33E-2 3.95E-2 4100. 6.10E-13 7.91E-5 1.05E-3 2.91E-3 5.28E-3 7.96E-3 1.09E-2 1.39E-2 1.70E-2 2.03E-2 2 0.1 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5

" for 10th percentiles

For example, the 10th percentile of a beta(1.5, 10) distribution is 2.81E!2. The 90th percentile of the same distribution (from the table con

is 2.63E!1. If X has a beta(", $) distributions with " > $, use the relation X = 1 ! Y, where Y has a beta($, ") distribution. Thus, for e

percentiles of a beta(10, 1.5) distribution are 1 ! 2.63E!1 = 0.737 and 1 ! 2.81E!2 = 0.9719, respectively.

For a beta(", $) distribution with $ >> ", the p quantile is approximated by . For example, the 10th p

χ α βχ α p p 2 2 ( ) /[ ( )] 22 2 +

distribution, shown above as 5.28E!3, is approximated by = 1.064/201.064 = 5.29E χ χ !3. The 90th percentile

0 10 2 0 10 2 . . ( ) / [ ( )] 4 200 4 +

by 7.770/207.779 = 3.74E!2.
C-9

Table C.4 90th and 10th percentiles of beta (", $) distribution (continued).

$ for 90th percentiles

6. 7. 8. 9. 10. 12.5 15. 20. 30. 50. 1

4.67E-2 3.98E-2 3.46E-2 3.06E-2 2.75E-2 2.18E-2 1.81E-2 1.35E-2 8.97E-3 5.36E-3 2

2.09E-1 1.81E-1 1.60E-1 1.43E-1 1.29E-1 1.05E-1 8.76E-2 6.62E-2 4.45E-2 2.68E-2 1

3.19E-1 2.80E-1 2.50E-1 2.26E-1 2.06E-1 1.68E-1 1.42E-1 1.09E-1 7.39E-2 4.50E-2 23.94E-1 3.51E-1 3.16E-1 2.87E-1 2.63E-1 2.18E-1 1.85E-1 1.43E-1 9.82E-2 6.03E-2 34.53E-1 4.06E-1 3.68E-1 3.37E-1 3.10E-1 2.59E-1 2.22E-1 1.73E-1 1.20E-1 7.41E-2 34.99E-1 4.52E-1 4.12E-1 3.79E-1 3.50E-1 2.95E-1 2.55E-1 2.00E-1 1.40E-1 8.70E-2 45.38E-1 4.90E-1 4.50E-1 4.15E-1 3.86E-1 3.27E-1 2.84E-1 2.24E-1 1.58E-1 9.91E-2 55.71E-1 5.23E-1 4.82E-1 4.47E-1 4.17E-1 3.56E-1 3.10E-1 2.47E-1 1.75E-1 1.11E-1 55.99E-1 5.52E-1 5.11E-1 4.75E-1 4.44E-1 3.82E-1 3.34E-1 2.68E-1 1.91E-1 1.22E-1 66.24E-1 5.77E-1 5.36E-1 5.01E-1 4.69E-1 4.06E-1 3.57E-1 2.87E-1 2.07E-1 1.32E-1 66.46E-1 5.99E-1 5.59E-1 5.23E-1 4.92E-1 4.27E-1 3.78E-1 3.06E-1 2.21E-1 1.43E-1 76.82E-1 6.38E-1 5.98E-1 5.63E-1 5.32E-1 4.66E-1 4.15E-1 3.40E-1 2.49E-1 1.62E-1 8

0.00E+ 6.69E-1 6.31E-1 5.96E-1 5.65E-1 5.00E-1 4.48E-1 3.70E-1 2.74E-1 1.80E-1 9

$for10th%iles

6. 3.18E-1 0.00E+ 6.58E-1 6.25E-1 5.94E-1 5.29E-1 4.77E-1 3.97E-1 2.98E-1 1.98E-1 1

7. 2.88E-1 3.31E-1 0.00E+ 6.50E-1 6.20E-1 5.56E-1 5.03E-1 4.22E-1 3.19E-1 2.14E-1 1

8. 2.64E-1 3.05E-1 3.42E-1 0.00E+ 6.42E-1 5.79E-1 5.26E-1 4.45E-1 3.40E-1 2.30E-1 1

9. 2.43E-1 2.82E-1 3.18E-1 3.50E-1 0.00E+ 6.27E-1 5.76E-1 4.95E-1 3.85E-1 2.66E-1 1

10. 2.26E-1 2.63E-1 2.97E-1 3.29E-1 3.58E-1 0.00E+ 6.16E-1 5.36E-1 4.25E-1 2.99E-1 1

12.5 1.91E-1 2.25E-1 2.56E-1 2.85E-1 3.12E-1 3.73E-1 0.00E+ 6.01E-1 4.89E-1 3.56E-1 2

15. 1.66E-1 1.96E-1 2.25E-1 2.52E-1 2.77E-1 3.34E-1 3.84E-1 0.00E+ 5.82E-1 4.45E-1 2

20. 1.31E-1 1.57E-1 1.81E-1 2.04E-1 2.26E-1 2.78E-1 3.23E-1 3.99E-1 0.00E+ 5.64E-1 3

30. 9.26E-2 1.12E-1 1.30E-1 1.48E-1 1.66E-1 2.07E-1 2.45E-1 3.12E-1 4.18E-1 0.00E+ 5

50. 5.83E-2 7.09E-2 8.35E-2 9.59E-2 1.08E-1 1.38E-1 1.66E-1 2.18E-1 3.06E-1 4.36E-1 0

100. 3.03E-2 3.71E-2 4.40E-2 5.09E-2 5.78E-2 7.50E-2 9.19E-2 1.25E-1 1.84E-1 2.85E-1 4

6. 7. 8. 9. 10. 12.5 15. 20. 30. 50. 1

" for 10th percentiles
C-10

Table C.5. 95th and 5th percentiles of beta (", $) distribution.

$ for 95th percentiles

0.1 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5

1.00E+0 9.24E-1 5.99E-1 4.09E-1 3.06E-1 2.44E-1 2.02E-1 1.72E-1 1.50E-1 1.33E-1 1

0.00 E+0 9.94E-1 9.02E-1 7.71E-1 6.58E-1 5.69E-1 4.99E-1 4.44E-1 3.99E-1 3.62E-1 3

$for5th%iles

0.1 8.66E-11 0.00E 0 9.50E-1 8.64E-1 7.76E-1 6.98E-1 6.32E-1 5.75E-1 5.27E-1 4.86E-1 4

0.5 3.38E-13 6.16E-3 0.00 +0 9.03E-1 8.32E-1 7.64E-1 7.04E-1 6.51E-1 6.04E-1 5.63E-1 5

1.0 9.77E-14 2.50E-3 5.00E-2 0.00 +0 8.65E-1 8.06E-1 7.51E-1 7.02E-1 6.57E-1 6.18E-1 5

1.5 5.46E-14 1.54E-3 3.36E-2 9.73E-2 0.0 E+0 8.35E-1 7.85E-1 7.39E-1 6.97E-1 6.59E-1 6

2.0 3.77E-14 1.11E-3 2.53E-2 7.60E-2 1.35E-1 0.00 +0 8.11E-1 7.68E-1 7.29E-1 6.92E-1 6

2.5 2.87E-14 8.68E-4 2.03E-2 6.24E-2 1.13E-1 1.65E-1 0.0 E+0 7.91E-1 7.54E-1 7.19E-1 6

3.0 2.31E-14 7.12E-4 1.70E-2 5.30E-2 9.76E-2 1.44E-1 1.89E-1 0.00 +0 7.75E-1 7.42E-1 7

3.5 1.94E-14 6.03E-4 1.45E-2 4.60E-2 8.57E-2 1.28E-1 1.69E-1 2.09E-1 0.00 +0 7.61E-1 7

4.0 1.67E-14 5.23E-4 1.27E-2 4.07E-2 7.64E-2 1.15E-1 1.53E-1 1.90E-1 2.25E-1 0.00 +0 7

4.5 1.46E-14 4.62E-4 1.13E-2 3.64E-2 6.90E-2 1.04E-1 1.40E-1 1.75E-1 2.08E-1 2.39E-1 0

5. 1.30E-14 4.13E-4 1.02E-2 3.30E-2 6.28E-2 9.55E-2 1.29E-1 1.61E-1 1.93E-1 2.23E-1 2

6. 1.07E-14 3.42E-4 8.51E-3 2.78E-2 5.34E-2 8.18E-2 1.11E-1 1.40E-1 1.69E-1 1.96E-1 2

7. 9.05E-15 2.91E-4 7.30E-3 2.40E-2 4.64E-2 7.15E-2 9.77E-2 1.24E-1 1.50E-1 1.75E-1 2

8. 7.85E-15 2.54E-4 6.39E-3 2.11E-2 4.10E-2 6.36E-2 8.73E-2 1.11E-1 1.35E-1 1.58E-1 1

9. 6.93E-15 2.25E-4 5.68E-3 1.89E-2 3.68E-2 5.72E-2 7.88E-2 1.01E-1 1.23E-1 1.45E-1 1

10. 6.21E-15 2.02E-4 5.12E-3 1.70E-2 3.33E-2 5.20E-2 7.19E-2 9.22E-2 1.13E-1 1.33E-1 1

12.5 4.92E-15 1.60E-4 4.10E-3 1.37E-2 2.70E-2 4.24E-2 5.89E-2 7.60E-2 9.33E-2 1.11E-1 1

15. 4.08E-15 1.33E-4 3.41E-3 1.15E-2 2.27E-2 3.57E-2 4.99E-2 6.47E-2 7.97E-2 9.48E-2 1

20. 3.03E-15 9.95E-5 2.56E-3 8.65E-3 1.72E-2 2.72E-2 3.82E-2 4.98E-2 6.17E-2 7.37E-2 8

30. 2.01E-15 6.61E-5 1.71E-3 5.80E-3 1.16E-2 1.85E-2 2.60E-2 3.41E-2 4.25E-2 5.11E-2 5

50. 1.20E-15 3.95E-5 1.03E-3 3.49E-3 7.01E-3 1.12E-2 1.59E-2 2.09E-2 2.62E-2 3.16E-2 3

100. 5.96E-16 1.97E-5 5.13E-4 1.75E-3 3.53E-3 5.67E-3 8.06E-3 1.06E-2 1.34E-2 1.62E-2 1

 0.1 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5

" for 5th percentiles

For example, the 5th percentile of a beta(1.5, 10) distribution is 1.70E!2. The 95th percentile of the same distribution (from the table con

is 3.17E!1. If X has a beta(", $) distributions with " > $, use the relation X = 1 ! Y, where Y has a beta($, ") distribution. Thus, for 

percentiles of a beta(10, 1.5) distribution are 1 ! 3.17E!1 = 0.683 and 1 ! 1.70E!2 = 0.9830, respectively.

For a beta(", $) distribution with $ >> ", the p quantile is approximated by . For example, th χ α βχ α e 5th percentile o

p p 2 2 ( ) / [ ( )] 22 2 +

shown here as 3.53E!3, is approximated by = 0.711/200.711 = 3.54E χ χ 0 05 !3. The 95th percentile, 4.61E!2, is appr

2 0 05 2 4 200 4 . . ( ) / [ ( )] +

= 4.53E!2.
C-11

Table C.5 95th and 5th percentiles of beta (", $) distribution (continued).

$ for 95th percentiles

6. 7. 8. 9. 10. 12.5 15. 20. 30. 50. 1

9.91E-2 8.47E-2 7.39E-2 6.56E-2 5.89E-2 4.70E-2 3.91E-2 2.92E-2 1.94E-2 1.16E-2 5

2.83E-1 2.47E-1 2.19E-1 1.97E-1 1.79E-1 1.45E-1 1.22E-1 9.27E-2 6.25E-2 3.79E-2 1

3.93E-1 3.48E-1 3.12E-1 2.83E-1 2.59E-1 2.13E-1 1.81E-1 1.39E-1 9.50E-2 5.82E-2 24.66E-1 4.17E-1 3.78E-1 3.45E-1 3.17E-1 2.64E-1 2.26E-1 1.76E-1 1.21E-1 7.48E-2 35.21E-1 4.71E-1 4.29E-1 3.94E-1 3.64E-1 3.06E-1 2.64E-1 2.07E-1 1.44E-1 8.97E-2 45.64E-1 5.14E-1 4.71E-1 4.35E-1 4.04E-1 3.42E-1 2.97E-1 2.34E-1 1.65E-1 1.03E-1 56.00E-1 5.50E-1 5.07E-1 4.70E-1 4.38E-1 3.74E-1 3.26E-1 2.59E-1 1.84E-1 1.16E-1 66.30E-1 5.80E-1 5.38E-1 5.01E-1 4.68E-1 4.02E-1 3.53E-1 2.82E-1 2.02E-1 1.28E-1 66.55E-1 6.07E-1 5.64E-1 5.27E-1 4.95E-1 4.28E-1 3.77E-1 3.04E-1 2.19E-1 1.40E-1 76.77E-1 6.30E-1 5.88E-1 5.51E-1 5.18E-1 4.51E-1 3.99E-1 3.23E-1 2.34E-1 1.51E-1 76.96E-1 6.50E-1 6.09E-1 5.73E-1 5.40E-1 4.72E-1 4.19E-1 3.42E-1 2.49E-1 1.62E-1 87.29E-1 6.85E-1 6.45E-1 6.10E-1 5.77E-1 5.10E-1 4.56E-1 3.75E-1 2.77E-1 1.82E-1 9

0.00E+ 7.13E-1 6.75E-1 6.40E-1 6.09E-1 5.42E-1 4.87E-1 4.05E-1 3.03E-1 2.01E-1 1

$for5th%iles

6. 2.71E-1 0.00+0 7.00E-1 6.67E-1 6.36E-1 5.70E-1 5.15E-1 4.32E-1 3.26E-1 2.18E-1 1

7. 2.45E-1 2.87E-1 0.00E0 6.89E-1 6.59E-1 5.94E-1 5.40E-1 4.57E-1 3.48E-1 2.35E-1 1

8. 2.24E-1 2.64E-1 3.00E-1 0.00E0 6.80E-1 6.16E-1 5.63E-1 4.79E-1 3.68E-1 2.51E-1 1

9. 2.06E-1 2.44E-1 2.79E-1 3.11E-1 0.00E0 6.62E-1 6.10E-1 5.27E-1 4.13E-1 2.88E-1 1

10. 1.91E-1 2.27E-1 2.60E-1 2.91E-1 3.20E-1 0.00E0 6.48E-1 5.67E-1 4.52E-1 3.21E-1 1

12.5 1.61E-1 1.93E-1 2.23E-1 2.52E-1 2.78E-1 3.38E-1 0.00E0 6.29E-1 5.15E-1 3.77E-1 2

15. 1.40E-1 1.68E-1 1.96E-1 2.22E-1 2.46E-1 3.03E-1 3.52E-1 0.00E0 6.05E-1 4.65E-1 2

20. 1.10E-1 1.34E-1 1.57E-1 1.79E-1 2.00E-1 2.50E-1 2.95E-1 3.71E-1 0.00E0 5.82E-1 3

30. 7.74E-2 9.50E-2 1.12E-1 1.29E-1 1.46E-1 1.86E-1 2.23E-1 2.89E-1 3.95E-1 0.00E0 5

50. 4.86E-2 6.02E-2 7.18E-2 8.34E-2 9.49E-2 1.23E-1 1.50E-1 2.01E-1 2.88E-1 4.18E-1 0

100. 2.52E-2 3.14E-2 3.77E-2 4.42E-2 5.06E-2 6.68E-2 8.29E-2 1.14E-1 1.73E-1 2.71E-1 4

6. 7. 8. 9. 10. 12.5 15. 20. 30. 50. 1

" for 5th percentiles
C-12

Table C.6. 97.5th and 2.5th percentiles of beta (", $) distribution.

$ for 97.5th percentiles

0.1 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5

1.00E+0 9.80E-1 7.76E-1 5.84E-1 4.58E-1 3.74E-1 3.16E-1 2.72E-1 2.40E-1 2.14E-1 1

0.00 E+0 9.98E-1 9.51E-1 8.53E-1 7.53E-1 6.67E-1 5.95E-1 5.36E-1 4.86E-1 4.45E-1 4

$for2.5th%iles

0.1 8.46E-14 0.00 +0 9.75E-1 9.15E-1 8.42E-1 7.71E-1 7.08E-1 6.51E-1 6.02E-1 5.59E-1 5

0.5 3.30E-16 1.54E-3 0.00 +0 9.39E-1 8.82E-1 8.23E-1 7.67E-1 7.16E-1 6.70E-1 6.29E-1 5

1.0 9.54E-17 6.25E-4 2.50E-2 0.00 +0 9.06E-1 8.55E-1 8.06E-1 7.59E-1 7.16E-1 6.77E-1 6

1.5 5.34E-17 3.86E-4 1.67E-2 6.08E-2 0.00 +0 8.77E-1 8.33E-1 7.91E-1 7.51E-1 7.14E-1 6

2.0 3.68E-17 2.78E-4 1.26E-2 4.73E-2 9.43E-2 0.00 +0 8.53E-1 8.14E-1 7.77E-1 7.42E-1 7

2.5 2.80E-17 2.17E-4 1.01E-2 3.87E-2 7.87E-2 1.23E-1 0.00 +0 8.33E-1 7.98E-1 7.65E-1 7

3.0 2.26E-17 1.78E-4 8.40E-3 3.28E-2 6.76E-2 1.07E-1 1.47E-1 0.00 +0 8.16E-1 7.85E-1 7

3.5 1.89E-17 1.51E-4 7.21E-3 2.85E-2 5.92E-2 9.44E-2 1.31E-1 1.67E-1 0.00 +0 8.01E-1 7

4.0 1.63E-17 1.31E-4 6.31E-3 2.51E-2 5.27E-2 8.47E-2 1.18E-1 1.52E-1 1.84E-1 0.00 +0 7

4.5 1.43E-17 1.15E-4 5.61E-3 2.25E-2 4.75E-2 7.68E-2 1.08E-1 1.39E-1 1.69E-1 1.99E-1 0

5. 1.27E-17 1.03E-4 5.05E-3 2.04E-2 4.33E-2 7.02E-2 9.90E-2 1.28E-1 1.57E-1 1.85E-1 26. 1.04E-17 8.53E-5 4.21E-3 1.71E-2 3.67E-2 6.00E-2 8.52E-2 1.11E-1 1.37E-1 1.62E-1 17. 8.83E-18 7.27E-5 3.61E-3 1.48E-2 3.19E-2 5.24E-2 7.49E-2 9.81E-2 1.22E-1 1.45E-1 18. 7.67E-18 6.33E-5 3.16E-3 1.30E-2 2.81E-2 4.65E-2 6.67E-2 8.78E-2 1.09E-1 1.31E-1 19. 6.77E-18 5.61E-5 2.81E-3 1.16E-2 2.52E-2 4.18E-2 6.02E-2 7.95E-2 9.92E-2 1.19E-1 110. 6.06E-18 5.03E-5 2.53E-3 1.05E-2 2.28E-2 3.80E-2 5.49E-2 7.27E-2 9.09E-2 1.09E-1 112.5 4.81E-18 4.01E-5 2.02E-3 8.43E-3 1.85E-2 3.09E-2 4.49E-2 5.98E-2 7.52E-2 9.08E-2 1

15. 3.98E-18 3.33E-5 1.69E-3 7.05E-3 1.55E-2 2.61E-2 3.80E-2 5.08E-2 6.41E-2 7.77E-2 920. 2.96E-18a 2.49E-5 1.27E-3 5.31E-3 1.17E-2 1.98E-2 2.91E-2 3.90E-2 4.95E-2 6.03E-2 7

30. 1.96E-18 1.65E-5 8.44E-4 3.56E-3 7.91E-3 1.34E-2 1.98E-2 2.67E-2 3.40E-2 4.17E-2 450. 1.17E-18 9.87E-6 5.06E-4 2.14E-3 4.78E-3 8.16E-3 1.21E-2 1.64E-2 2.09E-2 2.58E-2 3100. 5.82E-19a 4.92E-6 2.53E-4 1.08E-3 2.41E-3 4.12E-3 6.11E-3 8.31E-3 1.07E-2 1.32E-2 1 0.1 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5

" for 2.5th percentiles

a. May be inaccurate. Calculation had not converged after 100 iterations.

For example, the 2.5th percentile of a beta(1.5, 10) distribution is 1.05E!2. The 97.5th percentile of the same distribution (from the tab

page) is 3.67E!1. If X has a beta(", $) distributions with " > $, use the relation X = 1 ! Y, where Y has a beta($, ") distribution. Thus, 

97.5th percentiles of a beta(10, 1.5) distribution are 1 ! 3.67E!1 = 0.633 and 1 ! 1.05E!2 = 0.9895, respectively.

For a beta(", $) distribution with $ >> ", the p quantile is approximated by . For example, th χ α βχ α e 2.5th percentile o

p p 2 2 ( ) /[ ( )] 22 2 +

shown here as 2.41E!3, is approximated by = 0.484/200.484 = 2.41E χ χ 0 025 !3. The 97.5th percentile, 5.3

2 0 025 2 . . ( ) /[ ( )] 4 200 4 +

11.14/211.14 = 5.28E!2.
C-13

Table C.6. 97.5th and 2.5th percentiles of beta (", $) distribution (continued).

$ for 97.5th percentiles

6. 7. 8. 9. 10. 12.5 15. 20. 30. 50. 1

1.61E-1 1.38E-1 1.21E-1 1.08E-1 9.73E-2 7.79E-2 6.50E-2 4.88E-2 3.25E-2 1.95E-2 9

3.53E-1 3.10E-1 2.77E-1 2.49E-1 2.27E-1 1.85E-1 1.57E-1 1.19E-1 8.10E-2 4.92E-2 2

4.59E-1 4.10E-1 3.69E-1 3.36E-1 3.08E-1 2.56E-1 2.18E-1 1.68E-1 1.16E-1 7.11E-2 35.28E-1 4.76E-1 4.33E-1 3.97E-1 3.67E-1 3.07E-1 2.64E-1 2.06E-1 1.43E-1 8.88E-2 45.79E-1 5.27E-1 4.82E-1 4.45E-1 4.13E-1 3.49E-1 3.02E-1 2.38E-1 1.67E-1 1.04E-1 56.19E-1 5.67E-1 5.23E-1 4.84E-1 4.51E-1 3.85E-1 3.35E-1 2.66E-1 1.88E-1 1.19E-1 66.51E-1 6.00E-1 5.56E-1 5.18E-1 4.84E-1 4.16E-1 3.64E-1 2.92E-1 2.08E-1 1.32E-1 66.78E-1 6.28E-1 5.85E-1 5.47E-1 5.13E-1 4.44E-1 3.91E-1 3.15E-1 2.26E-1 1.45E-1 77.01E-1 6.52E-1 6.10E-1 5.72E-1 5.38E-1 4.68E-1 4.14E-1 3.36E-1 2.43E-1 1.57E-1 87.20E-1 6.74E-1 6.32E-1 5.94E-1 5.61E-1 4.91E-1 4.36E-1 3.56E-1 2.59E-1 1.68E-1 87.38E-1 6.92E-1 6.51E-1 6.14E-1 5.81E-1 5.11E-1 4.56E-1 3.74E-1 2.75E-1 1.79E-1 97.66E-1 7.23E-1 6.84E-1 6.49E-1 6.16E-1 5.47E-1 4.91E-1 4.07E-1 3.03E-1 2.00E-1 1

0.00E0 7.49E-1 7.11E-1 6.77E-1 6.46E-1 5.78E-1 5.22E-1 4.36E-1 3.28E-1 2.19E-1 1

$for2.5th%iles

6. 2.34E-1 0.00E+ 7.34E-1 7.01E-1 6.71E-1 6.04E-1 5.49E-1 4.63E-1 3.52E-1 2.37E-1 1

7. 2.11E-1 2.51E-1 0.00E+ 7.22E-1 6.92E-1 6.27E-1 5.73E-1 4.87E-1 3.73E-1 2.54E-1 1

8. 1.92E-1 2.30E-1 2.66E-1 0.00E+ 7.11E-1 6.48E-1 5.94E-1 5.08E-1 3.93E-1 2.70E-1 1

9. 1.77E-1 2.13E-1 2.47E-1 2.78E-1 0.00E+ 6.90E-1 6.39E-1 5.55E-1 4.38E-1 3.07E-1 1

10. 1.63E-1 1.98E-1 2.30E-1 2.60E-1 2.89E-1 0.00E+ 6.75E-1 5.93E-1 4.76E-1 3.40E-1 1

12.5 1.38E-1 1.68E-1 1.97E-1 2.24E-1 2.50E-1 3.10E-1 0.00E+ 6.52E-1 5.38E-1 3.96E-1 2

15. 1.19E-1 1.46E-1 1.72E-1 1.97E-1 2.21E-1 2.76E-1 3.25E-1 0.00E+ 6.25E-1 4.83E-1 3

20. 9.36E-2 1.16E-1 1.38E-1 1.59E-1 1.79E-1 2.28E-1 2.72E-1 3.48E-1 0.00E+ 5.97E-1 4

30. 6.56E-2 8.19E-2 9.83E-2 1.14E-1 1.30E-1 1.69E-1 2.05E-1 2.70E-1 3.75E-1 0.00E+ 5

50. 4.11E-2 5.18E-2 6.26E-2 7.35E-2 8.44E-2 1.11E-1 1.38E-1 1.87E-1 2.73E-1 4.03E-1 0

100. 2.13E-2 2.70E-2 3.28E-2 3.88E-2 4.49E-2 6.02E-2 7.56E-2 1.06E-1 1.63E-1 2.60E-1 4

6. 7. 8. 9. 10. 12.5 15. 20. 30. 50. 1

" for 2.5th percentiles
Statistical Tables

C-14

Table C.7. Acceptance limits for the Kolmogorov test of goodness of fit.

Sample size

(n)

Significance level

0.20 0.15 0.10 0.05 0.01

 1 0.900 0.925 0.950 0.975 0.995

 2 0.684 0.726 0.776 0.842 0.929

 3 0.565 0.596 0.636 0.708 0.829

 4 0.493 0.525 0.565 0.624 0.734

 5 0.447 0.474 0.509 0.563 0.669

 6 0.410 0.435 0.468 0.519 0.617

 7 0.381 0.405 0.436 0.483 0.576

 8 0.358 0.381 0.410 0.454 0.542

 9 0.339 0.360 0.387 0.430 0.513

10 0.323 0.343 0.369 0.409 0.489

11 0.308 0.327 0.352 0.391 0.468

12 0.296 0.314 0.338 0.375 0.449

13 0.285 0.302 0.325 0.361 0.432

14 0.275 0.292 0.314 0.349 0.418

15 0.266 0.282 0.304 0.338 0.404

16 0.258 0.274 0.295 0.327 0.392

17 0.250 0.266 0.286 0.318 0.381

18 0.244 0.259 0.279 0.309 0.371

19 0.237 0.252 0.271 0.301 0.361

20 0.232 0.246 0.265 0.294 0.352

25 0.208 0.221 0.238 0.264 0.317

30 0.190 0.202 0.218 0.242 0.290

35 0.177 0.187 0.202 0.224 0.269

40 0.165 0.176 0.189 0.210 0.252

50 0.148 0.158 0.170 0.188 0.226

60 0.136 0.144 0.155 0.172 0.207

70 0.126 0.134 0.144 0.160 0.192

80 0.118 0.125 0.135 0.150 0.179

Large n 1.07/ 1.14/ n n 1.22/ n 1.36/ n 1.63/ n

Reject the hypothesized distribution F(x) if D = max|Fn(x) − F(x)| exceeds the tabulated value.

The asymptotic formula gives values that are slightly too high ⎯ by 1% to 2% for n = 80.
Statistical Tables

C-15

Table C.8. Parameters of constrained noninformative prior for binomial p.

p0 b "

0.50

0.40

0.30

0.20

0.15

0.10

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.005

0.001

0

 0.

!0.8166

!1.748

!3.031

!4.027

!5.743

!6.295

!6.958

!7.821

!8.978

!10.61

!13.08

!17.22

!25.53

!50.52

!100.5

!500.5

!4

0.5000

0.4168

0.3590

0.3243

0.3211

0.3424

0.3522

0.3648

0.3802

0.3980

0.4171

0.4358

0.4531

0.4693

0.4848

0.4925

0.4985

0.5000

The table gives parameters of the constrained noninformative prior for a binomial parameter p, when the assumed

prior mean is p0. The exact constrained noninformative prior has the form

fprior(p) % ebpp!1/2(1 ! p)

!1/2 ,

with b tabulated above.

The tabulated value " is the first parameter of a beta distribution that has the same mean and variance as the constrained

noninformative prior. The second parameter of that beta distribution is obtained by solving "/(" + $) = p0. This results

in the formula $ = "(1 ! p0)/p0. Then a beta(", $) distribution with these parameters approximates the exact constrained

noninformative prior.

If p0 > 0.50, the exact constrained noninformative prior has positive b. For example, if p0 = 0.70, look in the table at the

row for 1 ! 0.70, and see that b there is !1.748. Therefore, the exact constrained noninformative prior is

fprior(p) % ebpp!1/2(1 ! p)

!1/2 ,

with b = +1.748.

If p0 > 0.50, the beta approximation is obtained by interchanging the roles of p0 and 1 ! p0 and of " and $ in the above

formulas. For example, if p0 were 0.30, the beta approximation would have " = 0.3590 and $ = 0.3590×0.70/0.30 =

0.8377. If instead p0 = 0.70, the beta approximation has " = 0.8377 and $ = 0.3590.
Statistical Tables

C-16
D-1

D. GLOSSARY

Arithmetic mean. See mean.

Bathtub curve. A plot of the failure rate as a function

of system age, showing a high failure rate when the

system is new (the burn-in period), then a low, roughly

constant failure rate, followed by a rising failure rate as

the system becomes old (the wear-out period). See

burn-in failure.

Bayesian inference. Statistical inference involving the

use of Bayesian methods. Bayesian inference uses

probability distributions to model uncertainty in unknown quantities. Thus, unknown parameters are

treated formally as if they were random variables. See

also frequentist inference and statistical inference.

Bias. The difference between the expected value of an

estimator and the true quantity being estimated. For

example, if Y is a function of the data that estimates a

parameter 2, the bias of Y is E(Y) ! 2.

Bin. A group of values of a continuous variable, used

to partition the data into subsets. For example, event

dates can be grouped so that each year is one bin, and

all the events during a single year form a subset of the

data.

Burn-in failure. Failures associated with the early

time-frame of a component’s life cycle, during which

the failure rate often starts from a maximum value and

decreases rapidly. The high failure rate early in the

component’s life cycle can be caused by poor quality

control practices and a natural wear-in or debugging

period. See bathtub curve.

c.d.f. See cumulative distribution function.

Cell. When the data are expressed in a table of counts,

a cell is the smallest element of the table. Each cell has

an observed count and, under some null hypothesis, an

expected count. Each cell can be analyzed on its own,

and then compared to the other cells to see if the data

show trends, patterns, or other forms of nonhomogeneity. In a 1 × J table, as with events in time, each cell

corresponds to one subset of the data. In a 2 × J table,

as with failures on demand, each data subset corresponds to two cells, one cell for failures and one for

successes.

Central moment. See moment.

Coefficient of variation. See relative standard

deviation.

Common-cause failure. A single event that causes

failure of two or more components at the same time

(also referred to as common-mode failure).

Confidence interval. In the frequentist approach, a

100p% confidence interval has a probability p of

containing the true unknown parameter. This is a

property of the procedure, not of any one particular

interval. Any one interval either does or does not

contain the true parameter. However, any random data

set leads to a confidence interval, and 100p% of these

contain the true parameter. Compare with credible

interval.

Conjugate. A family of prior distributions is conjugate, for data from a specified distribution, if a prior

distribution in the family results in the posterior distribution also being in the family. A prior distribution in

the conjugate family is called a conjugate prior. For

example, the gamma distributions are conjugate for

Poisson data, and the beta distributions are conjugate

for binomial data.

Credible interval. In the Bayesian approach, a 100p%

credible interval contains 100p% of the Bayesian

probability distribution. For example, if 8 has been

estimated by a posterior distribution, the 5th and 95th

percentiles of this distribution contain 90% of the

probability, so they form a (posterior) 90% credible

interval. It is not required to have equal probability in

the two tails (5% in this example), although it is very

common. For example, the interval bounded by 0 and

the 90th percentile would also be a 90% credible

interval, a one-sided interval. Bayes credible intervals

have the same intuitive purpose as frequentist confidence intervals, but their definitions and interpretations

are different.

Cumulative distribution function (c.d.f.). This

function gives the probability that the random variable

does not exceed a given value x. For a random variable

X, the c.d.f. F(x) = Pr(X # x). If X is discrete, such as a

count of events, the c.d.f. is a step function, with a jump

at each possible value of X. If X is continuous, such as

a duration time, the c.d.f. is continuous. See also

probability density function. Do not confuse the

statistics acronym c.d.f. with the PRA acronym CDF,

denoting core damage frequency!
Glossary

D-2

Density. See probability density function.

Disjoint. See mutually exclusive.

Duration. The time until something of interest happens. The thing of interest may be failure to run,

recovery from a failure, restoration of offsite power,

etc.

Error factor. A representation of one of the parameters of the lognormal distribution, defined as the 95th

percentile divided by the median. The error factor is a

measure of the spread of the distribution, and is denoted

by EF.

Estimate, estimator. In the frequentist approach, an

estimator is a function of random data, and an estimate

is the particular value taken by the estimator for a

particular data set. That is, the term estimator is used

for the random variable, and estimate is used for a

number. The usual convention of using upper-case

letters for random variables and lower-case letters for

numbers is often ignored in this setting, so the context

must be used to show whether a random variable or a

number is being discussed.

Event rate. See failure rate for repairable systems,

and replace the word “failure” by “event.”

Expected value. If X is discrete with p.d.f. f, the

expected value of X, denoted E(X), is Exi

f(xi). If

instead X is continuously distributed with density f, the

expected value is Ixf(x)dx. The expected value of X is

also called the mean of X. It is a measure of the center

of the distribution of X.

Exposure time. The length of time during which the

events of interest can possibly occur. The units must be

specified, such as reactor-critical-years, site-calendarhours, or system-operating-hours. Also called time at

risk.

Failure on demand. Failure when a standby system is

demanded, even though the system was apparently

ready to function just before the demand. It is modeled

as a random event, having some probability, but unpredictable on any one specific demand. Compare standby failure.

Failure rate. For a repairable system, the failure rate,

8, is such that 8)t is approximately the expected

number of failures in a short time period from t to t +

)t. If simultaneous failures do not occur, 8)t is also

approximately the probability that a failure will occur

in the period from t to t + )t. In this setting, 8 is also

called a failure frequency. For a nonrepairable

system, 8)t is approximately the probability that an

unfailed system at time t will fail in the time period

from t to t + )t. In this setting, 8 is also called the

hazard rate.

Fractile. See quantile.

Frequency. For a repairable system, frequency and

rate are two words with the same meaning, and are

used interchangeably. If simultaneous events do not

occur, the frequency 8(t) satisfies 8(t))t . Pr (an event

occurs between t and t + )t), for small )t.

Frequentist inference. Statistical inference that

interprets the probability of an event as the long-term

relative frequency of occurrence of the event, in many

repetitions of an experiment when the event may or may

not occur. Unknown parameters are regarded as fixed

numbers, not random. See also Bayesian inference

and statistical inference.

Geometric mean. The geometric mean is an estimator

of the location or center of a distribution. It is applicable only for positive data. The geometric mean, say , ~

t

for t1,t2, ..., tn, is defined as 

[ ]. ~

t nt = exp ( / ) ln 1 Σ i

It is always less than or equal to the arithmetic mean.

Goodness of fit. This term refers to a class of

nonparametric methods that are used to study whether

or not a given set of data follows a hypothesized

distribution. Both hypothesis tests and graphical

methods are used to investigate goodness of fit.

Hazard rate. For a nonrepairable system, hazard rate

and failure rate are two phrases with the same meaning, used interchangeably. The hazard rate h(t) satisfies

h(t))t . Pr(t < T # t + )t * T > t), where )t is small and

T denotes the duration time of interest. The hazard rate

is also called the hazard function.

Hypothesis. A statement about the model that generated the data. If the evidence against the null hypothesis, H0, is strong, H0 is rejected in favor of the alternative hypothesis, H1. If the evidence against H0 is not

strong, H0 is “accepted”; that is, it is not necessarily

believed, but it is given the benefit of the doubt and is

not rejected. 
Glossary

D-3

Improper distribution. A function that is treated as a

probability distribution function (p.d.f.), but which is

not a p.d.f. because it does not have a finite integral.

For example, a uniform distribution (constant p.d.f.) on

an infinite range is improper. Improper distributions

are sometimes useful prior distributions, as long as the

resulting posterior distribution is a proper distribution.

Independent. See statistical independence.

Inference. See statistical inference.

Initiating event. Any event, either internal or external

to the plant, that triggers a sequence of events that

challenge plant control and safety systems, whose

failure could potentially lead to core damage or large

early release.

Interval. The notation (a, b) denotes the interval of all

points from a to b. This is enough for all the applications in this handbook. However, sometimes an additional refinement is added, giving a degree of mathematical correctness that most readers may ignore: The

standard notation in mathematics is that (a, b) includes

the points between a and b, but not the two end points.

In set notation, it is {x | a < x < b}. Square brackets

show that the end points are included. Thus, (a, b)

includes b but not a, {x | a < x # b}.

Interval estimate. One way of estimating a parameter

is to identify that it falls in some interval (L, U) with a

specified degree of certainty, or confidence. The

interval (L, U) is referred to as an interval estimate of

the parameter. L and U are calculated from the random

data. The frequentist interval estimate is referred to as

a confidence interval. It does not give a probability

statement about the true parameter value. Rather, the

interpretation of a 100(1 ! ")% confidence interval is

that, if the random data were drawn many times,

100(1 ! ")% of the resulting interval estimates would

contain the true value. A Bayesian interval estimate is

referred to as a subjective probability interval, or

credible interval, and can be interpreted as giving a

subjective probability statement about the true parameter value being contained in the interval. Compare with

point estimate. See also confidence interval, credible

interval. 

Inverse c.d.f. algorithm. An algorithm for generating

random numbers (presented in Section 6.3.2.5.4).

Latin hypercube sampling (LHS). See Monte Carlo

simulation.

Likelihood. For discrete data, the likelihood is the

probability of the observations. For continuous data,

the likelihood is the joint density of the observations,

which is the product of the densities of the individual

observations if the observations are independent. When

some of the observations are discrete and some are

continuous, the likelihood is the product of the two

types. The likelihood is typically treated as a function

of the parameters, with the data regarded as fixed.

Markov Chain Monte Carlo (MCMC). See Monte

Carlo simulation.

Maximum likelihood estimator. For data generated

from a distribution with one unknown parameter, say 2,

the maximum likelihood estimate (MLE) of 2 is the

parameter value that maximizes the likelihood of the

data. It is a function of the data, and is commonly

denoted . The MLE is a popular frequentist estima- $

θ

tor for two reasons. (1) In commonly used models, the

MLE is an intuitively natural function of the data. (2)

Under certain, commonly valid, conditions, as the

number of observations becomes large the MLE is

approximately unbiased with approximately the minimum possible variance, and is approximately normally

distributed.

Mean. The mean, :, of a random variable X is the

weighted average of the outcomes, where the weights

are the probabilities of the outcomes. More precisely,

the mean of X is the expected value E(X), Exi

f(xi) if X is

discrete with p.d.f. f, and Ixf(x)dx if X is continuously

distributed with density f. See also expected value.

Mean square error or mean squared error. The

expected squared difference between an estimator and

the true quantity being estimated. For example, if Y is

a function of the data that estimates a parameter 2, the

mean squared error (MSE) of Y is E[(Y ! 2)

2

]. It can be

shown that the MSE(Y) = var(Y) + [bias(Y)]2

.

Median. For a random variable X with a continuous

distribution, the median is that value m for which

Pr(X < m) = 0.5, and thus also Pr(X > m) = 0.5. For a

sample of data values, or for a discrete random variable

X taking a finite number of values with equal probability, the median is the middle value in the ordered set

of values. The median m is the 50th percentile, x0.50.

See percentile for the general definition.

Mode. A mode of a distribution is a local maximum

value of the probability density or probability distribution function (p.d.f.). A normal distribution has a single

mode, which measures the center of the distribution. 
Glossary

D-4

Moment. The kth moment about a of a random variable X is the expected value of (X ! a)

k

. If X is discrete

with p.d.f. f, this is E(xi

!a)

k

f(xi). If X is continuous with

density f, the kth moment about a is I(x!a)

k

f(x)dx. The

moments about 0 are sometimes called simply “the”

moments. Moments about the mean are called central

moments. The first moment is the mean, often denoted

:. The second central moment is the variance.

Monte Carlo Sampling. See Monte Carlo simulation.

Monte Carlo simulation. Generally referred to as

Monte Carlo Sampling by probabilistic risk assessment

(PRA) analysts, Monte Carlo simulation uses a sample

of values from an uncertainty distribution to approximate the distribution. Moments and percentiles of the

distribution are approximated by the corresponding

moments and percentiles of the sample.

In the usual PRA application, the initiating event

frequencies and basic event probabilities have Bayesian

distributions. They are sampled by simple random

sampling or by Latin hypercube sampling (LHS). The

parameter values are propagated through a faulttree/event-tree model to produce a simulation of the

uncertainty distribution of a quantity of interest, such as

core damage frequency.

Monte Carlo simulation is also used to approximate the

posterior distributions of parameters, when direct

calculations are not feasible. In situations with a single

parameter and a nonconjugate prior, a simple random

sample from the posterior distribution can usually be

constructed. In more complicated situations, such as

those with multiple interrelated parameters, the unknown parameters can be simulated by Markov chain

Monte Carlo (MCMC) to produce a sequence, or chain,

of values of each parameter. The values are not

independent. However, if the initial portion of the

chain is discarded, the remainder of the terms mimic the

posterior distribution of the parameter.

Mutually exclusive. Events are mutually exclusive, or

disjoint, if no two of them have any elements in common. The intersection of any two of the events is the

empty set.

Nonparametric. In parametric inference, the data are

assumed to come from a known distributional form,

with only the parameters unknown. In nonparametric

inference, no distributional form is assumed. Not only

are the values of the parameters unknown, but the form

of the distribution is unknown as well. See parametric.

Nonrepairable system. A system that can only fail

once, after which data collection stops. An example is

a standby safety system, if the failure to run cannot be

recovered during the mission of the system. Data from

a nonrepairable system consist of data from identical

copies of the system. For example, data from a safety

system may be collected, with each run starting with the

system nominally operable, and the system either

running to completion of the mission or failing before

that time. The successive demands to run are regarded

as demands on identical copies of the system. See

repairable system.

Null hypothesis. See hypothesis.

Order statistics. The random values arranged from

smallest to largest. For example, suppose that three

times are observed, with t1 = 8.4, t2 = 3.0, and t3 = 5.1.

The order statistics are t(1) = 3.0, t(2) = 5.1, and t(3) =

8.4. Before the data are observed, one can consider the

order statistics as random variables, T(1), T(2), ..., T(n)

.

Outage, outage time. An outage is an event when a

system is unavailable, that is, out of service for some

reason. The outage time is the duration of the event.

Compare with unavailability.

Parameter. A parametric family of distributions is a

collection of distributions that is indexed by one or

more quantities called parameters. For example,

suppose that f(t; 8) = 8e!8t

, where t; 8 >0. For each

value of 8, f(t; 8) is a probability density function.

Here 8 is the parameter that identifies the particular

density in the family of exponential density functions.

The normal family has two parameters, the mean and

the variance.

Parametric. Parametric statistical inference is concerned with learning the values of unknown parameters

(and their associated properties) from sample data for

a given or assumed family of distributions. See

nonparametric.

p.d.f. See probability density function and probability distribution function. 

Percentile. Consider a continuous distribution with

density (p.d.f.) f and cumulative distribution function

(c.d.f.) F. The 100qth percentile is the value x such that

F(x) = q, or equivalently

f u du q .

x

( )

−∞

∫ =
Glossary

D-5

If the distribution is concentrated on the positive line,

the lower limit of integration may be replaced by 0.

The 100qth percentile is equal to the qth quantile. For

example, the 95th percentile equals the 0.95 quantile.

If X has a discrete distribution, a percentile may not be

unique. The 100qth percentile is defined in this case as

x such that Pr(X # x) $ 100q% and Pr(X $ x) $

100(1 ! q)%. 

Similarly, for a finite sample, the 100qth percentile is

defined as x such that at least 100q% of the values in

the sample are x or smaller, and at least 100(1 ! q)%

are x or larger. For example, if a sample is a set of

three numbers, {1.2, 2.5, 5.9}, the median (corresponding to q = 0.5) is 2.5, because at least half of the numbers are 2.5 or smaller and at least half are 2.5 or larger.

If the sample has four numbers, {1.2, 2.5, 2.8, 5.9},

then any number from 2.5 to 2.8 can be considered a

median. In this case, the average, (2.5 + 2.8)/2, is often

chosen.

Point estimate. An estimate of a parameter in the

form of a single number is called a point estimate of the

parameter. For example, the mean of a sample of

values of a random variable X is a commonly used

point estimate of the mean of the distribution. Compare

with interval estimate.

Poisson process. A process in which events (such as

failures) occur in a way such that the number of events

X in total time t is described by a Poisson distribution.

See Section 2.2.2, Section 7.2, or Appendix A.6.2 for

more details.

Pool. To combine data from distinct sources, ignoring

possible differences between the sources. Data are

sometimes pooled from distinct time periods, components, trains, and/or power plants.

Population. In the PRA setting, population refers to

the random distribution that generates data. Population

attributes, such as the population mean or population

median, are those attributes of the probability distribution. Compare with sample.

Posterior credible interval. See credible interval.

Posterior distribution. A distribution that quantifies,

in a Bayesian way, the belief about a parameter after

data have been observed. It reflects both the prior

belief and the observed data.

Power of a test. The probability that the test will reject

H0 when H0 is false. If many possible alternatives to H0

are considered, the power depends on the particular

alternative. See hypothesis.

Prior. A colloquial abbreviation for prior distribution.

Prior distribution. A distribution that quantifies, in a

Bayesian way, the belief about a parameter before any

data have been observed.

Probability model. A term for the set of mathematical

relationships which are used to define both cumulative

distribution functions and either probability distribution

functions (discrete case) or probability density functions (continuous case).

Probability density function (p.d.f.). For a continuous random variable X, the probability density function

f satisfies

Pr( ) ( ) a X b f x dx . a

b

≤≤= ∫

Properties of the density are

f(x) $ 0 for all x

f x dx ( )

− ∞

∞

∫ = 1

f(x))x . Pr(x < X # x + )x) for small )x.

The p.d.f. is related to the c.d.f. by

f(x) = FN(x), the derivative, 

and

F x f u du . x() () = −∞ ∫

See cumulative distribution function.

Probability distribution function (p.d.f.). For a

discrete random variable X, the p.d.f. f(x) = Pr(X = x).

p-value. In the context of testing, the p-value is the

significance level at which the data just barely cause H0

to be rejected. H0 is rejected when a test statistic is

extreme, and the p-value is the probability (under H0)

that the random test statistic would be at least as extreme as actually observed.

Quantile. Consider a continuous distribution with

density (p.d.f.) f and cumulative distribution function

(c.d.f.) F. The qth quantile is the value x such that F(x)

= q, or equivalently:

f u du q x( ) −∞ ∫ =
Glossary

D-6

If the distribution is concentrated on the positive line,

the lower limit of integration may be replaced by 0.

The qth quantile is equal to the (100q)th percentile. For

example, the 0.95 quantile equals the 95th percentile.

If X has a discrete distribution, a quantile may not be

unique. Some authors use the term fractile instead of

quantile. See percentile for a fuller explanation. 

Random sample. x1, ..., xn are a random sample if they

are the observed values of X1, ..., Xn, where the Xi

s are

statistically independent of each other and all have the

same distribution.

Random variable. A rule that assigns a number to

every outcome in a sample space. For example, if a

pump was demanded to start n times, the sample space

consists of all the possible outcomes, with their probabilities. A random variable of interest might be the

number of failures to start. If a stuck valve is repaired,

the sample space consists of all the possible outcomes

of the repair process, with their probabilities. A random variable of interest might be the time required for

repair, a number.

Range. The difference between the largest and smallest

values of a sample is called the range of the sample. 

Rate. See frequency.

Reactor critical year. 8760 hours during which a

reactor is critical.

Rejection-method algorithm. An algorithm for

generating a random sample from a particular distribution. Its general form is given in Section 6.2.2.6, and

applied in several places there and in Section 6.3.2.4. 

Relative standard deviation. The standard deviation,

expressed as a fraction of the mean. The relative

standard deviation of X is st.dev.(X)/E(X). Some

authors call it the coefficient of variation, and express

it as a percent.

Relative variance. The square of the relative standard deviation. The relative variance of X is

var(X)/[E(X)]2

.

Renewal process. A process in which events (such as

failures or restorations) occur in a way such that the

times between events are independent and identically

distributed. For example, if the process consists of

failures and nearly instantaneous repairs, each repair

restores the system to good-as-new condition.

Repairable system. A system that can fail repeatedly.

Each failure is followed by repair, and the possibility of

another failure sooner or later. An example is a power

plant, with initiating events counted as the “failures.”

After such an event, the plant is brought back up to its

operating condition, and more initiating events can

eventually occur. See nonrepairable system.

Residual. When a model is fitted to data, the residual

for a data point is the data value minus the fitted value

(the estimated mean). The residuals together can be

used to quantify the overall scatter of the data around

the fitted model. If the assumed model assigns different

variances to different data points, the standardized

residuals are sometimes constructed. A standardized

residual is the ordinary residual divided by its estimated standard deviation.

Return-to-service test. A test performed at the end of

maintenance, which must be successful. If the system

does not perform successfully on the test, the maintenance is resumed and the test is not counted as a returnto-service test. A return-to-service test can demonstrate

that no latent failed conditions exist (see standby

failure), but it provides absolutely no information about

the probability of failure on a later demand (see failure

on demand).

Sample. This term refers to data that are generated

randomly from some distribution. Sample attributes,

such as the sample mean or sample median, are those

attributes calculated from the sample. They may be

used as estimators of the corresponding population

attributes. The sample may be thought of as random,

before the data are generated, or as fixed, after the data

are generated. See also population, random sample,

sample mean, sample median, and sample variance.

Sample mean. The arithmetic average of the numbers

in a random sample. If the numbers are x1, ... , xn, the

sample mean is often denoted . It is an estimate of x

the population mean, that is, of the expected value

E(X).

Sample median. Let x(1), ... x(n)

 be the order statistics

from a random sample. The sample median is the

middle value. If n is odd, the sample median is x([n+1]/2).

If n is even, the sample median is any number between

x(n/2) and x(n/2 + 1) , although usually the average of these

two numbers is used.

Sample variance. Let x1, ... , xn be a random sample,

with sample mean . The sample variance, often x

denoted s

2

, is
Glossary

D-7

 . 1

1

2

n 1

x x i

i

n

− − =

∑ ( )

It is an estimate of the population variance, var(X).

Significance level of a test. The probability of making

a Type I error, that is, of rejecting H0 when H0 is true.

(If H0 includes a number of possibilities, so that the

probability of rejecting H0 varies, the significance level

is defined as the maximum of those probabilities.) The

significance level is denoted by ". Compare with pvalue and statistically significant.

Skewed distribution. A distribution that is not symmetrical. A distribution that is restricted to the range

from 0 to 4 is typically skewed to the right, or positively skewed. Its mean is larger than its median, and the

95th percentile is farther from the median than the 5th

percentile is. The Poisson, gamma, and lognormal

distributions are a few examples of positively skewed

distributions.

Standard deviation. The standard deviation of a

distribution is the square root of the variance. The

standard deviation and variance are two measures of

how much spread or dispersion there is in a distribution.

Standard error. The estimated standard deviation of

the estimator of a parameter, in the frequentist approach. For example, suppose that 8 is the parameter

to be estimated, and is the estimator. The estimator $

λ

depends on random data, and therefore is random, with

a standard deviation, s.d.( ). The estimated value of $

λ

this standard deviation is the standard error for 8.

Standardized residual. See residual.

Standby failure. For a standby system, failure to start

resulting from an existing, or latent, failed condition.

The system is in this failed condition for some time, but

the condition is not discovered until the demand.

Compare failure on demand.

Statistic. A function of the data, such as the sample

mean or the Pearson chi-squared statistic. Before the

data are observed, the statistic is a random variable

which can take many values, depending on the random

data. The observed value of a statistic is a number.

Statistical independence. Two events are statistically

independent if the probability of both occurring is the

product of their marginal (or individual) probabilities:

Pr(E11E2) = Pr(E1) × Pr(E2) . Three or more events are

statistically independent if the probability of any set of

the events is equal to the product of the probabilities of

those events. Two or more random variables are

statistically independent if their joint p.d.f. equals the

product of the marginal (or individual) p.d.f.s. For

brevity, the word statistically is often dropped.

It can be shown that two random variables are statistically independent if and only if any event defined in

terms of one random variable is statistically independent of any event defined in terms of the other random

variable. (A similar statement holds for more than two

random variables.) For example, suppose that X and Y

are independent continuously distributed random

variables, with joint density

f xy f xf y . XY X Y , (, ) () () =

Let A be the event a # X # b, and let B be the event c #

Y # d. Then

Pr(A1B) = Pr(a # X # b and c # Y # d)

= f x y dydx X Y c

d

a

b

, (, ) ∫ ∫

by the definition of a joint density

= f x f y dydx X Y c

d

a

b

() () ∫ ∫

because X and Y are independent

= f x dx f y dy X Y c

d

a

b

() () ∫ ∫

evaluating the integral

= Pr(a # X # b)Pr(c # Y # d)

by definition of the marginal densities

= Pr(A) × Pr(B).

Statistical inference. The area of statistics concerned

with using sample data to answer questions and make

statements about the distribution of a random variable

from which the sample data were obtained.

Statistically significant. A departure from a null

hypothesis is called statistically significant if the

hypothesis is rejected with some small significance

level, customarily set to 0.05. See p-value and

significance level of a test.
Glossary

D-8

Stochastic. Referring to a random, rather than a

deterministic, process. This is an elevated word for

random.

System. In this handbook, system is the general word

used to denote a collection of hardware for which data

are collected. The term can apply to a specific system

typically found in a nuclear power plant, such as the

auxiliary feedwater system, or to a train, or a component, or even a small piece part, as long as data for the

system are reported.

Time at risk. See exposure time.

Type I error. A rejection of the null hypothesis when

it is true.

Type II error. “Acceptance” of the null hypothesis

when it is false, that is, failure to reject the null hypothesis when it is false.

Unavailability. For a standby system, the probability

that the system is unavailable, out of service, when

demanded. This may be divided into different causes

— unavailability from planned maintenance and unavailability from unplanned maintenance. Unavailability is distinct from failure to start of a nominally available system. Compare outage.

Uncertainty. The imprecisions in the analyst’s knowledge or available information about the input parameters to PRA models, the PRA models themselves, and

the outputs from such models.

Variance. The variance of a random variable X,

denoted by F2

, is the second moment about the mean,

the average of the squared deviations from the mean,

E[(X!:)

2

]. It measures the dispersion in the distribution. Compare standard deviation.
I-1

INDEX

Page

aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-7

balanced data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-46

bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-74

basic event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

bathtub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-3, A-18

Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-8, 6-9, 6-33, 6-36, A-11, B-11

bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-4

BUGS computer package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-17,6-18, 6-39, 6-42, 6-60, 6-62, 6-63, 8-15

catastrophic failure

definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2

event classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4

component boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

conditional probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-3

confidence intervals

contrasted with simultaneous confidence band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8, 7-9

definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-5

event occurrence rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-6

event occurrence rate, time-dependent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8, 7-10, 7-21

event occurrence rate, with trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7

exponentially distributed durations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-61

for normal data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-5

for probability of failure on demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-32

one-sided and two-sided . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-5

plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-21

rate of failure to run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-53

conjugate prior

definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-13

for binomial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-35

for event occurrence rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-11

for exponential data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-61

for lognormal or normal data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-60

for Poisson data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-11

for probability of failure on demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-35

in empirical Bayes estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3

with large data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4, 7-18

constrained noninformative prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-19

contingency tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-45, 6-50, B-9

correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-10

covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-10

credible interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-11

criticality factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3

cumulative distribution function

confidence interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-58

definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-17, A-5

nonparametric estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-58

degradation, gradual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16

degraded failure

definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2

event classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-5
Index

I-2

Page

density

definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-17, A-5

histogram estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-58

kernel estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-74

deviance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-14, 7-23

distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-17

distributions

Bernoulli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-52

beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-35, 6-73, A-21, C-7

beta-binomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-50, 8-5

binomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8, A-12

Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-23

chi-squared . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-9, 7-14, A-18, C-4

conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-8

conjugate prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-11, 6-35, 6-36, 6-61, 7-18

constrained noninformative prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-15, 6-38, 6-40, B-14, C-15

Dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-25

exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14, 2-17, 6-53, 6-61, 6-70, 6-78, 8-16, 8-17, A-17

F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-79, 7-10, 7-21, A-23

gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-17, 6-11, 6-18, 6-61, 6-62, 8-17, A-18

gamma-Poisson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-30, 8-4

hyperprior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-12

inverted chi-squared . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-60, A-20

inverted gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-20

Jeffreys noninformative prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-14, 6-37, 6-60, 6-61, 6-83, B-13

joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-8

logistic-normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-40, 6-42, 8-18, A-22

lognormal . . . . . . . . . . . . . . . . . . 2-17, 6-16, 6-18, 6-31, 6-40, 6-42, 6-59, 6-65, 6-67, 6-69, 7-6, A-16, B-7

marginal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-8

maximum entropy prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-41, B-14

negative binomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-30, 8-4

normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-65, 6-67, 6-69, 6-82, 7-4, 7-19, A-15, C-1

Poisson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4, 6-78, A-13

population-variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1

reference prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-14

Snedecor's F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-79, 7-10, 7-21, A-23

Student's t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-59, 6-60, 6-82, 6-83, 7-10, 7-21, A-24, B-6, C-6

uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-14

Weibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-17, 6-62, A-18

Equipment Performance and Information Exchange (EPIX) System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4, 4-5

event-occurrence rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4

examples

AFW motor segment FTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2, 8-10, 8-16

AFW turbine FTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-31, 6-39, 6-42, 6-51

AFW turbine train FTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14, 2-15

causes of isolation valve failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7

compressor times of FTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16

confidence intervals from computer-generated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-7

EDG failure to start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7, 6-44

EDG failure-to-run events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14, 2-15

EDG failure-to-run times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-54

fire suppression times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16

gradual degradation until failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16
Index

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Page

examples

HPCI failure to start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7, 2-8

initiating events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3, 2-5

initiating events at many plants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2, 8-8, 8-16

initiating events with loss of heat sink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5, 6-12

LOSP recovery times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-56, 6-66, 6-71, B-7, B-10

moment-matching distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-18

recovery times from LOSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16

repair times for turbine-driven pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16

sensor/transmitter failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6

shutdown LOSP events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3, 2-5, 6-21

Simpson's paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-47

small-break LOCAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-18, 6-31

steam binding in AFW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7

temperature sensor/transmitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3

thermal-fatigue piping leaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

through-wall pipe leaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3, 2-6

times between LOSP events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-29, 6-70, 6-71

unavailability of CVC train because of maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-19, 6-77

unbalanced data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-47

unplanned HPCI demands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-25, 7-1, 7-6, 7-18

unplanned HPCI demands and failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-48, 7-17

unplanned reactor trips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3, 7-12, 7-25

exchangeability of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-10

expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-9

exposure time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2

failure severity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

gamma function, definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-19

hazard function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14, 6-54, A-6

cumulative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-54, 6-63

hazard rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-17

histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-58

hypotheses, one-sided or two-sided . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-27

hypothesis tests (see also tests) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-6

I-Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13

IEEE-500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-14

incipient failure

definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2

event classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-5

independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-3

initiating event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

inverse c.d.f. algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-41

joint probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-3

Kendall’s tau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-30

kernel

rectangular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-74

standard normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-76

triangular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-75

kernel estimator

of density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-74

of Poisson intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-25
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Page

least-squares fitting

iteratively reweighted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-10, 7-15, 7-21, 7-23

unweighted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-9, 7-20

likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5, 6-7, 6-11, 6-31

link function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-23

logistic regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17

logit function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-40, 7-17, A-22

marginal probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-3

maximum likelihood estimates

definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-4

event occurrence rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5

exponential parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-61

hierarchical model for failure on demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-4

hierarchical model for initiating events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3

parameters of normal or lognormal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-59

probability of failure on demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-31

rate of failure to run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-53

mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-8

mean squared error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-76

mean time to failure (MTTF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-77, A-19

mean time to repair (MTTR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-77, A-18

median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-8

mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-9

models

alternating renewal process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-19

binomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7

exponential event rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2

exponential duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-17

extended power-law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2

general linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6

generalized linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6, 7-17, 7-23

hierarchical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1

hierarchical Bayes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-12

homogeneous Poisson process (HPP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4, 2-6, 7-2

logit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17

loglinear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2, 7-6, 7-18

nonhomogeneous Poisson process (NHPP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2

Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4, A-13

power-law event rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2, 7-16

renewal process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16

waiting time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7, 2-10, 6-20, 6-43

Weibull process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2

moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-9

moment-matching

conjugate prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-18

gamma and lognormal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-18

nonrepairable system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13

Nuclear Plant Reliability Data System (NPRDS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4, 4-5

numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-16

offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-24

Offshore Reliability Data (OREDA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13

order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-57

p-value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-23
Index

I-5

Page

percentile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-8

plot types

box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-64, 6-84

box-and-whisker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-64, 6-84

confidence interval, for poolability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-21, 6-44

confidence interval, for trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-25, 6-48, 7-7, 7-13, 7-22

credible band, for trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6, 7-19

cumulative duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-67

cumulative event count . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-25, 7-14, 7-23

cumulative failure count . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-48, 6-55

cumulative hazard function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-55, 6-63

density estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-74 to 6-76

empirical c.d.f. (EDF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-58

empirical reliability function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-69

hazard function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-55, 6-63

histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-58, 6-68

Poisson intensity estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-26

quantile-quantile (Q-Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-69, 6-70, 6-79, 6-80, 6-82

residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-13, 7-14, 7-22

scatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-30, 6-66

Venn diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1

Poisson intensity function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2

Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4, 7-2, A-13

precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-60

prior distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-14

quantile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-8

quartile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-8

random number generation

adaptive rejection sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-15

from beta distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-41, 6-51

from logistic-normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-40

from lognormal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-31, 6-40

from maximum entropy distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-41

from posterior distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-17, 6-39

Gibbs sampler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-14

inverse c.d.f. method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-41

Markov chain Monte Carlo (MCMC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5, 8-13

Metropolis-Hastings method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-15

rejection method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-17, 6-39, 6-40

random sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-56, B-1

ranks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-29, 6-49, 6-65, 6-68

rates

event occurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4, 7-2

event occurrence, time-dependent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2

failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14

hazard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14, A-6

rejection method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-17

reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-6

Reliability Analysis Center (RAC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13

Reliability and Availability Data System (RADS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6

renewal process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16

repairable system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3, 2-13

residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-13, 7-23
Index

I-6

Page

return-to-service tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8

sample space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1

sample statistics

empirical distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-58, 6-66

interquartile range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-57

mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-56, B-1

median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-57

percentiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-57

quartile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-57

range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-57

reliability function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-58

skewness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1

survival function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-58

variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-56, 6-57, B-1

significance, statistical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-23

Simpson's paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-47

simultaneous confidence band

event occurrence rate, with trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8, 7-10, 7-21

probability on demand, with trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-22

skewness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-9

standard deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-9

standard error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-6, 6-32

statistical independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-3, A-8

survival function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-6

T-Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-12

tests

analysis of variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-65, 6-84

Anderson-Darling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-72, B-10

Bartlett’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-65

chi-squared . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-22, 6-45, 6-50, 6-70, 7-15, 7-23, 7-24, 8-8, 8-10, B-8

Cramer-von Mises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-72

definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-6

deviance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-15, 7-23

F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-65

Fisher's exact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-47, 6-50

for slope of zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7

goodness-of-fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-70, 7-14, 7-23, B-8

interpretation of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-23, 6-30, 6-67, 6-70

Kolmogorov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-72, B-10, C-14

Kolmogorov-Smirnov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-66, B-10

Kruskal-Wallis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-66, 6-84

Laplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-27

least-squares fitted trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-67

likelihood ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-65

Pearson-Hartley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-65

Shapiro-Wilk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-71

Student's t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-65

using weighted sum of squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-15, 7-23

Wilcoxon rank sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-49

Wilcoxon-Mann-Whitney . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-49, 6-65, 6-68, 7-20

time-related exposures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-9

unavailability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-19

unbalanced data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-46
Index

I-7

Page

uncertainty, not parametric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-3

variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-9

zero failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2, 6-6, 6-16, 6-33