* ~ ~ ~ ~ ~ ~

EGG-SSRE-M7

July 1990 R T

TECHNICAL REPORT

1.1, - , 11 11,

, , "r - ;,

DIdaho

Nationalo

Engin4eering' '' '.Laboratory',.

Usefrs Guide to PHAZE, a

Computer Program for Parametric Hazard Function Estimation

Corwin L Atwood Managed

.; .by the UJ.S.. - s.'Deparfretmm, -i

:-of Einergy .

Prepared for the

U. S. NUCLEAR REGULATORY COMMISSION

I., I .- . - nSEexssfthc

ol , .1 , - . I 1. ,

W04kpedon'ned nder 1

No.' DEAC0O7-76ID01O7I5IO
EGG-SSRE-9017

User's Guide to PHAZE,

a Computer Program for Parametric. Hazard Function Estimation

Corwin L. Atwood

Published July 1990

Idaho National Engineering Laboratory

EG&G Idaho

Idaho Falls, ID 83415-3421

Prepared for the

U.S. Nuclear Regulatory Commission

Washington, DC 20555

Under DOE Contract No. DE-AC07-761DO1570,.-
ABSTRACT

The program PHAZE (for Parametric HAZard function Estimation) performs statistical intfrence on a hazard function, based on reported failure times of components that are repaired and re.tored

to service. The inference includes parameter estimation, testing of hypotheses, and checking of the

model assumptions, under a choice of parametric models. This user's guide sketches only enough of the

theory so that PHAZE can be used, with a full presentation of the theory being given in a comnpalion

report. A typical PHAZE session is described. The format of a data file is given, and all the PIIAZE

commands are listed and explained. The program has been verified and validated, and this work is

summarized. Finally, some of the technical details are given, of interest to statisticians and

programmers. An appendix shows an entire PHAZE session, both the user's commands and the

program's responses.

FIN No. A6389--Aging Components and Systems IV:

Risk Evaluation and Aging Phenomena

..
SUNINMARY

The program PHAZE (for Parametric HAZard function Estimation) performs statistical inference on a hazard function (also called a failure rate or intensity function), based on reported failure

times of components that are repaired and restored to service. Three parametric models are allowed.

the exponential, linear, and Weibull hazard models. The inference includes estimation (maximum

likelihood estimators and confidence regions) of the parameters and of the hazard function itself. testing

of hypotheses such as of increasing failure rate, and checking of the model assumptions.

This user's guide sketches only enough of the theory so that PHAZE can be used, with a full

presentation of the theory being given in a companion report. A typical PHAZE session is described.

This consists of an initial exploratory phase, in which the various model assumptions are checked. and

a final estimation phase, in which the maximum likelihood estimator and a confidence interval are

found for the hazard function at time(s) of interest. The format of a data file is given, with examples.

PHAZE is an interactive command-based program; all the PHAZE commands are therefore listed and

explained.

The program has been verified and validated, and this work is summarized. Finally, some of

the technical details are given, of interest to statisticians and programmers. An appendix shows an

entire PHAZE session, both the user's commands and the program's responses. This appendix

illustrates virtually all of the PHAZE commands, and the resulting output.

iii
ACKNOWLEDG MENTS

WV. Scott Roesener used the program through many intermediate versions. discovering problems for me and showing patience while I corrected them. Andrew J. Wolford suggested features and

options.

iv
CONTENTS

ABSTR'ACT .......................................................... '...,,......,,..........;..........,,......................... . ii.........

SUM M ARY ...................................................................................... ................................................. iii

ACKNOWLEDGMENTS ..............................-.......................................... iv...........................

1. INTRODUCTION..................................................................................................................... 1I

1.1 Fundamental Assumptions .. 1

1.2 Full and Conditional Likelihood .................... 3

1.3 Typical PHAZE Sessin .3

1.4 Terminology'and Formats .4

2. DATA FILES .........................

3. PHAZE COMMANDS .. 9

3.1 Command Syntax .9

3.2 General Concepts . ; .. 11

3.3 Commands. 12

4. VERIFICATION AND VALIDATION....................................................................................... 28

4.1 Overview ........... . . . . . . . .. 28

4.2 Verification .......... 30

4.3 Validation .30

5. TECHNICAL DETAILS .................... 33

5.1 Language and Portability ................................ 33

5.2 Calculational Methods .33

5.3 Some Details of the Coding. 37

6. REFERENCES.......................................................................................................................... 39

APPENDIX-EXAMPLE PHAZE SESSION . . .A-I

TABLES

1. Summary of commands............................................................................................................... 10

2. Effect of switches on CI command.............................................................................................. 13

FIGURES

1. Data file with times entered as dates..............................................................................................7

2. Data file with single component, integer failure times....................................................................7

3. Data file with floating point failure times.....................................................................................7

4. Example output from CC B command .15

5. 90% confidence regions for (P, AO), based on exponential hazard model .19

v
6. 90% confidence regions for (j3, A0), based on Weibull hazard model ............................................. 19

7. 90% confidence regions for (,, A0), based on linear hazard model, uncentered data ...................... 20

8. 90% confidence regions for (,3, A0), based on linear hazard model, centered data .......................... 20

9. Q-Q plot, based on exponential hazard model.............................................................................. 25

vi
User's Guide to PHAZE,

a Computer Program for Parametric Hazard Function Estimation

1. INTRODUCTION

The program PHAZE carries out Parametric' HAZard function Estimation, based on reported

failure times of components tliat are repaired 'and 'restored to service. It is an interactive program.

intended for a personal computer, but adaptable to any' computer that runs Fortran. The theory

behind the method is presented in a companion report Estimating Hazard Functions for Repairable

Components (Atwood,' 1990). Thit'report, referred to from now on as EHF, is cited continually in this

user's guide. Fuller technical explanation of theimeithods'mentioned here can be found in EHF.

The outline of this guide is as follows. The data analysis setting is presented in Section 1. witlh

just enough of the theory so that PHAZE can be used, and with a summary of a typical data analysis

session. Section 2 describes the format for files that contain the failure data, files that are read as

inputs by PHAZE. Section 3 explains all the' PHAZE commands. Section 4 describes the work that

.was performed to verify and validate the correctness of PHAZE. Section 5 explains some of the

technical details of the algorithms and the program; this section is intended only for statisticians and

programmers. Section 6 lists the references, and Appendix A gives the complete record of an example

PHAZE session.

1.1 YFundamental Aisumptions

The concern is with the failure behavior of components. The failures are assumed to be

governed by a Poisson process with hazard function, or failure rate, A(t). Here, t is time, typically

measured 'from the' component's installation. The' Poisson assumption says that the probability of a

failure in a short period (t, t + 'At)'asymptoticaily"approaches A(t)At, and failure counts in nonoverlapping time intervals are independent. ' It is assumed that when a component fails, either it is

immediately repaired (made 'as good as old) and placed back in service, or else it is replaced by a new

component. -Failures of distinct-components are assuined to be independent. The data to be analyzed

therefore' coisist of seq'uences of failure times of similar independent 'components.

.. !' .. .- >. .' -. '' .t <. .* , '. ', ... T: .' .
Data for a component are called time censored if the component is observed for a fixed time

period, or plant records covering a fixed time period are examined, and the failure times are recorded.

The number of these failures is random. Data are called failure censored if the component is kept in

service until a predetermined number of failures has occurred, at which time the component is removed

from service. In this case the number of failures is fixed, but the end of the observation period equals

the final failure time, and is random. The mathematical formulas used for statistical analysis differ

slightly for time censored and failure censored data. They are given in EHF, but need not concern a

user of PHAZE. To perform the analysis, however, PHAZE must know whether each component is

time censored or failure censored. Therefore, the data file must state whether a component was replaced at the time of its final observed failure. If so, PHAZE treats the component as failure censored.

and otherwise as time censored.

There are three parametric forms assumed by PHAZE for the hazard function A(t). all of the

form A(t) = Aoh(t;1,3). The three forms are:

A(9) = Aoexp(,3t) (exponential hazard model)

A(i) = AO(l +16t) (linear hazard model)

A(t) = Ao(t/tco) (Weibull hazard model)

In all three models, AO has units 1/time. In the exponential and linear models, 13 has units 1/time. so

that fit is dimensionless. Under these models, AO is the value of the hazard function A(t) at time t = 0.

In the Weibull model to is an arbitrary normalizing time, and 1 is dimensionless. Under this model, AS

is the value of A(t) at t = to.

A typical PHAZE session consists of reading failure data from a file that was previously

prepared, selecting one of the above three models, and performing data analysis. Often the analyst

wants to decide whether A(i) is an increasing function, that is, whether failures tend to occur more

frequently as time goes on. This is done in PHAZE by constructing a confidence interval for 1, or by

testing whether 13 = 0; the hazard function A(t) is increasing if 13 > 0, is constant if 1 = 0, and is

decreasing if 13 < 0. In risk assessment it is also useful to estimate A(t) at any time t of interest; both

by a point estimate and by a confidence interval. This is done in PHAZE by assuming one of the three

models, estimating 3 and AO, and obtaining the corresponding estimate of A(t). In order to perform

any of the above tasks, certain assumptions must be made; these assumptions should be checked

2
against the data. The PHAZE commands, described in Section 3, include commands'for performing

these checks of the assumptions.

1.2 Full and Conditional Likelihood

Before a typical data analysis session can be described, the following jtwo approaches must be

mentioned. Statistical inference'is generally based on the likelihood, which depends on the data and on

the parameter(s). Inference for P is of primary interest in many investigations of A(t), because it is d

that determines whether A(t) is increasing. It is well known, and shown in Section 3 of EHF that the

conditional likelihood can be used to perform inference for 6, while not assuming that the components . . ... . . . . . .. . . .or the . . -

necessarily have a common value of A0, and without estimating or the AO s. (The conditional likelihood is defined as'the probability density of the nonreplacement failure times, given the failure counts

for time censored components an'd given the final failure times for failure censored components. The

full likelihood is this conditional likelihood times the probability of the failure counts for time censored

components and the probability density of the final failure times of the failure censored components.)

Therefore the first exploratory analysis, used to verify assumptions of the model, should be

based on the conditional likelihood. Later, when both parameters must be estimated simultaneously to

produce an estimate of the hazard function A.() at various times i, the full likelihood should be used.

1.3 Typical PHAZE Session

A typical analysis proceeds as follows. First the raw data (such as.plant maintenance records)

are read, interpreted, and encoded in a file that PHAZE can read. Then PHAZE is run to perform the

following steps. The capital letters in parentheses below are the PHAZE commands used to perform

each step. They are explained in Section 3.3, but are listed here to help guide a user who already has

some familiarity with PHAZE. For a fuller treatment of any of the steps, see the explanations of the

corresponding commands.

(1) Get and read a data file. (GD)

(2) 'Construct' a c'umuilative failure plot, to get a preliminary graphical picture of any trends

''-, ' ' that may be present.' (QQ) ' "

Steps (3) through (7) are exploratory, and'therefore'should be based on the conditional likelihood,

and' not use anvassurnption of bivariate asym'ptotic normality. (FL, AN)

(3) Investigate whether the components have a common value of P. If they do, continue.

3'
If they do not, try to find the reasons for the discrepancies, and consider splitting the data.

(CC B)

(4) Investigate whether A(t) is increasing. That is, test ,3 = 0 against the alternative 3 > 0.

As supplemental information, construct a confidence interval for 13, to see the range of

plausible values. If A(t) does not seem to be increasing, in some situations the analyst

would not be interested in continuing the study. (TE, CI B)

(5) Test whether the assumed model form is adequate. If it appears adequate, continue.

If instead the data show statistically significant lack of fit to the assumed model, try

to understand the reasons, and use a different model. (KS, QQ)

(6) Investigate whether the components have a common value of AO. If they do, continue. If

they do not, try to find the reasons, and consider splitting the data. (CC L)

(7) Generate a two-dimensional confidence region for (f3, AO), for comparison with the

confidence ellipse to be produced in Step (9). (Cl +)

The remaining steps should be based on the full likelihood, and on the assumption that

(/3, logAO), the maximum likelihood estimator (NILE), has approximately a bivariate normal

distribution. (FL, AN)

(8) Find the MLE for (1,, A0). (NIL +)

(9) Investigate the adequacy of the normal approximation for the distribution of (/3, logAO).

(Cl +)

(10) Get the MLE and a confidence interval for A(t) at various values of f. If the normal

approximation is deemed adequate in Step (9), generate an approximate confidence

interval; otherwise generate a conservative one. (NIL H, CI H, HF)

A sample PHAZE session, following the above outline, is given in Appendix A.

1.4 Terminology and Formats

The following terminology is used throughout the later explanations, because all input for

PHAZE is in free format. A separator is a blank, comma, or equal-sign. Free formal means the

following: Entries may appear anywhere on a line. However, every floating point number must be

immediately followed by a separator, and every integer must be immediately followed by a nonnumeric character. If successive character strings have a fixed length, they may optionally have

separators between them. A character string that does not have a fixed length must be followed by a

separator, and may not contain separators as part of the string.

4
Commands and their arguments may be entered in either upper case or lower case letters. The

only restriction is that two-letter commands must be entirely upper case or entirely lower case. Letter

arguments for a command do not have to be in the same case as the command.

2. DATA FILES

* A data file records failure information that arose as follows. Each component was installed, or

placed in service, at a certain time. The observation period, during which any failures were recorded,

began on or after the installation time, and continued until some final time. The final observation

time may simply be when no more data were recorded (for time censored data), or it may coincide

with a failure that resulted in the component's being removed or replaced (in which case the data are

treated as failure censored.)

The failure data are read from ASCII files, one file for each set of similar components. The file.

consists of a series of lines, with each component corresponding to one or more lines. The numbers for -'., -P;

a component are in free format, but they must appear in a specified order. The elements that must be

present for any component are, in order:

Component name (up to 14 characters, without embedded separators,' not beginning with')

Installation time (optional if times are dates and if installation date equals initial observation date,'

required otherwise) ' "

Initial observation time

Final observation time

Number of failures in the observation period, an integer N ' '

N failure times . - . ;

The letter .R (upper or lower case) if the component was removed or replaced-at the time of the final

failure. . - - . ; :

The times may be entered as dates, integers, or floating point numbers, but all the times in the

file must be of the same form. The form of the first time encountered in the file determines: whether

the times will all be. interpreted as dates,,integers, or floating point numbers. Dates are in the

yy'mmdd format. For example 840329 means March 29, 1984. If the first time in the file'is an integer

5
> 50010i, then all times are interpreted as dates (Jan. 1, 1950 or later). If it is a smaller integer, all

times are interpreted as integers. If it is a floating point number (that is, with a decimal point or in

scientific notation), all times are interpreted as floating point numbers. All times other than dates are

treated as if they are in units of hours.

Figure 1 shows an example of a data input file, using dates, with each line containing the

elements listed above. For example, look at the third line. It says that the component NlOV-1C was

installed on Dec. 1, 1972. It was observed from Aug. 24, 1977, through October 1, 1987. During that

time there were two events recorded as failures, on April 23, 1983 and on March 4, 1987. Note that

three of the components were replaced as a result of their final failure; for those components, the final

failure date and the end of the observation period agree, and the final 'R' tells PHAZE that this is not

a coincidence.

If the inputs are dates, then the installation dates may be omitted. In that case, the installation date is set equal to the initial observation date. In any file, the syntax must be the same for all

the components: either all have installation dates or none do.

A data record for a component may be continued on subsequent lines as follows. If fewer than

N failure times have been entered, the last failure time on the line should be followed by (optional

blanks and) a plus-sign. Then the next line will be interpreted as a continuation of the data for the

component. Any non-continuation line that begins with a # will be interpreted as a comment, and

ignored by PHAZE.

fn Figure 1, note also that three of the component records are continued onto a second line, as

indicated by the final plus signs. The installation dates are given; if the installation dates had all been

equal to the beginning of observation, as they are in only a few cases, then all the installation dates

could have been omitted. The comment at the end of the file is identified by the symbol #. The

entries are arranged in neat columns for legibility, but this is not necessary.

Figure 2 shows an example data file with integer failure times. PHAZE treats the times as

integers rather than dates because the first time is less than 500101. Even though the installation time

equals the beginning of observation, the two numbers must both be entered.

6
I

MOV-1A

31OV-1B

M1OV-1C

MOV-1D

51OV-1E

hIOV-1E(R)

MOV- 1F

MOV-iF(R)

NIOV-2A

MOV-2B

MOV-2C

MOV-2C(R)

AIOV-2D

MOV-2E

MOV-2E(R)

51OV-2F

721201

721201

721201

721201

721201

800220

721201

820815

730501

730501,1

730501

830427

730501

730501

800323

730501

770824

770824

770824

770824

770824

800220

770824

820815

77.0 824

*770824

770824

871001

871001

871001 ''

871001

800219

871001

820814

871001

871001

871001

830426

1

1

, 2 

810618

780706

830423 87C

830411 83C

'860131 87C

-' 1800219

3 810611

4 780605

; 1 850213

7 4,781015

5 800826

2 811207

!1 870226

83C

8ic

)304

)520

)304

)313

)325

840620 850814 860128 +

850626

801104

830426

870219

811001

..: C . .

850628

821218

r.

820814 R

851029

850620 861204

830427 871001 7782 8711 . .

770824 871001

..

6 780407 800513 800602 821218 850620+

770824

800323

770824

.. .' ' I- , t -860715

800322 .0 -.

871001 2 850620 i

871001 7 800509 E

350624

321218

350626

830424 830819 840412 +

850620 i

# File used as example by Atwood (1990) .!,

Figure 1. Data file with times entered as dates

Compl 1. 1 1000 5 212 414 605 883 912

Figure 2. -Data file with single component, integer'failure-times

validation-1 0. 3. 4. 1 3.7

validation-2 0. 3. 5. 3 3.6667 4.3333 5. r

#Data for validation test of PHAZE, 2 components, 4 failures, 1 replacement

Figure 3. Data file with floating point failure times.

7
Figure 3 shows an example of a data input file with floating point failure times. The first time

entered, in this example 0., must contain a decimal point or be in scientific notation (OEO). The subsequent times are not required to have decimal points, but will be interpreted as floating point numbers.

For example, the entry 4. could have been entered as an integer 4 without changing the way that

PHAZE interprets the entry.

The exact times of failure are interpreted slightly differently, depending on whether the input

times are dates, integers, or floating point numbers. Suppose for example that the observation period

was from 850101 to 851231, i.e., from January 1, 1985, to December 31, 1985. In this case. the observation period is considered to be 365 days long (1985 was not a leap year). That is, the observation

period goes from the beginning of the day January 1 (just after midnight) to the end of the day December 31 (just before midnight). Likewise, the installation time is interpreted as the beginning of the

installation date. Failures are assumed to occur at the midpoint (noon) of the failure date, the only

exception to this is that replacement failures are counted as occurring at the end of the day, coinciding

with the end of the observation period.

When the inputs are integers, the times are interpreted similarly. In the example of Figure 2.

suppose for simplicity that the integers stand for hours. The installation time and the beginning of

observation are at the beginning of hour 1, the end of the observation period is at the end of hour bOOO,

and the observation period lasts for 1000 hours (not for 1000 - 1 = 999). The failure times are

assumed to be at the middle of the hour. Therefore, for example, the first failure occurs 211.5 hours

after the start of the observation period.

When the inputs are floating point numbers, the times are interpreted just as they are written.

In the example of Figure 3, the second component has an observation period that lasts 2 hours

( =5. - 3.), and the first failure occurs 0.6667 hours after the start of the observation period.

8
3. PHAZE COMMANDS

3.1 Command Syntax

PHAZE has a command structurerather than ,a menu structure. That is, there is a set of

commands that the user must know. The program prompts the user for a command, executes this command, and then prompts the user for the next command.

Most of the commands-consist of two letters; only H and Q (for help and quit) consist of a

single letter. Some of the commands require arguments, either letters or numbers depending on the

command. Most of these arguments can be 'entered 'on the command line. If they are not entered.

PHAZE will request them from the user. Input is' in free format.

The commands are listed in Table 1, and presented below in alphabetical order. The.

commands are written in BOLD CAPITALS, and the arguments are written in italics. In each case.

the command, with its possible arguments, is given at the left, and the command in English is given in

the center. The bold capital letters in the English name of the command highlight the mnemonic

relation to the PHAZE commnand. Values for the arguments must be substituted by the user. For

example,

ML parameter

is the command to find the maximum likelihood estimate (MLE) of a parameter to be specified. If the

user enters

ml b

MLB

mlb

or any other variety of the command, PHAZE will find the MLE of /.

The commands CC, CI, and ML all take a parameter as an argument. With some restrictions

depending on the command, the parameter generally may take the values

B for /6 .

L for L0

+ for both / and A0

H for the hazard function A(t).

9
Table 1. Summary of commands

Parameter Meaning

AN

CC parameter

CD

CI parameter conf

DD

DU

FL

GD filename

II class

H1F time

HIF R timel time2

KS

LE

ML parameter

OU filename

PF

Q

QQ

SB value

SI

SM model

SN time

ST

TE value

*

Toggle option to treat (3, logAO) as having a bivariate

Asymptotically Normal distribution

Compare Components for parameter

Toggle option to Center Data, for linear hazard model

Find conf % Confidence Interval for parameter

Describe Data

DUmp values in storage, for code development only

Toggle option to use Full Likelihood

Get Data stored in filename

Show Help messages for a class of commands

Find Hazard Function at time

Find Hazard Function in Range timel to lime2

Kolmogorov-Smirnov test of the model

Toggle option to assume AO's Equal for all components

Find the Maximum Likelihood estimate (MILE) of parameter

Define OUtput file, filename, for echo of session

Toggle option to allow creation of Plot Files

Quit

Calculate values for a Q-Q (quantile-quantile) plot

Set 60 to value

See Data in concise display

Select Model model

Set Normalizing to to time, for Weibull model

Show STatus

TEst 61 = value

Comment

For code development only

10
' Certain commands have as an argument the name of a file, when this file is to be read or

written. The filename may contain disk and directory information, and must contain the DOS extension, if any.' Examples could be a:\data\mov.txt and \valves\chkval.dat. The filename may be at

most 22 characters long. Thiee filenames are reserved for special meanings: NONE, HELP, and QUIT. - - d !. *t'~ I

in upper case or lower case. If a command is given with the filename NONE, no file is used. (This can

be'used to turn off writing to an output file, or to' cancel the search for an input file which cannot be

found for some reason.) If the filename HELP'is entered, PHAZE briefly explains the need for a

filename, and the' user's po'ssible responses. If the name QUIT is entered, PHAZE stops executing and

returns control to DOS.

3.2,'General Concepts

* . -. *-,-~. ,. . ., * '

* PHAZE stores the current" environment. This environment consists of the'choices listed below.

In parentheses the initial value is shown, the value in effect at the start of the session. Also shown in

capital letters are the commands that directly affect that choice.

* the data set most recently read (no initial data set; GD)

* the selected model

- exponential, linear, or Weibull hazard modlel (no initial model; SM)

- components assumed to have common A' or not (initially no common A0; LE)

- data centered or not,'for linear hazard model (initially not centered; CD)

- value of normalizing to- for Weibull hazard model (see SN for initial value; SN)

* analysis options

use conditional likelihood or full likelihood (initially conditional; FL)

- use bivariate asymptotic normality of (A, logAO) or not' (initially do not; AN)

* output options

- echo session to a disk file or not (initially do not; OU) '

- allow writing of plot information to disk files or not (initially do not; 'PE)

.. .. ¶ .. . :,, , . , ,-. . , . ; :'. :.

PHAZE also stores a current value, denoted 60, as the default value of 1. This default value is

used when A0 is to be estimated by the MLE or by a confidence interval. (An estimate of AO depends

on 'the assumed value'of 68.) This default',60'also stpecifies the hypothesized value for' the commands

TE (test whether' P _ o),'KS (perform a kIolmngorov-Smirnov test of the assumed model, with

- - ! ' ' 'I ' '~~~~~1i -''In--
d = o0), and QQ (generate a Q-Q plot for the assumed model, with ;3 = '30). Whenever a data set.

model, or analysis option changes, the value of ,3 is set to 0; the only exception is that changing to

does not change the meaning of hi, and so does not cause l'o to be reset. Whenever the MILE 3 is

found, g3o is reset equal to 3; most applications, such as finding the NILE of AO, or checking the

goodness of fit of the model to the data, will be done when g3o ='3. The user may also set 30 manually

by the command SB. Sometimes the calculation of i, which results in resetting 3o, is invisible td the

user. For example, to compare components to see if they have a common value of 3, the MILE 3 must

be found. Therefore the command CC B changes ,30 from its previous value to 3. Commands that

use go will print the value used. To be sure in advance of the value of go, use the command SB or

ML B.

Some of the commands are toggle commands, that is they turn a switch on or off. The switch

determines which of two possible options PHAZE uses. For example, *PF is such a command. Each

time the command PF is entered, the plot-file switch changes, either from off to on or from on to off.

3.3 Commands

AN Toggle option to treat (j3, logAo) as having a bivariate

Asymptotically Normal distribution

This toggles the bivariate-asymptotic-normality switch on or off. If the switch is on, the confidence

region for (3, logAO) is constructed to have an elliptical shape. If the switch is off, the region is based

on a confidence interval for '3, and a confidence interval for A0 for each value in the confidence interval

for '3. The effect of AN on confidence regions is detailed in Table 2. The default value of the bivariate-asymptotic-normality switch is off.

Turning on the bivariate-asymptotic-normality switch may reset other switches: The equallambdas switch (command LE) will be turned on if it is off, and the full-likelihood switch (command

FL) will be turned on if it is off. These changes are announced to the user when they happen.

CC parameter Compare Components for parameter

The allowed parameter values are B, L, and +. This command compares the components to see if they

have a common value of i (when the parameter is B), or a common value of A0 assuming a common

value of ,3 (when the parameter is L). Using + as the parameter is equivalent to the calling CC B

12
Table 2. Effect of switches on CI command . . I ". .I . . :-- i - .

Asymptotic-Normality Switch Off

Full-Likelihood Switch Off

I ' , . .. I. I

Asymptotic-Normalitv Switch On

Parameter Full-Likelihood Switch On

13 .Use asymptotic normality of conditional 'Use asymptotic normality of (B. logA~).

log-likelihood. get confidence interval from marginal

:'distribution of 1-

A0 Use exact distribution of LO, assuming

6 = 13

o* NOTE, this is done whether

full-likelihood switch is on or off.

(pI, So) Find confidence interval for ,B, then for

each 60o in interval find confidence interval

for A0. Use (1 -a/2) confidence intervals

to get (1 -ar) region, for example 95%

intervals to get 90% 2-dimensional region.

A(t) Find maximum and minimum values of

A(t) for (6,; AO) in 2-dimensional confidence

region, yielding conservative interval

for A(t).

Use asymptotic normality of (0i, logAO),

get confidence interval from marginal

distribution of logAO.

Use "asymptotic bivariate normality of

(B, logLo) to get confidence ellipse for

(,6, logA0).

Use approximate normality of logA(t).

! I II .-, .7 11 I- ' I

13
'followed by CC L. If the parameter is not specified, the value used is the one remembered from the

most recent use of CC, Cl, or NIL.

Figure 4 shows a sample print-out generated by the command CC B. It is explained here. For

this comparison it is not assumed that the components all have the same value of d. Let :3j denote 3

for the jth component. The estimate of ,Bj is /j; this NILE uses only the data from the jth component.

Similarly, let Pi3 LO and Pi, UP denote the lower and upper ends of a 95% confidence interval for 43.

based only on data from the jth component. Finally, suppose that all the components except the jth

have a common value of 3, denoted by i

3..j and estimated by the NILE /3.. The first page of Figure 4

shows, for each component, the numerical values of (P3 LO, ip, P3i Up) and of 13..j. The second page of

Figure 4 shows the significance level, based on the difference j - /3,j and on the variance of 3., and

/3,,,. The exact method of calculation is explained in Section 6.1 of EHF, and in Section 5.2 of this

report. The significance level is the probability that 3j and 13..j would be as far apart as actually

observed, if in fact all the components have the same S. A small value indicates strong evidence that

the jth component has a different ,3 from the other components. Finally the print-out shows a plotted

.95% confidence interval, enclosed by parentheses, with /j shown by a star (*), and 3.. shown by an 1.

For several components in Figure 4, not all the symbols are shown. In these cases, one symbol was

printed over another symbol, thereby erasing it. The hierarchy is *, (,.

Although Figure 4 does not show this, there may be plots in which an interval continues off

the area shown; this is indicated at the end of the plotted portion by < or >. This occurs if a confidence interval is infinite (as can happen with the linear hazard model). In such a case, the numerical

printout of the intervals, as illustrated in the first page of Figure 4, should make clear what has

happened.

At the bottom of Figure 4 is the overall NILE and the overall 95% confidence interval,

calculated under the assumption that all the components have a common A. This is shown just as for

the individual components, except that ,

3 j is now meaningless, and the significance level is not shown.

To get an overall significance level, we must recognize that some components will appear extreme from

chance alone. This can be accounted for by using the Bonferroni inequality, discussed in many texts,

and by Alt (1982). In the present context, for any number c it says that

P( at least one of k significance levels is < c ) < kc.

14
-> ccb

95.0% conf .

comp.

MOV-1A

MOV-1B

MOV-1C

MOV-1D

MOV- 1E

1OYV-1E(R)

MOV-11F

MOV-1F(R)

blOV-2A

MOV-2B

MOV-2C

bIOV-2C( R)

MOV-2DMOV-2E

MOV-2E(R)

MOV-2F

OVERALL

intervals with MLEs for beta, by component and MLE without

(-l.lOE-04,-1.'72E-05,-6.;48E-05)

(-4.85E-04,-1.32E-04, 2.55E-05)

(-1.77E-05, 4.09E-'05; 1;08E-04)

( 4.06E-06, 3.90E-05, 7.52E-05)

not estimable

(-5.41E-05,-4.31E-'07, 5.32E-05)

(-6.39E-05, 1.88E-05, 1.03E-04)

(-1.67E-04,-3.35E-06, 1.59E-04)

(-2.04E-05, 1.67E-05, 5.46E-05)

(-2.35E-05, 9.12E-06, 4.20E-05)

(-8.71E-05, 7.48E-05, 3.04E-04)

(-7.88E-05, 1.91E-04, 6.82E-04)

(-3.47E-05, -4.86E-06, 2.49E-05)

not estimable

(-3.18E-05, 3.99E-05, 1.19E-04)

(-1.55E-05, -1.29E-05, 4.16E-05)

( 5.32E-07, 1.34E-05, 2.62E-05)

1.42E-05

1 .54E-05

1 .21E-05

8.57E-06

1.41E-05

1.33E-05

1.35E-05

1.30E-05

1.40E-05

1.29E-05

1.28E-05

1 .71E-05

1.26E-05

I .35E-05

Figure 4. Example output from CC B command. (Page 1 of 2)

:. I . . - 0 ,

15
Comparison

Component

MIOV-1A

MOV-1B

MOV-1C

MDOV-1D

MOV-1E

MOV-1E(R)

MOV-1F

OIDV-1F(R)

MOV-2A

SIOV-2B

MOV-2C

NIOV-2C(R)

MOV-2D

MOV-2E

MOV-2E(R)

MOV-2F

of beta values for components

Signif 95.0% Confidence Interval

0.48 (----*I-)

0.09 (---------------*------I)

0.44 (1I--)

0.14 I*-)

0.64

0.91

0.82

0.87

0.80

0.61

0.33

0.21

(-*I-)

( *)

(------*I------)

( ---- I--*__________-)

(*I)

0.54

0.97

(-1*---)

(*)

OVERALL

Overall significance

(*)

level for testing equality of betas = 1.00

Signif for jth component means

beta-hat(j) - beta-hat(others)

estimated probability that

is at least as extreme as observed

Overall significance level is PCat least one signif(j) <= observed min]

and is computed with Bonferroni inequality

Figure 4. (Page 2 of 2)

16
The inequality is close to equality when kc is small. Therefore the overall significance level for testing

equality of the 13j's is the number of components having calculated significance levels times the

minimum significance level calculated for a component. A small value of the attained overall signifi

cance level (say 0.05 or smaller) shows that there is strong evidence against the hypothesis that'all the

components have the same value of 6. The approximate overall attained significance level in Figure 4

is 14 (the number of components having calculated significance levels) times'0.09'(the smallest 's'ignificance level attained), which is larger than 1.0. Therefore it is printed as 1.00 in Figure 4.' 'This 'means

that there is no evidence against the assumption that all the components have the same value of 3.

PHAZE uses the Bonferroni multiplier k when k> 3, that is, when three or more components

have significance levels printed. When k= 2, the smallest number for which a comparison of compo- ., t ... ..

nents is possible, the Bonferroni multiplier is not used, and the overall significance level reported is the

smaller significance level' of the two. The reason for this is as follows:. Normally, the significance level

for -the first component is based on the difference while wB1-.&i, thesignificance level for the second is

based on 62 - 61. Therefore there is ~only one iignificance'level,' corresponding to the difference, 'and

multiplying it by two would be incorrect. Normally the significance levels printed 'for the two

components are the same, although they can be calculated as different if one component has one failure

while the other has more, as explained in Section 5.2.

CD Toggle option to Center Data, for linear hazard model

For the linear hazard model, time can be measured from any point, not only from the component's

installation time. A useful time to use as the origin is the center of the, observation period, t

m'd'

defined precisely in Section 5.2. This causes the estimators / and AO to be independent of each other,.

as explained in Section 4.3 of EHF. The CD command toggles the value of the center-data switch

between on and off. If on, the data are centered. If off, the data are uncentered, and times, are

measured from the component's installation. The default value is off.

In a future version of PHAZE, this command rmay be extended to CD value. The input' for

value will be a user-defined origin, from which all times are to be measured. 'The command CD

without any argument- will work as before, centering the' data at t

mid if the data 'were previously

uncentered, and uncentering the data if they were previously centered at any'value'other than 0. 'The

command will then work for both the exponential and the linear hazard models. When this extension

is implelnented, it will be shown in the output printed by the Help command.

-17
CI parameter conf Find conf% Confidence Interval for parameter

The parameter may be B, L, +, or H. The confidence level conf may be any number between 0 and

100. If the parameter is B or L, then a confidence interval is found for 3 or A0, respectively. If the

parameter is H, the command may be entered as

CI H conf time

PHAZE will read t, the value of time, or will prompt the user for t, and then will produce a confidence

interval for the hazard function A(t) at t. If the parameter is +, then a two-dimensional confidence

region is found for (t, A0). This region is also plotted, with A0 plotted on a logarithmic scale. If the

PF switch is on, the information for the two-dimensional region may be written to a disk file for later

analysis or plotting. If the arguments are not all specified, PHAZE will request them.

The form of all the confidence intervals and regions depends on whether the full-likelihood and

asymptotic-normality switches are turned on or off. The allowed combinations are shown in Table 2

with their effects.

The command CI + can be used to check the assumption that (j, logAO) has approximately a

bivariate normal distribution. First use the conditional likelihood and no assumption of bivariate

asymptotic normality, and generate a two-dimensional confidence region. Then use the full likelihood

and the asymptotic normality assumption to generate a confidence ellipse. The bivariate normal

approximation appears adequate if the two confidence regions cover mostly the same territory, that is.

they overlap well, and if the confidence ellipse is not truncated by some theoretical limit. Figures *;

through 8, taken from EHF, illustrate the concept. Figures 5 and 6, based on the exponential and

Weibull hazard models, respectively, show very good agreement between the two regions, indicating

that the bivariate normal approximation is quite acceptable for those models and that data set. Figure

7 shows the two regions based on the linear model and uncentered data. The overlap is terrible. and

the ellipse is truncated at the theoretical lower bound for A3. Figure 8 shows the two regions based on

the linear model with. centered data. The overlap is good, but the ellipse is truncated at the theoretical

upper bound for fl. These two pictures show that the bivariate normal approximation is inadequate for

the linear model, whether the data are centered or not.

Such overlays are not printed directly by PHAZE; output disk files from PHAZE were used as

inputs for a graphics package, which printed Figures 5 through 8. In a future version of PHAZE, the

command Cl may be extended to allow the parameter to be 0. (The 0 stands for overlay.) The

18
3.OE-5

2.5E-5

2.0E-5

. 1.5E-5

1 .OE-5

5.0E-6

O.OEO

Figure 5. 9

1.8

1.6

1.4

1.2

1.0

= 0.8

0.6

0.4

0.2

0.0

-0.2 -

2

3 4 5~ 6 7 8 9 23 -- - rgnf Abeox 1 0 X (ham)_ . . ..

90% confidence regions for (/3, At,), based on exponential hazard model

4 5 6

I

i

I

a A - . .-- a - -9 * ''--~'

1 0-4 - Xo (1 /h)

Figure 6. 90% confidence regions for (13, AO), based on Weibull hazard model

19
8.OOE-4

6.00E-4

4.OOE.4

2.OOE-4

O.OOEO

-2.OOE-4

4 5

Figure 7. 90%

1 .50E-5-

1 .25E-5

1 .OOE-5

_ 7.50E-6

5.00E-6

2.50E-6

O.OOEO -

2

Figure 8. 90%

2 3 4 5 2 3 4 5

1 0-6 Xo (I/h) 10

confidence regions for (13, AO), based on linear hazard model. uncentered data

el

3 4 5 6 7 a 9

Xo(1/h) 104

confidence regions for (1S, AO), based on linear hazard model. centered data

20
command CI 0 conf will result in two calls' of CI + con!, first with bivariate asymptotic normality

- turned off, then with it turned on. The two resulting confidence regions will be overlaid in a printer

plot on the terminal screen, instead of only being printed separately as they are now. When this

. change is implemented, the message printed by the Help command will show this.

, .- . ., S . * . . -, - S i . :

DD Describe Data

This command generates summary statistics for the current data set. For each component, PHAZE

prints the component's name, the number of nonreplacernent failures, the length of the observation,

period, the normalized mean of the nonreplacement failure times, whether the component.was replaced

upon failure at the end of the observation period, and the component's age at the start of the

observation period. The normalized mean of.the nonreplacement failure times means the following.

Let i be a nonreplacement failure time, and let so and s, be the start and end of the observation

period, respectively. Then the' normalized replacement time is (t - so)(s- so), and the mean is the

arithmetic. average of these terms. The output from DD is briefer than the output from SD, only one

line per component.

DU DUmp values in storage, for code development only

This command was used in the code development. It writes values stored in a multi-purpose common

block to the terminal. It is a command for programmers and debuggers only.

FL Toggle option to use Full Likelihood

When the full-likelihood switch is on, the full likelihood is used for maximum likelihood estimation of

16 and confidence intervals, for 6. When it is off, the conditional likelihood is used for these estimates

of 1. Estimation of A0 is always based on substituting the default Po into the full likelihood. (The

meaning of 60 is explained in Section 3.2.) The initial value of the full-likelihood switch is off.

The full and conditional likelihood are explained briefly in Section 1.2, and more thoroughly in

EHF. The conditional likelihood is appropriate -when.it is not 'certain whether 'all the 'components have

the same value of Ac, and when the primary interest of analysis is in P;. Therefore it is appropriate

when the. user initially is trying to decide if .if is positive, and.is not sure if 'the components have a

common Ac or if the model form is correct. The full likelihood is appropriate when' /3 and Aware to'be

used together to estimate the hazard function. ; -

21
GD filename Get Data stored in filename

This command causes PIIAZE to find and read the specified file. If the filename is not given. PHAZE

will prompt the user for it. The file must be in the format described in the Section 3. "Data Files for

Input." The data file is checked for internal consistency in many ways, and the results are announced

to the user. If fatal errors are found, PHAZE will not use the file.

H class Show Help messages for a class of commands

This command causes PHAZE to write information that approximately fills a terminal screen. The

printed information is a list of certain commands, each with its possible arguments, and a brief

explanation of the command. Use class D for commands that control Data or files, I for commands

that perform statistical Inference, and 0 for commands that set or show Options.

HF time Find Hazard Function at time

This command produces the MLE and 90% confidence interval for the hazard function \( I), where I is

the specified time. It is a short command to accomplish the same result as the sequence

ML H time

CI H 90 time

As mentioned for the CI command, the confidence intervals depend on the settings of certain switches.

as-shown in Table 2.

HF R timen time2 Find Hazard Function in Range timel to time2

This executes the command

HF t

for evenly spaced values of t in the range time) to time2. The default range is the total range of the

component observation periods, so that timel equals the minimum observed age and time2 equals the

maximum observed age. If no times are entered on the command line, PHAZE reports the default

values, and lets the user select them or other values. The number of increments in the range defaults

to 20, but is controlled by the user. If the bivariate-asymptotic-normality switch (command AN) is on.

the distribution of the MILE A(t) is approximately lognormal. In this case the mean and the error

factor (defined as the 95th percentile divided by the median) of the distribution are also printed, for

each requested time t in the range. If the PF switch is on, the user has the option of writing this information to a disk file, so that the hazard function A(t) (MLE and 90% confidence interval, and the

median and error factor in the lognormal case) can be plotted as a function of t or used as inputs to

other programs.
KS Kolmogorov-Smirnov test of the model '

This performs a Kolmogorov-Smirn6v 'test of the' hypothesis 'that the data come'from the assumed

model, with = 83 f0. This formal statistical test corresponds to the informal graphical test given bv the

QQ command. Here 6i0 is as described in Section 3.1 "Command Syntax,' either the initiail"vaIe of 0.

or the MLE A, or a value that the user has defined with the SB command. The test uses the

conditional distribution of the data, :given the:failure :counts and the final observation'times for the

components. This conditional distribution depends only 'on 6, not on the (possibly different) values of

the Ao's; it is written out at the beginning of Sectionr 6.3 of EHF.' As discussed in EHF,' when KS is

used to test the adequacy of the model form, setting fi 0='-' results in not 'rejecting the model often

enough, because the best fitting 6 --has been used; that is, using the best fitting f0 'violates the

theoretical assumptions, and therefore'the true'significance level is smaller 'than the printed significance

level. On the other hand, no better method'is known.- '

When 80 = 0, a second Kolmogorov-Smirnov test is also performed. of the hypothesis that all

the failures result from a Poisson process with constant failure rate. This hypothesis is true when all

the components have , = 0 and a'common Yaliue 'of\'O,, and the number of components being observed

is the same at 'all times (that is,' no components are introduced part way through the observation

period, a'nd'any component replacements are performed instantaneously.) For this hypothesis, times

are thought of in terms of the observation date, not in termms of the components' ages.

As discussed in Section 4.3 below, the significance level calculated by PHAZE may be inaccurate by 1% or more. This means that the theoretical significance level is not calculated accurately;.

this is not to be confused with the above comment that the actual significance level is smaller than the

theoretical one.

When the PF switch is turned on, the command KS offers 'the user the option of printing'

output to a plot file. - .'

LE .. .- Toggle option to assume AX0s Equal for all components

If the equal-lambdas switch is on, the 'components are assuimed to have a common' value of ',\. If the'

switch is off, they are not assumed to necessarily have a common value of O.- The initial value is off."''

23
Turning off the equal-lambdas switch may reset other switches: The bivariate-asymptoticnormality switch (command AN) will be turned off if it is on, and the full-likelihood switch (command

FL) will be turned off if it is on. Thus, turning off the equal-lambdas switch resets the analysis options

to their original values.

ML parameter Find the Maximum Likelihood estimate (MLE) of parameter

Here parameter may be B, L, +, or H, to obtain the NILE for 3, A\, the pair (p3, \O), or the hazard

function A(t), respectively. If the bivariate-asymptotic-normality switch is turned on, via the AN command, then NIL + produces not only the MLE for both parameters, but also the Fisher information

matrix and the asymptotic covariance matrix for (A, logA0). The formulas used, and the resulting

NILEs, depend on whether the full likelihood or the conditional likelihood is used, and whether the AOs

are assumed to be equal or not. These options are controlled by the commands FL, AN, and LE. If

the parameter is not specified, the value used is the one remembered from the most recent use of CC.

CI, or ML.

OU filename Define OUtput file, filename, for echo of session

This command allows the user to echo the current session to a disk file, for later printing or for documentation. To change the output file from a current file to no disk output, enter OU and enter NONE

for the filename. Initially there is no disk output file.

PF Toggle option to allow creation of Plot Files

The commands CC, CI +, HF R, and QQ all produce output that could be plotted graphically. If the

plot-file switch is turned on, PHAZE will ask the user if this output should be written to a disk file, for

possible later use by a graphics program. If the plot-file switch is turned off, PHAZE does not offer the

possibility of writing such a file. The initial value for the plot-file switch is off.

Q Quit

This command terminates the PHAZE session, and returns control to DOS.

QQ Calculate values for a Q-Q (quantile-quantile) plot

Q-Q plots are described in Section 6.3 of EHF and by Snee and Pfeifer (1983). A Q-Q plot provides a

graphical check of the correctness of the model, corresponding to the quantitative test of the KS

command. Figure 9, taken from EHF, shows an example of a Q-Q plot. The points fall approximately

on a straight line, indicating that the model fits the data well.

24
1.2E1 r

-1 .0125

'E 8.0E4 -- if

68.OE4

4.OE4 ~ ~ Us6.0E4 /1

: 4.E4 - 0 -/

2.0E4

O.OEO .

O.OE0 2.0E4 4.0E4 6.0E4 8.0E4 1.0E5 1.2ES

Observed Age at Failure (h)

Figure 9.Q-Q plot, bas on exponential hazard model

25
The output from the command QQ consists of a list of the observed failure times. and a

corresponding list of the expected failure times under the assumed model with ,3 = 30. These lists give

the z and y coordinates of the points in the Q-Q plot. Here So is as described in the Section 3.2. either

the initial value of 0, or the IfLE i3, or a value that the user has defined with the SB command. The

expected failure times are calculated based on the conditional distribution, given the failure counts and

the final observation times for the components. This conditional distribution depends only on 3, not

on the (possibly different) values of the AO's. It is the same distribution that is tested by the command

KS.

When 30 = 0, a second pair of lists of failure times is produced, based on the assumption that

all the failures result from a Poisson process with constant failure rate. This is assumption is true

when all the components have .3 = 0 and a common value of AO and when the number of components

being observed is the same at all times (that is, no components are introduced part way through the

observation period, and any component replacements are performed instantaneously.) For this assumption, times are thought of in terms of the observation date, not in terms of the components' ages. This

Q-Q plot is equivalent to a cumulative failure plot, that is, a plot showing the total number of failures

that have occurred by any time. Under the assumption, the expected failure times are uniform over

the observation period. If they are plotted on the vertical axis, they have equal spaces between them.

To make a cumulative failure plot instead of a Q-Q plot, relabel the expected failure times as 1, 2, etc.

instead of as times.

When the plot-file switch (command PF) is turned on, the command QQ offers the user the

option of printing the lists to a plot file.

At present, the command QQ does not cause a Q-Q plot to be printed on the terminal screen:

it only prints a pair of lists for use by a graphics package. A Q-Q plot on the terminal screen may be

added to a future version of PHAZE. When this capability is added, the command structure of QQ

will not change at all, only the form of the output.

SB value Set 60 to value

This command sets 60 to the specified number value. See the explanation of the use of 30 in Section

3.2. The number 80 is used by the commands CC L, CI L, KS, NIL L, QQ, and TE.

26
SD See Data in concise display

This command takes the current data set and, prints the failure information'for each component. with

all times measured from the component's installation. "Dates are converted to elapsed hours. The

information printed 'consists of: "the component's name, age' at the 'beginning' and at the end of the'

observation period, the age at the time 'of each nonreplacement failure, the average of these ages. and'-

the age at the component's replacement if the component 'was replaced upon failure at the end of the

observation period. This' command gives more output than DD, because it lists the 'individual failure

times. ':' "

SM model Select Model model

The value entered for Model must be E, L, or W, for the exponential, linear, or Weibull hazard model.

There is no default. , -

SN time Set Normalizing t

o to time, for Weibull model

This sets to to the specified time. The default value of to' is maxjrj, the largest observation range of

any component, where the observation range ri is 'defined as the the end time of the observation period

for the jth component minus the starting time of the observation period. A value for to such as t

defined for the command CD in Section 5.2, results in:MLEs #" and A0 that are less strongly correlated.

The SN command is meaningful only for the Weibull hazard model.

ST Show STatus ' '' ' '' '

This command causes PHAZE to show the status of the current environment, as defined in Section 3.2.

The current values of the data, the model, and the analysis options are all shown. The values of the

output options are shown if they are not the (write-nothing) defaults.

TE value TEst ft = value

This performs a statistical test of the hypothesis

Ho: 6 ='value ; '

against the one-sided alternative

HI: 13> value

To test for the presence of aging (,6 positive), value should be entered as 0. If no number is entered for

value, PHAZE 'uses the current value' of PO.' Note, t isicommand does not reset P0o regardless of the

number'entered as value. Regardless of the current value of the full-likelihood switch (command FL),

the test is based on' th'e conditional likelihood, is'explained in Secti6 n 5.2 below.

27 ' ' 
A test result is shorthand for part of the information given by a confidence interval. The relation is that the test accepts the values that are in a confidence interval, and rejects all other values.

More precisely, a test rejects L0 at the significance level cc if 30 is not in a 100(1 - a)% confidence

interval. Two-sided tests correspond to two-sided confidence intervals. The one-sided test performed

by the command TE corresponds to a one-sided confidence interval (OL3 0, co). Therefore, as a supplement to the command TE, it is helpful to construct a (two-sided) confidence interval for 03, with the

command CI B. This shows whether the range of plausible 3 is wide or narrow.

Comment

Any line beginning with # is treated as a comment. It is ignored as a command, but is echoed to any

output file that was established by the OU command. If the terminal is being copied to the printer

(via Ctrl-PrintScreen), the comment will be printed there.

* * For code development only

This character was used for testing the code. By changing the coding corresponding to the command .

PHAZE can be directed to execute various commands. The present coding merely causes PHAZE to

read the next command, so * can be used for a comment line.

4. VEIUFICATION AND VALIDATION

This section describes the work that was done to verify and validate the correctness of PHAZE.

4.1 Overview

ANSI/IEEE Std 730 (IEEE, 1984) states that software is logically produced in three steps:

* software requirements specification (SRS)

* software design

* code (in some computer language)

The SRS describes the requirements (functions, performances, design constraints, and attributes) of the

software and the external interfaces, while the software design specifies the algorithms, data bases,

internal-interfaces, etc. 'Verification" then means the ascertainment that the requirements of the SRS

are implemented in the software design, and that 'the design is correctly implemented in the code.

28
'Validation" means the demonstration that the code, when executed, complies with the requirements of

the SRS.

'The SRS is that the 'program should read data and 'then carry out the methods given by EHF

to analyze the data. All results should be accurate to at least two or three significant digits.

The software design was chosen for simplicity as follows. The main program consists primarily

of a long series of if-then-else blocks, with each block corresponding to one of the allowed commands.

When the user enters a command, the main program proceeds through the series 'of blocks until it

locates the cornimand,' at' which point it calls the appropriate' subprogram to execute the command.

When the command has been executed, control passes to the top of the 'main program; where the next

command is read. Where possible,' the subroutines 'were taken from' existing programs that had been

run successfully for years, or from published sources. 'About 5% of the lines of code were obtained this

way, with the rest being written especially for PHAZE.

The subprograms in PHAZE fall into several classes:

Class (1) Subprograms that evaluate formulas; for example, the function DV evaluates the

derivative V' according to the formulas in Table 1 of EHF.

Class (2) Subprograms that use these calculations to produce the desired output; for example,

the subroutine CONFB calculates a confidence interval for f3, and prints it if so

instructed.

Class (3) Subprograms that calculate standard quantities; for example; the functionn'ANORDF

calculates the' inormal cumulative distribution function at a specified point.'

Class (4) Subprograms thati decipher inpiit aiid' aniplate files;, forexample RDINT'starts

reading the'current'iiipui line'at a specified location'and finds th'e next integer on the

'line; GETNANI asks the user for the name of a file,'and either finds it or creates it.

as instructed.

The verification and validation activities differed somewhat for' subprograms in'"different classes, as

described below.

29
4.2 Verification

The mathematical formulas used are developed in EHF. In principle, they have long been

known; the only new developments are some of the exact algebraic forms taken for the various models.

These expressions were all either derived or confirmed using the symbolic computer program Mathematica (Wolfram, 1988). The formulas are given in Table 1 of EHF.

Subprograms in Classes (1) and (2) were verified by comparing them, line by line, with the

formulas or the methods given in EHF. Subprograms in Class (3) were taken either from printed

sources or directly from Fortran programs on disk; those copied from printed sources were verified by

line-by-line inspection of the code. The subprograms in Class (4) were originally written in 19S2.

They perform tasks for which wrong behavior would be apparent. Because they have been used

frequently in various applications since 1982, evolving only slightly, they are believed to be correct.

Before they were included in PHAZE, they were briefly inspected but not carefully verified. Tile

verification activity for all the classes of subprograms took place from December 1989 through March

1990. A few typographical errors that had been missed during the above verification were uncovered

during the validation exercises described below, and during a separate static analysis of the code.

4.3 Validation

When considering the accuracy of the calculations, we must distinguish between quantities that

are found by numerical iteration and quantities that are calculated from formulas. Typically, the MLE

and confidence limits for 8 are found by numerical iteration. The accuracy required can be specified in

advance, with more accuracy requiring more computing time. In PHAZE, these confidence bounds are

required to have a relative error of at most 0.001, i.e., a slight error in the third significant digit. The

MILE /3 is required to have relative error 0.00001, for reasons given below. To pass the validation tests

described below, PHAZE must correctly iterate to find the NILE and confidence limits for 3 to tie

required accuracy. In addition, the calculations that are simple evaluation of formulas must be

accurate enough so that the results of iterations are based on correct values.

Class (1)

The subprograms in Class (1) all involve only evaluation of formulas. To validate these

30
subprograms, a data set was created with one component and one failure. It was made this simple so

that correct results could be found with a hand calculator. For testing calculations with -the Weibull

model, two variations of the data set were used, one in which the component was observed from its

installation, the other in which the component was observed starting at a non-zero initial, age; these

cases correspond to two different sets of formulas in EHF. All calculated results were correct to D.

significant digits, except that under the exponential model when 13 was near -oo, the derivative of the

conditional log-likelihood was accurate to fewer places.

Many of the expressions must be evaluated using asymptotic approximations for certain values

of , (1 near 0 under the exponential or linear hazard model, or 6i near -1 under the Weibull hazard

model.) The subprograms therefore use the standard formula or the asymptotic approximation depending on whether 16 is greater than or less than some critical value. Many of the validation calculations

focused on 13 near these critical values, to make sure that the calculated expression varied continuously

as 13 moved from one side of the critical value to the other. In one case (calculating the information

for 6f under the linear hazard model), it was found that the second order asymptotic approximation

was inadequate; the fourth order approximation was therefore put into the Fortran coding, and into

Table 1 of EHF.

Class (2)

Some of the subroutines in Class (2) require more than one component, for example, subroutines that compare components to see if they have a common A. For these, a second data set was used,

having two components and four failures, and given in Figure 3 above. It was found.that sometimes

the covariance matrix for (1, logAO) was accurate only to two significant digits. This was the case

when , and logAO were strongly correlated, and when the information matrix (the inverse of the covariance matrix) was only accurate to four places. Two things were done to mitigate this problem. 'First,

the; MLE 83 is now required to be found to approximately .5-place accuracy (relative error 0.00001)

* - * , I., . -. . . '. --:.h

rather than 4-place accuracy. This means that the information matrix is now accurate to about five

significant digits. Second, a warning message is now printed when the correlation of 6 and logAo is

strong enough to greatly magnify the inaccuracy of the inverse matrix.

Class (3)- i, ;-: - -- -

To validate the members of Class (3), the subprograms were individually called with various

31
inputs, and the outputs were compared to values in published statistical tables. or values produced by

the computer package Minitab (1985). The tests showed that several internal checks were needed to

catch illegal inputs; these checks were added to the coding. The answers produced were correct. to

within the accuracy printed by the tables or by Minitab, with one exception: The calculated significance levels for the Kolmogorov-Smirnov (K-S) test may be off by more than + 0.01. This inaccuracy

seems unavoidable. For sample sizes in the range 20 to 35, published values only give the K-S statistic

to two decimal places, meaning that interpolated significance levels could be off by 0.01 or more. The

asymptotic formula for the significance level, when the sample size is in this range, also seems to have

only this degree of accuracy.

Class (4)

The subprograms in Class (4) were validated indirectly. It was verified that PIIAZE correctly

read data files, wrote requested output files, and interpreted the user's commands. Therefore the

subprograms in Class (4) seem to work correctly.

Other Validation Checks

A static analysis of the code was performed using the INEL software CRAFT. Two undefined

variables were identified-typographical errors that were corrected. Three unused variables were

identified, but two were left in the coding to facilitate possible future program enhancement. As a final

check, the program was compiled with three compilers: IBM Professional Fortran/2 Version 1.01,

Lahey Fortran 77 Version 4.00, and Microsoft Fortran Version 4.1. To compile the program under

Microsoft Fortran, it was necessary to reduce the length of the main driver program, making subroutines out of major sections. After this restructuring, the program was successfully compiled with all

three compilers. One useful option in the file-handling subroutines allows the user to append output to

an earlier file. This option works with Professional Fortran and Lahey Fortran, but not with Microsoft

Fortran. Therefore, two source-code versions of the file-handling subroutine GETNAM are kept, one

with the append-option and one without it.

The validation tests described above were carried out in March through May, 1990. The verification and validation history of each subprogram is documented in comments at the beginning of the

subprogram's source code.

32
5. TECHNICAL DETAILS

5.1 Language and Portability

PHAZE is written in ANSI standard Fortran 77, with the following two exceptions: (1) Lower

case letters are used freely in' comnments, and in Fortran statements used for debugging. (2) One

version of the subroutine GETNAM'includes6the'ihne

-OPEN (UNIT=IUN,FILE=FILNAM,STATUS='OLD),ACCESS='APPEND')

even though ACCESS='APPEND' is not ANSI standard.' This line works with'IBM Professional

Fortran/2 Version 1.01 and with Lahey Fortran 77 Version 4.00, but not with Microsoft Fortran

Version 4.1. Therefore, two versions of the source code for that subroutine are kept, one with the nonstandard line and one without it. -.'

Several machine-dependent constant 'are initialized in' the subroutine BLOCK DATA. The currently set'values aretacc'eptable for the three compilers on 'an IBM PC or clone, but may need to be

reinitialized if PHAZE 'is used on a different kind of computer. The-current values' of the three fundamental constants are

TINY = 1.2E-38, set to be slightly larger' than the rninimum positive real number,'

SMALL = 2.4E-7, such that 1. + SMALL approximnately equals 1. in single precision,

HUGE = 3.E38, a number almost as large as the'maximum possible number.

A few other constants, simple functions of these thre'e, are also initialized to the corresponding values.

No special libraries are required; all the necessary calculations are performed by ANSI Fortran

intrinsic functions or by subprograms of PHAZE.

5.2 Calculation Methods ' '

'This section is-a supplement to the explanations of the commands in Section 3.3. The material

here is organized'by' the commands, just as Section 3.3 is. This section does not repeat material

presented in Section 3.3 or in EHF, however, but only gives additio nal technical detail where required.

Most of the equations from EHF involve a notatiori that is interpreted for each model (exponential,

33
linear, and Weibull) in Table 1 of EHF.

CC B

Part of the comparison of components for 3 is to determine the attained significance level corresponding to component j. The significance level is defined as the probability that a test rejects the

hypothesis Ho:lj = 1_j in favor of the alternative HO:/3j : 93j. Two methods are used to test the hypothesis and to approximate the significance level.

When the jth component has more than one nonreplacement failure, the hypothesis is tested

using the difference of the MILEs 3,j and fp3 . The attained significance level is defined as twice the

smaller of the probabilities

P( 7J -. _j > observed difference ) and P( 3, 1 < observed difference),

where 1,j and 3_j are understood to be random and all the components are assumed to have a common

value of 3. The normal approximation is used for the distribution of both .j and ajL. The calculation

of Jj uses only the data from component j, while that for 3 j uses all the data except from component

j. Each NILE is based on the conditional likelihood, and is found by solving Eq. (6) of EHIF, while the

corresponding Fisher information I is found by evaluating Eq. (11) of EHF. The variance of each d is

the inverse of the Fisher information. Because the two MILEs are based on independent data sets, the

variance of their difference is the sum of their variances. Therefore, if all the components really have

the same ,3,

(0i - i j) / [r'(i3j) + F(103j) I,/2

is approximately normal(0,1). The approximate significance level can therefore be calculated. This

approximation method-is poorest if Bj is based on only two failures or if 3 j is based on few failures.

When the jth component has only one nonreplacement failure, the normal approximation is

clearly inadequate. The approximation used instead is to treat 3 j as known. This is reasonable if 3 J

is based on many more failures than J3j is, for. in this case iJ.j has much less variability than does 3j.

The single failure time t has a conditional density given below Eq. (5') of EHF. The test rejects H10 if t

is far from the value expected assuming that 1j equals the known !_p. Therefore the attained significance level is twice the smaller of the probabilities

P( t > observed value ) and P( t < observed value),

where i is understood to be random and governed by , = ! ,. This approximation method is poorest

if 3 j is based on very few failures.

34
CD ,. . .

Centering the datan means'that all times are measured from:a point t mid in the middle of the

observation periods. To define 'Md precisely, let'rj be'the'time range for 'the jth' component, that is.

the end time of the observation period minus the starting time of the observation period. Let' be the

midpoint of the observation period for the jth component.' Define the value imi d - 3j I /r .. (Note,

when there is just onecomponent,Imid reduc~s`to the 'midpoint of the component's observation period.)

The data are called centered if all times are measured from tId* 'Failures'that occurred before t

mid are

then counted as having occurred at'negative times, which is allowed.' This centering takes place within

the calculations and is not seen by the user;'times are still requested'from and reported to the user in

their original units.

Centering the data at some ic changes the meaning of parameters, and can effect the allo wed

values of those parameters. For example, suppose that a single component is observed from times so to

sl, with 0 < so < sl, and consider first the case when the data are 'not centered. 'The linear hazard

function is A (1 + Pt), and AO is the valu'e of A(O); The-hazard function' A() mi'st be positive for

so < t < sl; this means that ft must satisfy ',B >'' 1/sl. When instead 'the data "are' centered at 'icl the

linear hazard function, with i written'in' the original units, is A0(1 + tlt -ta)]. In thiscase, L is the

value of A(iQ). If I is between' s and 's, the requirement that A(t) remrain' positive for so < I < s,

translates to -1/(s - Q) < P < l/(tc-so). This is the reason why Figures 7 and 8, based'on the same

data set, are so different from each other.

CI

The procedure depends on the values of certain switches, as shown in Table 2. First consider a

confidence interval for (3 when the full-likelihood and bivariate-asymptotic-normality switches are both

off. As explained in Section 5.1 of EHF, define the standardized score SS by

SS = L'(,60) / [1(1o)11/2;

where L is the conditional log-likelihood, and I is the information for fi based on the conditional ti } l' . '!,v ,,;,,, . .. ...

likelihood. 'SS is'ioproximately normal(0,1) for large samples. Therefore an approximate confidence

interval for 13 is the set of all, PRO such that -c < SS < c, where c is the appropriate number from a

normal table, For example, c = 1.645 yields an approximate 90% confidence interval.

*3*5t s * ! f ., , .

35
A potential difficulty is that this may define a confidence set that is not an interval. This can

happen if SS is not monotone in AS. PHAZE finds a confidence interval by first acting as if SS is

monotone. Beginning at P = 0, PHAZE searches through larger or smaller values until it finds a value

pup where SS = c. Similarly it finds OLO where SS = -c. This is reported as the confidence interval.

but first PHAZE does some checking of monotonicity. It goes out beyond the upper limit, a distance 5

times (,PUp-A) above ,, and similarly below the lower limit. Of course it never goes beyond any

theoretical bounds of hi. In this range, PHAZE looks at evenly spaced values of 3, evaluates SS there.

and decides if SS appears monotone. If not, PHAZE prints a warning, and offers to print the values of

SS so that the user can examine them. Non-monotonicity is common under the linear model when the

confidence intervals are infinite. Therefore the warning message has been suppressed with the linear

model; it becomes a nuisance to see. Under the Weibull model, non-monotonicity may occur near the

theoretical lower bound of l3 = -1. Non-monotonicity has not yet been observed with the exponential

model.

For AO, when the bivariate-asymptotic-normality switch is off, a confidence interval is found.

given the assumed value /30 introduced in Section 3.2. The confidence limits use exact formulas given

below, not normal approximations. In EHF a quantity X is defined, depending on the model and on ;3.

Let n be the number of failures for time censored data (all non-replacement failures), and let m be the

number of failures for failure censored data (including the replacement failures). The confidence limits

on A0 for time censored data are then

X22n a/2/(2v)

X22(n + 1), (I - a/2)(2 V)

while for failure censored data the confidence limits are

I X22m a/2/(92v)

X22M (I _ /2) /(2v)

The confidence limits for the two cases agree except for the degrees of freedom. In PHAZE, it is possible for some components to be time censored while others are failure censored. PHAZE uses twice the

total number of failures as the degrees of freedom for the lower limit. If there are any failure censored

components, PHAZE uses this same number as the degrees of freedom for the upper limit; if instead all

the data are time censored, PHAZE uses 2(n + 1), where n is the number of failures. These assign36
ments agree with the correct values if all the data have one kind of censoring, and appear to be conservative for the other cases.

For (P, logAo) when the fulI-likelihcod and bivariate-asymptotic-normality switches are both

on, a confidence ellipse is based on the bivariate asymptotic normality of the pair (l, iogAo). The asymptotic covariance matrix is the inverse of the Fisher information matrix, which is estimated by the

sample information matrix'given by Equation (15) of EHF.

TE

This test uses the approximate normality of'the standardized score SS, introduced above for

the command CI. The null hypothesis is Ho: '` = fio, and the alternative is HI: l3 > I00. The test rejects Ho at significance level ot if th'e two-sided 100(1 - 2h)% confidence interval for 0 is entirely to the

right of Po, or equivalently, if the one-sided 100(1-a)% confidence interval (16LO0 cm) is entirely to the

right of 63. See the discussion below Expression (12) of EHF, and the examples in Section 5.3 of EIIF.

5.3 Some Details of the Coding

This section is for the benefit of anyone responsible for maintaining or modifying the source

code. Most of the' notation follows that of EHF. 'For example the two-dimensional array T stores the

failure' times ii:, R0 stores the observation ranges ro, and so forth. A difference is that the times in T.

for any component are measured from the beginning of the component's observation period, not from

the component's installation. This has historical rather than rational reasons, and is a minor quirk

that is not worth the trouble of changing. The rest-of this section mentions a few ways in which the

coding does not have the straightforward, obvious meaning.'

In writing PHAZE, much of the coding effort was concerned with preventing numerical overflovs or underflows. In typical examples with the exponential or linear hazard models, the values of

time are on the order of 104 hours or more, and 6 is on the order of 10-5 hours1l or less. Therefore, the

numbers are scaled internally, in order to keep the magnitudes nearer to 1.0. After a driver program

obtains a new data set by calling the subroutine GETDAT, the driver calls the subroutine PREP, to

prepare the data for analysis. PREP looks at the observation range, i.e., the length of the observation

period, for each component. It then defines TNORM as the maximum of these ranges, and divides

37
every time (failure time, observation beginning time, and observation end time) by T.NORNM. This is

effectively a change in the units of time. These renormalized times are the basis of the values stored

internally and used for calculations. This means that for the exponential and linear hazard models we

have

BETA = TNORM x 3,

where 3 is in the original units and BETA is in the units for internal calculations. The function of

TNORMN is invisible to the user: all times and parameters are reported to the user in the original

units.

For the Weibull model, to has the value of TNORNI. The command SN can be used with the

Weibull model to reset TNORM to any user-specified value. It works by calling PREP, which sets

TNORNI to the user-specified value and then divides all times (restored to their original input values)

by the new TNORM.

Under the Weibull model, when 3 is near -1 expressions must sometimes be evaluated using

asymptotic approximations in terms of (,3 + 1), as given by Table 1 of EliF. To avoid round-off error

in these cases, the value of , + 1 is stored internally, rather than the value of d3. That is, we have

BETA =3 1+ 1,

where 13 is in the original units and BETA is the value stored internally, when the Weibull model is in

effect. This change was added late in the development of PHAZE, which is why the variable is called

BETA; some subroutines use the less confusing name BARG (for '1-argument').

No problem is caused by giving BETA different meanings depending on the model in effect.

because whenever a new model is selected, the default value 30 is reset to 0. That is, the internal

variable BETAO is set to 0.0 if the exponential or linear model is in effect, and to 1.0 if the Weibull

model is in effect. Statistical inference then proceeds using the internal units. and all output is translated into the original units.

38
6. REFERENCES

F. B. Alt, 'Bonferroni Inequalities and Intervals," Encyclopedia of Statistical Sciences, Vol. 1. S. Kotz

and N. L. Johnson, eds., New York: John Wiley & Sons, 1982.

Corwin L. Atwood, Estimating Hazard Functions for Repairable Components, EGG-SSRE-8972. Slay

1990. (Referred to as EHF in this user's guide.)

IEEE, IEEE Standard for Software Quality Assurance Plans, ANSI/IEEE Std 730-1984, 1984.

Minitab Reference Manual, Release 5.1, State College, PA: Minitab, Inc., 1985.

Ronald D. Snee and Charles G. Pfeifer, "Graphical Representation of Data," Encyclopedia of Statistical

Sciences, Vol. 3, S. Kotz and N. L. Johnson, eds., New York: John Wiley & Sons, 1983.

Andrew J. Wolford, Corwin L. Atwood, and W. Scott Roesener, Aging Risk Assessment Methodology:

Development and Demonstration Study, NUREG/CR-5378, EGG-2567, DRAFT, Rev. 1, June 1990.

Stephen Wolfram, Mathematica: A System for Doing Mathematics by Computer, Redwood City, CA:

Addison-Wesley Publishing Company, 1988.

39
APPENDIX

SAMPLE PHAZE SESSION

A-1
APPENDIX

SAMPLE PHAZE SESSION ' '

The following printout is the echo of a PHAZE session. The very first command of the session

was

ou exampl.out

This caused all the remaining lines to be written'to'the disk output'file EXAMPL.OUT; which was

later captured electronically and inserted into this report.,The'very first lines'ofthe session, before the

command OU could be entered, appeared on -the terminalscreen'but'were'not echoed to a disk file.

Those few lines were copied manually to the beginning of the file, so that the reader of this report can

see how the entire session, proceeded. - -

Anything that was entered by the user is printed here in boid face. This highlighting is added

in this report, to make it easier for the reader to distinguish between commands by the-user and responses by the program.

The spacing of the program output has sometimes been changed slightlyfor reproduction here.

This is because the output of PHAZE can be up to 80 characters wide, while the word processor being

used for this report allows fixed-spacing lines only of 76 characters. -Also, the output from the program

was continuous, scrolling up the screen. In this appendix, however, page-breaks have been inserted at

logical places rather than at mechanical places after a fixed number of lines.

A-3 -....
C: \FORTRAN>phaze

Program PHAZE. for Parametric HAZard Estimation

Version 1.00

Enter command when prompted by ' ->'

For help, enter II

-> ou exampl.out

-> #############X##

-> # This is a sample PIIAZE session, with comments sprinkled

-> # liberally through the session.

-> W We will follow the steps (1) through (10) of Section 1.3.

-> # First, however, we demonstrate the help commands

-> ######tt##########

-> h

Htelp messages are given for groups of commands

Enter For commands that

H D Control Data or files

1I I Perform statistical Inference

II 0 Set or show Options ______________________________________

To quit the program, enter t

-> hd

Commands that control data or files are

Command Meaning

DD Describe Data (summary statistics)

GD x Get Data

OU x define disk file for OUtput

PF toggle option to allow creation of Plot Files.

Default is OFF

SD See Data in concise display

9 comment only (echoed to output file)

x is an (optional) DOS file name.

If none is given, you will be prompted.

-> h i

In the commands below, x identifies the parameter(s) of interest.

Replace x by B for inference on beta

by L for inference on lambdaO

by + for inference on both

or by HI for inference on the hazard function

Commands that perform statistical inference are

Command Meaning

CC x Compare Components for x values

CI x k find k% Confidence Interval for x, where

for example k = 90, 95, or another value

HF t find Hazard Function at time t (NILE and 90% int.)

HF R tl t2 find Hazard Function at times-in Range tl-t2

KS Kolmogorov-Smirnov test of model

ML x find Maximum Likelihood estimator for x

A-4
QQ calculate info for Q-Q plot

SB y Set BetaO to y, assumed value for tests or QQ plot or lambda

calculations. Default is 0; becomes NILE after ML command.

TE y TEst whether beta = y. If no value is given for y,

betaO is used.

x-retains its most recent value.-

For example, following the command ML B,. the command CC is interpreted as

CC B.

_> h o,

Commands that set options are

Command Meaning

AN Toggle option to treat MLE of (beta, log-lambdaO)

as jointly Asymptotically Normal.-

Default is asymptotic normality only for beta-hat

CD Toggle option to Center bata (with linear hazard function

only).

Default is no centering.

FL Toggle option to useFull Likelihood.

Default is to use conditional likelihood for inference about

beta

LE Toggle option to'assume LambdaOs Equal-for all components. -

Default is not to assume equality.

Sil x Select Model x, where x is E,-L,-.or W for exponential,-

linear, or Weibull hazard function. There is no default.

SN x Set Normalizing time to x, for Weibull model.

If x is missing, default time is used..

ST See current STatus of options.,

-> # Step 1

-> ##########$###

-> gd movdemo.dat

Data found for 16 components. -;

O warnings

o errors encountered in data

-> # This data set is shown in-Figure 1,of the text

-> dd

Component

_____- ______A_

MOV-1A

MOV-1B

MOV-1C

MOV-1E

AIOV-1E(R)

hO V-1F

MIOV-1F(R)

MOV-2A

MOV-2B

Nonrepl. Observed.,

Fails. Hrs. _ _ _ _ _ _ _ - - - - - - - -

1! 8.8584E+04

1 8.8584E+04

2 8.8584E+04

0 2.1840E+04

3 6.6744E+04

3_ 4.3608E+04

1 4.4976E+04

4 8.8584E+04.

5 8.8584E+04

Mean Failure

Time (Normed)

.0.378

0.086

0.752

0.743

0.498

0.568

0.487

0.619

0.567

I

of

leplaced - Initial

i Fail.? Age _ _ _ _ _ _ _ _ - - - - - - - -

4.1448E+04

- ~ .4.1448E+04

4.1448E+04

Y 4.1448E+04

O.OOOOE-0l

-YY :4.1448E+04

O. OOOOE-O1

:3.7824E+04

3.7824E+04

A-5
IOV-2C -

NIOV-2C(R)

NIOV-2D

DIOV-2E

NIOV-2E(R)

MIOV-2F

-> sd

1

1

6

0

2

4.9728E-'04

3. 8856E+04

8.8584E+04

2.2608E+04

6.5976E+04

8.8584E+04

0.756

0.866

0.464

0.698

0.593

Y 3.7824E-04

O.OOOOE-O1

3.7824E-04

3.7824E-04

O.OOOOE-01

3.7824E+04

The data set contains the following 16 components

MDOV-1A

Between component ages 4.1448E+04

7.4916E+04

Mean (non-replacement) failure age

NIOV-1B

Between component ages 4.1448E+04

4.9044E+04

Mean (non-replacement) failure age

MDOV-1C

Between component ages 4.1448E+04

9.1092E+04 1.2496E+05

Mean (non-replacement) failure age

and 1.3003E+05, there was 1 failure :

= 7.4916E+04

and 1.3003E+05, there was I failure :

= 4.9044E+04

and 1.3003E+05, there were 2 failures:

= 1.0802E+0.5

MIOV-ID

Between component ages 4.1448E+04 and 1.3003E+05. there were 7 failures:

9.0804E+04 9.1740E+04 1.0127E+05 1.1135E+05 1.1536E+05

1.1543E+05 1.2496E+05

Mean (non-replacement) failure age = 1.0727E+05

DIOV-1E

Between component ages 4.1448E+04 and 6.3288E+04, there were 0 failures

One more failure, at age 6.3288E+04, resulted in component's replacement

MOV-1E(R)

Between component ages O.OOOOE-01 and 6.6744E+04, there were 3 failures:

1.1460E+04 2.6820E+04 6.1356E+04

Mean (non-replacement) failure age = 3.3212E+04

.MIOV-IF

Between component ages 4.1448E+04 and 8.5056E+04, there were 3 failures:

4.8300E+04 7.2876E+04 7.7436E+04

Mean (non-replacement) failure age = 6.6204E+04

One more failure, at age 8.5056E+04, resulted in component's replacement

MDOV-1F(R)

Between component ages O.OOOOE-01 and 4.4976E+04, there was 1 failure

2.1924E+04

Mean (non-replacement) failure age = 2.1924E+04

NIOV-2A

A-6
Between component ages 3.T824E+04 and 1.2641E+05; there wer,

4.7844E+04 1.0655E+05 1.0660E+05 1.0955E+05

Mean (non-replacement) failure age = 9.2634E+04

MOV-2B

Between component ages 3.7824E+04 and 1.2641E+05, there wer

6.4188E+04 6.5868E+04 8.4444E+04 1.0640E+05 1

Mlean (non-replacement) failure age = 8.8015E+04-

MOV-2C

Between component ages .3.7824E+04 and 8.7552E+04, there was

7.5420E+04 -

Mean (non-replacement) failure age = 7.5420E+04,.

One more failure, at age 8.7552E+04, resulted in-component's

MIOV-2C(R)

Between component ages O.OOOOE-O1 and 3.8856E+04, there was

3.3636E+04

Mean (non-replacement) failure age = 3.3636E+04 ;- -

MOV-2D

Between component ages 3.7824E+04 and 1.2641E+05,:there wer

4.3260E+04 6.1668E+04 6*.2148E+04 8.4444E+04 I

1.1576E+05

Mtean (non-replacemenit) failure age = 7.8948E+04

e 4 failures:

I.f.,

e 5 failures:

.1917E+05

14failur6

replacement

1 failure

e 6 failures:

.0640E+05.

51OV-2E

Between component ages 3.7824E+04 and 6.0432E+O4, there were 0 failures

el I -!, I . I I . I.

I ,, r; ~ , ..- ..

4,' .there were 2 failures: -~ r.. .. .

AIOV-2E( R)

Between component ages O.OOOOE-O1 and 6.5976E+1

4.5972E+04 4.6068E+04

Mlean (non-replacement) failure age = 4.6020E+04

D

M1OV-2F

Between component-ages 3.7824E+04 and 1.2641E+05, there were 7.I

6.1572E+04 8.4444E+04 8.7492E+04 9.0300E+04 9.59881

1. 0640E+05 1. 0655E+05

Mean (non-replacement) failure age = 9.0393E+04 ; -

-> # Often, DD is enough, with no need for the long output of SD

_> ####1######## 4-

-> # Step 2, Construction of a cumulative failure plot.

-> #############- ; _> qq ..

Enter model type - Exponential failure rate

Linear failure rate

Weibull failure rate - *

Ea i l ures:

E1+04

- . I

I .

I , I, .

-> e ; .-

Now re-efdter command

-> # I forgot to enter the model, so PIIAZE prompted me.

-> # Now we do the cumulative failure plot

A-7

I
-> qq

QQ information when beta = O.OOOE-O1

based on conditional distribution of non-replacement failures

Calculated t Observed t

7.93E+03 1.15E+04

1.59E+04 2.19E+04

2.38E+04 2.68E+04

3.17E+04 3.36E+04

3.84E+04 4.33E+04

4.11E+04 4.60E+04

4.29E+04 4.61E+04

4.47E+04 4.78E+04

4.65E+04 4.83E+04

4.83E+04 4.90E+04

5.01E+04 6.14E+04

5.19E+04 6.16E+04

5.38E+04 6.17E+04

5.56E+04 6.21E+04

5.74E+04 6.42E+04

5.92E+04 6.59E+04

6.10E+04 7.29E+04

6.29E+04 7.49E+04

6.47E+04 7.54E+04

6.66E+04 7.74E+04

6.87E+04 8.44E+04

7.08E+04 8.44E+04

7.29E+04 8.44E+04

7.51E+04 8.75E+04

7.72E+04 9.03E+04

7.93E+04 9.08E+04

8.14E+04 9.1lE+04

8.35E+04 9.17E+04

8.58E+04 9.60E+04

8.82E+04 1.OlE+05

9.08E+04 1.06E+05

9.35E+04 1.06E+05

9.61E+04 1.06E+05

9.87E+04 1.07E+05

1.O1E+05 1.07E+05

1.04E+05 1.07E+05

1.07E+05 1.1OE+05

1.09E+05 l.l1E+05

1.12E+05 1.15E+05

1.14E+05 1.15E+05

1.17E+05 1.16E+05

1.20E+05 1.19E+05

1.22E+05 1.25E+05

1.25E+05 1.25E+05

A-8
QQ information when beta = 0 and all lambdaO values assumed equal

based bn observation times rather than component ages

This is relevant if the number of components was the same at all times

Calculated t Observed t

1.85E+03

3.69E+03

5.54E+03

7.38E+03

9.23E+03

i.11E+04

1.29E+04

1.48E+04

1.66E+04

1.85E+04

2.03E+04

2.21E+04

2.40E+04

2.58E+04

2.77E+04

2.95E+049

3.14E+04.

3.32E+04,

3.51E+04

3.69E+04

3.88E+04

4.06E+04

4.24E+04

4.43E+04

4.61E+04

4.80E+04

4.98E+04

5.17E+04

5.35E+04

5.54E+04

5.72E+04

5.91E+04

6.09E+04

6.27E+04

6.46E+04

6.64E+04

6.83E+04

7.01E+04

7.20E+04

7.38E+04

7.57E+04

7.75E+04

7.94E+04

8.12E+04

8.30E+04

8.49E+04

8.67E+04

5.44E+03

6.85E+03

7.60E+03

1.OOE+04

2.18E+04

2.37E+04

2.38E+04-

2.43E+04' :

2.64E+04:

2.80E+04 - 3.14E+04

3.33E+04-

3.35E+04'

I .; :, .

-( *3*.*. AI I - , " .

-I I - .-

3.60E+04 - '

3.76E+04

4.36E+04

t . .

4.66E+04; . -

4.66E+04

4.66E+04 .

4.87E+04

4.94E+04'^

4.96E+04 - ;

4.97E+04 -

4.97E+04

5.03E+04

5.25E+04 - -

5.82E+04 --

5.98E+04 A

6.55E+04

6.86E+04. -,

6.86E+04

6.86E+04 .

6.86E+04

6.87E+04 ---

6.87E+04

6.87E+04 :

.6.88E+04

6.99E+04 - --

7.17E+04

7.39E+04

7.40E+04 A

7.79E+04

8.13E+04

8.32E+04 ,

8.34E+04 <>t7: <,

8.35E+04

8.35E+04; -

A-9

. j , ,.

......

I , - ., f .

I .

. . I --. ,, i

. ill ', I I

I I - . ,

-I 11

A ; .. A I ' . : , I I  :. 1 ,, .
The second set of output is what is needed for a cumulative failure

-> # plot. It is only generated when beta = 0.

-> # If the PF switch had been on, PHAZE would have offered to write

-> # the output to a disk file, for us to use with a graphics package.

_> ####~###########

-> # Step 3, compare components for common beta

_> ############### -> ccb

95.0% conf. intervals with MILEs for beta, by component and MILE without

comp.

MOV-1A (-1.lOE-04,-1.72E-05, 6.48E-05) 1.42E-05

MIOV-1B (-4.85E-04,-1.32E-04, 2.55E-05) 1.54E-05

NIOV-1C (-1.77E-05, 4.09E-05, 1.08E-04) 1.21E-05

MOV-1D ( 4.06E-06, 3.90E-05, 7.52E-05) 8.o7E-06

MIOV-IE not estimable

NIOV-1E(R) (-5.41E-05,-4.31E-07, 5.32E-05) 1.41E-05

MOV-1F (-6.39E-05, 1.88E-05, 1.03E-04) 1.33E-05

NIOV-1F(R) (-1.67E-04,-3.35E-06, 1.59E-04) 1.35E-05

MIOV-2A (-2.04E-05, 1.67E-05, 5.46E-05) 1.30E-05

MOV-2B (-2.35E-05, 9.12E-06, 4.20E-05) l.40E-05

MOV-2C (-8.71E-05, 7.48E-05, 3.04E-04) 1.29E-05

NIOV-2C(R) (-7.88E-05, 1.91E-04, 6.82E-04) 1.28E-05

MOV-2D (-3.47E-05,-4.86E-06, 2.49E-05) 1.71E-05

NIOV-2E not estimable

1IOV-2E(R4 (-3.18E-05, 3.99E-05, 1.19E-04) 1.26E-05

MOV-2F (-1.55E-05, 1.29E-05, 4.16E-05) 1.35E-05

OVERALL ( 5.32E-07, 1.34E-05, 2.62E-05)

Comparison

Component

MOV- 1A

.MOV-1B

MOV- 1C

MOV- ID

MOV- 1E

MIOV- 1E(R)

NIOV-1F

MIOV-1F(R)

%IOV-2A

NIOV-2B

%1OV-2C

MIOV-2C( R)

MOV-2D

MIOV-2E

SIOV-2E(R)

NIOV-2F

of beta values for components

Signif 95.0% Confidence Interval

0.48 (----*I-)

0.09 (---------------*------I)

0.44 (1*--)

0.14 I*-)

0.64

0.91

0.82

0.87

0.80

0.61

0.33

0.21

0.54

0.97

(-*I-)

( * )

(------*I------)

( -)

( --- I-------g------------------ ___

(*I)

(-I* ---)

level for testing equality of betas = 1.00

OVERALL

Overall significance

Signif for jth component means estimated probability that

beta-hat(j) - beta-hat(others) is at least as extreme as observed

A-10
Overall significance level is P[at least one signif(j) <= observed min].

and is computed with Bonferroni inequality

-> # Overall significance is 1.00 -- no evidence of different beta's,

-> ##$######.

-> # Step 4, test for increasing failure rate

_> ###################-

-> te 0

For testing beta = O.OOOE-01, test statistic = 2.04

Approx. significance level, for testing against beta:> 0.OOOE-01, is 0.021

Significance level is based on normal approximation--

and distribution conditional on observed counts

-> # Small significance, i.e. strong evidence of increasing failure-rate

-> # Construct confidence interval to supplement the test

-> ml b -

Maximum Likelihood Estimate beta-hat is 1.336E-05

-> ci b 90

90.0% confidence interval for beta is -

( 2.56E-06, 2.42E-05)'

This is based on the normal approximation,

and the likelihood conditional on the number of nonreplacement failures.

-> ##################,;-

-> # Step 5, check goodness of fit of model to data

-> ################## '' -,';

-> ks

For testing model with beta = 1.336E-05

Kolmogorov-Smirnov test statistic is 0.090 -

Value of test statistic is attained at observation 4l

Significance level equals 0.869, based on 44 observed failures

-> # For the following Q-q plot info, I will turn on the plot-file switch.

-> # The resulting disk file is identical to the one used to produce -

-> # Figure 9 in the text, using a commercial graphics package.

-> pf .I

Option to allow creation of plot-files turned ON

-> qq

Plot information can be written to a file

Do you want a separate file for this?

-> y .

Please enter DOS filename -

-> qqdemo.out

Please enter identifier to copy to qqdemo.out

-> Q-Q plot info,'expo'nential model, data from movdemo.dat

QQ information whenrbeta-= i.336E-05 '

based on conditional distribution of non-replacement failures

Calculated t Observed t

1.1OE+04 1'15E+04 ' -' 

2.06E+04 2.19E+04

2.91E+04 2.68E+04

A-li
3.67E+04 3.36E+04

4.10E+04 4.33E+04

4.37E+04 4.60E+04

4.62E+04 4.61E+04

4.88E+04 4.78E+04

5.13E+04 4.83E+04

5.37E+04 4.90E+04

5.61E+04 6.14E+04

5.84E+04 6.16E+04

6.06E+04 6.17E+04

6.27E+04 6.21E+04

6.49E+04 6.42E+04

6.70E+04 6.59E+04

6.95E+04 7.29E+04

7.20E+04 7.49E+04

7.45E+04 7.54E+04

7.68E+04 7.74E+04

7.90E+04 8.44E+04

8.13E+04 8.44E+04

8.33E+04 8.44E+04

8.55E+04 8.75E+04

8.80E+04 9.03E+04

9.05E+04 9.08E+04

9.31E+04 9.11E+04

9.55E+04 9.17E+04

9.78E+04 9.60E+04

l.OOE+05 1.OlE+05

1.02E+05 1.06E+05

1.04E+05 1.06E+05

1.06E+05 1.06E+05

1.08E+05 1.07E+05

l.lOE+05 1.07E+05

1.12E+05 1.07E+05

1.14E+05 l.lOE+05

1.16E+05 l.llE+05

1.18E+05 1.15E+05

1.19E+05 1.15E+05

1.21E+05 1.16E+05

1.23E+05 1.19E+05

1.25E+05 1.25E+05

1.26E+05 1.25E+05

-> # The identifier is written as the first line of qqdemo.out

-> # Now, because this is only a demonstration, we turn off the

-> # plot-file option

_> pf

Option to allow creation of plot-files turned OFF

-> ####6##########

-> # Step 6, compare the components for a common lambdaO

_> ###############
-> ccl

95.0%-conf. intervals with MLEs for lambdaO, by component and MLE without

comp. .. .,

assuming beta

MOV-1A (

MOV-1B

AIOV-1C0

MOV- 1D

MOV-1E

AIOV-1E(R)

MOV- 1F

blOV-iF(R)

MOV-2A

MOV-2B

MOV-2C

AIOV-2C(R)

51OV-2D

AIOV-2E

51OV-2E( R)

MOV-2F

OVERALL

Comparison

.C

(

(

= 1.34E-O'

8.58E-08,

8.58E-08,

8.20E-07,

9.54E-06,

5.74E-07,

5.75E-06,

1.06E-05,

4.1 IE-07,'

3.88E-06,

5.78E-06,

2.07E-06,

4.97E-07,

7.83E-06,

0.00 9

2.29E-06,

1.OOE-05,

1. 17E-05,

3.39E-06,

3.39E706,

6.78E-06,

2.37E-05,

2.27E-05,

2.78E-05,

3.88E-05,

1 .'62E-05,

1.42E-05;

1.78E-05,

1.TlE-05,

'1.'96E-05,

2.13E-05,

0.00 ,

1.89E-05,

2.49E-05,

1 .60E-05,

1 .89E-05)

i. 89E-05)

2.45E-05)

4.89E-05)

8.36E-05)

,8.14E-05)

*8.51E-05)

,9.03E-05)

3.'64E-05)'

4.15E-05)

4. T6E-05)

1 .0osE-04)

4.64E-05)

8.43E-05)

6.82E-05)

5.13E-05)

2. 13E-05)

l.T4E-0;

1.T4E-05 1.70 E-05 l.T OE-05.

-1.51E-05.

1.59E-05

1.55E-05

1.52E-05

1.60E-05

1.62E-05

* 1.58E-05

1.59E-05

1.59E-05

1.54E-05

1.62E-05

1.59E-05

1. 50E-05

of lambdaO values for components, assuming beta =,1.34E-0-5

Component

AIOV-1A

MOV-1B

AIOV-1C

MOV- 1D

MOV- 1E

MOV-1E(R)

MOV-1F

MOV-1F(R)

AIOV-2A

AIOV-2B

AyOV-2C

MfOV-2C(R)

MOV-2D

MOV-2E

MOV-2E(R)

MAOV-2F

Signii

0.09

0.09

0.23

0.32

1.00

0.42

0.15

1.00

'1.00

0.80

0.90

1.00

0.45

0.65

1.00

0.21

F 95.07. Confidence Interval

(-*-----I . .-

(-*---- I -- -

(--*---- I---)

(I---*___________)-

(---------*-------________ --------- __ ------

(...---I----*---_--- _

(-I-------------------------- - __ 

(_-I------------)------

(-I---------------)

OVERALL

Overall significance level for testing equality of lambda0s = 1.00

Signif for jth component' means esti'mated probability that

lambdaO-hat(j) - lambdaO-hat(others).is at. least as'extreme as observed,

Overall significance level is P[at least one signif(j) <- observed. min],

and is c6mpu'ted'with':B6nferroni inequality ' '

-> # No evidence of different values of'lam'bdaO'

-> # Before I do the next step, I check what options are in effect

A-13 ' -
-> st

Model type: Exponential failure rate

Components are not assumed to have a common value of lambdaO.

Conditional likelihood is used for inference about beta.

Inference for lambdaO and beta does not use joint asymptotic normality.

The data are from the file movdemo.dat

The session is being echoed to file exampl.out

_> ####ff######l###f#

-> # Step 7, Generate 2-dimensional confidence region.

-> # If PF switch were on, the resulting disk file could be used to

-> #t generate one of the regions plotted in Figure 5 of the text.

-> ##X############## -, ci + 90

Setting switch to assume lambda0s equal for all components

Joint 90.0% confidence region for

with lambdaO plotted on log scale

(lambdaO,beta)

beta

2.62E-05

2.49E-05

2.36E-05

2.24E-05

2.1lE-05

1.98E-05

1.85E-05

1 .72E-05

1.59E-05

1.47E-05

1.34E-05

1.21E-05

1.08E-05

9.52E-06

8.24E-06

6.95E-06

5.67E-06

4.38E-06

3.1OE-06

1.82E-06

5.32E-07

lambdaO

(3.78E-06, 6.84E-06)

(4.26E-06, 7.70E-06)

(4.79E-06, 8.67E-06)

(5.38E-06, 9.75E-06)

(6.04E-06, 1.09E-05)

(6.78E-06, 1.23E-05)

(7.59E-06, 1.37E-05)

(8.48E-06, 1.53E-05)

(9.46E-06, 1.71E-05)

(1.05E-05, 1.91E-05)

(1.17E-05, 2.12E-05)

(1.30E-05, 2.36E-05)

(1.45E-05, 2.62E-05)

(1.60E-05, 2.90E-05)

(1.77E-05, 3.21E-05)

(1.96E-05, 3.54E-05)

(2.16E-05, 3.90E-05)

(2.37E-05, 4.29E-05)

(2.61E-05, 4.72E-05)

(2.86E-0, 5.17E-05)

(3.13E-05, o.67E-05)

* *** ***** *

*

*

W

* *

*

*

*

b

e

t

a

* *

*

* N

*

* #

*

#

*

****##** S -

lambdaO

-> # Note, the above command caused PHAZE to turn on the

-> #t lambdas-equal switch.

-> # Note also, a 90% conf. interval for lambdaO is

-> #t (3.78E-06, 5.67E-05)

-> # By contrast, the CI command gives a confidence interval, assuming

-> it that the value of beta is the MLE beta-hat, as follows

-> ci 1 90

90.0% confidence interval for lambdaO, assuming beta=1.336E-05. is

( 1.24E-05, 2.04E-05)

A- 1
-> # This interval is much shorter than the range of lambdaO in the

-> # .2-dimensional region, because it assumes a value of beta.

_> ################### -1

-> # From now on, we use bivariate asymptotic normality of

-> # (beta-hat, log lambda-hat)

_> ###~################

-> an -

Using asymptotic normality of MLE of (beta, log-lambdaO)

Setting switch to do joint inference with fufl likelihood-

-> # Note, turning on the AN switch also turnedon-the.FL switch.

-> st . X -

Model type: Exponential failure rate

Components assumed tothave a common..value of lambdaO. : _

Full likelihood is used for inference about beta and-lambdaO(s).-

Inference for lambdaO and beta uses joint asymptotic normality .

of MLEs of beta and log lambdaO. -;

The data are from the file movdemo.dat -.

The session is being echoed to file exampl.out . -

-> #################.-;.-

-> # Step 8, Find MLE

_> ##u############### -> .1+ . . .

Maximum Likelihood Estimate beta-hat is 1.071E-05

Maximum Likelihood Estimate lambdaO-hat is 1.984E-05 ; :

assuming that beta = 1.071E-05

Estimated Fisher information matrix of (beta, log-lambdaO)

3.469E+11 3.765E+06

3.765E+06 47.0

Estimated covariance matrix of MLE of (beta, log-lambdaO)

2.208E-11 -1.769E-06

-1.769E-06 0.163 . ' . r. ". . - *

_> ##########~########; --.- A

-> # Step 9, Investigate adequacy of bivariate normal approximation

-> # If the PF switch were on for the next command, the resulting

-> # disk file could be used to generate the ellipse in Figure-5' -

-> # of the text.

>#X############0#####; . *. _ ->ci + 90

A 15

-, S .1, ! . : .; - .I .; , .,- ._ -, :
Joint 90.0% confidence region for

with lambdaO plotted on log scale

(lambdaO.beta)

beta

2.08E-05

1.98E-05

1.88E-05

1.78E-05

1.68E-05

1 .58E-05

1.47E-05

1.37E-05

1. 7E-05

1.17E-05

1.07E-05

9.70E-06

8.69E-06

7.68E-06

6.68E-06

5.67E-06

4.66E-06

3.65E-06

2.64E-06

1.63E-06

6.24E-07

lambdaO

(8.85E-06, 8.85E-06)

(8.37E-06, l.1OE-05)

(8.62E-06, 1.25E-05)

(9.01E-06, 1.41E-05)

(9.51E-06, 1.57E-05)

(1.O1E-05, 1.74E-05)

(1.08E-05, 1.91E-05)

(1.16E-05, 2.10E-05)

(1.24E-05, 2.29E-05)

(1.34E-05, 2.50E-05)

(1.45E-05, 2.71E-05)

(1.58E-05, 2.94E-05)

(1.72E-05, 3.17E-05)

(1.88E-05, 3.41E-05)

(2.06E-05, 3.65E-0)

(2.27E-05, 3.90E-0-5)

(2.51E-05, 4.14E-05)

(2.79E-05, 4.37E-05)

(3.14E-05, 4.57E-05)

(3.58E-05, 4.71E-05)

(4.45E-05, 4.45E-05)

* *

*

*

*

*

*

*

#

*

b

e

t

a

*

*

* 5M W

*

*

*

*

*

.

:K

lambdaO

Figure 5 shows that the two regions overlay well.

-> # Therefore, we proceed with the hazard function estimation

-> # assuming normality of the MLE

_-> ##############

-> # Step 10

-> hf

Enter time (in hours after installation) at which hazard function

should be evaluated, or 'R' and two times for range

-> r

Enter two times (in hours after installation) defining range

over which hazard function should be evaluated

Default is the data window O.OOOOE-O1 to 1.3003E+05

Enter two times or RETURN for default

Time range is from O.OOOOE-01 to 1.3003E+05 hours

How many increments should this be divided into?

Default is 20 increments of 6502. hours each

Enter number of increments, or RETURN for default

-> 4

A-16
MLE and approximate 90.0% conf. int. for hazard function

Ti

o.ooa

3.25C

6.501

9.752

1.30C

-> #

ime Haz-lo Haz-MLE Haz-up

)OE-O1 ( 1.02E-05, 1.98E-05, 3.86E-05)

)8E+04 ( 1.81E-05, 2.81E-05, 4.36E-05)

.6E+04 ( 3.05E-05, 3.98E-05, 5.20E-05)

24E+04 ( 4.28E-05, 5.64E-05, 7.43E-05)

)3E+05 ('5.07E-05, 7.99E-05, 1.26E-04)

For a real plot, I would have turned on the

%lean

I 2.1E-05

2.91E-05

4.03E-05

5.72E-05

- 8.30E-05

PF switch, -

E.F.,

1.94

1 .5.5

1.31-

1.32

1. 5S

-> # and I would have used more increments.-

-> # By contrast, here is the hazard ,function if the AN:swit4

-> # We reset the options by the.LE command

->le

Components not assumed to'have common value for lambdaO

Setting switch to use conditional likelihood for inference f4

Setting switch to not use asymptotic normiality

->hf r

Enter two times (in hours after installation) defining range

over which hazard function should be evaluated

Default is the data window O.OOOOE-01 to,.1.3003E+05

Enter two times or RETURN for default . - ,

Setting switch to assume lambdaOs 'e'qual for all components

Time range'is from'O.OOOOE-Ol to.1.3003E+05 hours

How many increments'should th'is -bedivided into?

Default is 20 incr'ements 6f '6502. hours each

Enter number of increments', orlRETURN for default

;h is-off.

'r beta

. z

I t - . I I

->4

AILE and conservative 90.0% conf. int. for hazard function

Time Haz-lo Hiz-MLE Hiz-up

O.OOOOE-01 ( 3.78E-06, 1.60E-05, 5.67E-05)

3.2508E+'4 ( 8.86E-'06, 2.47E-05, 5.76E-05)

6.5016E+04 ( 2.08E-05, 3.81E-05, 5.86E-05) ..

9.7524E+04 ( 3.30E-05, 5.89E-05, 8.81E-05).

1.3003E+05 ( 3.35E-05, 9.09E-05, 2.07E-04) ; ;

-> # Each interval is conservative,' based on'the largest and smallest

-> # values attained by the hazard function in the non-elliptical

-> # 2-dimensional confidence region produced above.

-> ############## ': '' -> # Now we-demonstrate a few other commands . .

->'sm l .

Failure times centered at-' 69404-.5 hours '

,.. ,.... . .. :. ,,-:..

A-17
-> st

Model type: Linear failure rate

Components assumed to have a common value of lambdaO.

Conditional likelihood is used for inference about beta.

Inference for lambdaO and beta does not use joint asymptotic normality.

The failure times are centered at 69404.5

The data are from the file movdemo.dat

The session is being echoed to file exampl.out

-> # The options remain as before we changed models, but the parameter

-> # estimates are no longer in effect, until we re-estimate.

-> # This can be demonstrated by the KS command, for example.

-> ks

For testing model with beta = O.OOOE-01

Kolmogorov-Smirnov test statistic is 0.178

Value of test statistic is attained at observation 21

Significance level equals 0.125, based on 44 observed failures

For testing model with beta = 0 and all lambdas equal,

based on observed failure times rather than component ages

Kolmogorov-Smirnov test statistic is 0.186

Value of test statistic is attained at observation 17

This test is relevant if the number of components was the same at all timnes

Significance level equals 0.078, based on 47 observed failures

-> # See, PMAZE now considers beta to equal zero, not the value

-> # calculated as the MLE under the exponential model.

-> sm w

_> st

Model type: Weibull failure rate

Components assumed to have a common value of lambda0.

Conditional likelihood is used for inference about beta.

Inference for lambdaO and beta does not use joint asymptotic normality.

LambdaO is the value of the failure rate at time 88584.0

The data are from the file movdemo.dat

The session is being echoed to file exampl.out

-> # We can reset the normalizing time to the t-mid calculated for

-> #the linear model:

-> sn 69404.5

-> st

Model type: Weibull failure rate

Components assumed to have a common value of lambdaO.

Conditional likelihood is used for inference about'beta.

Inference for lambda0 and beta does not use joint asymptotic normality.

LambdaO is the value of the failure rate at time 69404.5

The data are from the file movdemo.dat

The session is being echoed to file exampl.out

-> # Enough. Let's quit.

-> q

A-18