Prediction of the Expected Safety 

Performance of Rural Two-Lane 

Highways

PUBLICATION NO. FHWA-RD-99-207 DECEMBER 2000

Research, Development, and Technology

Turner-Fairbank Highway Research Center

6300 Georgetown Pike

McLean, VA 22101-2296
FOREWORD

This report documents the algorithm for predicting the safety performance of rural two-lane

highways that forms the basis for the Crash Prediction Module of the Interactive Highway Safety

Design Model. The algorithm estimates the effect on safety performance of roadway segment

parameters including lane width, shoulder width, shoulder type, horizontal curves, grades,

driveway density, two-way left-turn lanes, passing lanes, and roadside design, and of intersection

parameters including skew angle, traffic control, exclusive left- and right-turn lanes, sight distance,

and driveways. The algorithm enables highway agencies to estimate the safety performance of

existing or proposed highways and to compare the expected safety performance of geometric

design alternatives. 

Michael F. Trentacoste

Director, Office of Safety

Research and Development

NOTICE

This document is disseminated under the sponsorship of the Department of Transportation in the

interest of information exchange. The United States Government assumes no liability for its

content or use thereof. This report does not constitute a standard, specification, or regulation.

The United States Government does not endorse products or manufacturers. Trade and

manufacturers’ names appear in this report only because they are considered essential to the

object of this document. 
Technical Report Documentation Page

1. Report No.

FHWA-RD-99-207

2. Government Accession No. 3. Recipient's Catalog No.

4. Title and Subtitle

PREDICTION OF THE EXPECTED SAFETY PERFORMANCE OF RURAL

TWO-LANE HIGHWAYS

5. Report Date

6. Performing Organization

 

7. Author(s)

D.W. Harwood, F.M. Council, E. Hauer, W.E. Hughes, and A. Vogt

8. Performing Organization Report No.

4584-09

9. Performing Organization Name and Address

Midwest Research Institute

425 Volker Boulevard

Kansas City, Missouri 64110-2299

10. Work Unit No. (TRAIS)

11. Contract or Grant No.

DTFH61-96-C-00055

12. Sponsoring Agency Name and Address

Office of Safety Research and Development

Federal Highway Administration

6300 Georgetown Pike

McLean, Virginia 22101-2296

13. Type of Report and Period Covered

Technical Report 

May 1997—September 2000

14. Sponsoring Agency Code

15. Supplementary Notes

Contracting Officer's Technical Representative (COTR): Michael S. Griffith, HRDS-06

16. Abstract

This report presents an algorithm for predicting the safety performance of a rural two-lane highway. The accident

prediction algorithm consists of base models and accident modification factors for both roadway segments and at-grade

intersections on rural two-lane highways. The base models provide an estimate of the safety performance of a roadway or

intersection for a set of assumed nominal or base conditions. The accident modification factors adjust the base model

predictions to account for the effects on safety for roadway segments of lane width, shoulder width, shoulder type,

horizontal curves, grades, driveway density, two-way left-turn lanes, passing lanes, roadside design and the effects on

safety for at-grade intersections of skew angle, traffic control, exclusive left- and right-turn lanes, sight distance, and

driveways.

The accident prediction algorithm is intended for application by highway agencies to estimate the safety performance

of an existing or proposed roadway. The algorithm can be used to compare the anticipated safety performance of two or

more geometric alternatives for a proposed highway improvement.

The accident prediction algorithm includes a calibration procedure that can be used to adapt the predicted results to

the safety conditions encountered by any particular highway agency on rural two-lane highways. The algorithm also

includes an Empirical Bayes procedure that can be applied to utilize the safety predictions provided by the algorithm

together with actual site-specific accident history data.

17. Key Words

Safety Accident Prediction

Accident Modeling Geometric Design

Two-Lane Highways Empirical Bayes Estimation

Roadway Segments At-Grade Intersections

18. Distribution Statement

No restrictions. This document is available to the public

through the National Technical Information Service,

Springfield, Virginia 22161.

19. Security Classif. (of this report)

Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of Pages

197

22. Price

Form DOT F 1700.7 (8-72) Reproduction of completed page authorized
iii
iii

Contents

1. INTRODUCTION ...................................................... 1

Estimates from Historical Accident Data .................................. 1

Estimates from Statistical Models ....................................... 2

Estimates from Before-and-After Studies ................................. 4

Estimates from Expert Judgment ........................................ 4

A New Approach .................................................. 4

Organization of this Report ............................................ 5

Units of Measure ................................................... 5

2. OVERVIEW OF THEACCIDENTPREDICTION ALGORITHM ................. 7

Accident Prediction Algorithm for Roadway Segments ....................... 7

Accident Prediction Algorithm for At-Grade Intersections ..................... 9

Predicted Accident Frequency for an Entire Project ......................... 9

Estimated Accident Severity and Accident Type Distributions ................. 10

Strengths and Weaknesses of this Approach .............................. 10

Structure of the Accident Prediction Algorithm ............................ 14

3. BASEMODELS ....................................................... 17

Base Model for Roadway Segments .................................... 17

Base Models for At-Grade Intersections ................................ 19

Calibration Procedure .............................................. 24

4. ACCIDENTMODIFICATION FACTORS .................................. 27

Development of Accident Modification Factors ........................... 27

Roadway Segments ................................................ 29

 Grades ......................................................... 40

At-Grade Intersections .............................................. 43

5. SENSITIVITY ANALYSIS RESULTS ..................................... 51

Roadway Segments ................................................ 51

Three-Leg STOP-Controlled Intersections ............................... 60

Four-Leg Signalized Intersections ...................................... 64

6. IMPLEMENTATION OF THEACCIDENTPREDICTION ALGORITHM WITHIN THE

IHSDM ............................................................ 69

Accident Prediction When Site-Specific Accident History Data are not Available . . 70

Accident Prediction When Site-Specific Accident History Data are Available . . 77

Situations in Which the EB Procedure Should and Should Not Be Applied ....... 78

Empirical Bayes Procedure .......................................... 79
iv

Example Application of the EB Procedure ............................... 82

Step-by-Step Methodology for Applying the Accident Prediction Algorithm Including

the EB Procedure ................................................ 90

7. CONCLUSIONS, RECOMMENDATIONS, AND FUTURE ENHANCEMENTS . . 105

Conclusions ..................................................... 105

Future Enhancement of the Accident Prediction Algorithm ................... 106

8. REFERENCES ....................................................... 111

9. BIBLIOGRAPHY ..................................................... 115

APPENDIX A EXPERT PANEL MEMBERSHIP .................................... 125

APPENDIX B DEVELOPMENT OF BASE MODELS ................................. 127

APPENDIX C CALIBRATION PROCEDURE TO ADAPT THE ACCIDENT PREDICTION ALGORITHM

TO THE DATA OF A PARTICULAR HIGHWAY AGENCY .................. 161

APPENDIX D DEFINITIONS OF ROADSIDE HAZARD RATINGS USED WITH THE ACCIDENT

PREDICTION ALGORITHM ....................................... 189
1

1. INTRODUCTION

One of the most critical gaps in the management of highway safety is the lack of a reliable

method for estimating the safety performance of an existing or planned roadway. Accident

record systems have been developed and maintained by highway agencies to monitor the safety

performance of their roadways, but these provide historical or retrospective data. Effective

management requires a prospective viewpoint. Highway engineers need to know not what the

safety performance of a roadway was in the recent or distant past, but what it is now and what it

is likely to be in the future if particular proposed actions are taken.

In the past, when current or future safety performance estimates for a roadway were

needed, they have been developed by one of four approaches: averages from historical accident

data, predictions from statistical models based on regression analysis, results of before-after

studies, and expert judgments made by experienced engineers. Each of these methods, used

alone, has significant weaknesses which are described below. A new approach combining

elements of each of these methods into an accident prediction algorithm is then described. This

new accident prediction algorithm, developed specifically for application to rural two-lane

highways, is the subject of this report.

Estimates from Historical Accident Data

Historical accident data are an important indicator of the safety performance of a roadway,

but they suffer from the weakness of being highly variable. Given this high variability, it is difficult

to estimate the long-term expected accident rate using a relatively short-duration sample of 1 to 3

years of accident data. This is especially true for rural roadway sections and intersections where

accidents are very rare events and many locations experience no accidents, or at most one

accident, over a period of several years. If a location has experienced no accidents in the past

several years, it is certainly not correct to think that it will never experience an accident, yet the

available data for that site alone provide an insufficient basis for estimating its long-term expected

safety performance.

Roadway improvement programs based on safety are often managed with accident

surveillance systems that use accident records to identify high-accident locations. A

high-accident location is a roadway section or intersection identified because it experienced more

than a specified threshold number of accidents during a recent period (typically 1 to 3 years). 

Each high-accident location is investigated by the engineering staff of the responsible highway

agency and, at locations where a particular accident pattern is clearly evident and an appropriate

countermeasure is feasible, an improvement project may be programmed and constructed. The

decisionmaking concerning such projects often involves a benefit-cost or cost-effectiveness

calculation based on the expected percentage reduction in accidents from the level of recent

accident experience found by the accident surveillance program. However, both statistical theory
2

and actual experience show that, because of the random nature of accidents, locations with high

short-term accident experience are likely to experience fewer accidents in the future even if no

improvement is made. This phenomenon, known as regression to the mean, makes it difficult

both to identify potential problem locations through accident surveillance and estimate the

potential (or actual) effectiveness of improvements made at such locations.

Estimates from Statistical Models

Safety analysts have, for many years, applied statistical techniques to develop models to

predict the accident experience of roadways and intersections. Such models are developed by

obtaining a database of accident and roadway characteristics (e.g., traffic volumes, geometric

design features, and traffic control features) data from highway agency records, selecting an

appropriate functional form for the model, and using regression analysis to estimate the values of

the coefficients or parameters in that model. Historically, most such models were developed with

multiple regression analysis. Recently, researchers have begun to use Poisson and negative

binomial regression analyses which are theoretically better suited to accident data based on small

counts (i.e., zero or nearly zero accidents at many sites). However, regardless of the statistical

technique used, accident prediction models never quite seem to meet the expectations of their

developers and potential users. 

Regression models are very accurate tools for predicting the expected total accident

experience for a location or a class of locations, but they have not proved satisfactory in isolating

the effects of individual geometric or traffic control features. There is a strong temptation to

interpret each coefficient in a regression model as representing the true effect of an incremental

change in its associated roadway feature. This is a reasonable assumption is some cases, but not

in others. A key drawback of regression models is that they are based on statistical correlations

between roadway characteristics and accidents that do not necessarily represent cause-andeffect relationships. Furthermore, if the independent variables in the model are strongly

correlated to one another, it is difficult to separate their individual effects. In addition, if a

variable in the model is strongly correlated to an important variable that happens not to be

included in the available data base, the coefficient of the variable in the model may represent the

effect of the unavailable variable rather than its own effect. Thus, the value of the coefficient of a

particular geometric feature may be a good estimate of the actual effect of that feature on safety,

or it may be merely an artifact of, or a surrogate for, its correlation to other variables.

As an example, consider the following negative binomial regression model developed in a

recent FHWA study to predict the accident experience at urban, four-leg intersections with

STOP control on the minor road:(1)
*

 Average lane width in this equation is specified in conventional units of measure (feet). 

See the explanation in the section entitled Units of Measure on page 5 of this report.

3

Y ' e

&5.073 (X1

)

0.635 (X2

)

0.294 exp(&0.969 X3

) exp(&0.518 X4)

(X5)

&0.091 exp(0.340 X6

) exp(0.087 X7) exp(&0.331 X8)

exp(&0.175 X9)

(1)

where:

 Y = expected number of total multiple-vehicle accidents in a 3-year period;

 X1 = average daily traffic on major road (veh/day);

 X2 = average daily traffic on minor road (veh/day);

 X3 = 1 if left-turn are prohibited on one or more major-road approaches; 0 otherwise;

 X4 = 1 if no access control is present along the major road approaches; 0 otherwise;

 X5 = average lane width on major road (ft)*;

 X6 = 1 if major road has three or fewer through lanes in both directions of travel

combined; 0 otherwise;

 X7 = 1 if major road has four or five through lanes in both directions of travel combined;

0 otherwise;

 X8 = 1 if there is no channelization for free right turns; 0 otherwise; and

 X9 = 1 if the intersection has no lighting; 0 otherwise.

This model, overall, provides quite reliable predictions of the total accident experience of urban,

four-leg, STOP-controlled intersections. In addition, the coefficients of many of the terms

appear to reasonably represent the expected effects of their associated variables. However, two

of the variables in the model have coefficients that are in a direction opposite to that which safety

engineers normally presume for those variables. Specifically, the negative coefficient of the

access control factor (X4) implies that more accidents would be expected at an intersection with

access-controlled approaches than at an intersection without access-controlled approaches. 

Furthermore, the negative coefficient of the lighting factor (X9), implies that lighted intersections

have more accidents than unlighted intersections. Such interpretations are unreasonable. The

negative signs for the access control and lighting variables in equation (1) could result merely from

correlations of access control and lighting with the variables already accounted for in the model,

such as traffic volumes, or with other important variables that are not included in the model

because no data for those variables are available. It is also possible that lighting has been

installed as an accident countermeasure at high-accident locations, so that lighting appears to be

associated with locations that have more accidents. Thus, while regression equations may

provide useful predictive models, their coefficients may be unreliable indicators of the incremental

effects of individual roadway features on safety.
4

Estimates from Before-and-After Studies

Before-and-after studies have been used for many years to evaluate the effectiveness of

highway improvements in reducing accidents. However, most before-and-after studies reported

in the literature have design flaws such that the study design cannot account for the effects of

regression to the mean. Therefore, the potential user of the before-and-after study results cannot

be certain whether they represent the true effectiveness of the potential improvement in reducing

accidents or an overoptimistic forecast that is biased by regression to the mean.

Safety experts are generally of the opinion that, if the potential bias caused by regression to

the mean can be overcome, a before-and-after study may provide the best method to quantify

the safety effects of roadway geometric and traffic control features. Hauer(2) has developed a

new approach that remedies the problem of regression to the mean that has, in the past, caused

before-and-after studies to provide unreliable results. However, very few of these well-designed

before-and-after studies have been conducted.

Estimates from Expert Judgment

Expert judgment, developed from many years of experience in the highway safety field, can

have an important role in making reliable safety estimates. Experts may have difficulty in making

quantitative estimates with no point of reference, but experts are usually very good at making

comparative judgments (e.g., A is likely to be less than B, or C is likely to be about 10 percent

larger than D). Thus, experts need a frame of reference based on historical accident data,

statistical models, or before-and-after study results to make useful judgments.

A New Approach

This report presents a new approach to accident prediction that combines the use of

historical accident data, regression analysis, before-and-after studies, and expert judgment to

make safety predictions that are better than those that could be made by any of these three

approaches alone. The recommended approach to accident prediction has its basis in published

safety literature, including both before-and-after evaluations and regression models, is sensitive to

the geometric features that are of greatest interest to highway designers, and incorporates

judgments made by a broadly based group of safety experts.

This report shows how this new approach can be implemented in an accident prediction

algorithm for rural two-lane highways. This same approach can potentially be adapted in the

future to rural multilane highways, urban arterial streets, and rural or urban freeways.
5

The Federal Highway Administration (FHWA) is currently developing an Interactive

Highway Safety Design Model (IHSDM) for use by highway designers to incorporate more

explicit consideration of safety into the highway design process. IHSDM will consist of a set of

computer tools that can work interactively with the Computer-Aided Design (CAD) systems

used by many agencies to design highway improvements. The components of the IHSDM will

include a Crash Prediction Module (CPM), Roadside Safety Module (RSM), Intersection

Diagnostic Review Module (DRM), Design Consistency Module (DCM), Policy Review

Module (PRM), Driver/Vehicle Module (D/VM), and Traffic Analysis Module (TAM). Initial

priority in IHSDM development is being given to evaluation of rural two-lane highways.

The accident prediction algorithm presented in this report has been developed for

incorporation in the IHSDM as the CPM for rural two-lane highways, but is also suitable for use

as a stand-alone model to predict the safety performance of rural two-lane highways. This report

documents how the accident prediction algorithm was developed and how it will function within

the IHSDM.

Organization of this Report

The remainder of this report is organized as follows. Section 2 presents an overview of the

accident prediction algorithm and its two primary components, base models and accident

modification factors. A more detailed description of the base models and accident modification

factors is presented in sections 3 and 4, respectively. Section 5 presents the results of sensitivity

analyses conducted with the accident prediction algorithm, and section 6 explains how the

accident prediction algorithm will be implemented within the IHSDM. The conclusions and

recommendations of the report are presented in section 7 and a list of references is presented in

section 8.

Appendix A identifies the members of the expert panels that developed the accident

modification factors. Appendix B documents the development of the base models. Appendix C

presents a calibration procedure that can be used by any highway agency to adapt the accident

prediction algorithm to their own local conditions and to the safety performance of their

highways. Appendix D documents the definitions of the roadside hazard ratings used in the

accident prediction algorithm to represent roadside design features.

Units of Measure

The text of this report presents all measured quantities in SI (metric) units with equivalent

quantities in conventional (English) units following in parentheses. However, virtually all of the

research on which the report is based was conducted using conventional units of measure. 

Therefore, all equations in the report, like equation (1) above, use conventional units. A metric
6

conversion chart is included for the convenience of readers. The software developed to

implement the accident prediction algorithm will allow users to provide input and obtain output at

their option in either SI or conventional units.
7

2. OVERVIEW OF THE ACCIDENT PREDICTIONALGORITHM

This section of the report presents an overview of the accident prediction algorithm for rural

two-lane highways. Separate accident prediction algorithms have been developed for roadway

segments and for three types of at-grade intersections. These separate algorithms can be used

together to predict the total accident experience for an entire highway section or improvement

project.

In the development of these algorithms, it was decided the roadway segment accident

prediction algorithm would predict all non-intersection-related accidents for each of the

segments that make up a highway project. Non-intersection-related accidents include accidents

that occur near an intersection but are not related to the intersections. For example, a ran-offroad accident or a head-on collision that occurs within 15 m (50 ft) of an intersection, but it

considered by the investigating officer to be unrelated to the intersection, would be classified as a

non-intersection-related accident. The intersection accident prediction algorithms predict the

additional intersection-related accidents that occur at or rear the intersection and occur because

of the presence of the intersection. For modeling purposes, only accidents that occurred within

76 m (250 ft) of the intersection and occurred because of the presence of the intersection were

considered to be intersection- related accidents. The total predicted accident frequency for

any highway project is the sum of the predicted frequency of non-intersection-related accidents

for each of the roadway segments and the predicted frequency of intersection-related accidents

for each of the at-grade intersections that make up the project.

The accident prediction algorithms for roadway segments and at-grade intersections are

each composed of two components: base models and accident modification factors. These

components and the manner in which they are combined are described below.

Accident Prediction Algorithm for Roadway Segments

The base model for roadway segments is the best available regression model for predicting

the total accident frequency of a roadway segment on a rural two-lane highway. The base

model, like all regression models, predicts the value of a dependent variable as a function of a set

of independent variables. For the roadway segment model, the dependent variable is the total

expected accident frequency on the roadway segment during a specified time period. The

independent variables used to predict accident frequency are descriptors of the traffic volumes,

geometric design features, and traffic control features of the roadway segment. The specific

regression model to be used as the base model for rural two-lane roadway segments is presented

in section 3 of this report.

As discussed in the introduction to this report, regression models like the base model are

useful in predicting overall accident frequency, but their coefficients cannot necessarily be relied
8

Nrs ' Nbr (AMF1r AMF2r@@@ AMFnr) (2)

upon to represent the incremental effects of individual geometric design and traffic control

features. Therefore, the base model will be used only to estimate the expected accident

frequency for a specified set of nominal base conditions, such as 3.6-m (12-ft) lane widths and

1.8-m (6-ft) shoulder widths. This base estimate of accident frequency will then be adjusted with

accident modification factors (AMFs) that represent the safety effects of individual geometric

design and traffic elements. The general formulation of the algorithm predicting roadway segment

accident frequency and combining the base models and AMFs is shown below:

where:

Nrs = predicted number of total roadway segment accidents per year after

application of accident modification factors;

Nbr = predicted number of total roadway segment accidents per year for

nominal or base conditions; and

AMF = accident modification factors for roadway segments. 1r@@@AMFnr

The AMFs are multiplicative factors used to adjust the base accident frequency for the effect of

individual geometric design and traffic control features. Each AMF is formulated so that the

nominal or base condition is represented by an AMF of 1.00. Conditions associated with higher

accident experience than the nominal or base condition will have AMFs greater than 1.00 and

conditions associated with lower accident experience than the nominal or base condition will have

AMFs less than 1.00. For example, if the accident frequency predicted by the base model for

roadway segments (Nbr) is based on 3.6-m (12-ft) lanes, but a particular roadway section of

interest has 3.3-m (11-ft) lanes, the AMF for lane width might have a value of 1.15. This AMF

implies that a two-lane roadway segment with 3.3-m (11-ft) lanes would be expected to

experience 15 percent more accidents than a comparable roadway section with 3.6-m (12-ft)

lanes.

The effect of average daily traffic (ADT) volume on predicted accident frequency is

incorporated through the base models, while the effects of geometric design and traffic control

features are incorporated through the AMFs.

The formulation shown in equation (2) allows the AMF for each geometric design and traffic

control element to be based solely on the most reliable information concerning the safety effects

of that particular element. The best method for considering the safety effects of lane width can be

selected as the basis for the AMF without being constrained by the treatment of lane width in the

base model or by the formulation of any other AMF. Thus, each AMF in the accident prediction
9

Nint ' Nbi (AMF1i AMF2i@@@ AMFni) (3)

algorithm can be based on the best and most applicable research available, as selected and

interpreted by knowledgeable experts. In fact, two panels of experts, identified in appendix A,

were formed to establish the AMFs presented in this report. Section 4 of the report presents the

AMFs used in predicting roadway segment accidents and documents their development.

Accident Prediction Algorithm for At-Grade Intersections

The structure of the accident prediction algorithm for at-grade intersections is similar to the

algorithm for roadway sections presented above. The predicted frequency of accidents that

occur at or are related to an at-grade intersection is determined as:

where: Nint = predicted number of total intersection-related accidents

per year after application of accident modification factors;

Nbi = predicted number of total intersection-related accidents

per year for nominal or base conditions; and 

 AMF = accident modification factors for intersections. 1i,@@@,AMFni

Separate base models have been formulated for three-leg intersections with STOP control, fourleg intersections with STOP control, and four-leg signalized intersections. The AMFs used in the

accident prediction algorithm for these three intersection types also differ, but the algorithms for

all three intersection types are structured as shown in equation (3). The base models for at-grade

intersections are presented in section 3 of this report, and the AMFs for at-grade intersections

are presented in section 4.

The effect of traffic volume on predicted accident frequency for at-grade intersections is

incorporated through the base models, while the effect of geometric and traffic control features

are incorporated through the AMFs. Each of the base models for at-grade intersections

incorporates separate effects for the ADTs on the major- and minor-road legs, respectively.

Predicted Accident Frequency for an Entire Project

The accident prediction algorithm will be applied in IHSDM to estimate the safety

performance of entire proposed projects or extended highway sections. The total predicted

accident frequency for an entire project or an extended highway section can be determined as:
10

Nt ' j

all

segments

Nrs% j

all

intersections

Nint

(4)

where: Nt = predicted accident frequency for an entire project or an

extended highway section.

Estimated Accident Severity and Accident Type Distributions

In addition to predictions of accident frequency based on equations (2) and (3), the accident

prediction algorithm will also provide estimates of the accident severity and accident type

distributions for roadway segments and at-grade intersections. Tables 1 and 2 present default

estimates of the accident severity and accident type distributions, respectively, that are used in the

accident prediction algorithm. The default accident severity and accident type distributions in

tables 1 and 2 are based on data from the FHWA Highway Safety Information System (HSIS)

for Illinois, Michigan, Minnesota, and North Carolina. These default distributions for accident

severity and accident type can be replaced with data suitable for the rural two-lane highway

system of a particular highway agency as part of the calibration process described in section 3

and appendix D of this report. Use of distributions applicable to a specific State or geographic

region is particularly appropriate because some percentages in the tables, such as the percentage

of animal-related accidents on roadway segments in table 2, clearly vary geographically.

Strengths and Weaknesses of this Approach

The strengths of the accident prediction algorithms formulated as shown in equations (2) and

(3) are as follows:

• The accident prediction algorithms make quantitative estimates of accident frequency.

• The base models serve as scale factors to assure that the magnitude of the predicted

accident frequency is appropriate, while the AMFs assure that the predicted accident

frequency is sensitive to site-specific geometric design and traffic control features.

• The use of AMFs that are separate from the base models assures that the effects of

individual geometric design and traffic control features are not dependent on

inappropriate regression coefficients that are too large, too small, or in the wrong

direction. Each AMF has been developed by a panel of experts to represent the 
11

Table 1. Default Distribution for Accident Severity Level on Rural Two-Lane

Highways.

Accident severity level

Percentage of total accidents

Roadway

segmentsa

Three-leg STOPcontrolled

intersectionsb

Four-leg

STOP-controlled

intersectionsb

Four-leg

signalized

intersectionsb

Fatal 1.3 1.1 1.9 0.4

Incapacitating Injury 5.4 5.0 6.3 4.1

Nonincapacitating injury 10.9 15.2 12.8 12.0

Possible injury 14.5 18.5 20.7 21.2

Total fatal plus injury 32.1 39.8 41.7 37.7

Property damage only 67.9 60.2 58.3 62.3

TOTAL 100.0 100.0 100.0 100.0

a Based on HSIS data for Illinois (1992), Michigan (1995), Minnesota (1996), and North Carolina

(1995).

b Based on HSIS data for Michigan (1995) and Minnesota (1996).
12

Table 2. Default Distribution for Accident Type and Manner of Collision on 

Rural Two-Lane Highways.

Accident type and 

manner of collision

Percentage of total accidents

Roadway

segments a

Three-leg

STOP-controlled

intersections b

Four-leg STOPcontrolled

intersections b

Four-leg

signalized

intersections b

SINGLE-VEHICLE ACCIDENTS

Collision with animal 30.9 2.1 0.6 0.3

Collision with bicycle 0.3 0.7 0.3 1.0

Collision with parked vehicle 0.7 0.1 0.1 0.1

Collision with pedestrian 0.5 0.4 0.2 1.3

Overturned 2.3 2.1 0.6 0.4

Ran off road 28.1 10.4 4.5 1.9

Other single-vehicle accident 3.6 3.9 1.4 1.6

Total single-vehicle accidents 66.3 19.7 7.7 6.6

MULTIPLE-VEHICLE ACCIDENTS

Angle collision 3.9 29.8 51.4 28.5

Head-on collision 1.9 2.0 1.4 1.8

Left-turn collision 4.2 6.4 5.9 9.0

Right-turn collision 0.6 0.4 0.2 0.4

Rear-end collision 13.9 26.2 17.2 36.2

Sideswipe opposite-direction

collision 2.4 2.9 1.7 2.0

Sideswipe same-direction

collision 2.6 4.5 4.4 5.5

Other multiple-vehicle collision 4.1 8.1 10.1 10.0

Total multiple-vehicle accidents 33.7 80.3 92.3 93.4

TOTAL ACCIDENTS 100.0 100.0 100.0 100.0

a Based on HSIS data for Illinois (1992), Michigan (1995), Minnesota (1996), and North Carolina

(1995).

b Based on HSIS data for Michigan (1995) and Minnesota (1996).
13

best information currently available on the safety effects of that particular geometric

design or traffic control feature.

• Equations (2) and (3) give the accident prediction algorithm a modular structure. 

Individual base models or AMFs can be easily replaced as better information becomes

available.

The potential weaknesses of the structure for the accident prediction algorithms based on

equations (2) and (3) are as follows:

• Accident frequencies are known to vary from agency to agency, even between roads

that are nominally similar, because of differences in factors such as accident reporting

thresholds, accident reporting practices, animal populations, driver population, and

climate. However, such variations are not reflected in the base models that were each

developed with data for only one or two States. For this reason, a calibration

procedure has been provided in this report to allow highway agencies to adapt the

accident prediction algorithm to their own local safety conditions. This calibration

procedure involves the estimation of calibration factors appropriate for a particular

highway agency that can be incorporated directly in equations (2) and (3).

• The accident prediction algorithm, as formulated in equations (2) and (3), is based on

data for many locations and on expert judgment, but does not take advantage of

knowledge of the actual accident history of the location being evaluated. Actual

accident history data should be available for many existing locations evaluated with the

accident prediction algorithm. For this reason, a procedure based on the Empirical

Bayes (EB) approach is provided in this report to combine the results of the accident

prediction algorithm with actual site-specific accident history data.

• The use of separate AMFs for each geometric design and traffic control element treats

the safety effects of these individual elements as independent and ignores potential

interactions between them. It is likely that such interactions exist and, ideally, they

should be accounted for in the accident prediction algorithm. However, such

interactions are poorly understood and none could be quantified by the expert panels

that participated in this research. It is the assessment of the expert panels that the

AMFs presented in this report represent the current state of knowledge about the

safety effects of geometric design and traffic control elements and cannot be improved

without further research. If future research leads to a better understanding of

interactions between the safety effects of various geometric features, those research

results can then be incorporated in the accident prediction algorithm.
14

Structure of the Accident Prediction Algorithm

The structure of the accident prediction algorithm, including base models, accident

modification factors, calibration factors, and EB procedure is illustrated in figure 1. The flow

diagram in figure 1 addresses the application of the accident prediction algorithm to a single

roadway segment or at-grade intersection. Section 6 of the report illustrates the application of

the algorithm to a project made up of numerous roadway segments and intersections.
15

Determine predicted

accident frequency, accident

severity distribution, and

accident type distribution

Select a roadway segment

or intersection

Apply base model

Apply calibration factor

Apply AMFs

Actual site-specific

accident history (if available)

Apply EB procedure

Present final predicted

values to user

Figure 1. Flow Diagram of the Accident Prediction Algorithm 

for a Single Roadway Segment or Intersection.
16
17

Nbr ' EXPO exp(0.6409 % 0.1388STATE & 0.0846LW & 0.0591SW % 0.0668RHR % 0.0084DD)

(ÓWHiexp(0.0450DEGi)) (ÓWViexp (0.4652Vi)) (ÓWGiexp(0.1048 GRi))

(5)

3. BASE MODELS

This section of the report presents the base models used as part of the accident prediction

algorithm. The base models for roadway segments and at-grade intersection are addressed

separately in the following discussion. The base models were developed in separate studies by

Vogt and Bared.(3,4,5) The development of the base models and the choices made among

alternative models that were considered are presented in appendix B.

Base Model for Roadway Segments

The base model for roadway segments is presented below:

where: Nbr = predicted number of total accidents per year on a particular roadway

segment;

EXPO = exposure in million vehicle-miles of travel per year = (ADT)(365)(L)(10-6);

ADT = average daily traffic volume (veh/day) on roadway segment;

L = length of roadway segment (mi);

STATE = location of roadway segment (0 in Minnesota, 1 in Washington);

LW = lane width (ft); average lane width if the two directions of travel differ;

SW = shoulder width (ft); average shoulder width if the two directions of travel

differ;

RHR = roadside hazard rating; this measure takes integer values from 1 to 7 and

represents the average level of hazard in the roadside environment along the

roadway segment. (For definitions of the roadside hazard rating categories,

see appendix D; for the development of the roadside hazard ratings, see

Zegeer.(6));

DD = driveway density (driveways per mi) on the roadway segment;
18

WHi = weight factor for the ith horizontal curve in the roadway segment; the

proportion of the total roadway segment length represented by the portion of

the ith horizontal curve that lies within the segment. (The weights, WHi, must

sum to 1.0.);

DEGi = degree of curvature for the ith horizontal curve in the roadway segment

(degrees per 100 ft);

WVj = weight factor for the jth crest vertical curve in the roadway segment; the

proportion of the total roadway segment length represented by the portion of

the jth crest vertical curve that lies within the segment. (The weights, WVj,

must sum to 1.0.);

Vj = crest vertical curve grade rate for the jth crest vertical curve within the

roadway segment in percent change in grade per 31 m (100 ft) = |gj2-gj1|/lj;

gjl, gj2 = roadway grades at the beginning and end of the jth vertical curve (percent);

lj = length of the jth vertical curve (in hundreds of feet);

WGk = weight factor for the kth straight grade segment; the proportion of the total

roadway segment length represented by the portion of the kth straight grade

segment that lies within the segment. (The weights, WGk, must sum to 1.0.);

and

GRk = absolute value of grade for the kth straight grade on the segment (percent).

This model was developed with negative binomial regression analysis for data from 619 rural

two-lane highway segments in Minnesota and 712 roadway segments in Washington obtained

from the FHWA HSIS. These roadway segments including approximately 1,130 km (700 mi) of

two-lane roadways in Minnesota and 850 km (530 mi) of roadways in Washington. The

database available for model development included 5 years of accident data (1985-1989) for

each roadway segment in Minnesota and 3 years of accident data (1993-1995) for each

roadway segment in Washington. The model predicts the total non-intersection accident

frequency for any roadway segment for which the independent variables shown in equation (5)

are known. The model predictions are reliable only within the ranges of independent variables

for which data were available in the database used to develop the model (see table 30 in

appendix B).

When the accident prediction model is employed to predict the expected accident

experience for any specified roadway section, equation (5) is used in the following manner:
19

Nbr ' (ADT) (L) (365) (10&6) exp(&0.4865) (6)

• The exposure variable (EXPO) in million vehicle-miles of travel is computed using the

actual ADT and segment length (L) for the roadway section and a duration of 1 year

(365 days). This assures the accident frequency predicted by the base model has units

of accidents per year.

• The STATE variable in base model is set equal to zero, representing Minnesota

conditions. This is done for consistency with the base models for three- and four-leg

STOP-controlled intersections, both of which are based solely on Minnesota data. It

should be noted that the calibration procedure described later in this chapter can be

used to adapt the base models to the safety conditions of any State other than

Minnesota. Calibration would even be desirable to apply the algorithm in Minnesota to

a time period other than the period for which the base models were developed.

• The remaining variables in the model are set to the following nominal or base conditions:

Lane width (LW) 3.6 m (12 ft)

Shoulder width (SW) 1.8 m (6 ft)

Roadside hazard rating (RHR) 3

Driveway density (DD) 3 driveways per km

(5 driveways per mi)

Horizontal curvature None

Vertical curvature None

Grade Level (0 percent)

With the default values given above, the base model in equation (5) reduces to:

Base Models for At-Grade Intersections

Base models have been developed for three types of at-grade intersections on rural twolane highways. These are:

• Three-leg intersections with STOP control on the minor-road approach.

• Four-leg intersections with STOP control on the minor-road approach.

• Four-leg signalized intersections.

The base models for each of these intersection types predict total accident frequency per year for

intersection-related accidents within 76 m (250 ft) of a particular intersection.
20

Nbi ' exp(&11.28 % 0.79ln ADT1% 0.49ln ADT2% 0.19RHRI % 0.28RT) (7)

These models address intersections that have only two lanes on both the major- and minorroad legs. Intersections on multilane highways or intersections between a two-lane highway and

a multilane highway may be addressed in a future improvement to the accident prediction

algorithm. The base models for each of the three intersection types are presented below.

Three-Leg STOP-Controlled Intersections

The base model for three-leg intersections with STOP control on the minor-road leg is

presented below:

where: ADT1 = average daily traffic volume (veh/day) on the major road;

ADT2 = average daily traffic volume (veh/day) on the minor road;

RHRI = roadside hazard rating within 76 m (250 ft) of the intersection on the

major road [see description of the variable RHR in equation (5)];

and 

RT = presence of right-turn lane on the major road (0 = no right-turn lane

present; 1 = right-turn lane present).

This model was developed with negative binominal regression analysis from data for 382

three-leg STOP-controlled intersections in Minnesota. The data base available for model

development included 5 years of accident data (1985-1989) for each intersection. The model

predicts the total intersection-related accident frequency for any three-leg STOP-controlled

intersection for which the independent variables shown in equation (7) are known. The model

predictions are reliable only within the ranges of independent variables for which data were

available in the data base used to develop the model (see table 31 in appendix B).

When the accident prediction model is employed to predict the expected accident frequency

for any specified three-leg STOP-controlled intersection on a two-lane highway, equation (7) is

used in the following manner:

• The traffic volume variables (ADT1 and ADT2) are set equal to the actual ADTs of the

major- and minor-road legs. If the ADTs differ between the two major-road legs, they

should be averaged.

• The remaining variables in the model should be set equal to the following nominal or

base conditions:
21

Nbi ' exp(&9.34%0.60ln ADT1% 0.61ln ADT2% 0.13 ND1& 0.0054SKEW4) (9)

Nbi ' exp(&10.9 % 0.79ln ADT1% 0.49ln ADT2) (8)

Roadside hazard rating (RHRI) 2

Presence of right-turn lane None present (0)

on the major road (RT)

With the default values of given above, the base model in equation (7) reduces to:

Four-Leg STOP-Controlled Intersections

The base model for four-leg intersections with STOP control is presented below:

where:

ND1 = number of driveways on the major-road legs within 76 m (250 ft) of the

intersection; and 

SKEW4 = intersection angle (degrees) expressed as one-half of the angle to the right

minus one-half of the angle to the left for the angles between the major-road

leg in the direction of increasing stations and the right and left legs,

respectively.

This model was developed with negative binominal regression from data for 324 four-leg

STOP-controlled intersections in Minnesota. The database available for model development

included 5 years of accident data (1985-1989) for each intersection. The model predicts the

total intersection-related accident frequency for any four-leg STOP-controlled intersection for

which the independent variables shown in equation (9) are known. The model predictions are

reliable only within the ranges of independent variables for which data were available in the

database used to develop the model (see table 38 in appendix B).

When the accident prediction model is employed to predict the expected accident frequency

for any specified four-leg STOP-controlled intersection on a two-lane highway, equation (9) is

used in the following manner:

• The traffic volume variables (ADT1 and ADT2) are set equal to the actual ADTs of the

major- and minor-road legs, respectively. If the ADTs differ between either the two

major- or minor-road legs, they should be averaged.
22

Nbi ' exp(&5.46 % 0.60ln ADT1% 0.20ln ADT2& 0.40PROTLT & 0.018PCTLEFT2

% 0.11VEICOM % 0.026PTRUCK % 0.041ND1)

(11)

Nbi ' exp(&9.34 % 0.60ln ADT1% 0.61ln ADT2) (10)

• The remaining variables in the model should be set equal to the following nominal or

base conditions:

Number of driveways within No driveways

 76 m (250 ft) of the intersection

 on the major road (ND1)

Intersection skew angle (SKEW4) 0 degrees

With the default values of ND1 and SKEW4 given above, the base model in equation (9)

reduces to:

Four-Leg Signalized Intersections

The base model for four-leg signalized intersections is presented below:

where:

PROTLT = presence of protected left-turn signal phase on one or more major-road

approaches; = 1 if present; = 0 if not present

PCTLEFT2 = percentage of minor-road traffic that turns left at the signal during the

morning and evening hours combined

VEICOM = grade rate for all vertical curves (crests and sags) within 76 m (250 ft) of

the intersection along the major and minor roads

PTRUCK = percentage of trucks (vehicles with more than four wheels) entering the

intersection for the morning and evening peak hours combined

ND1 = number of driveways within 76 m (250 ft) of the intersection on the major

road.

This model was developed with negative binominal regression from data for 49 four-leg

signalized intersections, 18 in California and 31 in Michigan. The data base available for model

development included three years of accident data (1993-1995) for each intersection. The

model predicts total intersection-related accident frequency for any four-leg signalized
23

Nbi ' exp(&5.73 % 0.60ln ADT1% 0.20ln ADT2) (12)

intersection for which the independent variables shown in equation (11) are known. The model

predictions are reliable only within the ranges of independent variables for which data were

available in the data base used to develop the model (see table 44 in appendix B).

When the accident prediction model is employed to predict the expected accident frequency

for any specified four-leg intersection on a two-lane highway, equation (11) is used in the

following manner:

• The traffic volume variables (ADT1 and ADT2) are set equal to the actual ADTs of the

major- and minor-road legs, respectively. If the ADTs differ between either the majoror minor-road legs, they should be averaged.

• The remaining variables in the model should be set equal to the following nominal or

base conditions:

Presence of protected left-turn No left-turn phase

 signal phase (PROTLT)

Percentage of minor-road traffic 28.4 percent

 turning left (PCTLEFT2)

Grade rate for vertical curves No vertical curves

 within 76 m (250 ft) of the

 intersection (VEICOM)

Percentage of trucks entering 9.0 percent

 the intersection (PTRUCK)

Number of driveways within 0 driveways

 76 m (250 ft) of the intersection

 on the major road (ND1)

With the nominal or base values of PROTLT, PCTLEFT2, VEICOM, and PTRUCK given

above, the base model in equation (11) reduces to:
24

Nrs ' Nbr Cr(AMF1r AMF2r... AMFnr) (13)

Nint ' Nbi Ci(AMF1i AMF2i... AMFni) (14)

Calibration Procedure

The accident prediction algorithm is intended for use by highway agencies throughout the

United States. Accident frequencies, even for nominally similar roadway sections or

intersections, are known to vary widely from agency to agency. These variations are of two

types, those that can be directly accounted for by the accident prediction algorithm and those that

cannot.

States differ markedly both in terrain and in the history of the development of their highway

system, resulting in state-to-state differences in roadway alignment, cross section, and

intersection design. However, differences of this type can be accounted for by the AMFs in the

accident prediction algorithm.

States also differ markedly in climate, animal population, driver populations, accident

reporting threshold, and accident reporting practices. These variations may result in some States

experiencing substantially more reported traffic accidents on rural two-lane highways than others. 

Such variations cannot be directly accounted for by the accident prediction algorithm. Therefore,

a calibration procedure has been developed to allow highway agencies to adjust the accident

prediction algorithm to suit the safety conditions present in their State.

The calibration procedure is implemented by a highway agency by determining the value of

calibration factors for roadway segments and at-grade intersections from comparison of their

own data to estimates from the accident prediction algorithm. The calibration factors are

incorporated in equations (13) and (14) in the following fashion for roadway segments and atgrade intersections, respectively:

where:

Cr = calibration factor for roadway segments developed for use by a particular

highway agency; and

Ci = calibration factor for at-grade intersections developed for use by a particular

highway agency.

The calibration factors (Cr and Ci) will have values greater than 1.0 for highway agencies

whose roadways, on the average, experience more accidents than the roadways used in the
25

development of the accident prediction algorithm. The calibration factors for highway agencies

whose roadways, on the average, experience fewer accidents than the roadways used in the

development of the accident prediction algorithm will have values less than 1.0. The calibration

factor for at-grade intersections (Ci) may have different values for each of the three intersection

types for which base models have been developed. The calibration procedures for application

by highway agencies is presented in appendix C.

It is generally expected that the calibration factors (Cr and Ci) would be determined by

highway agencies based on statewide data. In larger and more diverse States, a highway agency

might choose to develop separate calibration factors for individual highway districts or climate

regions. It is also possible for users to provide a local calibration factor for smaller areas with

distinct driver populations or climate conditions. However, use of the local calibration factor

would require a special study to determine the safety performance of roads in that specific local

area relative to the statewide or district-wide expected values.

In addition to estimates of accident frequency, the accident prediction algorithm includes

default distributions of accident severity and accident type for rural two-lane highway roadway

sections and intersections. These default distributions have been presented in tables 1 and 2 of

this report. The calibration procedure presented in appendix C includes a capability for highway

agencies who use the accident prediction algorithm to modify the default distributions of accident

severity and accident type to match their own experience on rural two-lane highways.
26
27

4. ACCIDENT MODIFICATIONFACTORS

The incremental effects of individual geometric design and traffic control elements are

represented in the accident prediction algorithm by AMFs. This section describes the

development of the AMFs and documents the AMFs for each geometric design and traffic

control factor considered in the algorithm.

Development of Accident Modification Factors

AMFs are used in the accident prediction algorithm to represent the effects on safety of

specific geometric design and traffic control features. The AMF for the nominal or base value of

each geometric design traffic control feature has a value of 1.0. Any feature associated with

higher accident experience than the nominal or base condition has an AMF with a value greater

than 1.0; any feature associated with lower accident experience than the base condition has an

AMF with a value less than 1.0. The multiplicative nature of the AMFs is illustrated in equations

(13) and (14).

The AMFs were developed by two expert panels, one for roadway sections and one of atgrade intersections. These panels exercised expert judgment in reviewing the reported research

findings concerning each geometric design and traffic control feature of interest and selecting an

appropriate basis for an AMF. The members of the two expert panels that developed the AMFs

are identified in appendix A.

Each expert panel selected a set of geometric and traffic control elements, including both

roadway segment and at-grade intersection elements, as candidates for the development of

AMFs. The candidates were selected based on the panel’s initial assessments of those roadway

segment and intersection features that are generally considered to be related to safety. A critical

review of published and unpublished safety literature related to each geometric design and traffic

control element was then undertaken. Each panel met and used the findings of the literature

review as the basis for (1) selecting the final set of geometric and traffic control elements for

which AMFs could be developed; and (2) quantifying those AMFs. For roadway segments, the

final AMFs included all of the variables in the roadway segment base models plus additional

variables. For at-grade intersections, the final AMFs did not include all of the variables in the

intersection base models because the expert panel found that reliable estimates of safety effects

from the literature were lacking for some variables in the base models and others were judged to

be of relatively less importance.

For some geometric design or traffic control elements, the expert panel selected the results

of one particular study that they considered most credible to serve as the basis for the AMF. In

other cases, the expert panel combined the results of two or more studies to develop an AMF. 

In other cases, where reliable research results were lacking, the panel exercised its collective
28

judgment to estimate values for an appropriate AMF; this was done, for example, where the

panel felt comfortable in establishing a bound, such as an appropriate maximum or minimum

value, for an AMF.

AMFs were based on a variety of sources including results of before-and-after accident

evaluations, coefficients or parameter values from regression models, and expert judgment. The

expert panel considered well-designed before-and-after evaluations to be the best source for

AMFs. However, relatively few well-designed before-and-after studies of geometric design

elements were found in the literature and, therefore, the expert panel had to rely in many cases on

other types of studies. Coefficients or parameter values from regression models are considered

less reliable, but were used when no before-and-after study results were available and the value

of the coefficient in question was considered by the panel to be credible. Expert judgment alone

was exercised in limited cases where no better results were available; even where an AMF was

based primarily on expert judgment, the panel used all relevant research results in making that

judgment. FHWA has many promising before-and-after evaluations underway, and it is hoped

that the AMFs recommended here will be updated with those analysis results when they become

available.

The expert panels faced many difficult judgments in determining the appropriate values of the

AMFs. A broad range of literature was reviewed and many studies were assessed. Many of the

studies reviewed were similar in quality, and the selection of one study over another may have

depended as much on consistency with other selected AMFs as on the relative merits of the

studies. Thus, the omission of any particular study from the final AMFs should not necessarily be

interpreted as disparagement of that particular study. The sources on which each AMF is based

are documented in the following discussion. Section 8 of this report not only identifies the

sources cited in the text but also includes a complete bibliography of all sources consulted in the

development of the AMFs.

While the individual AMFs were based on the panel’s best judgment about the relative

merits of the available research findings, the credibility of the model is supported by a sensitivity

analysis whose results are presented in section 5 of this report.

The AMFs incorporated in the model include:

Roadway Segments

• Lane width.

• Shoulder width.

• Shoulder type.

• Horizontal curves:

! length;

! radius;

! presence or absence of spiral transitions;
29

! superelevation.

• Grades.

• Driveway density.

• Two-way left-turn lanes.

• Passing lanes/short four-lane sections.

• Roadside design.

At-Grade Intersections

• Skew angle.

• Traffic control.

• Exclusive left-turn lanes.

• Exclusive right-turn lanes.

• Intersection sight distance.

An explanation of each AMF for roadway sections and for at-grade intersections is

presented below.

Roadway Segments

The AMFs for geometric design and traffic control features of roadway segments are

presented below.

Lane Width

The nominal or base value of lane width is 3.6 m (12 ft). Thus, 3.6-m (12-ft) lanes are

assigned an AMF of 1.00. Figure 2 illustrates the recommended values of the AMF for lane

widths from 2.7 to 3.6 m (9 to 12 ft). The AMF for any lane widths within the range of 2.7 to

3.6 m (9 to 12 ft) would be interpolated between the lines shown in figure 2. Lanes less than 2.7

m (9 ft) in width would be assigned an AMF equal to that for 2.7-m (9-ft) lanes. Lanes greater

than 3.6 m (12 ft) in width would be assigned an AMF equal to that for 3.6 m (12-ft) lanes. As

shown in the figure, the AMFs for lanes less than 3.6 m (12 ft) in width would be constant for all

ADTs above 2,000 veh/day, but would decrease to a substantially smaller value over the range

of traffic volumes between 400 and 2,000 veh/day. The AMFs then have constant, but lower,

values in the range of ADT below 400 veh/day.

If the lane widths for the two directions of travel on a roadway segment differ, the AMF

should be determined separately for the lane width in each direction of travel and the resulting

AMFs should then be averaged.
30

The AMFs shown in figure 2 apply to single-vehicle run-off-the-road and multiple-vehicle

head-on, opposite-direction sideswipe, and same-direction sideswipe accidents. The AMFs

expressed on this basis must, therefore, be adjusted to total accidents within the accident

prediction algorithm. This can be accomplished with the following equation:
31

2000.424-2

500 1,000 1,500 2,000 2,500

1.00

1.10

1.20

1.30

1.40

1.50

1.60

1.70 This factor applies to single-vehicle run-off-road,

multiple-vehicle same direction sideswipe accidents,

and multiple-vehicle opposite-direction accidents

Average Daily Traffic Volume (veh/day)

1.00

1.05

1.30

1.50 9-ft lanes

10-ft lanes

11-ft lanes

12-ft lanes

1.01

1.02

1.05

Figure 2. Recommended Accident Modification Factor for Lane Width.
32

AMF ' (AMFra & 1.0) Pra % 1.0 (15)

where:

AMF = accident modification factor for total accidents;

AMFra = accident modification factor for related accidents (i.e., single-vehicle run-offthe-road and multiple-vehicle head-on, opposite-direction sideswipe, and

same-direction sideswipe accidents), such as the accident modification factor

for lane width shown in figure 2;

Pra = proportion of total accidents constituted by related accidents.

The proportion of related accidents (Pra) is estimated as 0.35 (i.e., 35 percent) based on the

default distribution of accident types presented in table 2. This default accident type distribution,

and therefore the value of Pra, may be changed by a highway agency as part of the calibration

process.

The AMFs for lane width for roadways with ADT greater than 2,000 veh/day are based on

the results of Zegeer et al.(6) Furthermore, these values are reasonably consistent with the results

of Zegeer et al.(7,8) and Miaou.(9) The AMF of 1.05 for 3.3-m (11-ft) lane widths on roadways

with ADTs over 2,000 veh/day has been adjusted to a lower value than that indicated by Zegeer

et al.(6) based on the expert panel’s assessment of a broader set of studies concerning the safety

performance of roadways with 3.3-m (11-ft) lanes. The AMFs for lane widths on roadways

with ADTs less than 400 veh/day are based on the results of Griffin and Mak.(10) The transition

lines in the ADT range from 400 to 2,000 veh/day are based on a judgment by the expert panel.

Shoulder Width and Type

The nominal or base value of shoulder width and type is a 1.8-m (6-ft) paved shoulder,

which is assigned an AMF value of 1.00. Figure 3 illustrates the recommended AMF for

shoulder widths that differ from 1.8 m (6 ft). Another AMF, presented below, adjusts for

differences between gravel, turf, or composite shoulders and paved shoulders. The modification

factors in figure 3 illustrate that, for ADTs above 2,000 veh/day, the effect of a 0.6-m (2-ft)

change in shoulder width is equivalent to the effect of a 0.3-m (1-ft) change in lane width. For

ADTs below 400 veh/day, the effect of a 0.3-m (1-ft) change in shoulder width is equivalent to

the effect of a 0.3-m (1-ft) change in lane width. A linear transition between these effects occurs

over the range of ADTs from 400 to 2,000 veh/day. AMFs for shoulder widths between 0 and

2.4 m (0 and 8 ft) should be interpolated between the lines in figure 3. Shoulders greater than
33

2.4 m (8 ft) in width should be assigned AMFs equal to those for 2.4 m (8 ft). The AMFs

shown in figure 3 apply only to single-vehicle run-off-the-road and opposite-direction accidents. 
34

500 1,000 1,500 2,000 2,500

0.80

0.90

1.00

1.10

1.20

1.30

1.40

1.50

This factor applies to single-vehicle run-off-road,

multiple-vehicle same direction sideswipe accidents,

and multiple-vehicle opposite-direction accidents

Average Daily Traffic Volume (veh/day)

0.87

1.15

1.30 2-ft shoulders

4-ft shoulders

8-ft shoulders

1.60

1.00 6-ft shoulders

1.50 0-ft shoulders

0.98

1.02

1.07

1.10

Figure 3. Accident Modification Factor for Shoulder Width.
35

The AMFs for shoulder width on rural two-lane highways with ADTs over 2,000 veh/day

are based primarily on the results of Zegeer et al.(6) which suggests the 2:1 ratio between the

effects of lane and shoulder widths. This finding is reasonably consistent with the results of

Miaou(9,11) and Rinde.(12) The AMFs for roadways with ADTs less than 400 veh/day, which

indicate a 1:1 ratio between the effects of lane width and shoulder width, are based on the work

of Zegeer et al.(8) for low-volume roads. The transition curves in the ADT range from 400 to

2,000 veh/day are based on a judgment by the expert panel.

The nominal or base condition for shoulder type is the paved shoulder. Table 3 presents the

recommended AMFs for gravel, turf, and composite shoulders as a function of shoulder width. 

The AMFs shown in table 3 apply to single-vehicle run-off-the-road and opposite-direction

accidents. 

Table 3. Accident Modification Factors for Shoulder Types on Two-Lane Highways.

Shoulder

type

Shoulder width (ft)

0 1 2 3 4 6 8 10

Paved 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

Gravel 1.00 1.00 1.01 1.01 1.01 1.02 1.02 1.03

Composite 1.00 1.01 1.02 1.02 1.03 1.04 1.06 1.07

Turf 1.00 1.01 1.03 1.04 1.05 1.08 1.11 1.14

Conversion: 1 ft = 0.305 m

Note: The values for composite shoulders in this table represent a shoulder for which 

50 percent of the shoulder width is paved and 50 percent of the shoulder width is turf.

The panel made a judgment that the shoulder type effect observed by Miaou(9) was

appropriate to characterize the difference between gravel and paved shoulders and that the effect

observed by Zegeer et al.(6) was appropriate to characterize the difference between turf and

paved shoulders. The composite shoulders represented by table 3 represent a shoulder for

which 50 percent of the shoulder width is paved and 50 percent of the shoulder width is turf. 

The AMFs for composite shoulders are averages of the AMFs for paved and turf shoulders.

Table 3 assumes that the base condition for shoulder type is a paved shoulder. In fact, the

data base used to develop the base model for roadway sections shown in equation (4) consisted

of approximately 67 percent paved and composite shoulders and 33 percent gravel shoulders. 

However, the difference in safety performance between paved and gravel shoulders is so small

that the magnitude of the AMFs is only minimally affected by the proportion of gravel shoulders

in the data base. Therefore, the use of table 3 without modification is recommended. If, at any

future time, the base model for roadway sections is replaced or updated, it is recommended that
36

AMF ' (AMFwra AMFtra & 1.0) Pra % 1.0 (16)

either (1) the model be based solely on roadway sections with paved shoulders; or (2) that

shoulder type appear explicitly in the base model.

If the shoulder types and/or widths for the two directions of travel on a roadway segment

differ, the AMF should be determined separately for the shoulder type and width in each

direction of travel and the resulting AMFs should then be averaged.

The AMFs for shoulder width and type apply only to single-vehicle run-off the-road and

multiple-vehicle head-on, opposite-direction sideswipe, and same-direction sideswipe accidents. 

The AMFs expressed on this basis must, therefore, be adjusted to total accidents within the

accident prediction algorithm. This can be accomplished with the following equation that is

analogous to equation (15):

where:

AMFwra = accident modification factor for related accidents based on shoulder

width (from figure 4); and

AMFtra = accident modification factor for related accidents based on shoulder type

(from table 3).

The proportion of related accidents (Pra) is estimated as 0.35 (i.e., 35 percent) based on the

default distribution of accident types presented in table 2. This default accident type distribution

and therefore the value of Pra, may be changed by a highway agency as part of the calibration

process.

Horizontal Curves

Length, Radius, and Presence or Absence of Spiral Transitions

The nominal or base condition for horizontal alignment is a tangent roadway section. An

AMF has been developed to represent the manner in which accident experience of curved

alignments differs from that of tangents. This AMF applies to total roadway segment accidents,

not just the related accident types considered above for lane and shoulder widths.

The AMF for horizontal curves has been determined from the regression model developed

by Zegeer et al.(13) The Zegeer model includes the effects on accidents of length of horizontal
37

AMF '

1.55Lc%

80.2

R

& 0.012S

1.55Lc

(17)

curve, degree of horizontal curve, and presence or absence of spiral transition curves. The

Zegeer model could also be used to introduce an effect of lane width on horizontal curves that

differs from the effect of lane width on tangents. No data are available to represent any

differential effect of shoulder width between horizontal curves and tangents.

The AMF for horizontal curvature is in the form of an equation and, thus, might be termed

an accident modification function rather than an accident modification factor. The AMF for

length, radius, and presence or absence of spiral transitions on horizontal curves is:

where: Lc = length of horizontal curve (mi);

R = radius of curvature (ft); and 

S = 1 if spiral transition curve is present

0 if spiral transition curve is not present.

In applying the accident modification functions for curves with spiral transitions, the length

variable (Lc) should represent the length of the circular portion of the curve.

Superelevation

The nominal or base condition for the AMF for the superelevation of a horizontal curve is

the amount of superelevation required by the AASHTO Green Book.(14) The superelevation

required by the AASHTO Green Book must be determined taking into account the value of

maximum superelevation rate, emax, established by highway agency policies. Policies concerning

maximum superelevation rates for horizontal curves vary between highway agencies based on

climate and other considerations. If no value of emax specified by the particular highway agency

has been incorporated in the IHSDM, then emax = 0.06 will be assumed by default. The AMF

for superelevation is based on the superelevation deficiency of a horizontal curve (i.e., the

difference between the actual superelevation and the superelevation required by AASHTO

policy). When the actual superelevation meets or exceeds that required by AASHTO policy, the

value of the superelevation AMF is 1.00. The expert panel made a judgment that there would be

no effect of superelevation deficiency on safety until the superelevation deficiency exceeds 0.01.

The general functional form of an AMF for superelevation is shown in figure 4, based on the

work of Zegeer et al.(13,15) For a horizontal curve with 12-ft lanes and no spiral transitions, the

Zegeer work suggests an AMF of the form:
38

Figure 4. Accident Modification Factor for Superelevation Deficiency.
39

AMF '

1.22 %

1604

R

% 9.52SD

1.22 %

1604

R

(18)

AMF ' 1.00 for SD < 0.01 (19)

AMF ' 1.06 % 3 (SD ! 0.02) for SD $ 0.02 (21)

AMF ' 1.00 % 6 (SD ! 0.01) for 0.01 # SD < 0.02 (20)

where:

SD = superelevation deficiency.

However, the AMF in the form shown in equation (18) is not directly usable because it

suggests that for any given superelevation deficiency, the value of AMF increases with increasing

radius of curvature. In fact, it seems likely that the opposite should occur, with superelevation

deficiencies being more important on curves with smaller radii.

Zegeer, et al. (15) indicate that the mean radius of horizontal curves in their study was 257.0

m (842.5 ft). For this mean radius, equation (18) indicates the following values of the AMF:

Superelevation

deficiency AMF

0.02 1.06

0.03 1.09

0.04 1.12

0.05 1.15

Based on these values and the expert panel’s judgment that there is no effect on safety until the

superelevation deficiency reaches 0.01, the following relationships which form the basis for

figure 5 can be derived:
40

This AMF applies to total roadway segment accidents for roadway segments located on

horizontal curves.

 Grades

The nominal or base condition for grade is a level roadway (0% grade). Table 4 presents

the accident modification factor for grades based on an analysis of two-lane highway grades in

Utah conducted by Miaou.(16) This analysis considered accident and geometric data for

approximately 4,000 km (2,500 mi) of two-lane roads with 88.5 km/h (55-mi/h) speed limits,

3.6-m (12-ft) lanes, and tangent alignment. Two analysis approaches were used: univariate

analysis using smoothing techniques and negative binomial regression modeling. Both methods

estimated the effect of vertical grade on accidents as approximately a 1.6 percent increase in

accidents per 1 percent increase in grade. However, both studies found this effect to be not

statistically significant.

Despite the lack of statistical significance of the results, a decision was reached by the expert

panel to use the observed effect as the basis for an AMF because the result appeared reasonable

to the expert panel and because no more reliable results are available. Table 4 presents AMFs

for grade based on the observed 1.6 percent increase in accidents per 1 percent increase in

percent grade. The AMFs in table 4 are applied to each individual grade section on the

roadway being evaluated without respect to the sign of the grade. The sign of the grade is

irrelevant because each grade on a two-lane highway is an upgrade for one direction of travel

and a downgrade for the other. The grade factors are applied to the entire grade from one point

of vertical intersectional (PVI) to the next (i.e., there is no special account taken of vertical

curves). The AMFs in table 4 apply to total roadway segment accidents.

Table 4. Accident Modification Factors for Grade of Roadway Sections.

Grade (%)

0 2 4 6 8

1.00 1.03 1.07 1.10 1.14

Note: This factor can be expressed as an effect of 1.6 percent per percent grade.

Driveway Density

The nominal or base condition for driveway density is three driveways per km

(five driveways per mi). The AMF for driveway density is based on the following equation

derived from the work of Muskaug:(17)
41

AMF '

0.2 % [0.05 & 0.005 ln (ADT)] DD

0.2 % [0.05 & 0.005 ln (ADT)] (5)

(22)

where:

ADT = annual average daily traffic volume of the roadway being evaluated (veh/day);

and

DD = driveway density (driveways per mile).

The Muskaug study deals with injury accidents only but the expert panel made a judgment that

the AMF shown in equation (22) can be applied to total roadway accidents of all severity levels.

The expert panel considered the Norwegian study by Muskaug to be the best available

study on the safety effects of driveway density on rural two-lane highways. The panel was

concerned, however, about reliance on an international data source and undertook a further 

review of relevant U.S. literature. The most applicable U.S. reference appeared to be 

Transportation Research Circular 456, and an analysis concluded that its results were

consistent with the findings of Muskaug.(18) Still another review of HSIS data from Minnesota

concluded that the effect of driveway density in accidents is consistent with the results of the

Muskaug study. Therefore, the Muskaug results were retained as the basis for the driveway

density AMF.

Passing Lanes

The nominal or base condition for passing lanes is the absence of a lane (i.e., the normal

two-lane cross section). The AMF for a conventional passing or climbing lane added in one

direction of travel on a two-lane highway is 0.75 for total accidents in both directions of travel

over the length of the passing lane from the upstream end of the lane addition taper to the

downstream end of the lane drop taper. This value assumes that the passing lane is operationally

warranted and that the length of the passing lane is appropriate for the operational conditions on

the roadway. An IHSDM procedure other than the accident prediction algorithm should be used

to warn users if a passing lane is not operationally warranted or if an inappropriate passing lane

length is used. Passing lanes are known to have traffic operational effects that extend 5 to 13 km

(3 to 8 mi) downstream of the passing lane; while it might be presumed that these operational

effects provide analogous safety benefits over a similar length of highway, no such effect is

included in the accident prediction algorithm for lack of quantitative evidence of such a benefit.

The AMF for short four-lane sections (i.e., side-by-side passing lanes provided in opposite

directions on the same roadway section) is 0.65 for total accidents over the length of the short

four-lane section. This AMF applies to any portion of roadway where the cross section has four
42

AMF ' 1 ! 0.7PDPLT/D (23)

PD'

0.0047DD % 0.0024DD 2

1.199 % 0.0047DD % 0.0024DD 2

(24)

lanes and where both added lanes have been provided over a limited distance to increase passing

opportunities. This AMF does not apply to extended four-lane highway sections.

The AMF for passing lanes is based primarily on the work of Harwood and St. John, with

consideration also given to the results of Rinde and Nettleblad.(19, 12, 20) The AMF for short fourlane sections is based on the work of Harwood and St. John.(19) These AMFs apply to total

roadway segment accidents within the passing lane and short four-lane sections.

Two-Way Left-Turn Lanes

The installation of a center two-way left-turn lane (TWLTL) on a two-lane highway to

create a three-lane cross section can reduce accidents related to turning maneuvers at driveways. 

The AMF for installation of a TWLTL is:

where:

PD = driveway-related accidents as a proportion of total accidents; and

PLT/D = left-turn accidents susceptible to correction by a TWLTL as a proportion

of driveway-related accidents.

The value of PAP is estimated from the work of Hauer as:(21)

The value of PLT/D was estimated by the expert panel as 0.5.

The expert panel considers that equations (23) and (24) provides the best estimate of the

AMF for TWLTL installation that can be made without data on the left-turn volumes within the

TWLTL. Realistically, such volumes are seldom available to highway agencies for use in such

analyses. The AMF, as adjusted in equation (23), applies to total roadway segment accidents. 

Equation (24) was initially developed to represent total access point density (driveways plus

unsignalized intersections). However, it is used here to determine an AMF for driveway density

alone, because the effects of left-turn lanes at intersections are considered separately below.
43

AMF '

exp(!0.6869 % 0.0668RHR)

exp(!0.4865)

(25)

The AMF for TWLTL installation should not be applied unless the driveway density is

greater than or equal to three driveways per km (five driveways per mi). If the driveway density

is less than three driveways per km (five driveways per mi), the AMF for TWLTL installation is

1.00. TWLTL installation would, in any case, be inappropriate for roadway segments with

driveway densities lower than this threshold.

Roadside Design

For purposes of the accident prediction algorithm, the quality of roadside design is

represented by the roadside hazard rating (1 to 7 scale) developed by Zegeer et al. (6) No

studies were found in the literature that presented satisfactory relationships between the roadside

hazard rating and accident experience for two-lane highways. Therefore, the AMF for roadside

design was derived directly from the base model for roadway sections presented in equation (5). 

The nominal or base value of roadside hazard rating employed in the base model for roadway

sections is 3. The AMF is based on the ratio of the accident experience predicted by base

model using the actual roadway section in question to the accident experience predicted by the

base model using the nominal value of roadside hazard rating equal to 3. The AMF is:

This AMF applies to total roadway segment accidents. Photographic examples and quantitative

definitions for each roadside hazard rating (1 through 7) as a function of roadside design features

such as side slope and clear zone width are presented in appendix D.

The expert panel encourages future development of AMFs for specific roadside design

elements in as much detail as the roadway design factors in this accident prediction algorithm. 

For example, the algorithm could be made sensitive to the presence or absence of a guardrail at

specific roadside slopes and at individual roadside obstacles. The Roadside Safety Analysis

Program (RSAP) model currently being developed might be applied for this purpose rather than

just as a benefit/cost tool for comparing roadside design alternatives.(22) However, the RSAP

model is not yet complete and the consensus of the panel was that application of the RSAP

model would be beyond the scope of this initial effort to develop the accident prediction

algorithm.

At-Grade Intersections

The AMFs for geometric design and traffic control features of at-grade intersections are

presented below. As explained above, AMFs have been developed only for those geometric
44

AMF '

exp (!12.15 % 1.001ln ADT1% 0.406ln ADT2% 0.0040SKEW3)

exp (!12.15 % 1.001ln ADT1% 0.406ln ADT2)

(26)

design and traffic control features for which the expert panel found a suitable basis for quantifying

an AMF.

Number of Intersection Legs

There is no separate AMF for the number of intersection legs. Instead, the effect of the

number of intersection legs will be accounted for by the separate base models for three- and

four-leg intersections like those presented in section 3 of this report. Base models have been

developed for both three- and four-leg STOP-controlled intersections, while base models for

signalized intersections have been developed for four-leg intersections only. No base models will

be developed for intersections with more than four legs. Therefore, multi-leg intersections and

three-leg signalized intersections will not be addressed by the initial version of the accident

prediction algorithm.

Intersection Skew Angle

The nominal or base condition for intersection skew angle is 0 degrees of skew (i.e., an

intersection angle of 90 degrees). The skew angle for an intersection was defined as the

deviation from an intersection angle of 90 degrees and carries a positive or negative sign that

indicates whether the minor road intersects the major road at an acute or obtuse angle. This sign

was introduced into the base model because a Finnish study by Kulmala found that acute and

obtuse skew angles affected safety differently.(23)

STOP-Controlled Intersections

The AMF for intersection angle at three-leg STOP-controlled intersections is derived from

the base model for this intersection type. Because the intersection angle variable was not

statistically significant at the 0.015 significance level, it did not appear in the base model in

equation (7). Therefore, the base model presented in equation (49) in appendix B with additional

variables, including the intersection angle variable, was used to derive the following AMF:

where:

ADT1 = average daily traffic volume for the major road;
45

AMF '

exp(!9.15 % 0.534ln ADT1% 0.665ln ADT2! 0.0054SKEW4)

exp(!9.15 % 0.534ln ADT1% 0.665ln ADT2)

(28)

AMF ' exp(&0.0054 SKEW4) (29)

ADT2 = average daily traffic volume for the minor road; and

SKEW3 = intersection angle (degrees) minus 90 for the angle between the

major-road leg in the direction of increasing stations and a leg to the

right; 90 minus intersection angle (degrees) for the angle between the

major-road leg in the direction of increasing stations and a leg to the left.

Equation (26) reduces to:

AMF ' exp(0.0040 SKEW3) (27)

The AMF for intersection angle at four-leg STOP-controlled intersections is based directly

on the base model presented as equation (9) for this intersection type:

SKEW4 = intersection angle (degrees) expressed as one-half of the angle to the

right minus one-half of the angle to the left for the angles between the

major-road leg in the direction of increasing stations and the right and left

legs, respectively

Equation (28) reduces to:

The opposite signs of the coefficients of skew angle in equations (27) and (29) are a concern

because the difference in sign implies that positive and negative skew angles (as defined above for

available SKEW3 and SKEW4) have opposite effects on safety at three- and four-leg

intersections. On further review, the expert panel decided that the results of the Kulmala study

alone did not provide a sufficient basis for challenging the widely accepted view that any

intersection skew that departs from a 90-degree angle, whether positive or negative, is

detrimental to safety. Therefore, equations (27) and (29) have been recast as shown below. For

a three-leg STOP-controlled intersection:

AMF ' exp (0.0040 SKEW) (30)

For a four-leg STOP-controlled intersection:

AMF ' exp (0.0054 SKEW) (31)
46

where:

SKEW = intersection skew angle (degrees), expressed as the absolute value of the

difference between 90 degrees and the actual intersection angle.

These AMFs apply to total intersection accidents.

Signalized Intersections

Skew angle is a much less important factor in the operation of signalized intersection than in

the operation of STOP-controlled intersections. Since the traffic signal separates most

movements from conflicting approaches, the risk of collisions related to the skew angle between

the intersecting approaches is limited at a signalized intersection. Therefore, the AMF for skew

angle at four-leg signalized intersections is 1.00 for all cases.

Intersection Traffic Control

The safety differences between STOP-controlled and signalized intersections are accounted

for by use of separate base models rather than by an AMF. However, an AMF for the

difference between minor-leg and all-way STOP-controlled intersections has been developed

and is discussed below. The nominal base case for STOP-controlled intersections has STOP

signs on the minor leg(s) only. An AMF is provided for intersection with all-way STOP control. 

Minor-road YIELD controlled intersections are treated identically to minor-road STOPcontrolled intersections in the accident prediction algorithm.

All-way STOP control is most appropriate for lower-speed roadways with relatively equal

traffic volumes on all legs of the intersection. The Manual on Uniform Traffic Control Devices

(MUTCD) includes specific warrants for all-way STOP control.(24) All-way STOP control

should not be considered for an intersection unless these warrants are met. The AMF for

conversion from minor-road to all-way STOP-control is 0.53. This AMF applies to total

intersection-related accidents. The AMF value of 0.53 implies that an all-way STOP-controlled

intersection experiences 47 percent fewer accidents than a two-way STOP-controlled

intersection. This AMF is based on the findings of Lovell and Hauer.(25) They collected data for

three local agencies (San Francisco, Philadelphia, and Toronto) and one State agency

(Michigan). Of these, only the Michigan data pertains to low-volume, high-speed, rural roads. 

However, since the Michigan data set is small and its results are similar to those for the other

three agencies, the recommended AMF is based on the combined data for all four agencies. The

expert panel recommended that some other IHSDM module, possibly the diagnostic review

module for design of at-grade intersections, should make clear to IHSDM users that all-way

STOP-control should be used only when the established warrants are met. This is necessary to
47

discourage indiscriminate use of all-way STOP-control, because it is likely that the substantial

safety benefits of all-way STOP control shown below can be attained only when the warrants are

met.

Intersection Left-Turn Lanes

The nominal or base condition for intersection left-turn lanes is the absence of left-turn lanes

on the major-road approaches. The AMFs for presence of left-turn lanes on the major road are

presented in table 5. These AMFs apply to total intersection-related accidents. The expert

panel did not find any well-designed before-and-after studies on the accident reduction

effectiveness of left-turn lanes. Therefore, the AMFs in the table represent a judgment by the

expert panel combining results from several sources. The AMFs for installation of left-turn lanes

at STOP-controlled intersections are based on studies by Agent, Bauer and Harwood, California

Department of Public Works, Creasy and Agent, Dale, Ermer, Glennon, McCoy et al., McCoy

and Malone, and Smith et al. The AMFs for installation of left-turn lanes at signalized

intersections are based on studies by Agent, California Department of Public Works, Datta,

Smith et al., and McCoy and Malone.(26, 1, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37) The AMFs for installation

of left-turn lanes on both approaches to a four-leg intersection are equal to the square of the

corresponding AMF for installation of a left-turn lane on a single approach. No data are

available to quantify the effect on safety of left-turn lanes on a minor road, so these will not be

considered in the accident prediction algorithm.

Table 5. Accident Modification Factors for Installation of Left-turn Lanes on the

Major-Road Approaches to Intersection on Two-Lane Rural Highways.

Intersection type

Intersection traffic

control

Number of major-road approaches on which

left-turn lanes are installed

One approach Both approaches

Three-leg intersection STOP signa

Traffic signal

0.78

0.85

!

!

Four-leg intersection STOP signa

Traffic signal

0.76

0.82

0.58

0.67

a

 STOP signs on minor-road approach(es).

Intersection Right-Turn Lanes

The nominal or base condition for intersection right-turn lanes is the absence of right-turn

lanes on the major-road approaches. The AMF for the presence of right-turn lanes at STOPcontrolled intersections is 0.95 for a right-turn lane on one major-road approach and 0.90 for

right-turn lanes on both major road approaches. These AMFs apply to total intersection-related
48

accidents. The expert panel did not find any well-designed before-and-after studies on the

accident reduction effectiveness of right-turn lanes. Therefore, the values of the AMFs are based

on a judgment by the panel based on the work of Vogt and Bared, Kulmala, and Elvik.(3, 4, 5, 23,

38) No data are available to quantify the effect on safety of right-turn lanes on a minor road, so

these are not considered in the accident prediction algorithm. Also, no effect is considered for

the provision of a paved shoulder on an intersection approach unless that paved shoulder is

marked as a right-turn lane.

No studies that the expert panel considered appropriate were found concerning the accident

reduction effectiveness of right-turn lanes at signalized intersections. Therefore, the panel made a

judgment that the effectiveness of right-turn lanes at signalized intersections should be estimated

as half that found at STOP-controlled intersections. Thus, the AMF for the presence of rightturn lanes at signalized intersections is 0.975 for a right-turn lane on one major-road approach

and 0.95 for right-turn lanes on both major-road approaches. These AMFs also apply to total

intersection-related accidents.

Intersection Sight Distance

The nominal or base condition for intersection sight distance is the availability of adequate

intersection sight distance along the major road in all quadrants of the intersection. The AMFs

for intersection sight distance at intersections with STOP control on the minor leg(s) are:

• 1.05 if sight distance is limited in one quadrant of the intersection.

• 1.10 if sight distance is limited in two quadrants of the intersection.

• 1.15 if sight distance is limited in three quadrants of the intersection.

• 1.20 if sight distance is limited in four quadrants of the intersection.

These AMFs apply to total intersection-related accidents.

Sight distance in a quadrant is considered limited if the available sight distance is less than the

sight distance specified by AASHTO policy for a design speed of 20 km/h less than the majorroad design speed. Only sight distance restrictions due to roadway alignment and terrain are

considered by the accident prediction algorithm. Sight distance restrictions due to specific

obstructions (e.g., trees, bushes, poles, and buildings) are not available in the CAD system and,

therefore, are not to be considered by the accident prediction algorithm.

The AMFs for intersection sight distance apply only to two-way STOP-controlled or

YIELD-controlled intersections. An AMF of 1.00 is applicable to signal-controlled and all-way

STOP-controlled intersections.
49

There was no single evaluation of the effects of intersection sight distance on accidents that

the panel found to be most credible. Therefore, the recommended AMF was determined from

the panel’s best judgment based on the results of Kulmala, Brüde and Larsson, and Elvik.(39, 38)

These results were assumed to represent sight distance improvements in all quadrants of an

intersection. Therefore, this effect was proportioned on a per-quadrant basis for application in

the accident prediction algorithm.
50
51

5. SENSITIVITYANALYSIS RESULTS

A sensitivity analysis was performed to illustrate the estimated effect on safety of the various

parameters and factors in the accident prediction algorithm. The results of this sensitivity analysis

are present in this section.

All of the sensitivity analyses presented here were performed with the calibration factors for

both roadway segments (Cr) and intersections (Ci) set equal to 1.00. Thus, these results do not

represent the safety conditions experienced by any particular highway agency and should be

interpreted primarily in a relative sense.

Roadway Segments

The sensitivity analysis for roadway segments first established the variation of accident

frequency and accident rate with ADT for the base or nominal condition. Then specific AMFs in

the prediction algorithm for roadway segments were varied one at a time. Finally, the predicted

accident frequencies and accident rates for combinations of AMFs with extremely high and

extremely low accident experience were determined.

Nominal or Base Condition

The nominal or base condition for evaluating roadway segments consists of the following

combination of conditions:

• Lanes that are 3.6-m (12-ft).

• Paved shoulders that are 1.8-m (6-ft).

• Tangent roadway.

• Level grade (0 percent).

• Three driveways per km (5 driveways per mi).

• Roadside hazard rating = 3.

• No passing lanes or short four-lane sections.

Table 6 illustrates the variation of accident frequency and accident rate with the roadway

segment ADT for the nominal or base condition. The table shows that for the nominal or base

condition the accident frequency per mile per year increases linearly with increasing ADT, while

the accident rate per million veh-mi remains constant.

Table 6 and the other tables in this section of the report are presented in conventional units

because all of the equations and AMFs on which they are based are in conventional units (see

sections 3 and 4).
52

Table 6. Sensitivity of Safety to ADT for Nominal Conditions for Roadway Segments.

ADT

(veh/day) Accidents per mi per year Accidents per million veh-mi

400 0.09 0.61

1,000 0.22 0.61

3,000 0.67 0.61

5,000 1.12 0.61

10,000 2.24 0.61

Conversion: 1 mi = 1.61 km

Lane Width

Table 7 presents the sensitivity of safety to lane width while all other factors are held at their

nominal or base conditions. The table shows that under low-volume conditions there is very

limited sensitivity of safety to lane width, while the sensitivity is larger at higher volume levels. For

ADTs above 2,000 veh/day, accident frequency is 16.5 percent higher for 2.7 m (9 ft) lanes than

for 3.6 m (12 ft) lanes.

Table 7. Sensitivity of Safety to Lane Width on Roadway Segments.

ADT

(veh/day)

Lane Width (ft)

9 10 11 12

BASE

ACCIDENTS PER MILE PER YEAR

400 0.09 0.09 0.09 0.09

1,000 0.24 0.23 0.23 0.22

3,000 0.79 0.74 0.68 0.67

5,000 1.32 1.24 1.14 1.12

10,000 2.64 2.48 2.28 2.24

ACCIDENTS PER MILLION VEHICLE-MILES

400 0.63 0.62 0.62 0.61

1,000 0.66 0.64 0.62 0.61

3,000 0.72 0.68 0.63 0.61

5,000 0.72 0.68 0.63 0.61

10,000 0.72 0.68 0.63 0.61

Conversion: 1 mi = 1.61 km; 1 ft = 0.305 m

Shoulder Type and Width

Table 8 presents the sensitivity of safety to shoulder type and width while all other factors

are held at their nominal or base condition. Like the lane width effect, there is very limited

sensitivity of safety to shoulder type and width at low volume levels. For ADTs above 2,000
53

veh/day, accident frequency can differ by a maximum of 25 percent among various combinations

of shoulder type and width.

Table 8. Sensitivity of Safety to Shoulder Type and Width on Roadway Segments.

ADT

(veh/day)

Shoulder Type and Width (ft)

None Paved Gravel Turf

0 2 4 6 8 2 4 6 8 2 4 6 8

BASE

ACCIDENTS PER MILE PER YEAR

400 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09

1,000 0.24 0.24 0.23 0.22 0.22 0.24 0.23 0.23 0.22 0.24 0.23 0.23 0.23

3,000 0.79 0.74 0.71 0.67 0.64 0.75 0.71 0.68 0.65 0.75 0.72 0.69 0.67

5,000 1.32 1.24 1.18 1.12 1.07 1.24 1.18 1.13 1.08 1.25 1.20 1.15 1.11

10,000 2.64 2.48 2.36 2.24 2.14 2.49 2.37 2.26 2.16 2.51 2.40 2.31 2.22

ACCIDENTS PER MILLION VEHICLE-MILES

400 0.64 0.63 0.62 0.61 0.61 0.63 0.62 0.62 0.61 0.64 0.63 0.63 0.64

1,000 0.67 0.65 0.63 0.61 0.60 0.65 0.63 0.62 0.61 0.66 0.64 0.63 0.63

3,000 0.72 0.68 0.65 0.61 0.59 0.68 0.65 0.62 0.59 0.69 0.66 0.63 0.61

5,000 0.72 0.68 0.65 0.61 0.59 0.68 0.65 0.62 0.59 0.69 0.66 0.63 0.61

10,000 0.72 0.68 0.65 0.61 0.59 0.68 0.65 0.62 0.59 0.69 0.66 0.63 0.61

Conversion: 1 ft = 0.305 m; 1 mi = 1.61 km

Horizontal Curvature

Tables 9 and 10 present the sensitivity of safety to factors related to horizontal curvature. 

Table 9 compares the safety performance of a tangent roadway with various combinations of

horizontal curve length and radius with and without spiral transitions.

The values in the table are computed with all factors other than horizontal curvature set to

their nominal or base conditions. The table shows that the safety performance of long flat curves

is only slightly worse than the safety performance of a tangent roadway. However, short sharp

curves can have much higher accident rates. A horizontal curve with a length of 31 m (100 ft)

and a radius of 31 m (100 ft) on a roadway segment would be expected to have an accident rate

over 28 times as high as a tangent section on the same roadway. Addition of spiral transition

curves can reduce accident frequencies up to a maximum of 6.6 percent for the curves selected

for table 9.

Table 10 shows the expected effect on safety of superelevation deficiencies for horizontal

curves. the table shows, as indicated in equation (21) that a superelevation deficiency of 0.02
54

increases accidents on the curve by 6 percent and a deficiency of 0.04 increases accidents by 12

percent.

Table 9. Sensitivity of Safety to Horizontal Curve Length and Radius on Roadway

Segments.

Curve Length =

100 ft

Curve Length =

500 ft

Curve Length =

1,000 ft

Curve Length =

2,000 ft

ADT

(veh/day)

Radius (ft) Radius (ft) Radius (ft) Radius (ft)

Tangent 100 200 500 500 1,000 2,000 1,000 2,000 5,000 1,000 2,000 5,000

ACCIDENTS PER MILE PER YEAR—Curves without spiral transitions

400 0.09 2.55 1.32 0.58 0.19 0.14 0.11 0.11 0.10 0.09 0.10 0.10 0.09

1,000 0.22 6.37 3.30 1.45 0.47 0.35 0.29 0.29 0.26 0.24 0.26 0.24 0.23

3,000 0.67 19.11 9.89 4.36 1.41 1.04 0.86 0.86 0.77 0.71 0.77 0.72 0.69

5,000 1.12 31.84 16.48 7.27 2.35 1.74 1.43 1.43 1.28 1.18 1.28 1.20 1.15

10,000 2.24 63.69 32.97 14.53 4.70 3.47 2.86 2.86 2.55 2.37 2.55 2.40 2.31

ACCIDENTS PER MILLION VEHICLE-MILES—Curves without spiral transitions

400 0.61 17.45 9.03 3.98 1.29 0.95 0.78 0.78 0.70 0.65 0.70 0.66 0.63

1,000 0.61 17.45 9.03 3.98 1.29 0.95 0.78 0.78 0.70 0.65 0.70 0.66 0.63

3,000 0.61 17.45 9.03 3.98 1.29 0.95 0.78 0.78 0.70 0.65 0.70 0.66 0.63

5,000 0.61 17.45 9.03 3.98 1.29 0.95 0.78 0.78 0.70 0.65 0.70 0.66 0.63

10,000 0.61 17.45 9.03 3.98 1.29 0.95 0.78 0.78 0.70 0.65 0.70 0.66 0.63

ACCIDENTS PER MILE PER YEAR—Curves with spiral transitions

400 0.09 2.51 1.26 0.54 0.18 0.13 0.11 0.11 0.10 0.09 0.10 0.09 0.09

1,000 0.22 6.28 3.20 1.36 0.45 0.33 0.27 0.28 0.25 0.23 0.25 0.24 0.23

3,000 0.67 18.83 9.61 4.08 1.35 0.99 0.80 0.83 0.74 0.68 0.75 0.71 0.68

5,000 1.12 31.28 16.02 6.81 2.26 1.64 1.34 1.38 1.23 1.14 1.25 1.18 1.13

10,000 2.24 62.77 32.05 13.61 4.51 3.29 2.67 2.77 2.46 2.27 2.50 2.35 2.26

ACCIDENTS PER MILLION VEHICLE-MILES—Curves with spiral transitions

400 0.61 17.20 8.78 3.73 1.24 0.90 0.73 0.76 0.67 0.62 0.69 0.64 0.62

1,000 0.61 17.20 8.78 3.73 1.24 0.90 0.73 0.76 0.67 0.62 0.69 0.64 0.62

3,000 0.61 17.20 8.78 3.73 1.24 0.90 0.73 0.76 0.67 0.62 0.69 0.64 0.62

5,000 0.61 17.20 8.78 3.73 1.24 0.90 0.73 0.76 0.67 0.62 0.69 0.64 0.62

10,000 0.61 17.20 8.78 3.73 1.24 0.90 0.73 0.76 0.67 0.62 0.69 0.64 0.62

Conversion: 1 ft = 0.305 m; 1 mi = 1.61 km
55

Table 10. Sensitivity of Safety to Horizontal Curve Superelevation Deficiency on

Roadway Segments.

Curve Length =

 100 ft

Curve Length =

 500 ft

 Curve Length =

1,000 ft

Curve Length = 2,000

ft

Curve Radius = 

200 ft

Curve Radius =

1,000 ft

Curve Radius =

2,000 ft

Curve Radius = 2,000

ft

ADT

(veh/day)

Superelevation

Deficiency

Superelevation

Deficiency

Superelevation

Deficiency

Superelevation

 Deficiency

0.00 0.02 0.04 0.00 0.02 0.04 0 0.02 0.04 0 0.02 0.04

ACCIDENTS PER MILE PER YEAR

400 1.32 1.40 1.47 0.14 0.15 0.16 0.10 0.11 0.11 0.10 0.10 0.11

1,000 3.29 3.49 3.68 0.35 0.37 0.39 0.26 0.27 0.29 0.24 0.25 0.27

3,000 9.87 10.46 11.05 1.04 1.10 1.17 0.77 0.81 0.86 0.72 0.76 0.81

5,000 16.45 17.44 18.42 1.74 1.84 1.94 1.28 1.35 1.43 1.20 1.27 1.34

10,000 32.90 34.88 36.85 3.47 3.68 3.89 2.55 2.70 2.86 2.40 2.54 2.68

ACCIDENTS PER MILLION VEHICLE-MILES

400 9.01 9.55 10.10 0.95 1.01 1.06 0.70 0.74 0.78 0.66 0.70 0.74

1,000 9.01 9.55 10.10 0.95 1.01 1.06 0.70 0.74 0.78 0.66 0.70 0.74

3,000 9.01 9.55 10.10 0.95 1.01 1.06 0.70 0.74 0.78 0.66 0.70 0.74

5,000 9.01 9.55 10.10 0.95 1.01 1.06 0.70 0.74 0.78 0.66 0.70 0.74

10,000 9.01 9.55 10.10 0.95 1.01 1.06 0.70 0.74 0.78 0.66 0.70 0.74

Conversion: 1 ft = 0.305 m; 1 mi = 1.61 km

Grade

Table 11 illustrates the sensitivity of safety to roadway grades. The table shows that, as also

indicated in table 4, steeper grades increase accidents by 1.6 percent per 1-percent increase in

grade.
56

Table 11. Sensitivity of Safety to Percent Grade on Roadway Segments.

Percent Grade

ADT

(veh/day)

0 2 4 6 8

BASE

ACCIDENTS PER MILE PER YEAR

400 0.09 0.09 0.10 0.10 0.10

1,000 0.22 0.23 0.21 0.25 0.25

3,000 0.67 0.69 0.72 0.74 0.76

5,000 1.12 1.16 1.20 1.23 1.27

10,000 2.24 2.32 2.39 2.47 2.55

ACCIDENTS PER MILLION VEHICLE-MILES

400 0.61 0.63 0.66 0.68 0.70

1,000 0.61 0.63 0.66 0.68 0.70

3,000 0.61 0.63 0.66 0.68 0.70

5,000 0.61 0.63 0.66 0.68 0.70

10,000 0.61 0.63 0.66 0.68 0.70

Conversion: 1 mi = 1.61 km

Driveway Density

Table 12 presents the sensitivity of safety to driveway density for roadway segments while all

other factors remain at their nominal or base conditions. The table shows that a roadway segment

with 19 driveways per km (30 driveways per mi) can experience up to four times as many

accidents as a similar roadway segment with no driveways. The sensitivity of safety to driveway

density is greater at lower ADTs than at higher ADTs, although the absolute magnitudes of the

predicted accident frequencies at low ADT are very low. Nevertheless, it might be more

reasonable to expect greater sensitivity of accidents to driveways at higher ADTs than at lower

ADTs. Further research on this issue would be desirable.

Table 12 also shows the predicted accident frequency and accident rate for two-lane highway

sections with two-way left-turn lanes (TWLTLs). The AMF for TWLTLs is based on equations

(23) and (24). The accident reduction effectiveness of a TWLTL ranges from 2 to 23 percent as a

function of driveway density.
57

Table 12. Sensitivity of Safety to Driveway Density on Roadway Segments.

Driveway Density (driveways per mi)

ADT 0 5 10 15 20 25 30

(veh/day) BASE

ACCIDENTS PER MILE PER YEAR

400 0.06 0.09 0.12 0.15 0.18 0.21 0.24

1,000 0.16 0.22 0.29 0.35 0.41 0.47 0.54

3,000 0.54 0.67 0.81 0.94 1.08 1.21 1.34

5,000 0.95 1.12 1.30 1.47 1.65 1.82 2.00

10,000 2.04 2.24 2.45 2.65 2.85 3.05 3.25

ACCIDENTS PER MILLION VEHICLE-MILES

400 0.41 0.61 0.82 1.03 1.23 1.44 1.64

1,000 0.44 0.61 0.79 0.96 1.13 1.30 1.47

3,000 0.49 0.61 0.74 0.86 0.98 1.11 1.23

5,000 0.52 0.61 0.71 0.81 0.90 1.00 1.10

10,000 0.56 0.61 0.67 0.73 0.78 0.84 0.89

ACCIDENTS PER MILE PER YEAR—WITH TWLTLa

400 0.06 0.09 0.11 0.13 0.15 0.17 0.18

1,000 0.16 0.22 0.27 0.31 0.34 0.38 0.41

3,000 0.54 0.66 0.75 0.83 0.90 0.97 1.04

5,000 0.95 1.10 1.21 1.30 1.38 1.46 1.54

10,000 2.04 2.19 2.28 2.33 2.38 2.44 2.50

ACCIDENTS PER MILLION VEHICLE—MILESa

400 0.41 0.60 0.76 0.90 1.03 1.15 1.26

1,000 0.44 0.60 0.73 0.84 0.94 1.04 1.13

3,000 0.49 0.60 0.69 0.76 0.82 0.88 0.95

5,000 0.52 0.60 0.66 0.71 0.76 0.80 0.84

10,000 0.56 0.60 0.62 0.64 0.65 0.67 0.69

Conversion: 1 mi = 1.61 km

Passing Lanes

Table 13 presents the sensitivity of safety to passing lanes and short four-lane sections on

roadway segments. The table shows that, as explained in section 4 of this report, installation of

passing lanes to increase passing opportunities reduces accidents by 25 percent and installation of

short four-lane sections to increase passing opportunities reduces accidents by 35 percent.
58

Table 13. Sensitivity of Safety to Presence of Passing Lanes and Short Four-Lane

Sections on Roadway Segments.

ADT

(veh/day)

Passing Lane 

Present ?

Short Four-Lane

Section Present ?

No Yes No Yes

BASE BASE

ACCIDENTS PER MILE PER YEAR

400 0.09 0.07 0.09 0.06

1,000 0.22 0.17 0.22 0.15

3,000 0.67 0.50 0.67 0.44

5,000 1.12 0.84 1.12 0.73

10,000 2.24 1.68 2.24 1.46

ACCIDENTS PER MILLION VEHICLE-MILES

400 0.61 0.46 0.61 0.40

1,000 0.61 0.46 0.61 0.40

3,000 0.61 0.46 0.61 0.40

5,000 0.61 0.46 0.61 0.40

10,000 0.61 0.46 0.61 0.40

Conversion: 1 mi = 1.61 km

Roadside Design

Table 14 presents the sensitivity of safety to roadside hazard rating on roadway segments

while all other factors are held at their nominal or base conditions. The table shows that roadside

hazard rating can increase total accident frequency by up to 50 percent over the full range of

roadside hazard ratings.

Table 14. Sensitivity of Safety to Roadside Hazard Rating on Roadway Segments.

Roadside hazard rating

ADT 1 2 3 4 5 6 7

(veh/day) BASE

ACCIDENTS PER MILE PER YEAR

400 0.08 0.08 0.09 0.10 0.10 0.11 0.12

1,000 0.20 0.21 0.22 0.24 0.26 0.27 0.29

3,000 0.59 0.63 0.67 0.72 0.77 0.82 0.88

5,000 0.98 1.05 1.12 1.20 1.26 1.37 1.47

10,000 1.96 2.10 2.24 2.40 2.56 2.74 2.93

ACCIDENTS PER MILLION VEHICLE-MILES

400 0.54 0.58 0.61 0.66 0.70 0.75 0.80

1,000 0.54 0.58 0.61 0.66 0.70 0.75 0.80

3,000 0.54 0.58 0.61 0.66 0.70 0.75 0.80

5,000 0.54 0.58 0.61 0.66 0.70 0.75 0.80

10,000 0.54 0.58 0.61 0.66 0.70 0.75 0.80

Conversion: 1 mi = 1.61 km
59

Combinations of Geometric Design and Traffic Control Features

Table 15 presents the sensitivity of safety to extreme combinations of geometric design and

traffic control features. The low accident frequency combination represents the “best” combination

of features considered in the previous sensitivity analyses. Specifically, this low accident frequency

combination includes:

• Lanes that are 3.6-m (12-ft).

• Paved shoulders that are 2.4-m (8-ft).

• Tangent roadway.

• Level grade (0 percent).

• No driveways.

• Roadside hazard rating = 1.

• Short four-lane sections used to increase passing opportunities.

The accident frequencies and rates shown in the table represent levels that are unlikely to be

improved further through geometric design or traffic control modifications.

By contrast, the high accident frequency combination represents the “worst” combination of

features considered in the previous sensitivity analyses. Specifically, the high accident frequency

combination includes:

• Lanes that are 2.7-m (9-ft).

• No shoulders.

• Horizontal curve with length of 31 m (100 ft), radius of 31 m (100 ft), no spiral transition

curve, and a superelevation deficiency of 0.04.

• An 8-percent grade.

• Nineteen driveways per km (30 driveways per mi).

• Roadside hazard rating = 7.

• No passing lanes or short four-lane sections.

The accident frequencies and rates shown in the table are extremely high, but the combination of

geometric and traffic control features they represent is so extreme that it is unlikely to exist in the

real world.
60

Table 15. Sensitivity of Safety to Extreme Combinations of Geometric Design and Traffic

Control Features.

ADT

(veh/day)

Low-Accident Frequency

Combinationa

High-Accident Frequency

Combinationa

ACCIDENTS PER MILE PER YEAR

400 0.03 11.87

1,000 0.09 29.59

3,000 0.29 87.35

5,000 0.51 129.87

10,000 1.11 211.25

ACCIDENTS PER MILLION VEHICLE-MILES

400 0.23 81.31

1,000 0.25 81.07

3,000 0.27 79.77

5,000 0.28 71.16

10,000 0.30 57.88

Conversion: 1 mi = 1.61 km

a These combinations of geometric design and traffic control features are defined in the accompanying

text.

Three-Leg STOP-Controlled Intersections

A sensitivity analysis was performed with the accident prediction algorithm for three-leg

STOP-controlled intersections. The nominal or base condition for this analysis consisted of the

following geometric design conditions:

• No major-road left- or right-turn lanes.

• No skew angle (90-degree intersection angles).

• No intersection sight distance deficiencies.

The accident frequencies per year for this condition for various combinations of major- and

minor-road ADT are shown in table 16. Table 16 also shows the predicted accident frequencies

for various combinations of major-road left- and right-turn lanes. As indicated in table 5 installation

of a major-road left-turn lane at a three-leg STOP-controlled intersection is expected to reduce

accident frequency by 22 percent. Installation of a major-road right-turn lane is expected to

reduce accident frequency by 5 percent.

Table 17 presents the sensitivity of safety to skew-angle for three-leg STOP-controlled

intersections. A skew angle of 10 degrees results in an accident frequency 4 percent higher than a

90-degree intersection, while a skew angle of 45 degrees results in an accident frequency 20

percent higher than a 90-degree intersection.
61

Table 18 presents the sensitivity of safety to of intersection sight distance limitations at three-let

STOP-controlled intersections. As indicated in section 4 of this report, intersection sight distance

limitations can increase accident frequency by 5 percent per quadrant.

Table 16. Sensitivity of Safety to Major-Road Turn Lanes at Three-Leg

STOP-Controlled Intersections.

Major-road ADT

(veh/day)

Minor-road

ADT (veh/day)

No TLs One LTL One RTL

One LTL &

one RTL

BASE

ACCIDENTS PER YEAR

400 50 0.01 0.01 0.01 0.01

100 0.02 0.02 0.02 0.01

400 0.04 0.03 0.04 0.03

1,000 100 0.04 0.03 0.04 0.03

500 0.09 0.07 0.09 0.07

1,000 0.13 0.10 0.12 0.09

3,000 100 0.10 0.08 0.09 0.07

500 0.22 0.17 0.21 0.16

1,000 0.30 0.24 0.29 0.23

3,000 0.52 0.41 0.50 0.39

5,000 100 0.15 0.11 0.14 0.11

500 0.32 0.25 0.31 0.24

1,000 0.46 0.36 0.43 0.34

3,000 0.78 0.61 0.74 0.58

5,000 1.00 0.78 0.95 0.74

10,000 100 0.25 0.20 0.24 0.19

500 0.56 0.44 0.53 0.42

1,000 0.79 0.61 0.75 0.58

3,000 1.35 1.05 1.28 1.00

5,000 1.73 1.35 1.65 1.28

10,000 2.43 1.90 2.31 1.80

Note: TL=turn lane; LTL=left-turn lane; RTL=right-turn lane.
62

Table 17. Sensitivity of Safety to Skew Angles at Three-Leg STOP-Controlled

Intersections.

Major-road

ADT

(veh/day)

Minor-road

ADT

(veh/day)

Skew angle (degrees)

0 10 15 30 45

BASE

ACCIDENTS PER YEAR

400 50 0.01 0.01 0.01 0.01 0.01

100 0.02 0.02 0.02 0.02 0.02

400 0.04 0.04 0.04 0.05 0.05

1,000 100 0.04 0.04 0.04 0.05 0.05

500 0.09 0.09 0.10 0.10 0.11

1,000 0.13 0.14 0.14 0.15 0.16

3,000 100 0.10 0.10 0.11 0.11 0.12

500 0.22 0.23 0.23 0.25 0.26

1,000 0.30 0.31 0.32 0.34 0.36

3,000 0.52 0.54 0.55 0.59 0.62

5,000 100 0.15 0.16 0.16 0.17 0.18

500 0.32 0.34 0.34 0.37 0.39

1,000 0.46 0.47 0.48 0.51 0.55

3,000 0.78 0.81 0.83 0.88 0.94

5,000 1.00 1.04 1.06 1.13 1.20

10,000 100 0.25 0.26 0.27 0.28 0.30

500 0.56 0.58 0.59 0.63 0.67

1,000 0.79 0.82 0.84 0.89 0.94

3,000 1.35 1.40 1.43 1.52 1.61

5,000 1.73 1.80 1.84 1.95 2.07

10,000 2.43 2.53 2.58 2.74 2.91
63

Table 18. Sensitivity of Safety to Limited Intersection Sight Distance at Three-Leg

STOP-Controlled Intersections.

Major-road

ADT

(veh/day)

Minor-road

ADT

(veh/day)

Number of quadrants with limited ISD 

0 1 2

BASE

ACCIDENTS PER YEAR

400 50 0.01 0.02 0.02

100 0.02 0.02 0.02

400 0.04 0.04 0.04

1,000 100 0.04 0.04 0.05

500 0.09 0.10 0.10

1,000 0.13 0.13 0.14

3,000 100 0.10 0.10 0.11

500 0.22 0.23 0.24

1,000 0.30 0.32 0.33

3,000 0.52 0.55 0.57

5,000 100 0.15 0.15 0.16

500 0.32 0.34 0.36

1,000 0.46 0.48 0.50

3,000 0.78 0.82 0.86

5,000 1.00 1.05 1.10

10,000 100 0.25 0.27 0.28

500 0.56 0.59 0.62

1,000 0.79 0.83 0.87

3,000 1.35 1.42 1.48

5,000 1.73 1.82 1.91

10,000 2.43 2.55 2.68

Note: ISD = intersection sight distance.
64

Four-Leg STOP-Controlled Intersections

A sensitivity analysis was performed with the accident prediction algorithm for four-leg STOPcontrolled intersections. The nominal or base condition for this analysis consisted of the following

geometric design conditions:

• No major-road left- or right-turn lanes.

• No skew-angle (90-degree intersection angle).

• No intersection sight distance deficiencies.

The accident frequencies per year for this condition for various combinations of major- and

minor-road ADT are shown in table 19. Table 19 also shows the predicted accident frequencies

for various combinations of major-road left- and right turn lanes. As indicated in Table 5, a single

major-road left-turn lane at a four-leg STOP-controlled intersection is expected to reduce accident

frequency by 24 percent and two major-road left turn lanes are expected to reduce accident

frequency by 42 percent. A single major-road right-turn lane would reduce accident frequency by

5 percent and two major-road right-turn lanes would reduce accident frequency by 10 percent.

Table 20 presents the sensitivity of safety to intersection skew angle for four-leg STOPcontrolled intersections. A skew angle of 10 degrees results in an accident frequency 6 percent

higher than a 90-degree intersection, while a skew angle of 45 degrees results in an accident

frequency 28 percent higher than a 90-degree intersection.

Table 21 presents the sensitivity of safety to deficiencies of intersection sight distance at fourleg STOP-controlled intersections. As indicated in section 4 of this report, intersection sight

distance deficiencies can increase accident frequency by 5 percent per quadrant.

Four-Leg Signalized Intersections

A sensitivity was performed with the accident prediction algorithm for four-leg signalized

intersections. The nominal or base condition for this analysis consisted of a four-leg signalized

intersection with no major-road left- or right-turn lanes. The accident frequencies per year for this

condition for various combinations of major- and minor-road ADT are shown in table 22. Table

22 also shows predicted accident frequencies for various combinations of major-road left- and

right-turn lanes. As indicated in table 5, a single major-road left-turn lane at a four-leg signalized

intersection is expected to reduce accident frequency by 18 percent and two major-road left-turn

lanes are expected to reduce accident frequency by 33 percent. By contrast, a single major-road

right-turn lane would reduce accident frequency by 2.5 percent and two major-road right-turn

lanes would reduce accident frequency by 5 percent.

The predicted accident frequency at a four-leg signalized intersection is not sensitive to

intersection skew angle or intersection sight distance limitations.
65

Table 19. Sensitivity of Safety to Major-Road Turn Lanes at Four-Leg STOP-Controlled

Intersections.

Major-Road

ADT

(veh/day)

Minor-Road

ADT

(veh/day)

No TLs

BASE

One

LTL

Two

LTLs

One

RTL

Two

RTLs

One

LTL &

One

RTL

One

LTL &

Two

RTLs

One

RTL &

Two

LTLs

Two

LTLs &

Two

RTLs

ACCIDENTS PER YEAR

400 50 0.03 0.02 0.02 0.03 0.03 0.02 0.02 0.02 0.02

100 0.05 0.04 0.03 0.05 0.05 0.04 0.03 0.03 0.03

400 0.12 0.09 0.07 0.12 0.11 0.09 0.08 0.07 0.06

1,000 100 0.09 0.07 0.05 0.09 0.08 0.06 0.06 0.05 0.05

500 0.25 0.19 0.15 0.23 0.22 0.18 0.17 0.14 0.13

1,000 0.37 0.28 0.21 0.36 0.34 0.27 0.25 0.20 0.19

3,000 100 0.18 0.14 0.10 0.17 0.16 0.13 0.12 0.10 0.09

500 0.47 0.36 0.27 0.45 0.43 0.34 0.32 0.26 0.25

1,000 0.72 0.55 0.42 0.69 0.65 0.52 0.49 0.40 0.38

3,000 1.42 1.08 0.82 1.35 1.27 1.03 0.97 0.78 0.74

5,000 100 0.24 0.18 0.14 0.23 0.22 0.17 0.16 0.13 0.13

500 0.64 0.46 0.37 0.61 0.58 0.46 0.44 0.35 0.33

1,000 0.98 0.74 0.57 0.94 0.89 0.71 0.67 0.54 0.51

3,000 1.92 1.46 1.11 1.83 1.73 1.39 1.31 1.06 1.00

5,000 2.63 2.00 1.53 2.50 2.36 1.90 1.80 1.45 1.37

10,000 100 0.37 0.28 0.21 0.35 0.33 0.27 0.25 0.20 0.19

500 0.98 0.74 0.57 0.93 0.88 0.71 0.67 0.54 0.51

1,000 1.49 1.13 0.86 1.42 1.34 1.08 1.02 0.82 0.78

3,000 2.92 2.22 1.69 2.77 2.62 2.11 2.00 1.61 1.52

5,000 3.98 3.02 2.31 3.78 3.58 2.87 2.72 2.19 2.08

10,000 6.08 4.63 3.53 5.77 5.47 4.39 4.16 3.35 3.17

Note: TL=turn lane; LTL=left-turn lane; RTL=right-turn lane.
66

Table 20. Sensitivity of Safety to Skew Angle at Four-Leg STOP-Controlled

Intersections.

Major-Road ADT

(veh/day)

Minor-Road ADT

(veh/day)

Skew Angle (degrees)

0 10 15 30 45

BASE

ACCIDENTS PER YEAR

400 50 0.03 0.04 0.04 0.04 0.04

100 0.05 0.06 0.06 0.06 0.07

400 0.12 0.13 0.13 0.15 0.16

1,000 100 0.09 0.10 0.10 0.11 0.12

500 0.25 0.26 0.27 0.29 0.31

1,000 0.37 0.40 0.41 0.44 0.48

3,000 100 0.18 0.19 0.19 0.21 0.23

500 0.47 0.50 0.51 0.56 0.61

1,000 0.72 0.76 0.74 0.85 0.92

3,000 1.42 1.49 1.54 1.66 1.81

5,000 100 0.24 0.25 0.26 0.28 0.31

500 0.64 0.68 0.70 0.76 0.82

1,000 0.98 1.04 1.07 1.16 1.25

3,000 1.92 2.03 2.09 2.26 2.45

5,000 2.63 2.71 2.85 3.09 3.35

10,000 100 0.37 0.39 0.40 0.43 0.47

500 0.98 1.03 1.06 1.15 1.25

1,000 1.49 1.57 1.62 1.75 1.90

3,000 2.92 3.08 3.16 3.43 3.72

5,000 3.98 4.20 4.32 4.68 5.08

10,000 6.08 6.41 6.59 7.15 7.75
67

Table 21. Sensitivity of Safety to Limited Intersection Sight Distance Deficiencies at

Four-Leg STOP-Controlled Intersections.

Major-Road

 ADT

 (veh/day)

Minor-Road

ADT

(veh/day)

Number of quadrants with limited ISD 

0 1 2 3 4

BASE

ACCIDENTS PER YEAR

400 50 0.03 0.04 0.04 0.04 0.04

100 0.05 0.06 0.06 0.06 0.06

400 0.12 0.13 0.14 0.14 0.15

1,000 100 0.09 0.10 0.10 0.11 0.11

500 0.25 0.26 0.27 0.28 0.29

1,000 0.37 0.39 0.41 0.43 0.45

3,000 100 0.18 0.19 0.20 0.20 0.21

500 0.47 0.50 0.52 0.55 0.57

1,000 0.72 0.76 0.80 0.83 0.87

3,000 1.42 1.49 1.56 1.63 1.70

5,000 100 0.24 0.25 0.27 0.28 0.29

500 0.64 0.68 0.71 0.74 0.77

1,000 0.98 1.03 1.08 1.13 1.18

3,000 1.92 2.02 2.12 2.21 2.31

5,000 2.63 2.76 2.89 3.02 3.15

10,000 100 0.37 0.38 0.40 0.42 0.44

500 0.98 1.03 1.08 1.12 1.17

1,000 1.49 1.57 1.64 1.72 1.79

3,000 2.92 3.06 3.21 3.35 3.50

5,000 3.98 4.18 4.38 4.58 4.78

10,000 6.08 6.38 6.68 6.99 7.29
68

Table 22. Sensitivity of Safety to Major-Road Turn Lanes at Four-Leg Signalized

Intersections.

Major-Road

ADT

(veh/day)

Minor-Road

ADT

(veh/day)

No TLs

BASE

One

LTL

Two

LTLs

One

RTL

Two

RTLs

One

LTL &

One

RTL

One

LTL &

Two

RTLs

One

RTL &

Two

LTLs

Two

LTLs &

Two

RTLs

ACCIDENTS PER YEAR

400 50 0.26 0.21 0.17 0.25 0.25 0.21 0.20 0.17 0.16

100 0.30 0.24 0.20 0.29 0.28 0.24 0.23 0.19 0.19

400 0.39 0.32 0.26 0.38 0.37 0.31 0.31 0.26 0.25

1,000 100 0.51 0.42 0.34 0.50 0.49 0.41 0.40 0.34 0.33

500 0.71 0.58 0.48 0.69 0.67 0.57 0.55 0.46 0.45

1,000 0.82 0.67 0.55 0.80 0.77 0.65 0.64 0.53 0.52

3,000 100 0.99 0.82 0.67 0.97 0.95 0.80 0.78 0.65 0.63

500 1.37 1.13 0.92 1.34 1.30 1.10 1.07 0.90 0.87

1,000 1.58 1.29 1.06 1.54 1.50 1.26 1.23 1.03 1.00

3,000 1.96 1.61 1.32 1.92 1.87 1.57 1.53 1.28 1.25

5,000 100 1.35 1.11 0.91 1.32 1.28 1.08 1.05 0.88 0.86

500 1.87 1.53 1.25 1.82 1.77 1.49 1.45 1.22 1.19

1,000 2.14 1.76 1.44 2.09 2.04 1.71 1.67 1.40 1.36

3,000 2.67 2.19 1.79 2.60 2.54 2.13 2.08 1.74 1.70

5,000 2.96 2.42 1.98 2.88 2.81 2.36 2.30 1.93 1.88

10,000 100 2.05 1.68 1.37 2.00 1.95 1.64 1.60 1.34 1.30

500 2.83 2.32 1.89 2.76 2.69 2.26 2.20 1.85 1.80

1,000 3.25 2.66 2.18 3.17 3.08 2.60 2.53 2.12 2.07

3,000 4.05 3.32 2.71 3.94 3.84 3.23 3.15 2.64 2.57

5,000 4.48 3.67 3.00 4.37 4.26 3.58 3.49 2.93 2.85

10,000 5.15 4.22 3.45 5.02 4.89 4.11 4.01 3.36 3.28

Note: TL=turn lane; LTL=left-turn lane; RTL=right-turn lane.
69

6. IMPLEMENTATIONOF THE ACCIDENT PREDICTION

ALGORITHM WITHIN THE IHSDM

The accident prediction algorithm is intended to help the user to make unbiased estimates of

the expected safety performance for any given geometric design alternative for a specific highway

improvement project. Complete evaluation of one or more proposed geometric design alternatives

for a particular project will require the user to determine, for comparative purposes, both the safety

performance of the current design and the expected future safety performance of that current design

if nothing is done to change the roadway (the “do nothing” or “baseline” alternative).

This section of the report describes the implementation of the accident prediction algorithm

within the IHSDM. Procedures are presented by which the algorithm can be used to make

unbiased estimates of the:

• Expected safety performance of one or more geometric design alternatives for a planned

roadway that has not yet been constructed.

• Recent or current safety performance of an existing roadway considering both the

predicted safety performance of the roadway and its observed accident history.

• Expected safety performance of the existing roadway in the future if the geometrics are

left unchanged (the ADT may change, of course).

• Expected safety performance of one or more proposed geometric design alternatives for

improving the existing roadway.

This section describes two methods for producing these estimates—one without and one with

consideration of site-specific accident history data for the project of interest. The first procedure

described is used when no site-specific accident history data are available. This first procedure is

applicable to planned roadways that have not yet been constructed and to existing roadways

where, for whatever reason, site-specific accident history data are not available to the analyst. The

second procedure described is used when site-specific accident history are available. This

procedure incorporates an Empirical Bayes (EB) approach to combining estimates from the

accident prediction algorithm and site-specific accident history data. Examples of the EB

procedure are presented later in this section.

Site-specific accident history data make an important contribution to increasing the accuracy

of predictions of the expected safety performance of highway facilities. Therefore, the analyst

should seek to obtain and use site-specific accident history data and apply the EB procedure

whenever possible.

The procedures presented here can be applied to any existing two-lane highway or to any

two-lane highway improvement project that retains the basic two-lane character of the facility. 
70

Two-lane highway improvements evaluated with these procedures can include addition of a third or

fourth lane over a short distance to improve passing opportunities on the highway. Thus, the

procedures can evaluate passing lanes that create a three-lane cross section and short four-lane

sections that are operationally equivalent to side-by-side passing lanes. Such added lanes do not

normally exceed 3.2 km (2 mi) in length. The procedures do not address widening of a two-lane

highway to a four-lane cross section for an extended length. It is hoped that appropriate

procedures for four-lane highways will be developed in the future so that analysis of two-lane to

four-lane widening projects will be possible within the IHSDM.

Accident Prediction When Site-Specific Accident History Data

are not Available

The accident prediction algorithm is intended to estimate the expected accident frequency for

any specified geometric design alternative and for any specified evaluation period. The specified

geometric design alternative to which the algorithm is applied can be either the existing roadway

(i.e., the “do-nothing” alternative), a proposed geometric design improvement to the existing

roadway, or a proposed roadway that has not yet been constructed. This first procedure is

applicable only to a project for which no site-specific accident history data are available. The

algorithm can be used to compare the expected safety performance of several geometric

alternatives by applying the algorithm separately to each alternative for the same evaluation period

and comparing the results. When no site-specific accident history data are available, the duration

of the evaluation period may be one year or any multiple of one year. The accident prediction

algorithm is applied in IHSDM to any specific geometric design alternative in a series of straight

forward steps, as follows:

• Step 1—Define the limits of the project and determine the geometrics of the project for

which the expected safety performance is to be predicted.

• Step 2—Divide the project into individual homogeneous roadway segments and

intersections.

• Step 3—Determine the geometric design and traffic control features for each individual

roadway segment and intersection.

• Step 4—Determine the ADTs for each roadway segment and intersection during each

year for which the expected safety performance is to be predicted.

• Step 5—Select an individual roadway segment or intersection for evaluation. If there are

no more roadway segments or intersections to be evaluated, go to step 13.
71

• Step 6—Select a particular year of the specified evaluation period for the roadway

segment or intersection of interest. If there are no more years to be evaluated for that

roadway segment or intersection, go to step 12.

• Step 7—Apply the appropriate base model to determine the predicted accident

frequency for nominal or base conditions for the selected year.

• Step 8—Multiply the result obtained in step 7 by the appropriate calibration factor for a

specific State or geographical region.

• Step 9—Multiply the result obtained in step 8 by the appropriate AMFs representing

safety differences between the nominal or base conditions and the actual geometrics and

traffic control of the roadway segment or intersection.

• Step 10—Estimate the expected distribution accident severities and accident types for the

roadway segment or intersection from the default distributions of accident severity and

accident type.

• Step 11—If there is another year to be evaluated for the selected roadway segment or

intersection, return to step 6. Otherwise, proceed to step 12.

• Step 12—If there is another roadway segment or intersection to be evaluated, return to

step 5. Otherwise, proceed to step 13.

• Step 13—Summarize and present the predictions in useful formats for the IHSDM user.

Figure 5 presents a flow diagram of the accident prediction algorithm incorporating these

steps. Each of these steps is described below:

Step 1—Define the limits of the project and determine the geometrics of the project for

which the expected safety performance is to be predicted.

The project evaluated can represent either an existing roadway or a design alternative for a

proposed improvement project. The geometric design features of the project and the traffic control

at each intersection must be documented. The geometric design features are determined from

either a plan of the existing roadway available in the CAD system or from data entered by the user. 

If CAD data are to be used, a program must be developed to interrogate the CAD file, determine

the geometrics of the project, and store those geometric data in a format that can be used by

IHSDM.
72

YES

Define project limits

Divide project into roadway

segments and intersections

Select a roadway segment

or intersection

Apply base model

Apply calibration factor

YES

NO

Select a year of the

evaluation period

to be processed

NO

Apply AMFs

Estimate accident

severity distribution and

accident type distribution

Summarize and present final

predicted values to user

Step 1

Step 2

Step 5

Step 6

Step 7

Step 8

Step 9

Step 10

Step 13

Step 11

Step 12

Step 3

Determine geometric

design and traffic control

data for roadway

segments

and intersections

Determine ADTs for

roadway segments

and intersections

Step 4

Figure 5. Flow Diagram of the Accident Prediction Algorithm When No Site-Specific

Accident History Data Are Available.
73

Step 2—Divide the project into individual homogeneous roadway segments and

intersections.

The next step is to divide the project into individual homogeneous roadway segments and

intersections. The roadway must be divided into homogeneous segments. A new homogeneous

segment begins at each intersection where the value of one of the following variables changes:

C Average daily traffic volume (veh/day).

C Lane width.

C Shoulder width.

C Shoulder type.

C Driveway density (driveways per mile).

C Roadside hazard rating.

Also, a new analysis section starts at any of the following locations:

C Intersection.

C Beginning or end of a horizontal curve.

C Point of vertical intersection (PVI) for a crest vertical curve, a sag vertical curve, or an

angle point at which two different roadway grades meet.

C Beginning or end of a passing lane or short four-lane section provided for the purpose of

increasing passing opportunities.

C Beginning or end of a center two-way left-turn lane.

Step 3—Determine the geometric design and traffic control features for each individual

roadway segment and intersection.

For each roadway segment, the following geometric and traffic control features must be

quantified:

C Length of segment (mi).

C ADT (veh/day).

C Lane width (ft).

C Shoulder width (ft).

C Shoulder type (paved/gravel/composite/turf).

C Presence or absence of horizontal curve (curve/tangent).

C Length of horizontal curve (mi), if the segment is located on a curve. [This represents the

total length of the horizontal curve, even if the curve extends beyond the limits of the

roadway segment being analyzed.]

C Radius of horizontal curve (ft), if the segment is located on a curve.

C Presence or absence of spiral transition curve, if the segment is located on a curve. [This

represents the presence or absence of a spiral transition curve at the beginning and end of
74

the horizontal curve, even if the beginning and/or end of the horizontal curve are beyond

the limits of the segment being analyzed.]

C Superelevation of horizontal curve, if the segment is located on a horizontal curve.

C Grade (percent), considering each grade as a straight grade from PVI to PVI (i.e.,

ignoring the presence of vertical curves).

C Driveway density (driveways per mi).

C Presence or absence of a passing lane to increase passing opportunities.

C Presence or absence of a short four-lane Section to increase passing opportunities.

C Presence or absence of a two-way left-turn lane.

C Roadside hazard rating.

For each intersection, the following geometric and traffic control features must be quantified:

C Number of intersection legs (3 or 4).

C Type of traffic control (minor-road STOP, all-way STOP, minor-road YIELD control, or

signal).

C Intersection skew angle (degrees departure from 90 degrees, with a + or - sign indicating

the direction of the departure).

C Number of major-road approaches with intersection left-turn lanes (0, 1, or 2).

C Number of major-road approaches with intersection right-turn lanes (0, 1, or 2).

C Number of intersection quadrants with deficient intersection sight distance (0, 1, 2, 3, or

4).

The values of these geometric and traffic control parameters for roadway segments and

intersections will be determined from the CAD system, from existing data files, or from data

supplied by the user.

Step 4—Determine the ADTs for each roadway segment and intersection during each year

for which the expected safety performance is to be predicted.

For each roadway segment and for the major- and minor-road approaches to each

intersection, ADT data are needed for each year of the period to be evaluated. Ideally, these ADT

data will already be available in a file or they will be entered by the user. If ADTs are available for

every roadway segment, the major-road ADTs for intersection approaches can be determined

without additional data being supplied by the user. If the ADTs on the two major-road legs of an

intersection differ, the average of the two ADT values should be used for the intersection. For a

three-leg intersection, the user should enter the ADT of the minor-road leg. For a four-leg

intersection, the user should enter the average of the ADTs for the two minor-road legs.

In many cases, it is expected that ADT data will not be available for all years of the evaluation

period. In that case, the analyst should interpolate or extrapolate as appropriate to obtain an

estimate of ADT for each year of the evaluation period. If the analyst does not do this, the
75

following default rules are applied within the accident prediction algorithm to estimate the ADTs for

years for which the data are not available. If ADT data are available for only a single year, that

same value is assumed to apply to all years of the before period. If two or more years of ADT

data are available, the ADTs for intervening years are computed by interpolation. ADTs for years

before the first year for which data are available are assumed to be equal to the ADT for that first

year; ADTs for years after the last year for which data are available are assumed to be equal to the

last year (i.e., no extrapolation is used by the algorithm).

Step 5—Select an individual roadway segment or intersection for evaluation. If there are no

more roadway segments or intersections to be evaluated, go to step 13.

Roadway segments and intersections are evaluated one at a time. Steps 6 through 11,

described below, are repeated for each roadway segment and intersection.

Step 6—Select a particular year of the specified evaluation period for the roadway segment

or intersection of interest. If there are no more years to be evaluated for that roadway

segment or intersection, go to step 12.

The individual years of the evaluation period are evaluated one year at a time for any particular

roadway segment or intersection. Separate estimates are made for each year because several of

the AMFs considered in step 9 are dependent on the ADT of the roadway segment or intersection,

which may change from year to year. Steps 7 through 10, described below, are repeated for each

year of the evaluation period as part of the evaluation of any particular roadway segment or

intersection.

Step 7—Apply the appropriate base model to determine the predicted accident frequency for

nominal or base conditions for the selected year.

The predicted accident frequency for nominal or base conditions is determined with one of the

following base models:

• Roadway segments .......................... Equation (6).

• Three-leg STOP-controlled intersections .......... Equation (8).

• Four-leg STOP-controlled intersections .......... Equation (10).

• Four-leg signalized intersections ................ Equation (12).

The ADT(s) used in the base model should be the ADT(s) for the selected year of the evaluation

period.
76

Step 8—Multiply the result obtained in step 7 by the appropriate calibration factor.

The calibration factors used in step 8 are the calibration factor for roadway segments (Cr) and

the calibration factor for intersections (Ci) discussed in section 3 and appendix C of this report.

Step 9—Multiply the result obtained in step 8 by the appropriate AMFs representing safety

differences between the nominal or base conditions and the actual geometrics and traffic

control of the roadway segment or intersection.

The AMFs for roadway segments and intersections are those described in Section 4 of this

report. Steps 8 and 9 together implement equations (13) and (14).

Step 10—Estimate the expected distribution of accident severities and accident types for the

roadway segment or intersection of interest from the default distributions of accident

severity and accident type.

The predictions of accident frequencies are supplemented by breaking down those frequencies

by accident severity and by accident type. This can provide the IHSDM user with greater insight

about safety conditions within the project. The accident severity and accident type estimates are

based on the default distributions of accident severity and accident type presented in Tables 1 and

2, respectively. These default distributions may be changed by IHSDM users as part of the

calibration process.

Step 11—If there is another year to be evaluated for the selected roadway segment or

intersection, return to step 6. Otherwise, proceed to step 12.

This step creates a loop through steps 7 to 10 that is repeated for each year of the evaluation

period for each of the individual roadway segments and intersections within the project.

Step 12—If there is another roadway segment or intersection to be evaluated, return to

step 5. Otherwise, proceed to step 13.

This step creates a loop through Steps 6 to 11 that is repeated for each roadway segment and

intersection within the project.
77

Step 13—Summarize and present the predictions in useful formats for the IHSDM user.

In the final step, the predicted accident frequencies are summarized and presented in useful

formats for IHSDM users. The data presented include, at a minimum:

C Accident frequencies for the project as a whole including:

-- Total accident frequency.

-- Accident frequency by severity level.

-- Accident frequency by accident type.

C Accident frequencies for individual roadway segments and intersections, expressed as

accident rates per mi per year or accident rate per million veh-mi for roadway segments

and accident rates per million entering vehicles for intersections, so that accident “hot

spots” that might be corrected through design improvements are evident.

Estimated accident frequencies could also be broken down by individual years of the

evaluation period. However, this is not normally done because the combined estimates across all

years of the evaluation period are generally of greatest interest to safety analysts. Predicted

accident frequencies for a multiyear period are likely to be more accurate than predicted accident

frequencies for any particular year.

Accident Prediction When Site-Specific Accident History Data

are Available

Consideration of site-specific accident history data in the accident prediction algorithm

increases the accuracy of the predicted accident frequencies. When at least 2 years of site-specific

accident history data are available for the project being evaluated, and when the project meets

certain criteria discussed below, the accident history data should be used. When considering sitespecific accident history data, the algorithm must consider both the existing geometric design and

traffic control for the project (i.e., the conditions that existed during the before period while the

accident history was accumulated) and the proposed geometric design and traffic control for the

project (i.e., the conditions that will exist during the after period, the future period for which

accident predictions are being made). The EB procedure discussed below provides a method to

combine predictions from the algorithm with site-specific accident history data.
78

Situations in Which the EB Procedure Should and Should Not

Be Applied

The applicability of the EB procedure depends on the availability of observed accident history

data and the type of improvement project being evaluated. If no observed accident history data

are available, application of the EB procedure is infeasible and should not be considered. If

observed accident history data are available, the applicability of the EB procedure depends on the

type of improvement project being evaluated.

The EB procedure should be applied for the following improvement types whenever observed

accident history data are available:

C Sites at which the roadway geometrics and traffic control are not being changed (e.g., the

“do-nothing” alternative). 

C Projects in which the roadway cross Section is modified but the basic number of lanes

remains the same. This would include, for example, projects for which lanes or shoulders

were widened or the roadside was improved, but the roadway remained a rural two-lane

highway.

• Projects in which minor changes in alignment are made, such as flattening individual

horizontal curves while leaving most of the alignment intact.

C Projects in which a passing lane or a short four-lane section is added to a rural two-lane

highway to increase passing opportunities. 

C Any combination of the above improvements.

The EB procedure is not applicable to the following types of improvements:

C Projects in which a new alignment is developed for at least 50 percent of the project

length. In this case, the procedure used when no site-specific accident history data are

available, as described above, should be applied because there is no reason why the

accident history of the old alignment should be used as a predictor of future accident

frequency on the new alignment. In others words, there is no reason to think that the new

roadway will have substantially higher (or lower), accident experience, simply because

the existing roadway has high (or low) accident experience. For cases in which the user

is concerned that a particular geographic area or corridor has higher or lower accident

experience than expected, a special study may be performed to revise the calibration

factor accordingly.
79

w '

1

1 % k(Np)

(33)

Ep' w (Np) % (1&w) O (32)

C Individual intersections at which the basic number of intersection legs or type of traffic

control is changed as part of a project. The EB procedure can be applied to the rest of

any project containing such an intersection, but the intersection itself should be omitted.

The reason that the EB procedure is not used for these project types is that the observed accident

data for a past time period is not necessarily indicative of the accident experience that is likely to

occur in the future, after such a major geometric improvement. When the EB procedure does not

apply, the accident prediction algorithm without the EB procedure (described earlier in this section

of the report) is used to determine the expected safety performance for a project during some

future time period. When the EB procedure does apply, the accident prediction algorithm,

including the EB procedure, is applied as described below.

Empirical Bayes Procedure

The EB procedure provides a methodology to combine the accident frequencies predicted by

the accident prediction algorithm (Np) with the accident frequency from the site-specific accident

history data (O). The EB procedure uses a weighted average of Np and O. This procedure

constitutes Step 9 of the step-by-step methodology presented below. A previous application of the

EB methodology in before-and-after safety evaluations of intersections converted from STOP to

signalized control is presented by Griffith.(40)

The expected accident frequency considering both the predicted and observed accident

frequencies is computed as:

where:

Ep = expected accident frequency based on a weighted average of Np and No;

Np = number of accidents predicted by the accident prediction algorithm during a

specified period of time (equal to Nrs for a roadway segment or Nint for an

intersection);

w = weight to be placed on the accident frequency predicted by the accident

prediction algorithm; and

O = number of accidents observed during a specified period of time.

The weight placed on the predicted accident frequency is determined in the EB procedure as:
*

 Equation (33) follows from the theoretical development of the EB approach by Hauer.(2)

Hauer defines the weight in the EB procedure as (1+Var{ê}/E{ê})!1, where E{ê} is the

expected mean accident frequency and Var{ê} is the variance of accident frequency. The

expected mean accident frequency is best estimated by the model prediction, Np. Negative

binomial regression modeling is based on the assumption that Var{ê}=k(E{ê})2. Therefore, it

follows that the ratio Var{ê}/E{ê} can be estimated by k/Np, which leads to Equation (33).

80

where:

 k = overdispersion parameter of the relevant base model of the accident prediction

algorithm.

The formulation of Equation (33) shows an inverse relationship between the magnitude of the

accident frequency predicted by the algorithm, Np and the weight, w. Therefore, as the value of Np

predicted by the algorithm increases, the weight placed on Np decreases. This relationship implies

that the higher the expected accident frequency predicted by the algorithm for a particular location,

the more the reliance that should be placed on the observed site-specific accident history and the

less the reliance that should be placed on the model prediction itself. By contrast, when the model

prediction is smaller, less reliance should be placed on the observed site-specific accident history

and greater reliance should be placed on the model prediction.*

Table 23 shows the values of the overdispersion parameters (k) for the four base models used

in the accident prediction algorithm.

The EB procedure works best if the roadway segments and intersections to which it is applied

contain at least a specified minimum number of predicted accidents. The minimum accident

frequency needed for application of the EB procedure is generally 1/k, where k is the

overdispersion parameter of the relevant base model. In the accident prediction algorithm, this l/k

criterion is normally applied to the accident frequency for fatal and injury accidents because the

frequency of fatal and injury accidents is usually less than the frequency of property-damage-only

accidents and is always less than or equal to the total accident frequency. Where the fatal and

injury accident frequency of particular roadway segments or intersections is less than 1/k, such

segments and intersections may be aggregated into larger analysis units for application of the EB

procedure.

In the accident prediction algorithm, the EB procedure is applied separately to the total

predicted and observed accident frequencies and to the predicted and observed frequencies for

two accident severity levels: fatal and injury accidents and property-damage-only accidents. 

Because the EB procedure is applied separately, the predicted fatal and injury accident and

property-damage-only accident frequencies may not sum to the predicted total accident

frequencies. A proportional adjustment to the predicted accident frequencies for the individual

severity levels is made to correct this discrepancy.
81

w0'

1

1 %

krsN

2

rs%ki1N

2

i1%ki2N

2

i2%ki3N

2

i3

Nrs%Ni1%Ni2%Ni3

(34)

w1'

1

1 %

krsNrs% ki1Ni1% ki2Ni2% ki3Ni3

Nrs%Ni1%Ni2%Ni3

(36)

E0' w0(Nrs%Ni1%Ni2%Ni3) % (1&w0)(Ors%Oi1%Oi2%Oi3) (35)

Table 23. Overdispersion Parameters for Base Models and Minimum Accident

Frequencies for EB Procedure.

Geometric element Overdispersion Parameter for

base model (k)

Minimum accident frequency

for EB procedurea (1/k)

Roadway segment 0.31 3

Three-leg STOP-controlled

intersection 0.54 2

Four-leg STOP-controlled

intersection 0.24 4

Four-leg signalized

intersection 0.11 9

a Rounded for application in the EB procedure. Normally, this minimum accident frequency

criterion is applied in the EB procedure to the predicted fatal and injury accident frequency.

If an EB analysis unit combines roadway segments and intersections, or more than one type of

intersection, two additional factors must be accounted for. First, the overdispersion parameter, k,

in the denominator of equation (33) is no longer uniquely defined, because two or more base

models with differing overdispersion parameters must be considered. Second, it can no longer be

assumed, as is normally done, that the expected numbers of accidents for the roadway segments

and intersections, or for the different types of intersections, are statistically correlated with one

another. Rather, an estimate of expected accidents should be computed based on the assumption

that the different entities are statistically independent (ñ=0) and on the alternative assumption that

they are perfectly correlated (ñ=1). The expected accident frequency is then estimated as the

average of the estimates for ñ=0 and ñ=1. The following equations implement this approach:
82

E1' w1(Nrs%Ni1%Ni2%Ni3) % (1&w1)(Ors%Oi1%Oi2%Oi3) (37)

Eau '

E0% E1

2

(38)

where:

 w0 = weight placed on predicted accident frequency when accident frequencies for

different roadway elements are statistically independent (ñ=0);

 w1 = weight placed on predicted accident frequency when accident frequencies for

different roadway elements are perfectly correlated (ñ=1);

 E0 = expected accident frequency based on the assumption that different roadway

elements are statistically independent (ñ=0);

 E1 = expected accident frequency based on the assumption that different roadway

elements are perfectly correlated (ñ=1);

 Eau = expected accident frequency for an analysis unit made up of two or more roadway

segments or intersections;

 krs = overdispersion parameter for roadway segments (analogously, a subscript of i1

represents three-leg STOP-controlled intersections, i2 represents four-leg STOPcontrolled intersections, and i3 represents four-leg signalized intersections);

 Nrs = predicted total number of accidents for all roadway segments within the EB analysis

unit (analogous for subscripts i1, i2, and i3);

 Ors = observed total number of accidents for all roadway segments within the EB analysis

unit (analogous for subscripts i1, i2, and i3).

Example Application of the EB Procedure

The following discussion presents a numerical example to illustrate the application of the EB

procedure. The example shows the application of the EB procedure to combine predicted

accident frequencies for the period from 1989 through 1997 for roadway segments and

intersections within a project site with the observed site-specific accident history for that same time

period. The combined estimates of accident frequency resulting from application of the EB

procedure represent an unbiased estimate of safety conditions during the period prior to

construction of a proposed project at this site (the “before period”) and can subsequently be used

in estimating the effect on safety of proposed alternative geometric design improvements to this site

(the “after” period). The example illustrates the EB procedure, but does not illustrate the entire
83

accident prediction methodology that incorporates the EB procedure; that methodology is

presented later in this section of the report.

The example addresses two hypothetical roadway segments and one hypothetical intersection. 

Roadway segment 1 is a two-lane highway segment 1.6 km (1 mi) in length with an initial ADT of

2,000 veh/day. Over the 9-year period (1989-1997), the ADT varies up and down reaching

2,200 veh/day in 1997. Roadway segment 2 is a longer 8-km (5-mi) two-lane highway segment

with a lower, but faster growing, ADT than segment 1. Intersection 1 is a four-leg STOPcontrolled intersection located on segment 1. The intersection has a minor-road volume that

increases from 500 veh/day in 1989 to 600 veh/day in 1997. 

The first portion of the example shows the application of the EB procedure to roadway

segment 1. This roadway segment was predicted to experience 4.2 accidents during the 9-year

period (1989-1997), but actually experienced 6 accidents. Table 24 shows the application of the

EB procedure which determines that the expected accident frequency for the roadway segment,

considering both the predicted and observed values, is 5.2 accidents. Of these, it is expected that

3.3 would be fatal and injury accidents and 1.9 would be property-damage-only accidents.

While the computations in table 24 follow the EB procedure, the table shows that the

predicted fatal and injury accident frequency for roadway segment 1 before application of the EB

procedure was 1.4 accidents during the 9-year period. This does not meet the minimum accident

frequency criterion (l/k) of three fatal and injury accidents for roadway segments shown in table 23. 

Thus, rather than relying on the results shown in table 24, it would be better to combine roadway

segment 1 with another roadway segment before applying the EB procedure.

Table 25 presents the application of the EB procedure to roadway segment 2 in a manner

analogous to table 24. Roadway segment 2 was predicted to experience 10.3 accidents during the

9-year period, but actually experienced 14 accidents. The table shows that the expected accident

frequency for roadway segment 2, considering both the predicted and observed values, is 13.1

accidents during the 9-year period.

Roadway segment 2 was originally predicted to experience 3.3 fatal and injury accidents per

year, so the minimum accident criterion for application of the EB procedure shown in table 23 is

met. However, because the criterion was not met for roadway segment 1, it would be desirable to

combine roadway segments 1 and 2 into a single EB analysis unit. Table 26 shows that if roadway

segments 1 and 2 were combined into a single analysis unit for application of the EB procedure,

table 26 shows that the expected accident frequency for the combined roadway segment during the

9-year period would be 19 accidents, including 9.8 fatal and injury accidents and 9.2 propertydamage-only accidents. Simple proportions based on the original predicted accident frequencies

show
84

Table 24. Application of Empirical Bayes Procedure to Roadway Segment 1.

Year 89 90 91 92 93 94 95 96 97

Sum

(1989-1997) Weightsc

Expected

Before Period

Accident

Frequencyd

Expected

Before Period

Accident

Frequency

(Corrected)

Calibration Factor (Cr) 1.01 1.01 1.01 0.98 0.98 0.98 1.05 1.05 1.05

AADT (veh/day) 2,000 1,800 1,800 1,900 2,000 2,100 2,200 2,300 2,200

ACCIDENT FREQUENCY (1989 - 1997)

Predicted Totala 0.461 0.415 0.415 0.425 0.447 0.469 0.527 0.551 0.527 4.234 0.432 5.236 5.236

Predicted F+ Ib 0.148 0.133 0.133 0.136 0.143 0.151 0.169 0.177 0.169 1.359 0.704 2.735 3.366e

Predicted PDOb 0.313 0.281 0.281 0.288 0.303 0.319 0.358 0.374 0.358 2.875 0.529 1.520 1.871

Observed Total 1 0 1 1 0 0 1 1 1 6

Observed F + I 1 0 1 1 0 0 1 1 1 6

Observed PDO 0 0 0 0 0 0 0 0 0 0

NOTE: F + I = Fatal and injury accidents; PDO = property-damage-only accidents.

a

 from equation (13).

b

 based on 32.1 percent fatal and injury accidents and 67.9 percent property-damage-only accidents for roadway segments from table 1.

c

 from equation (33) with k = 0.31 for roadway segments from table 23.

d

 from equation (32).

e

 from equation (40).

f

 from equation (41).
85

Table 25. Application of Empirical Bayes Procedure to Roadway Segment 2.

Year 89 90 91 92 93 94 95 96 97

Sum

(1989-1997) Weights c

Expected

Before Period

Accident

Frequencyd

Expected

Before Period

Accident

Frequency

(Corrected)

Calibration Factor (Cr) 1.01 1.01 1.01 0.98 0.98 0.98 1.05 1.05 1.05

AADT (veh/day) 700 600 600 650 900 1,000 1,100 1,200 1,250

ACCIDENT FREQUENCY (1989 - 1997)

Predicted Totala 0.891 0.764 0.764 0.803 1.111 1.235 1.455 1.588 1.654 10.263 0.239 13.106 13.106

Predicted F+ Ib 0.286 0.245 0.245 0.258 0.357 0.396 0.467 0.510 0.531 3.295 0.495 4.662 4.953e

Predicted PDOb 0.605 0.518 0.518 0.545 0.755 0.838 0.988 1.078 1.123 6.969 0.316 7.674 8.153f

Observed Total 3 1 0 1 2 1 4 1 1 14

Observed F + I 1 1 0 0 1 1 1 1 0 6

Observed PDO 2 0 0 1 1 0 3 0 1 8

NOTE: F + I = Fatal and injury accidents; PDO = property-damage-only accidents.

a

 from equation (13).

b

 based on 32.1 percent fatal and injury accidents and 67.9 percent property-damage-only accidents for roadway segments from table 1.

c

 from equation (33) with k = 0.31 for roadway segments from table 23.

d

 from equation (32).

e

 from equation (40).

f

 from equation (41).
86

Table 26. Application of Empirical Bayes Procedure to Roadway Segments 1 and 2 Combined.

Year 89 90 91 92 93 94 95 96 97

Sum

(1989-1997) Weightsc

Expected

Before Period

Accident

Frequencyd

Expected

Before Period

Accident

Frequency

(Corrected)

Calibration Factor (Cr) 1.01 1.01 1.01 0.98 0.98 0.98 1.05 1.05 1.05

ACCIDENT FREQUENCY (1989 - 1997)

Predicted Total (Seg 1)a 0.461 0.415 0.415 0.425 0.447 0.469 0.527 0.551 0.527

Predicted Total (Seg 2)a 0.891 0.764 0.764 0.803 1.111 1.235 1.455 1.588 1.654

Predicted Total (Combined) 1.351 1.178 1.178 1.227 1.558 1.704 1.982 2.138 2.180 14.498 0.182 18.998 18.998

Predicted F+ I (Combined)b 0.434 0.378 0.378 0.394 0.500 0.547 0.636 0.686 0.700 4.654 0.409 8.992 9.792e

Predicted PDO (Combined)b 0.918 0.800 0.800 0.833 1.058 1.157 1.346 1.452 1.481 9.844 0.247 8.455 9.207f

Observed Total (Combined) 4 1 1 2 2 1 5 2 2 20

Observed F + I (Combined) 2 1 1 1 1 1 2 2 1 12

Observed PDO (Combined) 2 0 0 1 1 0 3 0 1 8

NOTE: F + I = Fatal and injury accidents; PDO = property-damage-only accidents.

a

 from equation (13).

b

 based on 32.1 percent fatal and injury accidents and 67.9 percent property-damage-only accidents for roadway segments from table 1.

c

 from equation (33) with k = 0.31 for roadway segments from table 23.

d

 from equation (32).

e

 from equation (40).

f

 from equation (41).
87

that roadway segment 1 would be expected to experience 5.5 accidents during the 9-year period

(including 2.9 fatal and injury accidents and 2.6 property-damage-only accidents) and roadway

segment 2 would be expected to experience 13.5 accidents (including 6.9 fatal and injury

accidents and 6.6 property-damage-only accidents). The procedure for performing this

proportional allocation is described below in step 10 of the methodology incorporating the EB

procedure.

Table 27 illustrates the application of the EB procedure to a four-leg STOP-controlled

intersection, designated as intersection 1. The intersection was predicted by the accident

prediction algorithm to experience 3.9 accidents in 9 years, but actually experienced 3 accidents. 

The table shows that the expected accident frequency for the 9-year period, combining both

predicted and observed values, is 3.4 accidents, including 1.4 fatal and injury accidents and 2.0

property-damage-only accidents. However, table 27 also shows that the original predicted fatal

and injury accident frequency for the intersection is 1.2 accidents, which is less than the minimum

accident frequency criterion (l/k) shown in table 23 as 4 accidents for a four-leg STOPcontrolled intersection. Therefore, it would be desirable to combine intersection 1 into a larger

analysis unit with another intersection for application of the EB procedure. Since there are no

other intersections available, intersection 1 should be combined into an analysis unit with roadway

sections 1 and 2. Combining intersections with roadways sections is less desirable than

combining them with other intersections, but is still acceptable.

Table 28 illustrates the application of the EB procedure to the combined data for an entire

project consisting of roadway segments 1 and 2 and intersection 1. The table 

indicates that the expected accident frequency for the entire project for a 9-year period is 21.0

accidents including 9.9 fatal and injury accidents and 11.1 property-damage-only accidents. This

estimate uses computation based on equations (32) through (36) because roadway segments and

four-leg STOP-controlled intersections have base models with different overdispersion

parameters (0.31 and 0.24, respectively). Using proportional allocation back to the original

roadway segments and intersections, roadway segment 1 would be expected to experience 4.8

accidents in the 9-year period (including 2.2 fatal and injury accidents and 2.5 property-damageonly accidents), roadway segment 2 would be expected to experience 11.8 accidents (including

5.5 fatal and injury accidents and 6.3 property-damage-only accidents), and intersection 1 would

experience 4.4 accidents (including 2.2 fatal and injury accidents and 2.3 accidents). The

procedure for performing this proportional allocation is described below in step 10 of the

methodology incorporating the EB procedure.

As noted above, this example illustrates the use of the EB procedure in estimating expected

accident frequencies for conditions in the period before construction of an improvement at the site

in question. The procedure described in step 11 of the methodology presented below illustrates

how the results obtained in the example can be used in estimating the expected future (“after”

period) accident frequencies for one or more alternative geometric design improvements.
88

Table 27. Application of Empirical Bayes Procedure to Intersection 1 (Four-Leg STOP-Controlled Intersection).

Year 89 90 91 92 93 94 95 96 97

Sum

(1989-1997) Weightsc

Expected

Before Period

Accident

Frequencyd

Expected

Before Period

Accident

Frequency

(Corrected)

Calibration Factor(Ci) 1.03 1.03 1.03 0.96 0.96 0.96 1.04 1.04 1.04

Major-Road AADT (veh/day) 2,000 1,800 1,800 1,900 2,000 2,100 2,200 2,300 2,200

Minor-Road AADT (veh/day) 500 550 550 530 550 580 600 620 600

ACCIDENT FREQUENCY (1989 - 1997)

Predicted Totala 0.402 0.400 0.400 0.377 0.398 0.423 0.481 0.504 0.481 3.866 0.519 3.449 3.449

Predicted F+ Ib 0.129 0.129 0.129 0.121 0.128 0.136 0.154 0.162 0.154 1.241 0.771 1.415 1.431e

Predicted PDOb 0.273 0.272 0.272 0.256 0.270 0.287 0.327 0.342 0.327 2.625 0.613 1.997 2.019f

Observed Total 0 0 1 0 0 1 0 0 1 3

Observed F + I 0 0 1 0 0 0 0 0 1 2

Observed PDO 0 0 0 0 0 1 0 0 0 1

NOTE: F + I = Fatal and injury accidents; PDO = property-damage-only accidents.

a

 from equation (13).

b

 based on 41.7 percent fatal and injury accidents and 58.3 percent property-damage-only accidents for four-leg STOP-controlled intersections

from table 1.

c

 from equation (33) with k = 0.24 for four-leg STOP-controlled intersections from table 23.

d

 from equation (32).

e

 from equation (40).

f

 from equation (41).
89

Table 28. Application of Empirical Bayes Procedure to Roadway Segments 1 and 2 and Intersection 1 Combined.

Year 89 90 91 92 93 94 95 96 97

Sum

(1989-1997)

Weightd

w 0

Expected

Before

Period

Accident

Frequency e

E0

Weightf

w 1

Expected

Before

Period

Accident

Frequencyg

E1

Expected

Before

Period

Accident

Frequencyh

Eau

Expected

Before Period

Accident

Frequency

ACCIDENT FREQUENCY (1989 - 1997) (Corrected)

Predicted Total (Seg 1)a 0.461 0.415 0.415 0.425 0.447 0.469 0.527 0.551 0.527 4.234

Predicted Total (Seg 2)a 0.891 0.764 0.764 0.803 1.111 1.235 1.455 1.588 1.654 10.263

Predicted Total (Int 1)b 0.402 0.400 0.400 0.377 0.398 0.423 0.481 0.504 0.481 3.866

Predicted Total (Combined) 1.754 1.578 1.578 1.604 1.956 2.127 2.463 2.642 2.611 18.364 0.211 22.023 0.648 19.995 21.009 21.009

Predicted F+ I (Seg 1)c 0.148 0.133 0.133 0.136 0.143 0.151 0.169 0.177 0.169 1.359

Predicted F+ I (Seg 2)c 0.286 0.245 0.245 0.258 0.357 0.396 0.467 0.510 0.531 3.295

Predicted F + I (Int 1)c 0.129 0.129 0.129 0.121 0.128 0.136 0.154 0.162 0.154 1.241

Predicted F+ I (Combined) 0.563 0.507 0.507 0.515 0.628 0.683 0.791 0.848 0.854 5.895 0.454 10.319 0.648 8.746 9.532 9.940i

Predicted PDO (Seg 1)c 0.313 0.281 0.281 0.288 0.303 0.319 0.358 0.374 0.358 2.875

Predicted PDO (Seg 2)c 0.605 0.518 0.518 0.545 0.755 0.838 0.988 1.078 1.123 6.969

Predicted PDOI (Int 1)c 0.273 0.272 0.272 0.256 0.270 0.287 0.327 0.342 0.327 2.625

Predicted PDO (Combined) 1.191 1.072 1.072 1.089 1.328 1.444 1.672 1.794 1.807 12.469 0.282 9.979 0.648 11.249 10.614 11.068j

Observed Total (Combined) 4 1 2 2 2 2 5 2 3 23

Observed F + I (Combined) 2 1 2 1 1 1 2 2 2 14

Observed PDO (Combined) 2 0 0 1 1 1 3 0 1 9

NOTE: F + I = Fatal and injury accidents; PDO = property-damage-only accidents.

a

 from equation (13).

b

 from equation (14).

c

 based on 32.1 percent fatal and injury accidents and 67.9 percent property-damage-only accidents for roadway segments and 41.7 percent fatal and injury accidents and

 58.3 percent property-damage-only accidents from four-leg STOP-controlled intersections from table 1.

d

 from equation (34).

e

 from equation (35).

f

 from equation (36).

g

 from equation (37).

h

 from equation (38).

i

 from equation (40).

j

 from equation (41).
90

Step-by-Step Methodology for Applying the Accident

Prediction Algorithm Including the EB Procedure

The accident prediction algorithm including the EB procedure, like the algorithm presented

that does not include the EB procedure, is intended to estimate the expected accident frequency

for any specified geometric design alternative for a two-lane highway project and for any

specified evaluation period. The specified geometric design alternative to which the algorithm is

applied can be either the existing roadway (i.e., the “do-nothing” or “baseline” alternative) or a

proposed geometric design improvement. For any given geometric alternative, the algorithm

incorporates consideration of actual reported accident frequencies for the existing roadway for a

previous time period designated as the “before” period. The accident data for this same before

period can be used in evaluating the expected safety performance of one or more proposed

geometric alternatives for the same project. The algorithm is employed separately to each

alternative for the same evaluation period, and the results for these various alternatives are then

compared. The accident prediction algorithm including the EB procedure is applied to any

specified geometric alternative for any specified evaluation period in 12 steps as follows:

• Step 1—Define the limits of the project and determine the geometrics of the project

during the previous period for which observed accident history data are available (the

before period) and for the future period for which the expected safety performance of

the project is to be predicted (the after period).

• Step 2—Divide the project into individual homogeneous roadway segments and

intersections

• Step 3—Determine the geometric design and traffic control features for each individual

roadway segment and intersection for both the existing and proposed roadway.

• Step 4—Determine the ADTs for each roadway segment and the major- and minorroad ADTs for each intersection during each year of the before period for which

observed accident history data are available and for each year of the after period for

which the expected safety performance is to be predicted.

• Step 5—Apply the accident prediction algorithm to each of the individual roadway

segments and intersections that make up the existing roadway.

• Step 6—Aggregate roadway segments to join together complete horizontal curve and

tangent sections.

• Step 7—Determine the observed accident history during the before period for each of

the aggregated roadway segments from step 6 and for each intersection. If accident

locations are not available in sufficient detail to identify the individual accident history for
91

each aggregated roadway segment and each intersection, aggregate roadway segments

and/or intersections further, as necessary, until EB analysis units are obtained for which

observed accident frequencies can be determined.

• Step 8—Aggregate roadway segments and/or intersection further, if necessary, into

larger EB analysis units to ensure a minimum accident sample size for each analysis unit.

• Step 9—For each EB analysis unit, as joined in step 8, apply the EB procedure by

computing the expected accident frequency for the before period as a weighted average

of the predicted and observed accident frequencies.

• Step 10—For each EB analysis unit, disaggregate the expected total accident

experience during the before period back to the original roadway segments and

intersections identified in step 2.

• Step 11—Apply ADT growth factors and/or AMFs for geometric changes to convert

the expected accident frequency for the before period to an expected accident

frequency for the proposed project during the specified future time period.

• Step 12—Summarize and present the predictions in useful formats for the IHSDM

user.

Figure 6 presents a flow chart illustrating the steps in the accident prediction algorithm

incorporating consideration of site-specific accident history data. These steps are described

below in greater detail.

Step 1—Define the limits of the project and determine the geometrics of the project during

the previous period for which observed accident history data are available (the before

period) and for the future period for which the expected safety performance of the project

is to be predicted (the after period).

The geometric design features of the project and the traffic control at each intersection must

be documented both before-and-after the planned improvement. The geometrics of the existing

roadway, which were in place during the before period, are determined from either a plan of the

existing roadway available in the CAD system or from data entered by the user. If CAD data

are to be used, a program must be developed to interrogate the CAD file, determine the

geometrics of the project, and store those geometric data in a format that can be used by the

IHSDM. The same sort of data must be obtained for the proposed improvement which, if

constructed, will be in place during the after period whose safety performance is to be estimated.
92

Step 5

Step 6

Step 7

Step 8

Step 9

Step 10

Step 11

Step 12

Apply EB procedure

Aggregate roadway segments

to join together complete

horizontal curves and tangents

Apply ADT growth factors

and AMFs for differences

in geometrics between

the existing and

proposed roadways

Disaggregate results to

original roadway segments

and intersections

Apply accident prediction

algorithm to each roadway

segment and intersection

on the existing roadway

Aggregate roadway

segments and intersections

further as necessary

Summarize and present final

predicted values to user

Step 1

Step 2

Divide project into roadway

segments and intersections

Define project limits

Step3

Determine geometric

design and traffic control

data for roadway

segments

and intersections

Determine ADTs for

roadway segments

and intersections

Step 4

Determine the actual site-specific

accident history for the existing

roadway for each roadway

segment and intersection

Figure 6. Flow Diagram of the Accident Prediction Algorithm When Site-Specific

Accident History Data Are Available.
93

Step 2—Divide the project into individual homogeneous roadway segments and

intersections

The next step is to divide the project into individual homogeneous roadway segments and

intersections. The roadway must be divided into homogeneous segments. A new homogeneous

segment begins at each intersection where the value of one of the following variables changes:

• Average daily traffic volume (veh/day).

• Lane width.

• Shoulder width.

• Shoulder type.

• Driveway density (driveways per mile).

• Roadside hazard rating.

Also, a new analysis section starts at any of the following locations:

• Intersection.

• Beginning or end of a horizontal curve.

• Point of vertical intersection (PVI) for a crest vertical curve, a sag vertical curve, or an

angle point at which two different roadway grades meet.

• Beginning or end of a passing lane or short four-lane section provided for the purpose

of increasing passing opportunities.

• Beginning or end of a center two-way left-turn lane.

This segmentation process is applied to both the existing (before period) roadway and the

proposed (after period) roadway. Each station that represents a division point for either the

existing or proposed roadway must be used as the beginning point of anew segment for purposes

of the analysis. In addition, each intersection is treated as a separate entity for analysis.

Step 3—Determine the geometric design and traffic control features for each individual

roadway segment and intersection for both the existing and proposed roadway.

For each roadway segment, the following geometric and traffic control features must be

quantified for both the existing and proposed roadways:

• Length of segment (mi).

• ADT (veh/day).

• Lane width (ft).

• Shoulder width (ft).

• Shoulder type (paved/gravel/composite/turf).

• Presence or absence of horizontal curve (curve/tangent).
94

• Length of horizontal curve (mi), if the segment is located on a curve. [This represents

the total length of the horizontal curve, even if the curve extends beyond the limits of the

roadway segment being analyzed.]

• Radius of horizontal curve (ft), if the segment is located on a curve.

• Presence or absence of spiral transition curve, if the segment is located on a curve.

[This represents the presence or absence of a spiral transition curve at the beginning

and end of the horizontal curve, even if the beginning and/or end of the horizontal curve

are beyond the limits of the segment being analyzed.]

• Superelevation of horizontal curve, if the segment is located on a horizontal curve.

• Grade (percent), considering each grade as a straight grade from PVI to PVI (i.e.,

ignoring the presence of vertical curves).

• Driveway density (driveways per mi).

• Presence or absence of a passing lane to increase passing opportunities.

• Presence or absence of a short four-lane section to increase passing opportunities.

• Presence or absence of a two-way left-turn lane.

• Roadside hazard rating.

For each intersection, the following geometric and traffic control features must be quantified for

both the existing and proposed roadways:

• Number of intersection legs (3 or 4).

• Type of traffic control (minor-road STOP, all-way STOP, minor-road YIELD control,

or signal).

• Intersection skew angle (degrees departure from 90 degrees, with a + or - sign

indicating the direction of the departure).

• Number of major-road approaches with intersection left-turn lanes (0, 1, or 2).

• Number of major-road approaches with intersection right-turn lanes (0, 1, or 2).

• Number of intersection quadrants with deficient intersection sight distance (0, 1, 2, 3, or

4).

The values of these geometric and traffic control parameters for roadway segments and

intersections, for both the existing and proposed designs, will be determined from the CAD

system, from existing data files for those designs, or from data supplied by the user.

Step 4—Determine the ADTs for each roadway segment and the major- and minor-road

ADTs for each intersection during each year of the before period for which observed

accident history data are available and for each year of the after period for which the

expected safety performance is to be predicted.

For each roadway segment and for the major- and minor-road approaches to each

intersection, ADT data are needed for each year of the before-and-after periods. Ideally, these

ADT data will already be available in a file or they will be entered by the user. If ADTs are
95

available for every roadway segment, the major-road ADTs for intersection approaches can be

determined without additional data being supplied by the user. If the ADTs on the two majorroad legs of an intersection differ, the average of the two ADT values should be used for the

intersection. For a three-leg intersection, the user should enter the ADT of the minor-road leg. 

For a four-leg intersection, the user should enter the average of the ADTs for the two minor-road

legs.

In many cases, it is expected that ADT data will not be available for all years of the before

or after period. In that case, the analyst should interpolate or extrapolate as appropriate to

obtain an estimate of ADT for each year of the evaluation period. If the analyst does not do this,

the following default rules are applied within the accident prediction algorithm to estimate the

ADTs for years for which the data are not available. If ADT data are available for only a single

year of the before period, that same value is assumed to apply to all years of the before period. 

If two or more years of ADT data for the before period are available, the ADTs for intervening

years is computed by interpolation. ADTs for years before the first year for which data are

available are assumed to be equal to the ADT for that first year; ADTs for years after the last

year for which data are available should be assumed to be equal to the last year (i.e., no

extrapolation is used by the algorithm). The same approach should be used to determine ADT

data for the after period.

Step 5—Apply the accident prediction algorithm to each of the individual roadway

segments and intersections that make up the existing roadway.

The accident prediction algorithm should be applied to each individual roadway segment and

intersection that makes up the existing roadway. This step is equivalent to steps 5 through 12 of

the algorithm applied when site-specific accident data are not available, as presented earlier in

this section of the report. The accident prediction algorithm is applied to roadway segments

using equation (13), which incorporates the base model in equation (6), the calibration factor for

roadway segments, and the roadway segment AMFs. The accident prediction algorithm for

intersections is applied using equation (14), which incorporates the base models in equations (8),

(10), or (12), depending on the type of intersection being evaluated, the calibration factor for

intersections, and the intersection AMFs. The accident prediction algorithm is applied to each

roadway segment or intersection individually for each year of the before period, using the

appropriate ADT(s) for that year, and the results for that segment or intersection are summed

over all the years to obtain a total predicted accident frequency for the before period. The total

predicted accident frequency is allocated to two severity levels (fatal or injury accidents and

PDO accidents) based on default proportions from table 1 in the main text of this report.

The predicted accident frequencies obtained from this process are designated Nrs for a

roadway segment, Ni1 for a three-leg STOP-controlled intersection, Ni2 for a four-leg STOPcontrolled intersection, and Ni3 for a four-leg signalized intersection. Each of these predicted
96

frequencies has associated with it predicted frequencies by severity level that sum to the

predicted total accident frequency. 

Step 6—Aggregate roadway segments to join together complete horizontal curve and

tangent sections.

The individual roadway segments are often so short, and have so few predicted and

observed accidents, that it would not be meaningful to apply the EB procedure to each individual

roadway segment. Therefore, aggregation of short roadway segments into longer segments,

which will be called EB analysis units, is desirable. 

The first stage of this process is to joint together complete horizontal curve and tangent

sections of roadway. Proceed in geographical order from one end of the project (e.g., in order

of increasing centerline stations) and join roadway segments together leaving breaks between

segments only at the following locations:

• Intersections.

• Other points at which the ADT changes by 20 percent or more.

• Beginnings and ends of horizontal curves.

• Beginnings and ends of grades of 5 percent or more that fall with a tangent roadway

section.

• Beginnings and ends of passing lanes.

• Beginnings and ends of short, four-lane sections.

• Beginnings and ends of two-way left-turn lanes.

As the segments are aggregated, their predicted accident frequencies (total and for each severity

level) should be added together.

Step 6 deals only with the aggregation of roadway segments. For the present, each

individual intersection serves as a separate EB analysis unit.

Step 7—Determine the observed accident history during the before period for each of the

aggregated roadway segments from step 6 and for each intersection. If accident locations

are not available in sufficient detail to identify the individual accident history for each

aggregated roadway section and each intersection, aggregate roadway sections and/or

intersections further, as necessary, until EB analysis units are obtained for which observed

accident frequencies can be determined.

The observed accident history for the before period is obtained from one of two sources: (1)

an available accident data file available that can be queried on-line; or (2) key entry of accident

data for individual EB analysis units by the IHSDM user. If an accident history file is used, the
97

user will have supply data on the correspondence between the accident location system used in

the file (e.g., mileposts) and the stationing system used for the project. This can consist, for

example, of the milepost corresponding to the end of the project with the lowest station and an

indication of whether mileposts and stations increase in the same direction or in opposite

directions. Then, the accident prediction algorithm can calculate the milepost limits of each

aggregated roadway segment and the milepost of each intersection and retrieve accident data for

each. Intersection-related accidents with 76 m (250 ft) of the intersection should be attributed to

that intersection. Non-intersection-related accidents and intersection-related accidents that are

located more than 76 m (250 ft) from an intersection should be attributed to the roadway

segment within which the accident falls. It should be noted that there are very few intersectionrelated accidents more than 76 m (250 ft) from an intersection on rural two-lane highways. 

Intersection-related and non-intersection-related accidents can be distinguished in accident data

from most states by the investigating officer’s assessment of whether a given accident was

intersection related. The available accident file should be used to determine both total observed

accidents and accidents by severity level.

If no accident file is available, the accident prediction algorithm will prompt the user to

supply total accidents and accidents by severity level for each roadway segment (identified by a

range of centerline stations) and each intersection (identified by station and, if possible, by the

name of the intersecting minor road). For a large project, this data entry may be tedious, so it

will be advantageous to have a accident file whenever possible.

If accident locations are not available in sufficient detail to identify the individual accident

history for each aggregated roadway section and each intersection, it will be necessary to

aggregate roadway sections and/or intersections further, as necessary, until EB analysis units are

obtained for which observed accident frequencies can be determined. The IHSDM user will be

asked to identify situations in which further aggregation is necessary because of limitations in the

availability or precision of accident locations. Roadway segments and intersections and

intersections of different types can be joined together, if necessary, if the accident histories of

roadway segments and intersections, or the accident histories of closely-spaced intersections,

cannot be distinguished. If the available accident data consist only of totals at the project level,

then the entire project can be aggregated into one EB analysis unit, although this should be done

only if necessary.

The observed accident frequencies for the before period for a given analysis unit are

designated Ors for a roadway segment, Oi1 for a three-leg STOP-controlled intersection, Oi2 for

a four-leg STOP-controlled intersection, and Oi3 for a four-leg signalized intersection. Each of

these observed frequencies has associated with it observed frequencies by severity level that sum

to the observed total accident frequency. 

Step 8—Aggregate roadway segments and/or intersection further, if necessary, into larger

EB analysis units to ensure a minimum accident sample size for each analysis unit.
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N ' Nrs % Ni1 % Ni2 % Ni3 (39)

It is desirable for an EB analysis unit (i.e., an aggregated roadway segment, an intersection

or intersections of a specific type, a combination of roadway segments and intersections, or a

combination of intersections of different types) to have at least a minimum number of predicted

accidents for the EB procedure to be applied as intended. An individual roadway segment or

intersection can constitute an EB analysis unit (i.e., can be evaluated by itself in the EB

procedure) if observed accident history data are available for the analysis unit and the accident

frequency predicted by the model for the before period is equal to at least 1/k, where k is the

overdispersion parameter of the relevant base model. Table 23 shows the desirable minimum

numbers of predicted accidents for application of the EB procedure to roadway segments and

each type of intersection.

The predicted injury accident frequencies for EB analysis units after any further aggregation

in step 7 should be reviewed to determine if there are any analysis units for which the predicted

injury accident frequency is less than 1/k, based on table 23. If roadway segments and

intersections or intersections of different types are mixed together, use the value of largest

appropriate value of 1/k from table 23 for purposes of step 8 (a more sophisticated procedure

for dealing with disparate k values will be used in step 9). Any EB analysis unit for which the

predicted accident frequency is less than 1/k is a candidate for further aggregation.

Since, an EB analysis unit can combine roadway segments and intersections of different

types, its total predicted accident frequency is the sum of the predicted accident frequencies of its

constituent segments and intersections:

where:

 Ni1 = predicted accident frequency for three-leg STOP-controlled intersections;

 Ni2 = predicted accident frequency for four-leg STOP-controlled intersections; and

 Ni3 = predicted accident frequency for four-leg signalized intersections.

Similarly, the total observed accident frequency for an EB analysis unit is the sum of Ors, Oi1, Oi2,

and Oi3.

The aggregation process is conducted as follows:

Step 8A—Select as candidates for further aggregation all analysis units consisting exclusively

of tangent roadway segments with grades less than 5 percent for which the predicted injury

accident frequency is less than 1/k.

Step 8B—For each of the candidate analysis units, compute the ratio of the predicted total

accident frequency to the observed total accident frequency [i.e., N/O, or

(Nrs+Ni1+Ni2+Ni3)/(Ors+Oi1+Oi2+Oi3)].
99

Step 8C—Select the candidate with the lowest predicted accident frequency. If no more

candidates remain, go to step 8F.

Step 8D—Identify and select another candidate whose ratio of predicted to total accident

frequency is closest to that of the candidate selected in step 8C. The two selected candidate

do not necessarily need to be geographically adjacent. If there are no remaining candidates

other than the candidate selected in step 8C, then select the analysis unit of the same type

(i.e., a tangent roadway segment with grades less than 5 percent) with predicted injury

accident frequency greater or equal to 1/k whose ratio of predicted to total accident

frequency is closest to that of the selected candidate. If there are no eligible analysis units,

including even those with predicted injury accident frequency greater than 1/k, proceed to

step 8F.

Step 8E—Combine the analysis unit selected in step 8C with the analysis unit selected in

step 8D, add together their predicted and observed accident frequencies (i.e., sum the

values of Nrs, Ni1, Ni2, Ni3, Ors, Oi1, Oi2, and Oi3, for total accidents and by accident

severity level), and recompute the ratio of predicted and observed total accident frequency. 

If the predicted injury accident frequency for the combined section is greater than 1/k, then

go to step 8F. If the predicted injury accident frequency is still less than 1/k, then go back

to step 8D and identify another section to combine.

Step 8F—Repeat steps 8A through 8E for each of the following types of analysis units, in

turn:

• Tangent segments with grades greater than or equal to 5 percent.

• Horizontal curve segments.

• Tangent segments with grades less than 5 percent in passing lanes.

• Tangent segments with grades greater than or equal to 5 percent in passing lanes.

• Horizontal curve segments in passing lanes.

• Tangent segments with grades less than 5 percent in short four-lane sections.

• Tangent segments with grades greater than or equal to 5 percent in short four-lane

sections.

• Horizontal curve segments in short four-lane sections.

• Tangent segments with grades less than 5 percent in two-way left-turn lanes.

• Tangent segments with grades greater than or equal to 5 percent in two-way leftturn lanes.

• Horizontal curve segments in two-way left-turn lanes.

• Three-leg STOP-controlled intersections.

• Four-leg STOP-controlled intersections.

• Four-leg signalized intersections.
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Step 8G—If analysis units with predicted injury accident frequency less than 1/k remain,

repeat steps 8A through 8E joining together analysis units of disparate types in the following

order of descending priority:

• Join tangent roadway segments of any type to other tangent roadway segments of

any type.

• Join horizontal curve segments of any type to horizontal curve segments of any

type.

• Join three-leg STOP-controlled intersections to other three-leg STOP-controlled

intersections.

• Join four-leg STOP-controlled intersections to other four-leg STOP-controlled

intersections.

• Join four-leg signalized intersections to other four-leg signalized intersections.

• Joint tangent roadway segments with horizontal curves.

• Join three-leg STOP-controlled intersections to four-leg STOP-controlled

intersections.

• Join signalized intersection with STOP-controlled intersections.

• Join roadway segments and intersections.

The aggregation process can be stopped whenever no analysis units with predicted injury

accident frequency less than 1/k remain or whenever the entire project has been joined into

a single analysis unit.

Step 9—For each EB analysis unit, as joined in Step 8, apply the EB procedure by

computing the expected accident frequency for the before period as a weighted average of

the predicted and observed accident frequencies.

The next step is to apply the EB procedure by computing a weighted average of the

predicted and observed accident frequencies for the before period. For any EB analysis unit that

is composed of entirely of roadway segments or composed entirely of a single intersection type,

compute the weight to be placed on predicted accident frequency using equation (33) and then

compute the expected accident frequency using equation (32).

If an EB analysis unit combines roadway segments and intersections, or more than one type

of intersection, two additional factors must be accounted for. First, the overdispersion

parameter, k, in the denominator of equation (33) is no longer uniquely defined, because two or

more base models with differing overdispersion parameters must be considered. Second, it can

no longer be assumed, as is normally done, that the expected numbers of accidents for the

roadway segments and intersections, or for the different types of intersections are statistically

correlated with one another. Rather, an estimate of expected accidents should be computed

based on the assumption that the different entities are statistically independent (ñ=0) and on the

alternative assumption that they are perfectly correlated (ñ=1). The expected accident frequency
101

Epdo/corr ' Etot

Epdo

Efi%Epdo

(41)

Ep' Eau

Np

Nau

(42)

Efi/corr ' Etot

Efi

Efi%Epdo

(40)

is then estimated as the average of the estimates for ñ=0 and ñ=1. This approach is implemented

with equations (34) through (38).

Step 9 is applied both to predicted and observed total accident frequencies and to predicted

and observed accident frequencies by accident severity level. Since these computations are

independent, the expected accident frequencies by severity level may not sum to the expected

total accident frequency. A correction is made as follows so that the expected accident

frequencies for the individual severity levels do sum to the expected total accident frequency:

where:

Efi/corr = expected accident frequency for fatal and injury accidents (corrected);

Epdo = expected accident frequency for property-damage-only accidents (corrected);

Etot = expected accident frequency for total accidents as estimated with equations

(34) to (38);

Efi = expected accident frequency for fatal and injury accidents as estimated with

equations (34) to (38); and

 Epdo = expected accident frequency for property-damage-only accidents as estimated

with equations (34) to (38).

Step 10—For each EB analysis unit, disaggregate the expected total accident experience

during the before period back to the original roadway segments and intersections

identified in step 2.

If an EB analysis unit, consists of more than one roadway segment or intersection, the

expected accident frequency (Nxx) for the EB analysis unit should be disaggegated back to the

original roadway segments and intersections in proportion to their original predicted accident

frequencies. In other words:
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where:

 Ep = expected accident frequency for an individual homogeneous roadway segment or

intersection;

 Eau = expected accident frequency for an EB analysis unit combining two or more

roadway segments or intersections;

 Np = original predicted accident frequency for the individual homogeneous roadway

segment or intersection from step 5;

 Nau = original predicted accident frequency for all roadway segments and 

intersections that make up the analysis unit, as aggregated in steps 6 and 8.

Repeat this process for each of the original roadway segments and intersections, for total

accidents and for each severity level.

Step 11—Apply ADT growth factors and/or AMFs for geometric changes to convert the

expected accident frequency for the before period to an expected accident frequency for

the proposed project during the specified future time period.

At the conclusion of step 10, Ep represents the expected accident frequency for a given

roadway segment or intersection during the before period. To obtain an estimate of expected

accident frequency in a future period (the after period), the estimate must be corrected for (1)

any difference in the duration of the before-and-after periods; (2) any growth or decline in ADTs

between the before-and-after periods; and (3) any changes in geometric design or traffic control

features between the before-and-after periods that affect the values of the AMFs for the

roadway segment or intersection. The expected accident frequency for a roadway segment or

intersection in the after period can be estimated as:

Ef = Ep (Nbf / Nbp) (AMF1f / AMF1p) (AMF2f / AMF2p) ... (AMFnf / AMFnp) (43)

where:

Ef = expected accident frequency during the future time period for which accidents

are being forecast for the analysis segment or intersection in question;

Ep = expected accident frequency for the past time period for which accident history

data were available;

Nbf = number of accidents forecast by the base model using the future ADT data, the

specified nominal values for geometric parameters, and—in the case of an

analysis segment—the actual length of the analysis segment;
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Nbp = number of accidents forecast by the base model using the past ADT data, the

specified nominal values for geometric parameters, and—in the case of an

analysis segment—the actual length of the analysis segment;

AMFxf = value of the xth AMF for the geometric conditions planned for the future (i.e.,

proposed) design; and

AMFxp = value of the xth AMF for the geometric conditions for the past (i.e., existing)

design.

Because of the form of the base model for roadway segments, the ratio Nbf / Nbp is the same as

the ratio of the traffic volumes, ADTf / ADTp. However, for intersections, the ratio Nbf / Nbp

must be evaluated explicitly with the base models because the intersection base models

incorporate separate major- and minor-road ADT terms with differing coefficients. In applying

equation (43), the values of Nbp, Nbf, AMFxp, and AMFxf should be based on the average ADTs

during the entire before or after period, respectively.

Equation (43) is applied to total accident frequency. The expected future accident frequencies

by severity level should also be determined by multiplying the expected accident frequency from

the before period for each severity level by the ratio Ef/Ep.

In the case of minor changes in roadway alignment (i.e., flattening a horizontal curve), the

length of an analysis segment may change from the past to the future time period. In this case, the

length of the analysis segment for the existing condition is used to determine Nbp and the modified

length of the analysis segment for the planned condition is used to determine Nbf. This implicitly

incorporates the assumption that, if the length of the analysis segment is changed, its base

accident rate (per million-vehicle miles) remains constant and, therefore, the accident frequency

increases or decreases in proportion to length. Of the course, the AMF ratios that also appear in

equation (43) will account for any change in geometrics (i.e., reduction in radius of curvature) that

accompany a change in length.

Step 12—Summarize and present the predictions in useful formats for the IHSDM user.

In the final step, the predicted accident frequencies are summarized and presented in useful

formats for IHSDM users. The data presented include, at a minimum:

• Accident frequencies for the project as a whole including:

-- Total accident frequency.

-- Accident frequency by severity level.

-- Accident frequency by accident type.
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The accident severity and accident type distribution for each roadway segment and

intersection in the specified future time period is based on the default accident severity

and accident type distributions presented in tables 1 and 2.

• Accident frequencies for individual roadway segments and intersections, expressed as

accident rates per mi per year or million veh-mi for roadway segments and accident rates

per million entering vehicles for intersections, so that accident “hot spots” that might be

corrected through design improvements are evident.

Estimated accident frequencies could also be broken down by individual years of the

evaluation period. However, this is not normally done because the combined estimates across all

years of the evaluation period are generally of greatest interest to safety analysts. Predicted

accident frequencies for a multiyear period are likely to be more accurate than predicted accident

frequencies for any particular year.
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7. CONCLUSIONS, RECOMMENDATIONS, AND FUTURE

ENHANCEMENTS

This section presents the conclusions concerning the accident prediction algorithm developed

in this report and presents recommendations for possible future enhancements of the algorithm.

Conclusions

The primary conclusion of this report is that an accident prediction algorithm has been

developed and that this algorithm appears to be a useful tool for predicting the safety

performance of rural two-lane highways. The primary strengths of the algorithm are as follows:

• The algorithm makes quantitative estimates of accident frequency and of the accident

severity and accident type distributions for any two-lane highway section or project.

• The algorithm has been developed with a modular structure that combines base models

and AMFs. The base models serve as scale factors to assure that the magnitude of the

predicted accident frequency is appropriate, while the AMFs assure that the predicted

accident frequency is sensitive to site-specific geometric and traffic control features. 

• The use of AMFs that are separate from the base models assures that the effects of

individual geometric design and traffic control features are not dependent upon

inappropriate regression coefficients that are too large, too small, or in the wrong

direction. Each AMF has been developed by a panel of experts to represent the best

information currently available on the safety effects of that particular geometric design

or traffic control feature.

• The modular structure makes the algorithm easy to update as better information,

including new research results, become available.

• A calibration procedure is provided to allow individual highway agencies to adapt the

algorithm to the safety conditions present on their rural two-lane highway system. The

calibration procedure allows IHSDM users to adjust the predicted accident frequencies

for agency-to-agency and State-to-State differences in factors such as accident

reporting thresholds, accident reporting practices, animal populations, driver

populations, and climates.

• A procedure based on the EB method allows users to combine accident predictions

obtained from the algorithm with observed site-specific accident history data.
106

The major weakness of the algorithm is that it incorporates the effects on safety of most, but

not all, geometric and traffic control features of interest to highway agencies. The algorithm

incorporates only those features whose effects were considered by a panel of experts to be well

established in quantitative terms. Geometric and traffic control features that are poorly

understood, or not understood at all, have necessarily been omitted. The model generally treats

the effects of individual geometric design and traffic control features as independent of one

another and ignores potential interactions between them. It is likely that such interactions exist

and, ideally, they should be accounted for in the accident prediction algorithm. However, such

interactions are poorly understood and none could be quantified by the expert panels that

participated in the development of the algorithm. It is the assessment of the expert panels that the

base models and AMFs presented in this report represent the current state of knowledge about

safety on rural two-lane highways and cannot be improved without further research. The next

section of report discusses potential areas to which future research might be directed to improve

the model.

FHWA plans to incorporate the accident prediction algorithm for rural two-lane highways

presented in this report in software for implementation as part of the IHSDM. A stand-alone

version of the software may also be available for use independent of a CAD system.

Future Enhancement of the Accident Prediction Algorithm

It is recommended that future enhancements be made to the accident prediction algorithm as

further research is completed and that forthcoming research on rural two-lane highways be

structured so that results are obtained in a form that can be directly implemented in the accident

prediction algorithm. It is also recommended that a program of additional research be

undertaken with the specific goal of filling gaps in the accident prediction algorithm and expanding

its scope. Specific areas for future enhancement of the accident prediction algorithm are

discussed below.

Base Models

The base models for roadway sections and for three- and four-leg STOP-controlled

intersections appear to be well established and there is no immediate need for work to improve

them. By contrast, the base model for four-leg signalized intersections is based on a small sample

size (only 49 intersections split between two different States). The resulting models were not as

satisfactory as desired, and it was difficult to choose among the available candidates. Signalized

intersections are relatively rare on rural two-lane highways, so the limitations of the base model

for signalized intersections do not overly limit the utility of the algorithm, but it would be desirable

to assemble a larger database on signalized intersections on rural two-lane highways for the

purpose of developing an improved base model. It would also be desirable to develop a base

model for three-leg signalized intersections on rural two-lane highways, which are outside the

scope of the current accident prediction algorithm.
107

A minor drawback of the base model for roadway segments is that it was based on sites

with a mix of shoulder types, but the shoulder type itself was not a statistically significant factor in

the model. This is understandable given the small effect of shoulder type on safety based on the

expert panel’s assessment shown in table 3. However, for consistency, if the roadway segment

base model should be updated in the future, it would be desirable to (1) include an effect of

shoulder type in the base model if found to be statistically significant; or (2) omit the shoulder

type factor from the model and develop the model solely with data for sites with paved

shoulders.

Accident Modification Factors

The accident prediction algorithm omits AMFs for several geometric design and traffic

control features which the accident prediction algorithm should desirable address. Other AMFs

that are included in the model could be improved through further research. These improvement

needs are described below:

Roadway Segments

One of the greatest limitations of the roadway segment algorithm is the lack of an AMFs for

bridge width. The expert panel on roadway segments strongly desired to include a bridge width

factor in the algorithm because narrow bridges are known to be associated with accident

concentrations on two-lane highways. However, the panel found that even the best study of the

relationship between bridge width and safety had a major flaw that limited its use in the algorithm. 

The flaw was that the study included only bridges that had experienced one or more accidents

during the study period.(41) Omission of sites which have experienced no accidents is a known

source of bias in accident research. It is recommended that a well designed study of the

relationship between bridge width and accidents on rural two-lane highways be undertaken and

that its results, if found to be satisfactory, be incorporated in the accident prediction algorithm.

The roadway segment algorithm lacks an AMF to account for the effect on safety of vertical

curve design and stopping sight distance. This effect has never been satisfactorily quantified, and

recent research suggests that the safety effect of limited stopping sight distance at a crest vertical

curve is relatively small.(42) However, should this effect—even if small—be reliably quantified in

the future, it would be desirable to include it in the accident prediction algorithm.

It would be desirable to improve the representation of driveway effects in the accident

prediction algorithm for roadway effects. The algorithm currently bases driveway effects on the

driveway density (driveways per mi). The AMF for driveway density, based on a regression

equation, indicates that there is greater sensitivity of safety to driveway density at lower ADT

than at higher ADT. This appears to be the opposite of what might be expected. Furthermore, it

would be more desirable to develop a method to quantify the safety effect of each individual

driveway, but such a method does not currently exist. Such an approach would also require the
108

user to supply detailed data on individual driveway locations and types (e.g., commercial vs.

residential) and driveway traffic volumes which are not necessarily available to all users.

The effects of passing lanes and shoulder width are treated as independent in the accident

prediction algorithm, but there may in fact be an interaction between them. In the research that

established the safety effects of passing lanes, some of the sites at which passing lanes were

installed may also have had full shoulders provided as part of the same project. Other sites may

have had a portion of the shoulder converted to the passing lane. If further research quantifies the

separate effects of passing lanes and shoulder widths, or the interactions between them, it would

be desirable to incorporate these effects in the accident prediction algorithm.

The effect of center two-way left-turn lanes currently included in the accident prediction

algorithm is clearly an oversimplification of a much more complex effect. Many evaluations of

two-way left-turn lanes have been conducted, but little of this work is specific to rural two-lane

highways. Further research to improve the AMF for two-way left-turn lanes would be desirable.

The roadside design AMF incorporated in the model is based on a qualitative roadside

hazard rating system (a subjective 1 to 7 rating scale), rather than addressing the explicit effects

of specific roadside design features. It would be desirable to provide the capability for individual

roadside design features to be evaluated explicitly in situations where detailed data on roadside

design features are available. Research is currently underway as part of the NCHRP program

which may lead to such a capability.(22)

Intersections

Two undesirable omissions in the accident prediction algorithm for at-grade intersections are

the lack of effects for roadside design and driveways in the vicinity of an intersection. The

accident prediction algorithm for roadway segments includes the effect of both roadside design

and driveways over the entire length of a study section or project, but the intersection algorithm

places no special weight on restrictive roadside design at an intersection or driveways located

near an intersection. The omission of roadside design issues at intersections may be a minor

limitation, because run-off-the-road accidents are generally understood to be a roadway

segment, rather than an intersection, problem. For example, table 2 shows that the proportion of

intersection-related single-vehicle run-off-the-road accidents is small. However, poor roadside

design in the vicinity of an intersection could increase the severity of multiple-vehicle accidents in

which one or more of the involved vehicles leaves the traveled way. The lack of an effect for

driveways near intersections is an omission of greater concern, although it is not as serious an

omission on rural two-lane highways as it would be on urban or suburban arterials. Driveways

near intersections are known to be a safety-related access management concern, but the expert

panel found that the safety effects of driveways near intersections have not been well quantified. 

Research on this issue should be encouraged and should address, at a minimum, the type of

driveway (e.g., commercial vs. residential), the distance from the intersection to the driveway,
109

and the presence or absence of access control measures that restrict turning maneuvers

(e.g., medians or turn prohibitions).

It would be desirable to improve the AMFs for left- and right-turn lanes at intersections. 

The expert panel found substantial past research on the effects of left- and right-turn lanes, but no

definitive results. The AMFs included in the accident prediction algorithm relied heavily upon

expert judgment in interpreting the available results and additional research to improve these

AMFs would be desirable. FHWA has such research underway, and this research may lead to

an update of the accident prediction algorithm.

A final concern with the accident prediction algorithm for intersections is that it does not

address before-and-after evaluations of improvement projects in which an existing STOPcontrolled intersection is signalized. It would be desirable to develop a specific AMF for such

projects based on well-designed before-and-after studies.

Accident Prediction Algorithms for Other Facility Types

The accident prediction algorithm presented in this report applies only to rural two-lane

highways. Rural two-lane highways were selected for the initial development of the IHSDM

because existing rural two-lane highways have a wide range of variation in the quality of their

design features and, therefore, present substantial opportunities for improvement of safety. Most

freeways, by contrast, have been built much more recently and are more consistent in their design

features.

It would be desirable to expand the accident prediction algorithm to other facility types,

following the general approach developed for the accident prediction algorithm for rural two-lane

highways presented in this report. It is recommended that the scope of the accident prediction

algorithm be expanded in the following priority order: rural multilane highways, urban and

suburban arterials, and, finally, freeways.
110
111

8. REFERENCES

1. Bauer, K. M., and D. W. Harwood, Statistical Models of At-Grade Intersection

Accidents, Report No. FHWA-RD-96-125, Federal Highway Administration, 1996.

2. Hauer, E., Observational Before-After Studies in Traffic Safety, Elsevier Science, 1997.

3. Vogt, A., and J.G. Bared, Accident Models for Two-Lane Rural Roads: Segments and

Intersections, Report No. FHWA-RD-98-133, Federal Highway Administration, October

1998.

4. Vogt, A., and J.G. Bared, “Accident Models for Two-Lane Rural Segments and

Intersection,” Transportation Research Record 1635, Transportation Research Board,

1998.

5. Vogt, A., Crash Models for Rural Intersections: 4-Lane by 2-Lane Stop-Controlled and

2-Lane by 2-Lane Signalized, Report No. FHWA-RD-99-128, Federal Highway

Administration, October 1999.

6. Zegeer, C. V., D. W. Reinfurt, J. Hummer, L. Herf, and W. Hunter, “Safety Effects of

Cross-Section Design for Two-Lane Roads,” Transportation Research Record 1195,

Transportation Research Board, 1988.

7. Zegeer, C. V., R. Stewart, F. Council, and T. R. Neuman, “Accident Relationships of

Roadway Width on Low-Volume Roads,” Transportation Research Record 1445,

Transportation Research Board, 1994.

8. Zegeer, C. V., R. C. Deen, and J. G. Mayes, “Effect of Lane and Shoulder Width on

Accident Reduction on Rural, Two-Lane Roads,” Transportation Research Record 806,

Transportation Research Board, 1981.

9. Miaou, S-P., Measuring the Goodness-of-Fit of Accident Prediction Models, Report

No. FHWA-RD-96-040, Federal Highway Administration, December 1996.

10. Griffin, L. I., and K. K. Mak, The Benefits to Be Achieved from Widening Rural, TwoLane Farm-to-Market Roads in Texas, Report No. IAC(86-87) - 1039, Texas

Transportation Institute, College Station, TX, April 1987.

11. Miaou, S-P., “Development of Adjustment Factors for Single-Vehicle Run-Off-Road

Accidents Rates by Horizontal Curvature and Grade,” Oak Ridge National Laboratory,

draft, August 3, 1995.

12. Rinde, E. A., Accident Rates vs. Shoulder Width, Report No. CA-DOT-TR-3147-1-77-

01, California Department of Transportation, 1977.
112

13. Zegeer, C. V., J. R. Stewart, F. M. Council, D. W. Reinfurt, and E. Hamilton, “Safety

Effects of Geometric Improvements on Horizontal Curves,” Transportation Research

Record 1356, Transportation Research Board, 1992.

14. American Association of State Highway and Transportation Officials, Policy on Geometric

Design of Highways and Streets, Washington, DC, 1994.

15. Zegeer, C., R. Stewart, D. Reinfurt, F. Council, T. Neuman, E. Hamilton, T. Miller, and W.

Hunter, Cost-Effective Geometric Improvements for Safety Upgrading of Horizontal

Curves, Report No. FHWA-R0-90-021, Federal Highway Administration, October 1991.

16. Miaou, S-P., “Vertical Grade Analysis Summary,” unpublished, May 1998.

17. Muskaug, R., Accident Rates on National Roads, Institute of Transport Economics, Oslo,

Norway, 1985.

18. Transportation Research Board, “Driveway and Street Intersection Spacing,”

Transportation Research Circular 456, March 1996.

19. Harwood, D. W., and A. D. St. John, Passing Lanes and Other Operational

Improvements on Two-Lane Highways, Report No. FHWA/RD-85/028, Federal

Highway Administration, July 1984.

20. Nettelblad, P., Traffic Safety Effects of Passing (Climbing) Lanes: An Accident

Analysis Based on Data for 1972-1977, Meddelande TU 1979-5, Swedish National

Road Administration, 1979.

21. Hauer, E., “Two-Way Left-Turn Lanes: Review and Interpretation of Published Literature,”

unpublished, 1999.

22. Texas Transportation Institute, University of Nebraska-Lincoln, “Improved Procedures for

Cost-Effectiveness Analysis of Roadside Safety Features,” NCHRP Project 22-9, ongoing.

23. Kulmala, R., Safety at Three- and Four-Arm Junctions: Development and Application

of Accident Prediction Models, VTT Publication 233, Technical Research Centre of

Finland, Espoo, 1995.

24. Federal Highway Administration, Manual on Uniform Traffic Control Devices for

Streets and Highways, 1988.

25. Lovell, J., and E. Hauer, “The Safety Effect of Conversion to All-Way STOP Control,”

Transportation Research Record 1068, Transportation Research Board, 1986.
113

26. Agent, K. R., “Development of Warrants for Left-Turn Lanes,” Research Report 526,

Bureau of Highways, Kentucky Department of Transportation, Lexington, KY, July 1989.

27. California Department of Public Works, Evaluation of Minor Improvements: Flashing

Beacons, Safety Lighting, Left-Turn Channelization, Traffic Department, California of

Public Works, 1967 (as cited in Reference 27).

28. Hagenauer, G. F., et al., “Chapter 5. Intersections,” in Synthesis of Safety Research

Related to Traffic Control and Roadway Elements, Volume 1, Report No. FHWA-TS82-232, Federal Highway Administration, December 1982.

29. Creasey, T., and K. R. Agent, “Development of Accident Reduction Factors,” Research

Report UKTRP-85-6, Kentucky Transportation Research Program, College of Engineering,

University of Kentucky, Lexington, KY, March 1985.

30. Dale, C. W., Cost-Effectiveness of Safety Improvement Projects, Federal Highway

Administration, Washington, D.C., May 1973.

31. Ermer, O. J., Accident Reduction Factors for Indiana, Report No. FHWA/IN/JHRP91/11, Joint Highway Research Program, Purdue University School of Engineering, Indiana

Department of Transportation, May 1991.

32. Glennon, J. C., et al., Technical Guidelines for the Control of Direct Access to Arterial

Highways, Volume II: Detailed Description of Access Control Techniques, Report

No. FHWA-RD-76-87, Federal Highway Administration, 1975.

33. Glennon, J. C., et al., Evaluation of Techniques for the Control of Direct Access to

Arterial Highways, Report No. FHWA-RD-76-85, Federal Highway Administration,

1976.

34. McCoy, P. T., W. J. Hoppe, and D. V. Dvorak, “Benefit-Cost Evaluation of Left-Turn

Lanes on Uncontrolled Approaches of Rural Intersections (Abridgement),” Transportation

Research Record 1026, Transportation Research Board, Washington, DC, 1985.

35. McCoy, P. T., and M. S. Malone, “Safety Effects of Left-Turn Lanes on Urban Four-Lane

Roadways,” Transportation Research Record 1239, Transportation Research Board,

1989.

36. Smith, S. A., et al., Identification, Quantification, and Structuring of Two-Lane Rural

Highway Safety Problems and Solutions, Volume II—Technical Report, Federal

Highway Administration, July 1981.
114

37. Datta, T. K., “Head-On, Left-Turn Accidents at Intersections with Newly Installed Traffic

Signals,” Transportation Research Record 1318, Transportation Research Board,

Washington, DC, 1991.

38. Elvik, R., Meta Analysis of Intersection Accidents, Institute of Transport Economics,

Oslo, Norway.

39. Brüde, U., and J. Larsson, Countermeasures at Junctions Taken As Part of the

Regional Road Authorities’ Traffic Safety Program, VTI-Rapport 292, Swedish Road

and Traffic Research Institute, Linkoping, 1985.

40. Griffith, M. S., The Application of an Improved Accident Analysis Method for Highway

Safety Evaluation, Report No. FHWA-RD-94-082, Federal Highway Administration,

October, 1994.

41. Turner, D. S., “Prediction of Bridge Accident Rates,” Journal of Transportation

Engineering, American Society of Civil Engineers, Vol. 110, No. 1, January 1984.

42. Fambro, D. B., K. Fitzpatrick, and R. Koppa, Determining of Stopping Sight Distances,

NCHRP Report 400, Transportation Research Board, 1997.

43. Fridstrøm, L., et al., “Measuring the Contribution of Randomness, Exposure, Weather, and

Daylight to the Variation in Road Accident Counts,” Accident Analysis and Prevention,

Vol. 27, 1995.
115

9. BIBLIOGRAPHY

The bibliography presented below includes all sources consulted in the development of the

AMFs.

Access Points

Hauer, E., “Two-Way Left-Turn Lanes: Review and Interpretation of Published Literature,”

unpublished, 1999.

Li, J., “Joint Effects of Access and Geometry on Two-Lane Rural Highway Safety in British

Columbia,” Canadian Journal of Civil Engineering, 1994/12, XXI(vi), 1994.

Muskaug, R., Accident Rates on National Roads, Institute of Transport Economics, Oslo,

Norway, 1985.

Transportation Research Board, “Driveway and Street Intersection Spacing,” Transportation

Research Circular 456, March 1996.

Cross Section (Lane Width, Shoulder Width, and Shoulder Type)

Benekohal and Lee, “Comparison of Safety Effects of Roadside Versus Roadway Improvements

on Two-Lane Rural Highways,” Transportation Research Record 1303, Transportation

Research Board, 1991.

Brüde, U., and J. Larsson, Traffic Safety Effect of Wide Driving Lanes, VTI-Meddelande

807, Swedish Road and Traffic Institute, Linkoping, Sweden, 1996.

Busbee, C.B., “Cost-Effectiveness of a Two-Way Left-Turn Lane,” Traffic Engineering,

Institute of Traffic Engineers, December 1975.

Conradson, B.A., and N. Al-Ashari, Evaluation of Four Safety Projects, Traffic and Safety

Division, Michigan Department of State Highways, 1972.

Datta, T.K., Assessment of the Benefits of Two-Way Center Left-Turn Lanes, Report No.

FHWA-MI-RD-94-06, Michigan Department of Transportation, Federal Highway

Administration, 1994.

Goldstine, R., “Influence of Road Width on Accident Rates by Traffic Volume,” Transportation

Research Record 1318, Transportation Research Board, 1991.
116

Griffin, L. I., and K. K. Mak, The Benefits to Be Achieved from Widening Rural, Two-Lane

Farm-to-Market Roads in Texas, Report No. IAC(86-87) - 1039, Texas Transportation

Institute, College Station, TX, April 1987.

Harwood, D.W., and J.C. Glennon, “Selection of Median Treatments for Existing Arterial

Highways,” Transportation Research Record 681, Transportation Research Board, 1978.

Harwood, D.W., Multilane Design Alternatives for Improving Suburban Highways,

NCHRP Report 282, Transportation Research Board, 1986.

Heimbach, C.L., W.W. Hunter, and G.C. Chao , et al., “Paved Highway Shoulders and

Accident Experience,” Transportation Engineering Journal, American Society of Civil

Engineers, Vol 4, November 1974.

Hoffman, M.R., “Two-Way Left-Turn Lanes Work!,” Traffic Engineering, Institute of Traffic

Engineers, August 1974.

Nemeth, Z.A., “Two-Way Left-Turn Lanes: State of the Art Overview and Implementation

Guide,” Transportation Research Record 681, Transportation Research Board, 1983.

Pendleton, O.J., Evaluation of Accident Analysis Methodology, Report No. FHWA-RD-96-

039, Federal Highway Administration, November 1996 (particularly Chapter 5, “Illinois

Roadway High Hazard Analysis”).

Rinde, E. A., Accident Rates vs. Shoulder Width, Report No. CA-DOT-TR-3147-1-77-01,

California Department of Transportation, 1977.

Sawhill, R.B., and D.R. Neuzil, “Accidents and Operational Characteristics on Arterial Streets

with Two-Way Left-Turn Lanes,” Highway Research Record 31, Highway Research Board,

1963.

Thakkar, J.S., “A Study of the Effect of Two-Way Left-Turn Lanes on Traffic Accidents,”

Transportation Research Record 960, Transportation Research Board, 1984.

Turner, D.S., D.B. Fambro, and R.O. Rogness, “Effects of Paved Shoulders on Accident Rates

for Rural Texas Highways,” Transportation Research Record 819, Transportation Research

Board, 1981.

Walton, C.M., R.B. Machemehel, T. Horne, and W. Fung, “Accident and Operational guidelines

for Continuous Two-way Left-Turn Median Lanes,” Transportation Research Record 737,

Transportation Research Board, 1979.
117

Yagar, S., and M. Van Aerde, “Reducing Accidents with Two-Way Left Turn Lanes,”

Proceedings, Canadian Multidisciplinary Road Safety Conference III, 1984.

Zegeer, C.V., and F.M. Council, Safety Effectiveness of Highway Design Features:

Volume III—Cross Sections, Report No. FHWA-RD-91-048, Federal Highway

Administration, 1991.

Zegeer, C.V., and F.M. Council, “Safety Relationships Associated with Cross-Sectional

Elements,” Transportation Research Record 1512, Transportation Research Board, 1994. 

Zegeer, C. V., R. C. Deen, and J. G. Mayes, “Effect of Lane and Shoulder Width on Accident

Reduction on Rural, Two-Lane Roads,” Transportation Research Record 806, Transportation

Research Board, 1981.

Zegeer, C. V., D. W. Reinfurt, J. Hummer, L. Herf, and W. Hunter, “Safety Effects of CrossSection Design for Two-Lane Roads,” Transportation Research Record 1195, Transportation

Research Board, 1988.

Zegeer, C. V., R. Stewart, F. Council, and T. R. Neuman, “Accident Relationships of Roadway

Width on Low-Volume Roads,” Transportation Research Record 1445, Transportation

Research Board, 1994.

General or Other Issues

American Association of State Highway and Transportation Officials, Policy on

Geometric Design of Highways and Streets, Washington, DC, 1994.

Choueri, E.M., et al., “Safety Aspects of Individual Design Elements and Their Interactions on

Two-Lane Highways: International Perspective,” Transportation Research Record 1445,

Transportation Research Board, 1994.

Federal Highway Administration, Manual on Uniform Traffic Control Devices for Streets

and Highways, 1988.

Gunnerson, H.E., “Iowa Narrow Bridge Study,” Highway Research Abstracts, Vol. 31, No. 7,

July 1961.

Hanley, K.E., A. R. Gibby, and T. C. Ferrara, The Evaluation of Accident Reduction Factors

on California State Highways, Report No. FHWA/CA-TO-96- , California Department of

Transportation, Federal Highway Administration, December 1996.
118

Mak, K.K., “Effects of Bridge Width on Highway Safety,” in Relationship Between Safety and

Key Highway Elements, State of the Art Report 6, Transportation Research Board, 1987.

Roy Jorgensen and Associates, Evaluation of Criteria for Safety Improvements on the

Highway, Bureau of Public Roads, October 1966.

Smith, S. A., et al., Identification, Quantification, and Structuring of Two-Lane Rural

Highway Safety Problems and Solutions, Volume II—Technical Report, Federal Highway

Administration, July 1981.

Texas Transportation Institute, University of Nebraska-Lincoln, “Improved Procedures for CostEffectiveness Analysis of Roadside Safety Features,” NCHRP Project 22-9, ongoing

Turner, D. S., “Prediction of Bridge Accident Rates,” Journal of Transportation Engineering,

American Society of Civil Engineers, Vol. 110, No. 1, January 1984.

Woods, D.L., et al., “Remedial Safety Treatment of Narrow Bridges,” Traffic Engineering,

March 1976.

Horizontal and Vertical Alignment

Council, F.M., “The Safety Effects of Spiral Transitions on Horizontal Curves,”

Ph.D. Dissertation, North Carolina State University, 1992.

Deacon, J., “Relationship Between Accidents and Horizontal Curvature,” in Designing Safer

Roads: Practices for Resurfacing, Restoration, and Rehabilitation (Appendix D), Special

Report 214, Transportation Research Board, 1987.

Fambro, D.B., K. Fitzpatrick, and R. Koppa, Determination of Stopping Sight Distances,

NCHRP Report 400, Transportation Research Board, 1997.

Glennon, J.C., T.R. Neuman, and J.E. Leisch., Safety and Operational Considerations for

Design of Rural Highway Curves, Report No. FHWA-RD-86/035, Federal Highway

Administration, December 1985.

Hanley, K.E., A.R. Gibby, and T.C. Ferrara, The Evaluation of Accident Reduction Factors

on California State Highways, Report No. FHWA/CA-TO-96, California Department of

Transportation, Federal Highway Administration, December 1996.

Hadi, et al., “Estimating Safety Effects of Cross Section Design for Various Highway Types

Using Negative Binomial Regression,” Transportation Research Record 1500, Transportation

Research Board, 1995.
119

Lamm, R., and E. M. Choueri, “Accident Rates on Curves as Influenced by Highway Design

Elements Based on Investigations in the State of New York,” presented at the 66th annual

meeting of the Transportation Research Board, January 1987.

Li, J., W. Abdelwahad, and G. Brown, “Joint Effect of Access and Geometry on Two-Lane

Rural Highway Safety in British Columbia,” Canadian Journal of Civil Engineering, XXI (vi),

1994.

Lin, F.B., “Flattening Horizontal Curves on Two-Lane Highways,” Transportation Engineering

Journal, American Society of Civil Engineers, March/April 1990.

Matthews, L. R., and J. W. Barnes, “Relation Between Road Environment and Curve

Accidents,” Proceedings, 14th ARRB Conference, Part 4, 1988.

Miaou, S-P., “Development of Adjustment Factors for Single-Vehicle Run-Off-Road Accidents

Rates by Horizontal Curvature and Grade,” Oak Ridge National Laboratory, draft, August 3,

1995.

Miaou, S-P., “Vertical Grade Analysis Summary,” unpublished, May 1998.

Miaou, S-P., Measuring the Goodness-of-Fit of Accident Prediction Models, Report

No. FHWA-RD-96-040, Federal Highway Administration, December 1996.

Neuman, T.R., and J.C. Glennon, “Cost-Effectiveness of Improvements to Stopping-SightDistance Safety Problems,” Transportation Research Record 923, Transportation Research

Board, 1983.

Tom, G.K.J., “Accidents on Spiral Transition Curves,” ITE Journal, Vol. 65(9), September

1995.

Urbanik, T., W. Hinshaw, and D.B. Fambro, “Safety Effects of Limited Sight Distance on Crest

Vertical Curves,” Transportation Research Record 1208, Transportation Research Board,

1989.

Zador, et al., Superelevation and Roadway Geometry: Deficiency at Crash Sites and

Grades, Insurance Institute for Highway Safety, 1995.

Zegeer, C.V., et al., Safety Effectiveness of Highway Design Features: Volume II—

Alignment, Report No. FHWA-RD-91-045, Federal Highway Administration, 1991. 

Zegeer, C. V., J. R. Stewart, F. M. Council, D. W. Reinfurt, and E. Hamilton, “Safety Effects of

Geometric Improvements on Horizontal Curves,” Transportation Research Record 1356,

Transportation Research Board, 1992.
120

Zegeer, C., R. Stewart, D. Reinfurt, F. Council, T. Neuman, E. Hamilton, T. Miller, and W.

Hunter, Cost-Effective Geometric Improvements for Safety Upgrading of Horizontal

Curves, Report No. FHWA-RD-90-021, Federal Highway Administration, October 1991.

Intersections

Agent, K. R., “Development of Warrants for Left-Turn Lanes,” Research Report 526, Bureau

of Highways, Kentucky Department of Transportation, Lexington, KY, July 1989.

Agent, K.R., and R.C. Deen, “Relationship Between Roadway Geometrics and Accidents,”

Transportation Research Record 541, Transportation Research Board, 1975.

Bauer, K. M., and D. W. Harwood, Statistical Models of At-Grade Intersection Accidents,

Report No. FHWA-RD-96-125, Federal Highway Administration, 1996.

Brüde, U., and J. Larsson, Countermeasures at Junctions Taken As Part of the Regional

Road Authorities’ Traffic Safety Program, VTI-Rapport 292, Swedish Road and Traffic

Research Institute, Linkoping, 1985.

Brüde, U., and J. Larsson, Effects on Traffic Safety of Road Improvements at Junctions,

VTI-Rapport 310, Swedish Road and Traffic Institute, Linkoping, Sweden, 1985.

Brüde, U., and J. Larsson, Traffic Safety Effects of LHOVRA Signals, VTI-Meddelande 575,

Swedish Road and Traffic Institute, Linkoping, Sweden, 1988

Brüde, U., and J. Larsson, Conversion from Stop to Yield. Effect on the Number of

Personal Injury Accidents, VTI-Meddelande 695, Swedish Road and Traffic Institute,

Linkoping, Sweden, 1992.

California Department of Public Works, Evaluation of Minor Improvements: Flashing

Beacons, Safety Lighting, Left-Turn Channelization, Traffic Department, California of Public

Works, 1967 (as cited in Reference 27).

Creasey, T., and K. R. Agent, “Development of Accident Reduction Factors,” Research Report

UKTRP-85-6, Kentucky Transportation Research Program, College of Engineering, University

of Kentucky, Lexington, KY, March 1985.

Cribbins, P., and J. Walton, “Traffic Signals and Overhead Flashers at Rural Intersections: Their

Effectiveness in Reducing Accidents,” Transportation Research Record 325, Transportation

Research Board, 1970.
121

Dale, C. W., Cost-Effectiveness of Safety Improvement Projects, Federal Highway

Administration, Washington, DC, May 1973.

Datta, T. K., “Head-On, Left-Turn Accidents at Intersections with Newly Installed Traffic

Signals,” Transportation Research Record 1318, Transportation Research Board, Washington,

DC, 1991.

David, N.A., and J.R. Norman, Motor Vehicle Accidents in Relation to Geometric and

Traffic Features of Highway Intersections: Vol. II—Research Report, Report No. FHWARD-76-129, Federal Highway Administration, 1976.

Elvik, R., Meta Analysis of Intersection Accidents, Institute of Transport Economics, Oslo,

Norway.

Ermer, O. J., “Accident Reduction Factors for Indiana,” Report No. FHWA/IN/JHRP-91/11,

Joint Highway Research Program, Purdue University School of Engineering, Indiana Department

of Transportation, May 1991.

Exnicios, H.F., “Accident Reduction Through Channelization of Complex Intersections,” in

Improved Street Utilization Through Traffic Engineering, Special Report 93, Highway

research Board, 1967.

Glennon, J. C., et al., Technical Guidelines for the Control of Direct Access to Arterial

Highways, Volume II: Detailed Description of Access Control Techniques, Report No.

FHWA-RD-76-87, Federal Highway Administration, 1975.

Glennon, J. C., et al., Evaluation of Techniques for the Control of Direct Access to Arterial

Highways, Report No. FHWA-RD-76-85, Federal Highway Administration, 1976.

Hagenauer, G. F., et al., “Chapter 5. Intersections,” in Synthesis of Safety Research Related

to Traffic Control and Roadway Elements, Volume 1, Report No. FHWA-TS-82-232,

Federal Highway Administration, December 1982.

Hall, D.L., K.C. Sinha, and H. Michael, “Comprehensive Evaluation of Nonsignalized Control at

Low Volume Intersections,” Transportation Research Record 541, Transportation research

Board, 1978.

Hanna, J.T., T.E. Flynn, and L.T. Webb, “Characteristics of Accidents in Rural Municipalities,”

Transportation Research Record 601, Transportation Research Board, 1976.

Harwood, D.W., M.T. Pietrucha, M.D. Wooldridge, R.E. Brydia, and K. Fitzpatrick, Median

Intersection Design, NCHRP Report 375, Transportation Research Board, 1995.
122

Hauer, E., J.C.N. Ng, and J. Lovell, “Estimation of Safety at Signalized Intersections,”

Transportation Research Record, Transportation Research Board, 1988.

Helberg, N., S. Hemdorff, H. Hojgaard, H. Lund, and H. Ludvigsen, Effect of Stop Signs—

Report of an Experiment at 4-Arm Junctions in Rural Areas, Working Report 8/1996,

Danish Council for Road Safety Research, Gentofte, Denmark, 1996.

King, G.F., and R.B. Goldblatt, “Relationship of Accident Patterns to Type of Intersection

Control,” Transportation Research Record 540, Transportation Research Board, 1975.

Kuciemba, S. R., and J. A. Cirillo, Safety Effectiveness of Highway Design Features: 

Intersections, Report No. FHWA-RD-91-048, Federal Highway Administration,

November 1992.

Kulmala, R., Safety at Three- and Four-Arm Junctions: Development and Application of

Accident Prediction Models, VTT Publication 233, Technical Research Centre of Finland,

Espoo, 1995.

Lau, M.Y., and A.D. May, Accident Prediction Model Development: Signalized

Intersections, Research Report No. UCB-ITS-RR-88-7, Institute of Transportation Studies,

University of California-Berkeley, December 1988.

Lau, M.Y., and A.D. May, Accident Prediction Model Development: Unsignalized

Intersections, Research Report No. UCB-ITS-RR-89-12, Institute of Transportation Studies,

University of California-Berkeley, May 1989.

Leckie, D, Traffic Signalization—Before and After Accident Analysis, Ontario Department of

Highways, Canada, 1971.

Lovell, J., and E. Hauer, “The Safety Effect of Conversion to All-Way STOP Control,”

Transportation Research Record 1068, Transportation Research Board, 1986.

Maze, T. H., J. L. Henderson, and S. Sankar, “Impacts on Safety of Left-Turn Treatment at

High-Speed Signalized Intersections,” Iowa Highway Research Board Project HR-347, Iowa

State University, January 1994.

McCoy, P. T., W. J. Hoppe, and D. V. Dvorak, “Benefit-Cost Evaluation of Left-Turn Lanes on

Uncontrolled Approaches of Rural Intersections (Abridgement),” Transportation Research

Record 1026, Transportation Research Board, Washington, D.C., 1985.

McCoy, P. T., and M. S. Malone, “Safety Effects of Left-Turn Lanes on Urban Four-Lane

Roadways,” Transportation Research Record 1239, Transportation Research Board, 1989.
123

McCoy, P. T., E. J. Tripi, and J. A. Bonneson, “Guidelines for Realignment of Skewed

Intersections,” Final Report of Research Project No. RES2(0099) P471, Nebraska Department

of Roads, November 1994.

New York State Department of Transportation, Effect of Signalization, Traffic and Safety

Division, 1982.

Parker, M.R., et al., Geometric Treatments for Reducing Passing Accidents at Rural

Intersections on Two-Lane Rural Highways: Vol I—Final Report, Report No. FHWA/RD83/074, Federal Highway Administration, 1983.

Poch, M., “Geometric and Traffic-Related Elements that Influence Total, Rear-End, and

Approach-Turn Intersection Approach Accidents,” M. Sc. Thesis, University of Washington,

1994.

Poch, M., and F. Mannering, ‘Negative Binomial Analysis of Intersection Accident

Frequencies,” Journal of Transportation Engineering, American Society of Civil Engineers,

Vol. 122, No. 2., March/April 1996.

Schoene, G.W., and H.L. Michael, Effects of a Change in Traffic Control Device on

Intersection Accidents, Purdue University, Engineering Reprint CE 237, 1968.

Short, M.S., G.A. Woelfe, and C. Chang, “Effects of Traffic Signal Installation on Accidents,”

Accident Analysis and Prevention, 14, 1982.

Solomon, D., “Traffic Signals and Accidents in Michigan,” Public Roads, 30, 1959.

Washington, S.P., et al., Evaluation of High-Speed Isolated Intersections in California,

Report No. FHWA/CA/TO/91-2, California Department of Transportation, Federal Highway

Administration, June 1991.

Passing Lanes and Short Four-Lane Sections

Harwood, D.W., and C.J. Hoban, Low-Cost Methods for Improving Traffic Operations on

Two-Lane Roads, Report No. FHWA-IP-87-2, Federal Highway Administration, January

1987. 

Harwood, D. W., and A. D. St. John, Passing Lanes and Other Operational Improvements

on Two-Lane Highways, Report No. FHWA/RD-85/028, Federal Highway Administration,

July 1984.
124

Harwood, D.W., A.D. St. John, and D.L. Warren, “Operational and Safety Effectiveness of

Passing Lanes on Two-Lane Highways.” Transportation Research Record 1026,

Transportation Research Board, 1985.

Nettelblad, P., Traffic Safety Effects of Passing (Climbing) Lanes: An Accident Analysis

Based on Data for 1972-1977, Meddelande TU 1979-5, Swedish National Road

Administration, 1979.

Rinde, E. A., Accident Rates vs. Shoulder Width, Report No. CA-DOT-TR-3147-1-77-01,

California Department of Transportation, 1977.

Taylor, W.C., and M. Jain, “Warrants for Passing Lanes,” Transportation Research Record

1303, Transportation Research Board, 1991.
125

APPENDIX A

EXPERT PANEL MEMBERSHIP

The following individuals participated in the expert panels that established the accident

modification factors:

EXPERT PANEL ON ROADWAY SECTIONS

Forrest M. Council University of North Carolina

Douglas W. Harwood Midwest Research Institute

Ezra Hauer University of Toronto

Timothy R. Neuman CH2M/Hill

Andrew Vogt Pragmatics, Inc.

Charles V. Zegeer University of North Carolina

EXPERT PANEL ON AT-GRADE INTERSECTIONS

Forrest M. Council University of North Carolina

Kay Fitzpatrick Texas A&M University

Douglas W. Harwood Midwest Research Institute

Ezra Hauer University of Toronto

Warren E. Hughes Bellomo-McGee, Inc.

Patrick T. McCoy University of Nebraska

Timothy R. Neuman CH2M/Hill

Andrew Vogt Pragmatics, Inc.

INTERNATIONAL RESOURCE PERSONS

Bruce Corben Monash University (Melbourne, Australia)

Rune Elvik Institute of Transport Economics (Oslo, Norway)

FHWA PARTICIPANTS

Joe G. Bared Federal Highway Administration

Michael Dimiauta A/E Group, Inc.

Ann Do Federal Highway Administration
126

Gregory L. Giering A/E Group, Inc.

Michael S. Griffith Federal Highway Administration

Raymond A. Krammes Federal Highway Administration

Jeffrey F. Paniati Federal Highway Administration

Justin G. True Federal Highway Administration

Davey L. Warren Federal Highway Administration

These expert panels were central to the approach used in developing the accident prediction

algorithm, and the contribution of these panel members to the results presented here is gratefully

acknowledged.
127

APPENDIX B

DEVELOPMENT OF BASE MODELS

The base models for the accident prediction algorithm were developed as part of the

preparation of two FHWA reports, Accident Models for Two-Lane Rural Roads: Segments

and Intersections, and Accident Models for Rural Intersections: 4-Lane by 2-Lane StopControlled and 2-Lane by 2-Lane Signalized.

(3,5) This appendix describes the data base

development, the base model for roadway segments, and the base models for at-grade

intersections.

Data Base Development

The base models were developed with geometric design, traffic control, traffic volume, and

accident data on roadway sections and intersections on rural two-lane highways in California,

Michigan, Minnesota, and Washington. These data were obtained from the FHWA Highway

Safety Information System (HSIS). The geometric design data in the HSIS files, from databases

maintained by the States identified above, were supplemented with additional data obtained by

Vogt and Bared from field measurements and photolog review. These efforts are described more

fully in two FHWA reports and a published paper.(3,4,5)

Base Model for Roadway Segments

In the modeling of roadway segment accidents, the dependent variable included five years of

accident data (1985-1989) for 619 rural two-lane roadway segments in Minnesota and 3 years

of accident data (1993-1995) for 712 roadway segments in Washington. The model

development excluded roadway segments within 76 m (250 ft) of an at-grade intersection and

excluded the (relatively few) accidents that occurred more than 76 m (250 ft) from an

intersections but were identified by the investigating officer as related to an intersection. 

The independent variables representing geometric, design traffic control. and traffic volume

used in modeling included:

• Exposure (million vehicle-miles of travel).

• State in which the roadway section is located (Minnesota/Washington).

• Lane width.

• Shoulder width.

• Roadside hazard rating.

• Driveway density.
128

Nbr ' EXPO exp(0.6409 % 0.1388STATE & 0.0846LW & 0.0591SW % 0.0668RHR % 0.0084DD)

(ÓWHiexp(0.0450DEGi)) (ÓWViexp (0.4652Vi)) (ÓWGiexp(0.1048GRi))

(42)

• Horizontal curvature.

• Grade rate for crest vertical curves.

• Percent grade for straight grades.

All of these independent variables were found to have a statistically significant relationship to

roadway section accidents.

The base model for roadway segments was developed from the HSIS roadway segment data

for rural two-lane highways in Minnesota and Washington. The base model is presented below:

where: Nbr = predicted number of total accidents per year on a particular roadway

segment;

EXPO = exposure in million vehicle-miles of travel per year = (ADT)(365)(L)(10-6);

ADT = average daily traffic volume (veh/day) on roadway segment;

L = length of roadway segment (mi);

STATE = location of roadway segment (0 in Minnesota, 1 in Washington);

LW = lane width (ft); average lane width if the two directions of travel differ;

SW = shoulder width (ft); average shoulder width if the two directions of travel

differ;

RHR = roadside hazard rating; this measure takes integer values from 1 to 7 and

represents the average level of hazard in the roadside environment along the

roadway segment. (For the development of the roadside hazard rating, see

Zegeer et al.; for definitions of individual rating levels, see Appendix D.);(6)

DD = driveway density (driveways per mi) on the roadway segment;

WHi = weight factor for the ith horizontal curve in the roadway segment; the

proportion of the total roadway segment length represented by the portion of

the ith horizontal curve that lies within the segment. (The weights, WHi, must

sum to 1.0.);
129

DEGi = degree of curvature for the ith horizontal curve in the roadway segment

(degrees per 100 ft);

WVj = weight factor for the jth crest vertical curve in the roadway segment; the

proportion of the total roadway segment length represented by the portion of

the jth crest vertical curve that lies within the segment. (The weights, WVj,

must sum to 1.0.);

Vj = crest vertical curve grade rate for the jth crest vertical curve within the

roadway segment in percent change in grade per 31 m (100 ft) = |gj2- gj1|/lj;

gjl, gj2 = roadway grades at the beginning and end of the jth vertical curve (percent);

lj = length of the jth vertical curve (in hundreds of feet);

WGk = weight factor for the kth straight grade segment; the proportion of the total

roadway segment length represented by the portion of the kth straight grade

segment that lies within the segment. (The weights, WGk, must sum to 1.0.);

and

GRk = absolute value of grade for the kth straight grade on the segment (percent).

The model was developed with extended negative binomial regression analysis. This

extension of the standard negative binomial regression analysis technique was devised by

Miaou.(3) In negative binomial models for roadway segments, the mean number of accidents in a

specified time period is typically represented in the form:

EXPO exp(aX + bY + ...)

which is equivalent to:

EXPO exp(aX) exp(bY) ...

where EXPO is a measure of exposure, X and Y are measures of roadway segment

characteristics, and a and b are appropriate regression coefficients. 

The extended negative binomial regression analysis technique devised by Miaou replaces

some of the factors exp(aX) by expressions of the form:
130

R

2

k

' 1 &

k

kmax

(43)

w1 exp(aX1) + w2 exp(aX2) + ... +wm exp (aXm)

where X1, X2, ..., Xm are local variables along the roadway segment, characterizing for

subsegments what X attempts to measure for the entire segment. For example, X1, ..., Xm might

represent the degree of curvature for individual horizontal curves while X is the average degree of

curvature for the roadway segment as a whole. Thus, X is a composite variable, while the Xis

represent variation within the segment. Such variation occurs for many variables used in accident

modeling of roadway segments, even on supposedly homogeneous segments. Notable examples

are degree of curvature, grade, and change of grade per unit length. The variable wi is the

proportion of the segment length to which the value Xi applies. (The default value for Xi is

assumed to be zero if there is no horizontal or vertical curve or if the grade is level, and an

artificial subsegment with this value is added, if necessary, so that the weights, wi, always sum to

1.0.)(3)

The extended negative binomial regression model decomposes the roadway segment into

subsegments within which the roadway characteristic measured by X is constant. If this is done

for two or more variables (e.g., X, Y, ...), the method assumes that the variables are independent

of one another, so that the value Yj occurs with a particular value of Xi. Although such

independence cannot be assured, the extended negative binomial regression model attempts to

capture the effect of variation within a segment in an additive manner consistent with the basic

form of the model.(3)

Table 29 summarizes the model presented in equation (42) with the coefficient, standard

deviation, and significance level (p) for each independent variable and the overdispersion

parameter (k). The goodness-of-fit measures for the roadway segment base model include R2,

the traditional measure of the percentage of variation in accident frequency explained by the

independent variables in the model, as well as Rk

2

, defined as:

Where:

k = the overdispersion parameter for the regression model; and

kmax = the overdispersion parameter in a model with no covariables (the so-called

“zero model”).
131

This latter measure of goodness of fit, Rk

2

, has been proposed by Miaou.(9) For the roadway

segment base model in equation (42), the values of the goodness-of-fit measures are R2 = 0.6547

and Rk

2

 = 0.8291.
132

Table 29. Model Parameters and Goodness of Fit for Equation (42).

Independent Variable

Parameter Intercept State LW SW RHR DD DEG V GR

Overdispersionparameter (k)

Coefficient 0.6409 0.1388 -0.0846 -0.0591 0.0668 0.0084 0.0450 0.4652 0.1048 0.3056Standard deviation 0.5008 0.0659 0.0425 0.0114 0.0211 0.0026 0.0078 0.1260 0.0287 0.0331Significance level (p) 0.2006 0.0351 0.0465 0.0001 0.0015 0.0011 0.0001 0.0002 0.0003 0.0001Note: The values of the goodness-of-fit measures are R2 = 0.6547 and Rk2 = 0.8291
133

Table 30 presents descriptive statistics for the variables in the roadway segment model.

Table 30. Descriptive Statistics for Roadway Segments Used in Modeling.

Variable Mean

Standard

deviation Minimum

25th

percentile Median

75th

percentile Maximum

MINNESOTA (619 roadway segments)

ADT (veh/day) 2,402 1,937 208 1,176 1,866 2,900 15,162

L (mi) 1.14 1.30 0.10 0.26 0.66 1.50 8.24

LW (ft) 11.54 0.67 10.00 11.00 12.00 12.00 12.00

SW (ft) 7.08 2.44 0.00 6.00 8.00 8.00 12.00

RHR 2.14 0.98 1.00 1.00 2.00 3.00 6.00

DD (mi!1) 6.58 10.25 0.00 0.89 3.73 7.68 100.00

DEG (degrees/100 ft) 0.51 0.95 0.00 0.00 0.08 0.62 7.50

V (percent/100 ft) 0.066 0.092 0.00 0.007 0.037 0.086 0.888

GR (percent) 0.38 0.52 0.00 0.10 0.24 0.45 4.46

WASHINGTON (712 roadway segments)

ADT (veh/day) 3,352 3,199 159 1,261 2,239 4,455 17,766

L (mi) 0.75 0.83 0.10 0.27 0.554 0.948 13.23

LW (ft) 11.37 0.56 9.00 11.00 11.00 12.00 12.00

SW (ft) 5.01 2.35 0.00 3.00 5.00 7.00 10.00

RHR 3.67 1.57 1.00 2.00 3.00 6.00 7.00

DD (mi!1) 10.12 12.41 0.00 2.07 6.12 13.61 85.07

DEG (degrees/100 ft) 1.03 2.13 0.00 0.00 0.32 1.31 30.55

V (percent/100 ft 0.068 0.127 0.000 0.000 0.026 0.083 1.997

GR (percent) 0.92 1.17 0.00 0.20 0.49 1.13 6.92

Conversion: 1 ft = 0.305 m; 1 mi = 1.61 km

The exposure variable, EXPO, was treated as a scale factor in the development of the

model. Therefore, even though multiple years of accident data were used in developing the

model, the expected annual accident frequency can be determined from the model if EXPO is

determined as (ADT)(365)(L)(10-6).
134

Other variables investigated in the development of the roadway segment base models

included posted speed limit, truck percentage, and intersection density (i.e., number of

intersections per mile). Posted speed limit was found to be negatively correlated with accident

frequency, while truck percentage and intersection density were positively correlated with

accident frequency. However, none of these three variables was statistically significant in

regression models that included the variables listed above. Grade change per unit length of

roadway was also considered for sag vertical curves and for all vertical curves (i.e. both sags and

crests), but the version of the variable for crest vertical curves (Vj) had the greatest statistical

significance and was, therefore, retained in the final model. Two weather-related variables were

also investigated in the modeling for the Minnesota roadway segments—number of rain days and

number of snow days per year. These variables were based on the climate district in which each

roadway segment was located. These variables were found to be negatively correlated with

accident frequency. Because these variables were not sufficiently local (i.e., they represented

climate districts rather than the climate of individual roadway segments) and because they were

only marginally statistically significant, they were not collected for the Washington data and were

omitted from the final model.

Base Models for At-Grade Intersections

Base models have been developed for three types of at-grade intersections on rural two-lane

highways. These are:

• Three-leg intersections with STOP control on the minor-road approach.

• Four-leg intersections with STOP control on the minor-road approach.

• Four-leg signalized intersections.

Models for each of these intersection types were developed using three different modeling

approaches, described below. The available database used in modeling included 5 years of

accident data (1985-89) for selected STOP-controlled intersections in Minnesota and 3 years of

accident data (1993-1995) for selected signalized intersections in California and Michigan. 

Following the description of the modeling approaches, the models developed for each

intersection type are presented.

The available accident data for at-grade intersections generally included accidents within 76

m (250 ft) of the intersection on the major road, typically a State highway. For minor roads that

are also State highways, data were generally available for all accidents that occur within 76 m

(250 ft) of the intersection. For minor roads that are not State highways, accidents on the minor

road that are classified as intersection-related are typically assigned the milepost of the

intersection on the major road and are therefore included in the available data. In Michigan, all

minor-road accidents within 31 m (100 ft) of the intersection and, in California, all accidents

within 76 m (250 ft) of the intersection are included in the available data. In Minnesota, all
135

accidents that occurred on minor-road approaches and were identified as related to the

intersection are included in the available data.

Modeling Approaches

Three different modeling approaches were employed to develop candidate base models for

each intersection type. The approaches are:

• The dependent variable (number of accidents in a 5-year period) included only

accidents that occurred within the curbline limits of a particular intersection or within 76

m (250 ft) of that intersection and were identified by the investigating officer as being

related to the intersection.

• The dependent variable included only accidents that occurred within the curbline limits

of a particular intersection or within 76 m (250 ft) of that intersection and were of

accident types that are generally related to intersection operations.

• The dependent variable included all accidents that occurred within the curbline limits of

a particular intersection or within 76 m (250 ft) of that intersection, whether those

accidents were related to the intersection or not. However, the modeling technique

used an offset to the observed accident frequency applied iteratively to account for the

frequency of roadway section accidents that would be expected to occur on 152 m

(500 ft) of the major roadway (76 m or 250 ft on either side of the intersection), as

predicted by equation (37).

Each of these modeling techniques is described in greater detail in the following discussion. 

A fourth modeling technique that was considered but not used is explained at the end of this

appendix. 

The research reports by Vogt and Bared from the FHWA project that developed the base

models and the related paper include only the models based on accidents that were identified by

the investigating officer as being intersection related.(3,4,5) The models based on accident types

that are generally related to intersection operations and on the iterative offset approach were

developed as part of the same project but are documented only in this report.

Limitation of the Analysis to Intersection-Related Accidents

The first modeling approach used to develop base models for at-grade intersections was to

limit the dependent variable to those accidents that occurred within the curbline limits of the

intersection or within 76 m (250 ft) of the intersection on any approach and were identified by the
136

investigating officer as related to the intersection. This definition is generally reasonable because

the vast majority of intersection-related accidents on a rural two-lane highway would be expected

to occur within 76 m (250 ft) of the intersection to which they are related. For example, traffic

queues that extend more than 76 m (250 ft) from an intersection and which might lead to rear-end

collisions remote from the intersection are much less common in rural than in urban areas.

Not all accidents within 76 m (250 ft) of an intersection would be expected to be related to

the operation of that intersection. For example, collisions between motor vehicles and animals

and collisions involving vehicles entering or leaving a driveway may occur within 76 m (250 ft) of

an intersection but may have no particular relationship to the presence of that intersection. In

other words, some accidents would be expected to occur on the roadway near the intersection

whether the intersection were present or not. the first approach to modeling distinguishes

between intersection-related and non-intersection-related accidents based on judgments made by

the investigating officer or by an accident coder and recorded in the State’s computerized

accident record system. Nominally, such judgments make exactly the distinction that is desired,

but there is concern that investigating officers and accident coders may not make such judgments

consistently.

Once the dependent variable was defined, as described above, predictive models were

developed using extended negative binomial regression in a manner similar to the development of

the roadway segment model in equation (42).

Limitation of the Analysis to Selected Accident Types

The second modeling approach used was to limit the dependent variable to accidents that

occurred within the curbline limits of the intersection and are of accident types that are generally

related to intersection operations. The accident types classified as related to the intersection

were:

• Rear-end collisions.

• Right-angle collisions.

• Left-turn collisions.

• Right-turn collisions.

• Sideswipe, same-direction collisions.

All other accidents types were classified as unrelated to the intersection.

The list of accident types classified as intersection-related and non-intersection-related were

established through two special studies that involved a review of hard copy police accident

reports and classification of accidents as intersection-related or non-intersection-related based on

the judgment of safety experts.
137

Zi' exp(ki% aiX % biY % ...) (44)

This approach to modeling eliminates the concern raised in the first modeling approach about

the potential misjudgment in classifying accidents made by investigating officers or accident

coders. However, the approach based on classification of particular accident types as

intersection-related has a similar concern since not every accident of the types identified above is

actually related to an intersection and some accidents of other types are, in fact, intersectionrelated. For example, the proposed classification scheme based on accident types would classify

all reported right-turn accidents within 76 m (250 ft) of the intersection as intersection-related,

including accidents that are, in fact, related to turning movements at nearby driveways. The

effectiveness of this classification could be improved by an agency by utilizing other fields of the

accident record such as a driveway involvement indicator, if available, in classifying accidents.

Once the dependent variable was defined, as described above, predictive models were

developed using extended negative binomial regression in a manner similar to the development of

the roadway segment model in equation (42).

Use of All Accidents with an Offset for Expected Roadway-Segment Accidents

The third modeling technique uses as the dependent variable all accidents that occur within

the curbline limits of a particular intersection and all accidents that occur within 76 m (250 ft) of

that intersection, regardless of the accident type and regardless of the judgment made by the

investigating officer or accident coder concerning the relationship of the accident to the

intersection. Since this dependent variable includes some accidents that are clearly unrelated to

the intersection, the expected frequency of roadway segment accidents, as predicted by equation

(42), is used as an offset factor in the model development. This modeling used an iterative

technique that makes successive approximations to the model coefficients.

The iterative offset modeling approach was performed as follows. Let:

denote an equation with integer subscript i estimating the mean number of intersection-related

accidents per unit time, Zi, in terms of intersection characteristics (X, Y, ...). Let N denote the

mean number of non-intersection-related accidents on a 152 m (500 ft) section of roadway

containing the intersection (i.e., 76 m or 250 ft on either side of the intersection), as predicted by

a roadway section model such as equation (42), as applied to Minnesota (STATE=0).

Let A be the mean number of accidents of all kinds per unit time within 76 m (250 ft) of the

intersection. Let:
138

OFFSETi' ln

N % Zi

Zi

(45)

Zi%1' exp(ki%1% ai%1X % bi%1Y % ...) (46)

A ' exp(OFFSETi% ki%1% ai%1X % bi%1Y % ...) (47)

A ' exp(OFFSETi) exp(ki%1% ai%1X % bi%1Y) (48)

A ' exp(OFFSETi) Zi%1 (49)

A '

N % Zi

Zi

Zi%1 (50)

The modeling approach used is iterative. Equation (44) implies that the following relationship

must be valid for iteration i+1:

A negative binomial model for A is sought of the form:

which can be expressed as:

The initial negative binomial model for intersection-related accidents is represented by an

equation for Z0 in the form of equation (44). The offset technique represented by equations (47)

through (50) is applied repeatedly to obtain a sequence of new models for Z1, Z2, Z3, ... . This

process is continued until the coefficients kn, an, bn, ... cease to change appreciably; i.e., until Zn+1
139

Z ' exp(kn% anX % bnY % ...) (51)

is approximately the same as Zn. The appropriate model for the mean number of intersectionrelated accidents is then:

In principle, A = N + Z, or Z = A ! N. Thus from, the model for roadway segment accidents

used to estimate N and the initial model for intersection-related accidents Z0, the offset technique

yields a final model for intersection-related accidents, Z. This final model depends on the choice

of the model for N, but should not generally depend on the initial model for Z0. Thus, initial

estimates or starting values of the regression coefficients k0, a0, b0, ... can be selected through

engineering judgment or an alternative preliminary model.

Models Developed

The following discussion presents the candidate base models developed for three-leg STOPcontrolled intersections, four-leg STOP-controlled intersections, and four-leg signalized

intersections.

Three-Leg STOP-Controlled Intersections

Candidate base models were developed for three-leg intersections with STOP control on the

minor-road approach. The dependent variables used in these models have been described above

in the discussion of modeling techniques. All of the models for three-leg STOP-controlled

intersections used a data sets of approximately 382 intersections in Minnesota including 5 years of

accident data (1985-1989). There were minor variations in sample size from one model to the

next because of small amounts of missing data. The candidate independent variables considered

in predicting accidents at three-leg STOP-controlled intersections were:

• Average daily traffic volume on the major road.

• Average daily traffic volume on the minor road.

• Intersection skew angle.

• Posted speed limit on major road.

• Presence of a right-turn lane on the major road.

• Number of driveways on the major road within 76 m (250 ft) of the intersection.

• Grade rate for crest vertical curves on the major road within 76 m (250 ft) of the

intersection.

• Radius of horizontal curves on the major road within 76 m (250 ft) of the intersection.

• Roadside hazard rating along the major road within 76 m (250 ft) of the intersection.
140

Nbi ' exp(&12.99 % 0.805ln ADT1% 0.504ln ADT2% 0.290VCI % 0.034HI

% 0.029SPDI % 0.173RHRI % 0.27RT % 0.0045SKEW3)

(52)

Table 31 presents descriptive statistics for these variables.

A candidate model for three-leg STOP-controlled intersections developed using the

intersection-related accident definition based on the investigating officer’s assessment of each

accident is:

Table 31. Descriptive Statistics for 382 Three-Leg STOP Controlled Intersections in

Minnesota Used in Modeling.

Variable Mean

Standard

deviation Minimum

25th

percentile Median

75th

percentile Maximum

ADT1 (veh/day) 3,718 3,725 201 1,239 2,333 4,627 19,413

ADT2 (veh/day) 408 531 5 103 237 478 4206

ln ADT1 7.81 0.91 5.30 7.12 7.75 8.44 9.87

ln ADT2 5.40 1.14 1.51 4.64 5.47 6.17 8.34

VCI (percent/100 ft) 0.142 0.300 0 0 0 0 4.39

HI (degrees/100 ft) 1.22 2.52 0 0 0 2 29

SPDI (mph) 52.8 4.6 22.5 52.5 55 55 55

RHRI 2.10 0.88 1 1 2 3 5

RT 0.42 0.49 0 0 0 1 1

SKEW3 (degrees) -0.67 24.98 -90 0 0 0 85.1

ND1 1.24 1.44 0 0 1 2 9

Conversion: 1 ft = 0.305 m

where: Nbi = predicted number of total accidents per year at a particular intersection and

within 76 m (250 ft) in either direction along the major road;

ADT1 = average daily traffic volume (veh/day) on the major road; if the ADTs differ

between the major-road legs, they should be averaged;

ADT2 = average daily traffic volume (veh/day) on the minor road;

VI = crest vertical curve grade rate on the major road within 76 m (250 ft) of the

intersection = (1/m) ÓVi for all crest vertical curves wholly or partly within 76

m (250 ft) of the intersection;
141

m = number of crest vertical curves wholly or partly within 76 m (250 ft) of the

intersection;

HI = horizontal curvature change rate on the major road within 76 m (250 ft) of the

intersection = (1/n) ÓDEGi for all horizontal curves wholly or partly within 76

m (250 ft) of the intersection;

n = number of horizontal curves within 76 m (250 ft) of the intersection;

SPDI = posted speed limit on the major road (mi/h);

RHRI = roadside hazard rating within 76 m (250 ft) of the intersection on the major

road [see description of the variable RHR in Equation (42)];

RT = presence of right-turn lane on the major road (0 = no right-turn lane present;

1 = right-turn lane present); and

SKEW3 = intersection angle (degrees) minus 90 for the angle between the major-road 

leg in the direction of increasing stations and a leg to the right; 90 minus

intersection angle (degrees) for the angle between the major-road leg in the

direction of increasing stations and a leg to the left.

Equation (52) includes all of the candidate independent variables except the number of

driveways which was dropped because its coefficient was not statistically significant (p=0.5405)

and because coefficient had a negative sign, which is opposite to the direction expected. Table

32 summarizes the model parameters and goodness of fit for the model in equation (52). 

Goodness of fit for models like equation (52) is measured by R2 and Rk

2

, as noted earlier for

equation (42), as well as by RPD

2

. This last goodness-of-fit measure, RPD

2

, has been proposed by

Fridstrøm for use with negative binomial models.(43) A description of this goodness-of-fit measure

is also provided by Vogt and Bared.(3)

Table 32. Model Parameters and Goodness of Fit for Equation (52).

Independent variable

Overdispersion

parameter

Parameter Intercept ln ADT (k) 1ln ADT2 VCI HI SPDI RHRI RT SKEW3

Coefficient !12.99 0.805 0.504 0.290 0.034 0.029 0.173 0.27 0.0045 0.481

Standard deviation 1.15 0.064 0.071 0.294 0.033 0.018 0.068 0.140 0.0032 0.100

Significance level 0.000 0.0001 0.0001 0.323 0.3004 0.107 0.0108 0.0561 0.1578 0.0001

Note: The values of the goodness-of-fit measures are R2 = 0.4409, Rk

2

 = 0.7805, and RPD

2

 = 0.6279
142

Nbi ' exp(&11.28 % 0.79ln ADT1% 0.49ln ADT2% 0.19RHRI % 0.28RT)

(53)

Nbi ' exp(&12.82 % 1.001ln ADT1% 0.406ln ADT2% 0.22RHRI

% 0.33RT % 0.0040SKEW3)

(54)

The model in equation (52) was reevaluated including only those variables that were

statistically significant with a significance level (p) of 0.150 or less. This model is presented

below:

Table 33 summarizes the model parameters and goodness of fit for the model in equation

(53). 

A candidate model developed for three-leg STOP-controlled intersections using only those

accident types generally related to intersection operations is:

Equation (54) includes all of the candidate independent variables except speed limit

(p=0.41), number of driveways (p=0.56), horizontal curvature (p=0.62), and vertical curvature

(p=0.40) which were dropped because their coefficients were not statistically significant. Table

34 summarizes the model parameters and goodness of fit for the model in equation (54). 

Table 33. Model Parameters and Goodness of Fit for Equation (53).

Independent variable Overdispersion

Parameter Intercept ln ADT parameter (k) 1ln ADT2 RHRI RT

Coefficient !11.28 0.79 0.49 0.19 0.28 0.54

Standard deviation 0.063 0.062 0.068 0.067 0.14 0.102

Significance level (p) 0.0001 0.0001 0.0001 0.0035 0.0402 0.0001

Note: The values of the goodness-of-fit measures are R2 = 0.3955, Rk

2

 = 0.7546, and RPD

2

 = 0.6109.
143

Nbi ' exp(&12.40 % 0.74ln ADT1% 0.53ln ADT2% 0.36VCI % 0.028SPDI

% 0.14RHRI % 0.0063SKEW3)

(56)

Nbi ' exp(&13.01 % 1.015ln ADT1% 0.42ln ADT2% 0.23RHRI % 0.29RT) (55)

Table 34. Model Parameters and Goodness of Fit for Equation (54).

Independent variable Overdispersion

Parameter Intercept ln ADT parameter (k) 1ln ADT2 RHRI RT SKEW3

Coefficient !12.82 1.001 0.406 0.22 0.33 0.0040 0.46

Standard deviation 0.73 0.072 0.073 0.073 0.15 0.0029 0.106

Significance level 0.0001 0.0001 0.0001 0.0024 0.032 0.17 0.0001

Note: The values of the goodness-of-fit measures are R2 = 0.4181, Rk

2

 = 0.8070, and RPD

2

 = 0.7228.

The model in equation (54) was reevaluated including only those variables that were

statistically significant with a significance level of 0.150 or less. This model is presented below:

Table 35 summarizes the model parameters and goodness of fit for the model in

equation (55).

Table 35. Model Parameters and Goodness of Fit for Equation (55).

Independent variable Overdispersion

Parameter Intercept ln ADT parameter (k) 1ln ADT2 RHRI RT

Coefficient !13.01 1.015 0.42 0.23 0.29 0.49

Standard deviation 0.72 0.072 0.071 0.073 0.14 0.108

Significance level (p) 0.0001 0.0001 0.0001 0.0020 0.041 0.0001

Note: The values of the goodness-of-fit measures are R2 = 0.4008, Rk

2

 = 0.7953, and RPD

2

 = 0.7162

A candidate model for three-leg STOP-controlled intersections developed with the iterative

offset technique is:
144

Nbi ' exp(&12.25 % 0.75ln ADT1% 0.52ln ADT2% 0.026SPDI

% 0.15RHRI % 0.0059SKEW3)

(57)

Equation (56) includes all of the candidate independent variables except number of

driveways (p=0.35), presence of right-turn lane (p=0.47), and horizontal curvature (p=0.50)

which were dropped because their coefficients were not statistically significant. Table 36

summarizes the model parameters and goodness of fit for the model in equation (56). It should

be noted that the goodness-of-fit measure, RPD

2

, is not directly applicable to models developed

with the iterative offset technique and, therefore, is not presented in Table 36. Moreover, R2

must be interpreted with caution since it measures the goodness-of-fit for the combined roadway

segment and intersection model for all accidents with 76 m (250 ft) of the intersection. Since

most of these accidents are intersection-related, R2 at least roughly measures the goodness of fit

for intersection-related accidents

Table 36. Model Parameters and Goodness of Fit for Equation (56).

Independent variable

Overdispersion

parameter

Parameter (k) Intercept ln ADT1ln ADT2 VCI SPDI RHRI SKEW3

Coefficient !12.40 0.74 0.53 0.36 0.028 0.14 0.0063 0.52

Standard deviation 0.93 0.063 0.064 0.28 0.013 0.070 0.0029 0.090

Significance level (p) 0.0001 0.0001 0.0001 0.20 0.033 0.053 0.032 0.0001

Note: The values of the goodness-of-fit measures are R2 = 0.4163, and Rk

2

 = 0.5809.

The model in equation (56) was reevaluated including only those variables that were

statistically significant with a significance level of 0.150 or less. This model is presented below:

Table 37 summarizes the model parameters and goodness of fit for the model in equation

(57).

A decision was made to use in the accident prediction algorithm the models containing only

those independent variables that are statistically significant at a significance level of 0.15 or less,

like equations (53), (55), and (57).
145

Nbi ' exp(&10.43 % 0.603ln ADT1% 0.609ln ADT2% 0.29VCI

% 0.045HI % 0.019SPDI % 0.12ND1& .0049SKEW4)

(58)

Table 37. Model Parameters and Goodness of Fit for Equation (57).

Independent Variable

Overdispersion

parameter

Parameter Intercept ln ADT (k) 1ln ADT2 SPDI RHRI SKEW3

Coefficient !12.25 0.75 0.52 0.026 0.15 0.0059 0.52

Standard deviation 0.92 0.062 0.064 0.013 0.065 0.0029 0.089

Significance level (p) 0.0001 0.0001 0.0001 0.046 0.019 0.043 0.0001

Note: The value of goodness-of-fit measures are R2 = 0.4069 and Rk

2

 = 0.5765.

 

Four-Leg STOP-Controlled Intersections

Candidate base models were developed for four-leg intersections with STOP control on the

minor-road approaches. The dependent variables used in these models have been described

above in the discussion of modeling techniques. All of the models for four-leg STOP-controlled

intersections used a data sets of approximately 324 intersections in Minnesota including 5 years of

accident data (1985-1989). There were minor variations in sample size from one model to the

next because of small amounts of missing data. The candidate independent variables considered

in predicting accidents at four-leg STOP-controlled intersections were:

• Average daily traffic volume on the major road.

• Average daily traffic volume on the minor road.

• Intersection skew angle.

• Posted speed limit on major road.

• Presence of right-turn lanes on the major road.

• Number of driveways within 76 m (250 ft) of the intersection.

• Grade rate for crest vertical curves within 76 m (250 ft) of the intersection.

• Radius of horizontal curves within 76 m (250 ft) of the intersection.

• Roadside hazard rating.

Table 38 presents descriptive statistics for these variables.

A candidate model for four-leg STOP-controlled intersections developed using the

intersection-related accident definition is:
146

where:

ND1 = number of driveways on the major road with 76 m (250 ft) of the intersection;

and

SKEW4 = intersection angle (degrees) expressed as one-half of the angle to the right

minus one-half of the angle to the left for the angles between the major-road

leg in the direction of increasing stations and the right and left legs,

respectively.

Table 38. Descriptive Statistics for 324 Four-Leg 

STOP-Controlled Intersections in Minnesota.

Variable Mean

Standard

deviation Minimum

25th

percentile Median

75th

percentile Maximum

ADT1 (veh/day) 2,216 1,966 174 972 1,739 2,611 14,611

ADT2 (veh/day) 304 383 7 105 191 365 3,414

ln ADT1 7.42 0.75 5.16 6.88 7.46 7.87 9.59

ln ADT2 5.25 0.97 1.93 4.65 5.25 5.90 8.14

VCI (percent/100 ft) 0.146 0.280 0 0 0.023 0.207 2.942

HI (degrees/100 ft) 0.46 1.08 0 0 0 0.25 8.00

SPDI (mi/h) 54.0 3.3 30 55 55 55 55

ND1 0.61 1.14 0 0 0 1 6

SKEW4 (degrees) !0.14 18.34 -60.00 -0.44 0.00 0.58 75.00

Conversion: 1 ft = 0.305 m; 1 mi = 1.61 km

Equation (58) includes all of the candidate independent variables except roadside hazard

rating (p=0.28) and presence of right-turn lanes (p=0.66) which were not statistically significant. 

Table 39 summarizes the model parameters and goodness of fit for the model in equation (58).

Table 39. Model Parameters and Goodness of Fit for Equation (58).

Independent variable

Overdispersion

parameter

Parameter Intercept lnADT (k) 1lnADT2 VCI HI SPDI ND1 SKEW4

Coefficient !10.43 0.603 0.609 0.29 0.045 0.019 0.12 !0.0049 0.205

Standard deviation 1.32 0.084 0.069 0.26 0.047 0.018 0.05 0.0033 0.065

Significance level 0.000 0.000 0.000 0.26 0.34 0.29 0.01 0.13 0.0016

Note: The values of goodness-of-fit are R2 = 0.5944, Rk

2

 = 0.8336, and RPD

2

 = 0.7364
147

Nbi ' exp(&9.34 % 0.60ln ADT1% 0.61ln ADT2% 0.13ND1& .0054SKEW4)

(59)

Nbi ' exp(&9.40 % 0.55ln ADT1% 0.65ln ADT2% 0.31VCI

% 0.14ND1& 0.0049SKEW4)

(60)

The model in equation (58) was reevaluated including only those variables that were

statistically significant with a significance level of 0.150 or less. This model is presented below:

Table 40 summarizes the model parameters and goodness of fit for the model in

equation (59).

Table 40. Model Parameters and Goodness of Fit for Equation (59).

Independent variable Overdispersion

Parameter Intercept ln ADT parameter (k) 1ln ADT2 ND1 SKEW4

Coefficient !9.34 0.601 0.61 0.13 !0.0054 0.24

Standard deviation 0.72 0.078 0.069 0.039 0.0034 0.071

Significance level (p) 0.0001 0.0001 0.0001 0.0009 0.108 0.0008

Note: The values of goodness-of-fit measures are R2 = 0.5662, Rk

2

 = 0.8081, and RPD

2

 = 0.7326

A candidate model developed using only those accident types generally related to

intersection operations is:

Equation (60) includes all of the candidate independent variables except speed limit (p=0.25

to 0.50), presence of right-turn lane (p=0.59), and horizontal curvature (p=0.48) which were not

statistically significant, and roadside hazard rating (p=0.12) which was marginally statistically

significant but whose coefficient was negative, which is opposite to the direction expected. Table

41 summarizes the model parameters and goodness of fit for the model in equation (60).

The model in equation (60) was reevaluated including only those variables that were

statistically significant with a significance level of 0.150 or less. This model is presented below:

Nbi ' exp(&9.30 % 0.53ln ADT1% 0.67ln ADT2

% 0.15ND1& .0057SKEW4)

(61)
148

Nbi ' exp(&11.25 % 0.69ln ADT1% 0.56ln ADT2% 0.033SPDI

% 0.11ND1& 0.21RT & .0059SKEW4)

(62)

Table 41. Model Parameters and Goodness of Fit for Equation (60).

Independent variable Overdispersion

Parameter Intercept ln ADT parameter (k) 1ln ADT2 VCI ND1 SKEW4

Coefficient !9.40 0.55 0.65 0.31 0.14 !0.0049 0.253

Standard deviation 0.77 0.085 0.073 0.27 0.05 0.0038 0.079

Significance level (p) 0.0001 0.0001 0.0001 0.25 0.004 0.1997 0.0014

Note: The values of goodness-of-fit measures are R2 = 0.5495, Rk

2

 = 0.8131, and RPD

2

 = 0.7183

Table 42. Model Parameters and Goodness of Fit for Equation (61).

Independent variable

Over-dispersion

Parameter Intercept ln ADT parameter (k) 1ln ADT2 ND1 SKEW4

Coefficient !9.30 0.53 0.67 0.15 !0.0057 0.293

Standard deviation 0.78 0.085 0.074 0.043 0.0039 0.086

Significance level (p) 0.0001 0.0001 0.0001 0.0004 0.14 0.0007

Note: The values of goodness-of-fit measures are R2 = 0.5020, Rk

2

 = 0.7835, and RPD

2

 = 0.7047

Table 42 summarizes the model parameters and goodness of fit for the model in equation

(61).

A candidate model for four-leg STOP-controlled intersections developed with the

iterative offset technique is:

Equation (62) includes all of the candidate independent variables except horizontal

curvature (p=0.51) and vertical curvature (p=0.30 to 0.41) which were not statistically significant,

and roadside hazard rating (p=0.11) which was marginally statistically significant but whose

coefficient was negative, which is opposite to the direction expected. Table 43 summarizes the

model parameters and goodness of fit for the model in Equation (62). No value of RPD

2

 is
149

computed for equation (62) because this goodness-of-fit measure is not directly applicable to

iterative offset technique.

Table 43. Model Parameters and Goodness of Fit for Equation (62).

Independent variable Overdispersion

Parameter Intercept ln ADT parameter (k) 1ln ADT2 SPDI ND1 RT SKEW4

Coefficient !11.25 0.69 0.56 0.033 0.107 !0.21 !0.0059 0.203

Standard deviation 1.59 0.080 0.073 0.021 0.054 0.12 0.0031 0.065

Significance level 0.0001 0.0001 0.0001 0.11 0.048 0.082 0.057 0.0018

Note: The values of goodness-of-fit measures are R2 = 0.5563, and Rk

2

 = 0.7984

All of the variables in equation (62) are statistically significant at the 0.15 significance level. 

Therefore, no alternative version of equation (62) was developed.

Four-Leg Signalized Intersections

Candidate base models were developed for four-leg signalized intersections. The dependent

variables used in these models were those described above for the intersection-related and

selected-accident-type modeling techniques. All of the models for four-leg signalized

intersections used a data sets of 49 intersections, 18 in California and 31 in Michigan, with 3

years of accident data (1993-1995) for each intersection. The candidate independent variables

considered in predicting accidents at four-leg signalized intersections on rural two-lane highways

were:

• Average daily traffic volume on the major road.

• Average daily traffic volume on the minor road.

• Presence of protected left-turn signal phase on the major road.

• Grade rate for all vertical curves (including both crests and sags) within 244 m (800 ft)

of the intersection along the major and minor roads.

• Percentage of minor-road traffic turning left at the signal for the morning and evening

peak hours combined.

• Percentage of trucks (vehicles with more than four wheels) in the traffic stream entering

the intersection from all approaches for morning and evening peak hours combined.

• Number of driveways within 76 m (250 ft) of the intersection on the major-road leg.
150

Nbi ' exp(&6.12 % 0.46SUMLADT & 0.61PROTLT & 0.013PCTLEFT2

% 0.12VEICOM % 0.030PTRUCK)

(63)

Table 44 presents descriptive statistics for these variables.

Table 44. Descriptive Statistics for 49 Four-Leg Signalized Intersections in California

and Michigan Used in Modeling.

Variable Mean

Standard

deviation Minimum

25th

percentile Median

75th

percentile Maximum

ADT1 (veh/day) 10,491 4,331 4,917 7,568 8,900 13,133 25,133

ADT2 (veh/day) 4,367 2,369 940 2,800 3,670 5,080 12,478

ln ADT1 9.18 0.39 8.50 8.93 9.09 9.48 10.13

ln ADT2 8.26 0.49 6.85 7.93 8.21 8.53 9.43

SUMLADT 17.44 0.65 16.34 16.94 17.49 17.78 19.14

PCTLEFT2 (percent) 28.4 15.1 2.5 19.0 25.7 35.7 75.70

VEICOM (percent/100 ft)) 1.88 1.87 0.00 0.50 1.43 2.54 8.13

PTRUCK (percent) 9.0 6.7 2.7 5.0 7.7 11.2 45.4

ND1 3.00 3.00 0.00 0.00 3.00 4.00 15.00

PROTLTa ! ! ! ! ! ! !

Conversion: 1 ft = 0.305 m

a 43 percent of the four-leg signalized intersections have protected left-turn signal phases and 57 percent do not.

Negative binomial models were developed for predicting accident experience at four-leg

signalized intersections on rural two-lane highways using the modeling technique based on the

investigating officer’s identification of intersection-related accidents. A candidate model

developed using this approach is:

where:

SUMLADT = ln ADT1 + ln ADT2;

PROTLT = presence of protected left-turn signal phase on one or more major-road

approaches; = 1 if present; 0 = if not present;

PCTLEFT2 = percentage of minor-road left-turning traffic at the signal for the morning

and evening peak hours combined;
151

Nbi ' exp(&4.96 % 0.39SUMLADT & 0.38PROTLT & 0.015PCTLEFT2

% 0.103VEICOM % 0.027PTRUCK)

(65)

Nbi ' exp(&6.084 % 0.60ln ADT1% 0.29ln ADT2& 0.47PROTLT

& 0.017PCTLEFT2% 0.11VEICOM % 0.029PTRUCK)

(66)

Nbi ' exp(&6.95 % 0.62ln ADT1% 0.39ln ADT2& 0.68PROTLT

& 0.014PCTLEFT2% 0.13VEICOM % 0.032PTRUCK)

(64)

VEICOM = grade rate for all vertical curves (crests and sags) any portion of which is

within 244 m (800 ft) of the intersection averaged for the major- and

minor- road legs of the intersection;

PTRUCK = percentage of trucks (vehicles with more than four wheels) entering the

intersection for morning and evening peak hours combined;

The variable PROTLT indicates the presence of either a fully protected left-turn signal phase

or a protected-permitted phase.

An alternative to equation (63) using the major- and minor-road traffic volumes separately:

Tables 45 and 46 summarize the model parameters and goodness of fit for equations (63) and

(64), respectively.

Negative binomial models were developed for predicting accident experience at four-leg

signalized intersections on rural two-lane highways using the modeling technique based on the

accident types generally considered to be intersection-related. A model using the sum of the

major- and minor-road traffic volume variables is:

An alternative form of equation (65) using separate variables for the major- and minor-road

traffic volumes is:

Negative binomial models comparable to equations (65) and (66) including an independent

variable representing the number of driveways within 76 m (250 ft) of the intersection were also

developed. A model using the sum of the major- and minor-road traffic volume variables is:
152

Nbi ' exp(&4.11 % 0.33SUMLADT & 0.30PROTLT & 0.016PCTLEFT2

% 0.100VEICOM % 0.023PTRUCK % 0.035ND1)

(67)
153

Table 45. Model Parameters and Goodness of Fit for Equation (63).

Independent variable Overdispersion

parameter

Parameter Intercept SUMLADT PROTLT PCTLEFT (k) 2 VEICOM

PTRUC

K

Coefficient !6.12 0.46 !0.611 !0.013 0.12 0.030 0.12

Standard deviation 2.60 0.15 0.151 0.0048 0.051 0.014 0.032

Significance level (p) 0.018 0.0017 0.0001 .0052 0.014 0.033 0.0002

Note: The values of goodness-of-fit measures are R2 = 0.5208, Rk

2

 = 0.6414, and RPD

2

 = 0.2550

Table 46. Model Parameters and Goodness of Fit for Equation (64).

Independent variable

Overdispersion

Parameter Intercept ln ADT parameter (k) 1ln ADT2 PROTLT PCTLEFT2 VEICOM

PTRUC

K

Coefficient !6.95 0.62 0.39 !0.68 !0.014 0.13 0.032 0.12

Standard deviation 2.79 0.25 0.17 0.18 0.0047 0.045 0.014 0.032

Significance level 0.013 0.013 0.023 0.0002 .0023 0.0039 0.028 0.0003

Note: The values of goodness-of-fit measures are R2 = 0.5053, Rk

2

 = 0.6490, and RPD

2

 = 0.2362
154

Nbi ' exp(&5.46 % 0.60ln ADT1% 0.20ln ADT2

& 0.40PROTLT & 0.018PCTLEFT2% 0.11VEICOM

% 0.026PTRUCK % 0.041ND1)

(68)

Nbi ' ADT0.307

1

exp(0.0000376 ADT1) ADT0.461

2

LTLN1RTLN1

(69)

where:

ND1 = number of driveways with 76 m (250 ft) of the intersection on the major

road.

An alternate form of equation (67) using separate variables for the major- and minor-road traffic

volumes is:

Tables 47 through 50 summarize the model parameters and goodness of fit for equations

(65) through (68), respectively.

A negative binomial model for predicting accident experience at four-leg signalized

intersections on rural two-lane highways was also developed using the iterative offset technique. 

This model uses a different functional form for the major and minor road ADT variables than was

used in the preceding models. This model is:

where:

LTLN1 = factor for number of major-road left-turn lanes present at the intersection

(LTLN1 = 1.000 for zero left-turn lanes, 0.934 for one left-turn lane, and

0.737 for two left-turn lanes.); and

RTLN1 = factor for number of major-road right-turn lanes present at the intersection

(RTLN1 = 1.000 for zero right-turn lanes, 0.832 for one right-turn lane, and

0.871 for two right-turn lanes).

The value of the overdispersion parameter for this model is 0.26.
155

155

Table 47. Model Parameters and Goodness of Fit for Equation (65).

Independent variable

Overdispersion

Parameter Intercept SUMLADT PROTLT PCTLEFT parameter (k) 2 VEICOM

PTRUC

K

Coefficient !4.96 0.39 !0.38 !0.015 0.103 0.027 0.14

Standard deviation 3.078 0.18 0.17 0.006 0.042 0.013 0.039

Significance level (p) 0.107 0.0309 0.022 0.0101 0.013 0.040 0.0005

Note: The values of goodness-of-fit measures are R2 = 0.3913, Rk

2

 = 0.5382, and RPD

2

 = 0.1770

Table 48. Model Parameters and Goodness of Fit for Equation (66).

Independent variable

Overdispersion

Parameter Intercept ln ADT parameter (k) 1ln ADT2 PROTLT PCTLEFT2 VEICOM

PTRUC

K

Coefficient !6.084 0.60 0.29 !0.47 !0.017 0.113 0.029 0.13

Standard deviation 3.39 0.28 0.20 0.20 0.0057 0.037 0.013 0.039

Significance level 0.072 0.037 0.14 0.019 .0036 0.002 0.028 0.0008

Note: The values of goodness-of-fit measures are R2 = 0.3650, Rk

2

 = 0.5521, and RPD

2

 = 0.1816
156

156

Table 49. Model Parameters and Goodness of Fit for Equation (67).

Independent variable Overdispersion

Parameter Intercept SUMLADT PROTLT PCTLEFT parameter (k) 2 VEICOM PTRUCK ND1

Coefficient !4.11 0.332 !0.302 !0.016 0.100 0.023 0.035 0.12

Standard deviation 2.95 0.172 0.175 0.0055 0.038 0.012 0.027 0.037

Significance level (p) 0.163 0.053 0.083 0.0038 0.009 0.055 0.199 0.0011

Note: The values of goodness-of-fit measures are R2 = 0.4563, Rk

2

 = 0.2834, and RPD

2

 = 0.1822

Table 50. Model Parameters and Goodness of Fit for Equation (68).

Independent variable Overdispersion

parameter

Parameter Intercept ln ADT (k) 1ln ADT2 PROTLT PCTLEFT2 VEICOM PTRUCK ND1

Coefficient -5.46 0.599 0.202 -0.404 -0.018 0.111 0.026 0.098 0.11

Standard 3.19 0.279 0.192 0.188 0.005 0.033 0.012 0.041 0.040

Significance level 0.087 0.032 0.293 0.032 0.0005 0.0006 0.028 0.025 0.0043

Note: The values of goodness-of-fit measures are R2 = 0.4327, Rk

2

 = 0.6094, and RPD

2

 = 0.1816
157

Assessment of Alternative Base Models

A careful examination and comparison was made of the alternative base models presented

above to select the most suitable models for use in the accident prediction algorithm. The final

selections were made based on a sensitivity analysis and an assessment of the combined roadway

segment and intersection models to predict total accidents within 76 m (250 ft) of the set of study

intersections as a whole, including both intersection-related and non-intersection-related

accidents.

Equation (42) was the only candidate roadway segment model and that model was found to

be satisfactory for use as the base model for roadway segments. This model also appears as

equation (5) in the main text of the report.

Six candidate models were developed for three-leg STOP-controlled intersections. Of

these six models, the models developed using the officer’s assessment of intersection-related

accidents were found, together with the roadway segment model, to provide the best estimate of

total accidents with in 76 m (250 ft) of the 382 study intersections. In choosing between

equations (52) and (53), preference was given to equation (53) which includes only those

variables found to be statistically significant with a significance level (p) of 0.015 or less. 

Therefore, equation (53) was selected as the base model for three-leg STOP-controlled

intersections and it also appears as equation (7) in the main text of this report.

Similar logic was used to select equation (59) as the base model for four-leg STOPcontrolled intersections. Equation (59) also appears as equation (9) in the main text of this

report.

The assessment of the candidate models for four-leg signalized intersections was difficult

because all seven candidate models were based on a relatively small sample size

(49 intersections) and none of the models was found to be completely satisfactory. 

Equation (69) was appealing because it contains the same independent variables for which

AMFs have been provided in the accident prediction algorithm. However, the functional form

selected for equation (69) is such that the predicted accident frequency begins to decrease when

the major-road ADT exceeds approximately 10,000 veh/day. This does not appear reasonable,

so equation (69) was eliminated from consideration. The best of the remaining candidate models

was equation (68), which also appears in the main text of the report as equation (11).
158

Nt' (1&p) Nrs% (p) Nint (70)

p '

tminorADT2

86,400

, p # 1.0 (71)

Another Alternative Approach for Future Consideration

The accident prediction algorithm for two-lane highways is based on the concept that

accidents in the vicinity of an intersection can be predicted by separate models for intersectionrelated accidents and roadway-segment-related (or non-intersection-related) accidents. The

models used to predict roadway segment and intersection accidents are independent; this is

based on the assumption that there is a certain “background” level of roadway segment accidents

that would occur on the major-road legs of the intersection whether the intersection was present

or not. The accident prediction algorithm does not consider the possibility that the presence of

the intersection could affect the frequency, severity, or collision types for accidents that are

classified as non-intersection-related.

An alternative concept for considering the interaction described above was proposed during

the development of the accident prediction algorithm. While this approach was, in the end,

judged to have sufficient conceptual problems to be unworkable, this alternative approach is

documented here for future reference in case a workable variation can be developed in the

future.

The proposed concept was that a 152 m (500 ft) roadway segment along the major road

(76 m or 250 ft on either side of the intersection) operates as an intersection approach during the

time when traffic is present on the minor-road approach and operates as a roadway segment the

rest of the time. Thus, if one could estimate the proportions of time when the segment operated

as a roadway segment and as an intersection, its safety performance could be predicted as:

where:

Nt = Total predicted accident frequency for a 152 m (500 ft) roadway segment

containing an intersection; and

p = proportion of time during which the roadway segment operates as an

intersection (0#p#1).

Values for Nrs and Nint would could be determined from the accident prediction algorithm as

shown in Equations (13) and (14). Furthermore, it was proposed that the proportion p could be

estimated as follows:
159

where:

tminor = average time each minor-road vehicle is present at or near the

intersection (sec/veh); and

ADT2 = average daily traffic volume on the minor road (veh/day).

The average time each minor-road vehicle is present at or near the intersection, tminor, could

be estimated as some representatives value, such as 10 to 20 sec. The constant value of 86,400

in the denominator of equation (66) represents the number of seconds in a 24-hour day.

The approach represented by equations (70) and (71) has substantial conceptual appeal

because it incorporates a direct proportionality to the minor-road ADT. The approach could be

strengthened conceptually by accounting for traffic turning from the major road onto the minor

road and by adjusting for time periods when more than one minor-road vehicle is present at the

intersection at the same time. Furthermore, the concept might also be adapted to the prediction

of driveway accidents by assuming very low values of ADT2 for driveways. This could allow the

accident prediction algorithm to address driveway accidents much more explicitly than is

permitted by the driveway density AMF shown in equation (22).

The approach represented by equations (70) and (71) has conceptual disadvantages as

well. First, it is based on a presumption that when a minor-road vehicle is present at the

intersection there is no risk of an accident occurring elsewhere on the adjacent 152 m (500 ft)

roadway segment. This presumption appears unrealistic. There must be some risk of an

accident elsewhere on the major road when a minor-road vehicle is present at the intersection. 

Indeed, there is simply no evidence as to whether the presence of a minor-road vehicle at an

intersection affects the risk of accidents elsewhere on the major road at all. Second, the form of

equation (71) is such that, depending on the value of tminor that is selected, there is some level at

which p becomes equal to 1.0. At and above that ADT level, equation (70) implies that there

would be no roadway-section-related accidents within 152 m (500 ft) of the intersection. Such a

proposition is also unrealistic.

On balance, this approach to intersection accident prediction discussed above was found to

be unworkable and was abandoned. The concept has been presented here to document that it

was considered and to suggest an alternative that might be adapted for use in the future.

 
160
161

APPENDIX C

CALIBRATION PROCEDURE TO ADAPT THE ACCIDENT PREDICTION

ALGORITHM TO THE DATA OF A PARTICULAR HIGHWAY AGENCY

This appendix presents a calibration procedure for use with the accident prediction

algorithm. The purpose of the calibration procedure is to allow potential users of the algorithm to

scale the accident predictions to be suitable for the roads under the jurisdiction of their agency.

The base models that form a key element of the accident prediction algorithm are based on

data from several States. The roadway section base models were based on data from

Minnesota and Washington, but are implemented in the accident prediction algorithm for

Minnesota conditions (i.e., STATE=0 in Equation (42). The base models for STOP-controlled

intersections are based on Minnesota data, and the base models for signalized intersections are

based on combined data from California and Michigan. The development of these base models

has been documented in appendix B of this report.

The calibration procedure provides a method for highway agencies other than those

identified above to adapt the accident prediction algorithm to their own safety conditions. 

Because safety conditions change over time, even a state whose data were used in the

development of the base models should consider applying the calibration procedure every 2 or 3

years.

The accident prediction algorithm applies to rural two-lane highways. Safety conditions and

the resulting accident rates and distributions of accident severities and accident types, on rural

two-lane highways, vary substantially from one highway agency to another. Some of these

variations are due to geometric design factors, such as differences in the distributions of lane and

shoulder width, and differences in terrain, which lead to differences in horizontal and vertical

alignment. These geometric design factors are accounted for by the accident prediction algorithm

and should not require calibration to allow the algorithm to be used by different agencies.

By contrast, there are several factors that lead to safety differences between highway

agencies in different geographical areas that are not directly accounted for by the accident

prediction algorithm. These include:

• Differences in climate (i.e., exposure to wet pavement and snow-and-ice-covered

pavement conditions).

• Differences in animal populations that lead to higher frequencies of collision with animals

in some States than in others.
162

Nrs ' Nbr Cr(AMF1r AMF2r... AMFnr) (72)

Nint ' Nbi Ci(AMF1i AMF2i... AMFni) (73)

• Differences in driver populations and trip purposes (i.e., commuter vs. commercial

travel vs. recreational travel).

• Accident reporting thresholds established by State law (i.e., minimum property damage

threshold that requires reporting of an accident).

• Accident investigation practices (i.e., some police agencies are much more diligent

about investigating and reporting property-damage-only collisions than others).

The calibration procedure is intended to account for these differences and provide accident

predictions that are comparable to the estimates that a highway agency would obtain from its

own accident records system.

The tables for the calibration procedure are presented in this appendix using conventional

units, because all of the underlying equations and data are in conventional units. A metric

conversion chart is included for the convenience of those who wish to implement the calibration

procedure in metric units.

How Does the Calibration Procedure Affect the Accident

Predictions?

The calibration procedure is implemented by a highway agency by determining the value of

calibration factors for roadway sections and at-grade intersections from comparison of their own

data to estimates from the accident prediction algorithm. As shown in section 3 of this report, the

calibration factors are incorporated in the accident prediction algorithm in the following fashion

for roadway segments and at-grade intersections, respectively:

where:

Cr = calibration factor for roadway sections developed for use by a particular highway

agency; and

Ci = calibration factor for at-grade intersections developed for use by a particular

highway agency.
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The procedures for estimating values of Cr and Ci for a particular highway agency are described

later in this section.

The calibration procedure also permits a highway agency to modify the basic accident

severity distribution for rural two-lane highways presented in table 1 and the basic accident type

distribution for rural two-lane highways presented in table 2 based on their own data.

Who Should Perform the Calibration Procedure?

It is recommended that one division or office within a specific State highway agency should

calibrate the accident prediction algorithm for subsequent use by all other highway agency

employees and their contractors throughout the State. Because the calibration procedure

requires the use of historical accident, roadway, and traffic data for State highways, it is further

recommended that the calibration be performed by a division or office of the highway agency that

has the following:

• Unrestricted access to the State traffic, accident, and roadway record system(s). 

• Capability to generate estimates of roadway mileage and accident experience for

various ADT levels. 

• Expertise in highway safety analysis procedures.

It is not necessary to calibrate these models to different regions (or districts) within the State,

although that certainly can be done if the highway agency wishes by stratifying mileage and

accidents by region (or district) and applying the recommended calibration procedures for each

region (or district). 

Furthermore, since the accident analysis module is only applicable to rural two-lane

highways at this time, the models currently cannot be calibrated for different urban areas within

the State. As the models for multilane and urban roads are developed, calibration procedures

that apply to individual urban areas may also be desirable.

Should the Accident Prediction Algorithm Be Used Without

Calibration?

It is possible for a highway agency to use the accident prediction algorithm without

calibration, but this is not recommended. Using the accident prediction algorithm without

calibration requires the user to accept the assumption that for their agency Cr = 1.0, Ci = 1.0,
164

and the accident severity and accident type distributions for two-lane highways are those shown

in tables 1 and 2, respectively. These assumptions are unlikely to be correct for any highway

agency and are unlikely to remain correct over time. Even a minimal calibration effort (referred

to later in this appendix as level 1 calibration) is likely to produce far more satisfactory results

than using the algorithm without calibration.

How Often Should the Accident Prediction Algorithm Be

Calibrated?

The recommended calibration procedure uses the three most recent years of accident data

for the highway agency’s rural two-lane highway system. Recalibration every year is not

necessary because it is unlikely that adding a new year of data and dropping the oldest of the

three years used for calibration in the previous year will change the calibration factors

substantially. Recalibration every 2 or 3 years is recommended instead.

Recommended Calibration Procedures

It is neither necessary nor recommended for users of the accident prediction algorithm to

generate new accident prediction models using accident, roadway, and traffic data from their

State and appropriate statistical analysis procedures. It is definitely not recommended for users

to change the accident modification factors in the accident prediction algorithm. These AMFs

are based on a compilation of the best available research, and they will be updated by FHWA as

new research becomes available. Rather, the recommended calibration procedure estimates the

values of calibration factors that adapt the outputs from the accident prediction algorithm to the

safety conditions experienced by an individual highway agency.

Calibrating the Roadway Segment Accident Prediction

Algorithm

Optional levels are available to calibrate the roadway segment accident prediction algorithm. 

The levels vary in terms of the effort required and the availability of the following:

• Type of data elements maintained within the existing traffic records systems.

• Existence, quality and coverage of roadway, and traffic files.

• Availability and quality of accident data.
165

• Skills of personnel who will perform the calibration.

• Level of effort and personnel resources that a State is willing to commit to calibration.

The minimum data requirements and anticipated effort for each calibration process level are

specified in table 51. Level 1 is deemed the minimum and, compared to the other level, the

easier calibration to perform. Level 2 requires more effort, but with a corresponding gain in

applicability of the accident prediction algorithm to a particular State's accident experience. It is

strongly recommended that the models NOT be used without calibration. Attempting to calibrate

using procedures less stringent than the level 1 procedure is strongly discouraged.

Table 51. Minimum Requirements for Calibration Levels 1 and 2

Calibration

process* Minimum requirements

Anticipated effort

for calibration

Level 1 The State must have the ability to:

(1) Stratify all two-lane rural roads by ADT; and 

(2) Identify all non-intersection related accidents reported on

those two lane rural roads.

Minimal

Level 2 Level 1 requirements + the State must have the ability to:

(3) Stratify all two-lane rural roads by ADT, shoulder width

and lane width.

Moderate

* to be selected by the State.

While table 51 describes the minimum data requirements for each calibration level and shows

how the calibration levels relate to each other, table 52 identifies the data elements that must be

present within the State’s record systems to complete the calibration level. The procedures to

perform levels 1 and 2 are described on the following pages.

The basic procedure to be followed in each level involves six steps shown in figure 7. The

only difference in the two levels is that, in level 2, step 1 will develop estimates of miles of

roadway by lane and shoulder width, as well as by curve and grade.

Road System to be Used in Calibration Process

It is important to recognize that there are differences in State highway agency record

systems. Some States, such as California, include only roads under State jurisdiction in their

record systems. Others, such as Minnesota, include local roads that are maintained by counties

and municipalities. Calibration can be performed using whatever rural two-lane
166

Figure 7. Flow Diagram of Calibration Process.
167

Table 52. Data Needs for Calibration Levels 1 and 2.

Type of

data Data element Level 1 Level 2A

Required Accident

records

Relationship of

accidents to

intersections or

junctions

x x

Traffic

volume

files

ADT

x x

Roadway

inventory

files

Lane width

Shoulder width x

x

Desirable Alignment

inventory

files

Horizontal curve

data; 

Grade and vertical

curve data

x

x

x

x

Access

point

Driveway x x

road system the user wishes to apply the accident prediction algorithm to as long as the data are

accurate. Specifically, the following should exist:

• A reliable inventory of paved roads.

• Reliable counts of ADT and accidents that can be assigned to specific sections of a

highway.

• Ability to stratify sections of highways into different highway types (i.e., the ability to

distinguish rural two-lane highways from their highway types).

It should be noted that the base models used in the accident prediction algorithm were

developed for State primary routes that generally had 88-km/h (55-mi/h) speed limits.

If the State has accurate data for county highways and higher order routes but not for

municipal highways, then the calibration process should not use municipal or township highways. 

Rather, data for only the county highways and higher order routes should be used. For example,

Utah has data for 20,600 km (12,800 mi) of paved roadways under State jurisdiction and

another 51,000 km (32,000 mi) of roadways not under State jurisdiction. A complete roadway

inventory to which accidents can be linked is available for the 20,600 km (12,800 mi) system of

state roadways, but the other 51,000 km (32,000 mi) have only “zone” records, that do not
168

permit accident records to be linked to specific roadway sections. Consequently, any calibration

of the accident prediction model by Utah should use only the 20,600 km (12,800 mi) of roadway

under State jurisdiction.

Only paved two-lane highways should be considered in the calibration process.

Level 1 Calibration Process for Roadway Segment Accident Prediction Using

ADT Only

Step 1. Develop Estimates of Paved Rural Two-Lane Highway Mileage by Curve

and Grade. The State will first need to estimate the following for all paved rural, two-lane

highways in the State for each of five ADT groups:

• Number of miles of tangent roadway.

• Number of miles of roadway on horizontal curves.

• Average degree of curvature for horizontal curves.

• Number of miles of level roadway.

• Number of miles of roadway on grade.

• Average grade percent for roadway on grade.

For States with Alignment (Curve and Grade) Inventory Files

If the State has curve and grade inventory files for paved rural two-lane highways, then it will

be possible to calculate the necessary alignment data to perform the calibration. Using their

horizontal curve inventory data, the State should first calculate the number of miles of tangent

roadway, the number of miles of curved roadway, and the average degree of curve for horizontal

curves by ADT interval, using the format shown in table 53. Table 53, and subsequent tables in

this section, are presented to illustrate the format in which the calibration calculations should be

made; the blank spaces in the tables may be filled in by users of the calibration process. Using its

vertical alignment (grade) inventory data files, the State should then calculate the number of miles

of two-lane rural roads that are not on grade (e.g., level), the number of miles with non-zero

grades, an average percent grade for the miles with non-zero grades, and an overall average

percent grade as shown in table 54.

Table 53. Estimate Mileage by ADT Level and Horizontal Alignment.

ADT Interval

Number of

Tangent

Miles

Number of

Curved

Miles

Average Degree of

Curvature for

Curved Miles

(D)

Average

Radius of

Horizontal

Curvea

Average Length

of Curve

< 1000

1,001 - 3,000

3,001 - 5,000

5,001 - 10,000

 > 10,000

a

 Calculated as 5729.58 / (D).
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Table 54. Estimate Mileage by ADT Level and Vertical Alignment.

ADT Interval

Number of

Level Miles

(Ml)

Number of 

Miles on Grade

(Mg)

Average Percent Grade

for Miles on Grade

(Pg)

Average Percent

Gradea

< 1000

1,001 - 3,000

3,001 - 5,000

5,001 - 10,000

 > 10,000

a

 Calculated as [(Ml ) * (0) + (Mg) * (Pg)] / [Ml + Mg].

For States without Alignment (Curve and Grade) Inventory Files

For States without curve or grade files, they can use an estimation procedure that will

calculate “default” values for curve and grade mileages and average degree of curve and percent

grade values based on the percent of rural two-lane miles that fall into each of the three terrain

groups—flat, rolling, and mountainous. Thus, the State needs to estimate the mileage by ADT

interval and the percentage of mileage in flat, rolling, and mountainous terrains, which could be

derived from Highway Performance Monitoring System (HPMS) data. Then, the State will need

to determine the mileages by ADT intervals and the percentages of level, rolling, and

mountainous terrain. 

Default values for percent of tangent and curved miles and the average degree of curve for

the curved miles were based on analyses of States in the HSIS database which had both a curve

file and terrain information. Based on these analyses, default values for proportion of two-lane

rural miles that are curved and the average degree of curve for these miles are as follows:

• Flat: percent of non-tangent miles = 19 percent; average degree of curvature= 2E.

• Rolling: percent of non-tangent miles = 24 percent; average degree of curvature = 4E.

• Mountainous: percent of non-tangent miles = 38 percent; average degree of curvature =

8E.

Similarly, information on default values for percent of non-flat miles and average grade for

non-flat mileage from HSIS States were used to calculate the following default values:

• Flat: percent of non-flat miles = 87 percent; average grade = 1.5 percent.

• Rolling: percent of non-flat miles = 91 percent; average grade = 2.0 percent.

• Mountainous: percent of non-flat miles = 97 percent; average grade = 3.7 percent.

Tables 55 and 56 illustrate the data that are required. 
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Table 55. Estimate Mileage by ADT Interval.

ADT Interval

Estimated miles of paved,

two-lane rural highways

< 1,000

1,000 - 3,000

3,001 - 5,000

5,001-10,000

> 10,000

Table 56. Estimate Proportion of Mileage by Terrain.

Terrain

Proportion of paved, twolane rural highways

(percentage)

Flat

Rolling

Mountainous

Total 100%

These values in tables 55 and 56 will then be used by the calibration software to produce

estimates of the following:

• Number of miles of tangent roadway.

• Number of miles of roadway on horizontal curves.

• Average degree of curvature for horizontal curves.

• Average radius of horizontal curves.

• Average length of horizontal curves.

• Number of miles if level roadway.

• Number of miles of roadway on grade.

• Average percent grade for roadway on grade.

• Statewide average percent grade.

Step 2. Accept or Modify Default Values for Other Geometric Parameters. In

addition to the values of curvature and grade developed in the proceeding step, the calibration

procedure requires other geometric parameters. Using data from a sample of States during the

calibration procedure development, it has been found reasonable to use the following default

values for these parameters:

• Shoulder type = paved.
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• Driveway density = three driveways/km (five driveways/mi).

• Passing lane = not present.

• Short four-lane section = not present.

• Two-way left turn lanes = not present.

• Roadside hazard rating = 3.

• For horizontal curves:

no spiral transition present; and

superelevation is not deficient (e.g., AMF = 1.0).

For level 1 calibration (which does not require data to be input for lane width and shoulder

width), the following default values were estimated based on average values for two-lane

highways obtained from eight HSIS States and are recommended for use:

ADT Interval

(Vehicles/day)

DEFAULT VALUES FOR LEVEL 1 CALIBRATION

ADT

 (vehicles/day)

Lane Width

(ft)

Shoulder Width

 (ft)

< 1000 400 11 3.5

1,000 - 3,000 1800 11.5 5

3,001 - 5,000 3900 12 5

5,001 - 10,000 6900 12 6.25

> 10,000 13800 12 6.5

The State may accept these defaults as presented and proceed onward. However, if a

State has a better estimate for a value [e.g., if average driveway density is known to be equal to

four driveways/km (seven driveways/mi), rather than three driveways/km (five driveways/mi)], or

if the predominant two-lane roadway design used in a particular State differs from these assumed

defaults (e.g., if the predominant rural two-lane highway shoulder type is unpaved, rather than

paved), then the user should feel free to modify the defaults.

Step 3. Calculate Estimate of Annual Non-Intersection Accidents Using the

Accident Prediction Algorithm. Using the roadway segment accident prediction algorithm,

calculate the estimated annual number of non-intersection accidents for tangents and horizontal

curves. Then, sum the total. Table 57 illustrates how the predicted number of accidents per year

can be determined by the calibration software.
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Table 57. Calculate the Predicted Annual Number of Non-Intersection Accidents as a

Function of ADT.

ADT (veh/day)

Assumed

mean of ADT

interval*

Mileage of rural two lane

highways

Predicted number of nonintersection accident per year**

Tangent Curve Total Tangent Curve Total

< 1,000 400

1,000-3,000 1,800

3,001-5,000 3,800

5,001-10,000 7,000

>10,000 13,500

* Generally the average is less than the midpoint of the ADT interval.

** Determined with the uncalibrated accident prediction algorithm for roadway segments.

Step 4. Determine Actual Annual Non-Intersection Accidents from State Data.

Using data from the last 3 years, determine the actual number of non-intersection accidents per

year that were reported on the rural two-lane highways. First, determine the number of total

accidents on the selected rural two-lane highways. Then, deduct all accidents on those rural

two-lane highways that were identified by the investigating officer as being “at intersection” or

“intersection-related.”

With respect to this criterion, the State must determine the most appropriate field(s) to

establish intersection-relatedness. for example, a State may have a field for “type of event

location” or “site location” or “relationship to junction” on its accident report form that includes

categories such as intersection, junction area, non-junction area, driveway access, and alley

access. Some States have an explicit field for “relation to intersection” with categories for yes or

no. The decision on which field to use is left to the user. It is important to note that driveway

accidents are NOT considered intersection accidents and should NOT be excluded from the

calibration data for roadway segments. Driveway accidents were included in the data set from

which the roadway segment base model was developed and, therefore, driveway accidents

should not be excluded with intersection accidents.

Step 5. Calculate Calibration Factor Using Outputs of Steps 3 and 4. Calculate the

calibration factor (Cr) as the ratio of the total number of reported non-intersection accidents

(from step 3) to the total number of predicted non-intersection accidents (from step 2).

Step 6. Enter Final Calibration Factor into Software. The preceding five steps will

produce a calibration factor based on the user’s inputs. At this point, the user will usually accept

the calibration factor as calculated. However, if the resulting factor appears unreasonable

(perhaps because of data problems of which the user is aware), it may be modified before entry
173

into the system. Once accepted by the user, the value of the calibration factor (Cr) should be

entered into a file of default input values for the accident prediction algorithm. In subsequent

applications of the accident prediction algorithm, this value of Cr will be used in applying equation

(72). As discussed earlier, the user can, and should, update the calibration factor every 2 or 3

years.

Level 2 Calibration Process for Roadway Segment Accident Prediction Using

ADT, Lane Width, and Shoulder Width

Step 1. Develop Estimates of Mileage by Curve and Grade and Lane and Shoulder

Width. The State will first need to stratify the mileage of their paved rural, two-lane highways in

the State by the following factors and associated levels:

• ADT 

< 1000 veh/day

1,001 - 3,000 veh/day

3,001 - 5,000 veh/day

5,001 - 10,000 veh/day

>10,000 veh/day

• Lane Width

< 2.9 m (9.5 ft)

9.5-10.5 2.9-3.2 m (9.5-10.5 ft)

3.2-3.5 m (10.5-11.5 ft)

> 3.5 m (11.5 ft)

• Shoulder Width

< 0.9 m (3 ft)

0.9-1.5 m (3-5 ft)

1.5-2.1 m (5-7 ft)

> 2.1 m (7 ft)

• Horizontal Alignment 

tangent

curve

For States with Alignment (Curve and Grade) Inventory Files

If the State has curve and grade inventory files, then it will be possible to calculate the

necessary alignment data to perform the calibration. Using their horizontal curve inventory data,

they should first estimate the number of miles of tangent roadway, the number of miles of curved

roadway and the average degree of curve for horizontal curves for each unique combination of
174

ADT interval, lane width and shoulder width. Table 58 illustrates this for a portion of all possible

combinations.

Using their vertical alignment (grade) inventory data files, the State should then calculate the

number of miles of two-lane rural roads that are not on grade (e.g., level), the number of miles

with non-zero grades, an average percent grade for the miles with non-zero grades, and an

overall average percent grade as shown in table 59.

For simplification purposes, it can be assumed that the average percent grade computed for

an ADT interval is equally applicable across lane widths and shoulder widths within that interval. 

By way of an example, consider the case where 1.9 percent is computed as the average percent

grade for all two-lane rural roads within the <1000 ADT interval. The 1.9 percent grade can be

assumed for all lane and shoulder width combinations having ADTs less than 1,000 veh/day, as

illustrated in table 60.

For States without Alignment (Curve and Grade) Inventory Files

For States without curve or grade files, they can use an estimation procedure that will

calculate “default” values for curve and grade mileages and average degree of curve and percent

grade values based on the percent of rural two-lane miles that fall into each of the three terrain

groups—flat, rolling, and mountainous. Thus, the State needs to estimate the mileage by ADT

interval, lane width and shoulder width, and the percentage of mileage in flat, rolling, and

mountainous terrains, which could be derived from HPMS data. Then, the State will need to

enter the mileages by ADT intervals and the flat, rolling, and mountainous percentages into the

calibration software. Tables 61 and 62 illustrate the data that are required. 

Default values for percent of tangent and curved miles and the average degree of curve for

the curved miles were based on analyses of States in the HSIS database which had both a curve

file and terrain information. Based on these analyses, default values for proportion of two-lane

rural miles that are curved and the average degree of curve for these miles are as follows:

• Flat: percent of non-tangent miles = 19 percent; average degree of curvature = 2E. 

• Rolling: percent of non-tangent miles = 24 percent; average degree of curvature = 4E.

• Mountainous: percent of non-tangent miles = 38 percent; average degree of curvature =

8E.
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Table 58. Develop Estimates Required for Alignment Component of the Procedure.

ADT interval

Lane

width

(ft)

Shoulder

width (ft)

Number

of miles

on

tangent

Number of

miles on

horizontal

curves

Average

degree of

curvature

(D)

Average

radius of

horizontal

curvea

 (ft)

Average

length of

curve (mi)

< 1000 < 9.5 < 3

3 to 5

5 to 7

> 7

9.5 to

10.5

< 3

3 to 5

5 to 7

> 7

10.5 to

11.5

< 3

3 to 5

5 to 7

> 7

> 11.5 < 3

3 to 5

5 to 7

> 7

1,001 to 3,000 < 9.5 < 3

3 to 5

5 to 7

> 7

9.5 to

10.5

< 3

3 to 5

5 to 7

> 7

10.5 to

11.5

< 3

3 to 5

5 to 7

> 7

> 11.5 < 3

3 to 5

5 to 7

> 7

a

 Calculated as 5729.58 / (D).
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Table 59. Estimate Mileage by ADT Level and Vertical Alignment.

ADT interval

Number of

miles of level

roadway

(Ml)

Number of 

miles on grade

(Mg)

Average percent grade

for miles of roadway on

grade

(Pg)

Average percent

gradea

< 1000

1,001-3,000

3,001-5,000

5,001-10,000

> 10,000

a

 Calculated as [(Ml ) * (0) + (Mg) * (Pg)] / [Ml + Mg].

Table 60. Illustration of How Average Percent Grade Can Be Applied Across Lane and

Shoulder Width Combinations.

AADT

(Vehicles

per day)

Average Percent Grade (%)

Lane Width

< 9.5 ft 9.5 to 10.5 ft 10.5 to 11.5 ft > 11.5 ft

Shoulder Width (ft)

< 3 3-5 5-7 > 7 < 3 3-5 5-7 > 7 < 3 3-5 5-7 > 7 < 3 3-5 5-7 > 7

< 1,000 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9

Table 61. Estimate Proportion of Mileage by Terrain.

Terrain Proportion of paved, twolane rural highways

(percentage)

Flat

Rolling

Mountainous

Total 100%
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Table 62. Estimate Mileage by ADT Interval, Lane and Shoulder Widths.

AADT 

(Vehicles per

day)

Mileage of paved, two-lane rural roads

Lane Width <9.5 ft 9.5<Lane Width<10.5ft 10.5<LaneWidth<11.5 ft Lane Width > 11.5 ft

Shoulder Width (ft)

<3 3-5 5-7 >7 <3 3-5 5-7 >7 <3 3-5 5-7 >7 <3 3-5 5-7 >7

< 1,000

1,001 - 3,000

 3,001 - 5,000 

5,001 - 10,000

> 10,000

1 ft = 0.305 m

Similarly, information on default values for percent of non-flat miles and average grade for

non-flat mileage from HSIS States were used to calculate the following default values:

• Flat: percent of non-flat miles = 87 percent; average grade = 1.5 percent.

• Rolling: percent of non-flat miles = 91 percent; average grade = 2.0 percent.

• Mountainous: percent of non-flat miles = 97 percent; average grade = 3.7 percent.

These values in tables 61 and 62 will then be used by the calibration software to produce

estimates of the following:

• Number of tangent miles.

• Number of curved miles.

• Average degree of curvature for horizontal curves.

• Average radius of horizontal curve.

• Average length of curve.

• Number of level miles.

• Number of miles on grade.

• Average percent grade for miles on grade.

• Statewide average percent grade.

Step 2. Accept or Modify Default Values for Other Geometric Parameters. Next,

the State will need to develop values for other input parameters of the roadway segment accident

prediction algorithm. Values for average percent grade, average length of curve, and average

radius of curve will have been developed in step 1. For the purposes of calibration, it has been

found to be reasonable to assume the following default values:
178

• Shoulder type = paved.

• Driveway density = three driveways per km (five driveways per mile).

• Passing lane = not present.

• Short four-lane section = not present.

• Two-way left turn lanes = not present.

• Roadside hazard rating = 3.

• For horizontal curves:

no spiral transition present; and 

superelevation is not deficient (e.g., AMF = 1.0).

Step 3. Calculate Estimate of Annual Non-Intersection Accidents Using the

Accident Prediction Algorithm. Calculate the predicted annual number of non-intersection

related accidents for tangents and curves using the roadway segment accident prediction

algorithm. Then, sum the total. Table 63 illustrates the procedure. the predicted number of total

non-intersection accident rates will then be the sum of the number predicted in each line of the

table (i.e., sum all calculated values in the final table column for all combinations of ADT, lane

width, shoulder width, curve, and grade).

Step 4. Determine Actual Annual Non-Intersection Accidents from State Data.

Determine the number of non-intersection accidents on rural two-lane highways using 3 years of

accident data. This is the same as step 3 from the level 1 calibration process, which was

previously described.

Step 5. Calculate Calibration Factor Using Outputs of Steps 3 and 4. Calculate the

calibration factor (Cr) as the ratio total number of reported non-intersection accidents on rural

two-lane highways (from step 4) to the predicted total number of non-intersection accidents on

rural two-lane highways (from step 3).

Step 6. Enter Calibration Factor into Software. As in level 1, the preceding five steps

will produce a calibration factor based on the user’s inputs. At this point, the user will usually

accept the calibration factor as calculated. However, if the resulting factor appears unreasonable

(perhaps because of data problems of which the user is aware), it may be modified before entry

into the system. Once accepted by the user, the value of the calibration factor (Cr) should be

entered into a file of default input values for the accident prediction algorithm. In subsequent

applications of the accident prediction algorithm, this value of Cr will be used in applying equation

(72). As discussed earlier, the user can, and should, update the calibration factor every 2 or 3

years.
179

Table 63. Predicting Total Non-Intersection Accidents as a Function of ADT, Lane

Width, and Shoulder Width.

ADT 

(veh per

day)

Lane

width

(ft)

Shoulder

width

(ft)

Calculated mean ADT

for each combination

of ADT, lane width,

and shoulder width

Mileage of rural two lane

highways

Predicted number of nonintersection accidents per

year**

Tangent Curve Total Tangent Curve Total

< 1,000 < 9.5 < 3

3.1-5.0

5.1-7.0

> 7.0

9.5-10.5 < 3.0

3.1-5.0

5.1-7.0

> 7.0

10.5-11.5 < 3.0

3.1-5.0

5.1-7.0

> 7.0

> 11.5 < 3.0

3.1-5.0

5.1-7.0

> 7.0

Continue Table for Other AADT Intervals

Predicted total number of non-intersection accidents =

1 ft = 0.305 m 1 mi = 1.61 km

* To be calculated by accident prediction algorithm, although the State will have to enter the mean ADT values

for each interval.

** Determined with the uncalibrated accident prediction algorithm for roadway segments.

Calibrating the At-Grade Intersection Accident Prediction

Algorithm

A calibration procedure for at-grade intersections has also been developed. Several

alternative methodologies were developed, applied, and evaluated using California, Minnesota,

and Maine data obtained from FHWA’s HSIS. The objective was to develop a calibration

process that produces intuitively logical, accurate, reliable, and consistent calibration factors. 

Based on the results of the research, the following calibration process is recommended.

Step 1. Identify Intersection Sites. For each of the following intersection types, identify

a sample of intersections for which both intersecting roads are paved rural two-lane highways:

• Three-leg, STOP-controlled intersections.

• Four-leg, STOP-controlled intersections.
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• Four-leg, signalized intersections.

Separate values of the calibration factor (Ci) are obtained for each intersection type.

It should be recognized that the current IHSDM Accident Analysis Module does not

currently contain prediction algorithms for other types of rural road intersections, such as (1) fiveleg intersections; (2) traffic circles/rotaries; (3) three-leg signalized intersections; (4) intersections

of two-lane roads with multi-lane roads; (5) uncontrolled or YIELD-controlled intersections; (6)

intersections involving one-way streets; or (7) intersections (which are technically considered

ramp terminals in the AASHTO Green Book) formed by the junction of freeway ramps and twolane rural roads. In general, these types of intersections constitute a small percentage of all twolane rural road intersections. There was insufficient data to develop reliable accident prediction

algorithms for these types of intersections. As times goes on, more research may be conducted

to develop additional accident prediction algorithms, which can then be integrated into the

IHSDM. 

The desirable minimum sample sizes are suggested in table 64.

Table 64. Minimum Sample Sizes by Type of Intersection.

Type of Intersection

Suggested Minimum Sample

Size

Three-leg STOP-controlled intersections 100

Four-leg STOP-controlled intersections 100

Four-leg signalized intersections 25

The suggested minimum sample sizes are somewhat arbitrary in that they do not have a

statistical basis. Clearly, more intersections should produce a more accurate and reliable

calibration factor. However, more data collection translates into greater costs to the highway

agency. It is believed that the minimum sample sizes presented in table above represent a

reasonable compromise between accuracy and cost.

States should identify the sample of their two-lane rural road intersections based on selected

characteristics for each of the three intersection types (e.g., three-leg STOP-controlled, four-leg

STOP-controlled, and four-leg signalized intersections). The most important characteristic is

ADT

It should be understood that there are other parameters used in the accident prediction

algorithm such as intersection skew angle, number of major approaches with left-turn lanes, and

number of major approaches with right-turn lanes, among others. Moreover, data on several of

these variables may even be available within the State’s intersection inventory file. However,

because the research has found that the ADTs had the strongest predictive relationship with
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accidents compared to the other variables, it was deemed that the sampling strategy should be

based only on the ADTs.

For States with complete and accurate intersection inventory files, it may be possible to

automate the sample identification process. For States without intersection inventory files, it will

not be possible to sample in proportion to known intersection distributions. Hence, a different

sampling procedure applies. The two procedures are discussed separately in the following

sections.

Sample Selection for States with Intersection Inventory Files

If a State maintains an accurate and comprehensive intersection inventory file which includes

attribute data about the ADTs on the mainline and the cross road, then the following process is

applicable. 

For all three- and four-leg STOP-controlled intersections, stratify intersections by ADT on

the major road. Table 65 illustrates how this can be accomplished for the three- and four-leg

STOP-controlled intersections. As shown, the State will calculate the proportions of, say, threeleg STOP-controlled intersections which fall into each major-road ADT group (see “Relative

Percentage”). The needed sample of 100 intersections will be divided into groups using these

proportions (i.e., see “Desired Sample”).

For four-leg signalized intersections, both the major- and minor-road ADTs have a

significant effect on accidents. Hence, for four-leg signalized intersections, sampling should

consider both the major-road ADT and the minor-road ADT. Given that there are expected to

be relatively few rural signalized intersections in a State, the decision was made to combine

major- and minor-road ADTs into a single factor, which can be subsequently used for sampling

stratification. The factor is the sum of the average major-road ADT and the average minor-road

ADT. Table 66 presents a numerical example of how this might be done for four-leg signalized

intersections. 

It is important to note that States can do much more with respect to selecting an appropriate

sample of intersections. For example, additional factors can be considered and integrated into

the sampling stratification strategy. In some States, there are distinct differences in climate,

topography, and/or accident reporting practices such that the accident experience in certain parts

of the State may significantly differ from other parts of the States. For example, a State’s

western portion may be dominated by mountains while the eastern portion is primarily flat plains. 

In addition, there may be distinct differences in geometric design, maintenance practices, and

traffic control as a function of highway district (or region). For example, District A may be

“better” with respect to maintenance and traffic control devices than District Z, which may help to

explain differences in average accident rates between the two districts. In any event, the State

has the ability to
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Table 65. Example of Desired Sample Stratification of Three- and Four-Leg

STOP-Controlled Intersections Based on Major Road ADT.

Intersection type

ADT on major road

(vehicles/day)

Intersections of two-lane rural roads

Number

Relative

percentage

Desired

sample

Three-leg

STOP-controlled

intersections

< 1,000 1,095 18.3% 18

1,001 - 3,000 1,443 24.1% 24

3,001 - 5,000 1,641 27.4% 27

5,001 - 10,000 887 14.8% 15

10,001 - 15,000 574 9.6% 10

> 15,000 347 5.8% 6

Total 5,987 100.0% 100

Four-leg

STOP-controlled

intersections

< 1,000 874 21.0% 21

1,001 - 3,000 777 18.6% 19

3,001 - 5,000 1,219 29.2% 29

5,001 - 10,000 489 11.7% 12

10,001 - 15,000 544 13.0% 13

> 15,000 267 6.4% 6

Total 4,170 100.0% 100

Table 66. Example of Desired Sample Stratification of Four-Leg 

Signalized Intersections Based on Major Road ADT.

Four-leg, signalized

intersections

Sum of major- and

minor-road ADTs

Intersections of two-lane rural roads

Number

Relative

percentage

Desired

sample

< 15,000 49 36.0% 9

$ 15,000 87 64.0% 16

Total 136 100.0% 25

 integrate additional factors such as regional area, district, or terrain into the sampling strategy

(provided that they have data for these factors in their intersection inventories). 

By way of an example, consider a situation in which the State wants to sample intersections

based on ADT and highway district. Table 67 illustrates how highway districts can be
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considered in the development of a sampling strategy for three-leg STOP-controlled intersections

only in a State with 3 districts. The results of this table can be compared with the top portion of

table 65 to illustrate how the desired stratification scheme for the sample can be further refined.

Table 67. Example of Desired Sample Stratification of Three-Leg STOP-Controlled

 Intersections Based on Major-Road ADT and Highway District.

ADT on major roads

State

highway

district

Intersections of two

two-lane rural roads

Number

% within

State

Desired

sample

< 1,000 District 1 620 10.4% 10

District 2 339 5.7% 6

District 3 136 2.3% 2

Subtotal 1,095 18.3% 18

1,001 - 3,000 District 1 435 7.3% 7

District 2 674 11.3% 11

District 3 334 5.6% 6

Subtotal 1,443 24.1% 24

3,001 - 5,000 District 1 592 9.9% 10

District 2 527 8.8% 9

District 3 522 8.7% 9

Subtotal 1,641 27.4% 27

5,001 - 10,000 District 1 363 6.1% 6

District 2 446 7.4% 7

District 3 78 1.3% 1

Subtotal 887 14.8% 15

10,001 - 15,000 District 1 185 3.1% 3

District 2 334 5.6% 6

District 3 55 0.9% 1

Subtotal 574 9.6% 10

> 15,000 District 1 58 1.0% 1

District 2 289 4.8% 5

District 3 0 0.0% 0

Subtotal 347 5.8% 6

Total for entire State 5,987 100

After the sampling stratification is established, then the specific intersections within each of

the cells must be identified. A random selection process should be employed. Within this
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context, the term random means that all intersection sites of the same type (e.g., either three-leg

STOP-controlled intersections, four-leg STOP-controlled intersections, or four-leg signalized

intersections) must have the same chance of being selected. 

As an example, using the sampling stratification shown in table 65, the names of all three-leg

STOP-controlled intersections with major-road ADT less than 1,000 veh/day could be put into a

hat and the names of 18 intersections could be drawn at random. The same procedure would be

followed for each of the other ADT levels, and then repeated again for four-leg STOP-controlled

and signalized intersections. The same “random draw” could be accomplished with a

computerized sample by assigning a number to each three-leg STOP-controlled intersection with

major-road ADT less than 1,000 veh/day and determining a ratio by dividing the total number of

intersections in the file by 18. The first intersection selected would be the intersection whose

number is closest to that ratio, the second selected is the intersection whose number is closest to

twice that ratio, etc. For example, if there were 90 total intersection from which 18 were to be

selected every fifth intersection should be chosen (i.e., 90/18=5).

Sample Selection for States Without Intersection Inventory Files

For States without comprehensive or accurate intersection inventories, information about the

existing distribution of intersections within a State may not be known. Hence, the task of

identifying existing distributions on which to base the sampling strategy may be limited or even

non-existent. Moreover, it may not be possible to identify every intersection of two-lane rural

roads, although one expects traffic signal inventory data to be available to do so for signalized

intersections.

When a comprehensive list of intersections is not available or cannot be created, then it will

be necessary to identify and develop a sample of two-lane rural road segments. [It is assumed

that the State has the capability to identify, sort and develop a random sample of all two-lane

rural road segments.] Using an approach similar to one previously described for States with

intersection inventories, a sample of 100 two-lane rural road segments should be identified based

on either (1) ADT alone; or (2) ADT plus additional factors such as highway district that the

State deems necessary. (Thus, using the results shown in table 65 for three- and four-leg STOPcontrolled intersections in the lowest ADT groups, the State should sample approximately 20

intersections with ADTs less than 1,000 veh/day.)

The data collection teams should be dispatched to those two-lane rural road segments with

instructions to collect data at the first three-leg STOP-controlled intersection and the first four-leg

STOP-controlled intersection encountered on each segment. In this manner, data can be

gathered on a sample of 100 intersections for two of the three intersection types. 

As was noted earlier, it is reasonable to assume that traffic signal inventory information, even

if in hard copy format, will be available to identify the locations of four-leg signalized intersections

at which both the major and minor legs are two-lane rural roads. If the information is not

adequate (e.g., cannot determine whether the intersecting roads are two-lane rural roads), then it
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may be necessary to use the random selection process described for three- and four-leg STOPcontrolled intersections above. Based on the intersection inventory data from three HSIS States,

there are relatively few four-leg signalized intersections of two-lane rural roads. Thus, the

potential exists that the data collection team will have to drive long distances before encountering

their first four-leg signalized intersection at which both legs are two-lane rural roads.

Additional Constraints with Respect to Final Selection of the Sample

It is important to recognize that regardless of how the sample is identified, the State should

ensure that each selected site experienced no significant changes in geometry, traffic control,

roadside improvements, or other factors during the 3-year period corresponding the accident

data (which will be discussed in steps 4 and 5). In addition, it will be necessary to exclude

intersections that have undergone changes since the end of that 3-year period because

measurements of skew angle and intersection sight distance taken today may not be

representative of conditions that existed during the 3-year period. By way of an example,

assume that accident data from 1995, 1996, and 1997 are going to be used in the calibration. 

To be included in the sample, the intersection should not have undergone any major changes

since January 1, 1995.

If an intersection is known to have had experienced a major change during the study period,

then it should be excluded from the sample. Major changes at intersection sites include the

following: 

• Installation of traffic signal control.

• Widening to provide more approach lanes and/or turn lanes.

• Cchange from a two-way STOP control to an all-way STOP control.

• Changes in intersection geometry (e.g., alignment or cross-section).

Step 2. Collect Data at Selected Intersections. Collect the following information for

each intersection in the selected sample:

• Average ADT of major road.

• Average ADT of minor road.

• Intersection skew angle.

• Number of quadrants with deficient sight distance.

• Number of major-road approaches with left turn lanes.

• Number of major-road approaches with right turn lanes.

• Type of traffic control applies (e. g., minor-road or all-way STOP control).

Several of these variables are described below in more detail.

Average ADT of Major Road and Average ADT of Minor Road pertain to the bidirectional (i.e., two-way) ADT on the mainline and cross road, respectively. For intersections
186

where the ADT on one leg of the major road differs from the ADT of the other leg of the major

road, the average of the two values should be used. Similarly, if the ADT on one leg of a cross

road at a four-leg intersection differs from the ADT on the other leg of the crossroad, then the

average of the two values should be used.

Most importantly, the ADT should correspond to the accident data. For example, if the

calibration is to be performed using accident data from 1995, 1996, and 1997, then the ADTs to

be used in the accident prediction algorithm should be the average of the 1995, 1996, and 1997

ADTs for the major and minor roads for the selected intersection. When data for selected years

are missing or not available, then the State should use the mid-year if available, or the last year if

the mid-year is not available.

The skew angle is described in the main text of this report. The skew angle is the deviation

of the intersection angle from the nominal or base angle of 90 degrees (i.e., a right angle). For a

four-leg intersection where the angles of the intersection legs to the left and the right of the major

road differ, they are averaged. For example, if one leg forms a 60 degree angle and the other

intersects at 90 degrees, then the average would be 15 degrees [e.g., *(60-90)* / 2].

Step 3. Execute Accident Prediction Algorithm. The data collected in step 2 will be

entered into tables. Data for each intersection within each of the three intersection types will be

entered in a row of the table by itself. The accident prediction algorithm will then be used to

calculate the predicted number of annual intersection accidents for each intersection. The total

number of predicted accidents for each of the three types of intersections will be obtained by

summing the calculated predictions for all intersections of that type.

Step 4. Tally Reported Intersection Accidents. Determine the number of intersectionrelated accidents reported at these intersections over a period of 3 calendar years. An

intersection-related accident is an accident that occurs at the intersection itself or an accident that

occurs on an intersection approach with 76 m (250 ft) of the intersection and is related to the

presence of the intersection. The State should exercise its judgment to develop the criteria that

best apply this definition to their available data. Within each intersection type, the 3-year totals

for intersection-related accidents will be divided by three to obtain annual counts. These annual

counts will then be used in determining the calibration factor.

Step 5. Calculate Calibration Factors. For each intersection type, calculate the

calibration factor (Ci) as the total number of reported annual intersection accidents (from step 4)

divided by the total number of predicted annual intersection accidents (from step 3). As with the

earlier roadway segment calibration, the user will then have the option of accepting the calculated

factors or modifying them. Once accepted by the user, the values of the calibration factors for

each intersection type should be entered into a file of default input values for the accident

prediction algorithm. In subsequent applications of the accident prediction algorithm, these

values of Ci will be used in applying equation (73). Again, it is suggested that the calibration
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factors be updated every 2 to 3 years. The same sample of intersections can be used at 2- to 3-

year intervals for this purpose.

Calibration of Accident Severity Distribution

It is recommended that users of the accident prediction algorithm replace the default

accident severity distribution shown in table 1 with values specifically applicable to the rural twolane highways under a particular agency’s jurisdiction. In step 3 of the preceding calibration

procedures for roadway segments and step 4 of the procedures for intersections, the user will

have identified appropriate files of accidents for roadway segments and intersections. It is

recommended that these accidents be used in determining the accident severity distributions to

update table 1.

Calibration of Accident Type Distribution

The default accident type distribution shown in table 2 can be calibrated by a user to obtain

values specifically applicable to the rural two-lane highways under a particular agency’s

jurisdiction using a procedure that is entirely analogous to the procedure for accident severity

distributions described above. The accident type distributions for both roadway segments and

at-grade intersections in table 2 should be calibrated in this manner using the same accident data

used to update table 1. It should be noted that the accident type distribution for roadway

segments influences the AMFs for lane width, shoulder width, and shoulder type presented in

section 4 of this report.

Calibration Factors for Subareas within a State

The calibration procedures presented earlier in this appendix for the roadway segment and

intersection accident prediction algorithms will result in statewide calibration factors. If the State

deems it necessary, then they can develop calibration factors for subareas in the State. For

example, States may feel that it is necessary to developing unique sets of calibration factors for

subareas within the State stratified by the following:

• Geographic area (e.g., upstate vs. downstate, plains vs. mountains, etc.).

• Terrain (e.g., flat areas, rolling areas, or mountainous areas).

• Highway district or region.

The calibration procedures previously described can be applied to develop multiple sets of

calibration factors within the same State. If it is possible to stratify the State’s mileage of all twolane rural highways into the associated categories or levels that define the subareas, then sets of

calibration factors can be developed for each unique category or level. Of course, the effort

required will also increase. As more cells are added to the stratification matrix, sample sizes for
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individual cells will decrease. If the sample size is too small, then the resulting calibration factors

may not be reliable. It is recommended that calibration factors only be used for the following

absolute minimum sample sizes:

Type of intersection

Absolute

minimum sample size

(number of intersections)

STOP-controlled intersections 50

Signalized intersections 25

Local Calibration Factors 

In addition to the calibration process described in this appendix, which can be thought of as

a “global” calibration of the algorithms, another form of calibration is possible. Specifically,

predictions made with the algorithm [including the calibration factors (Cr and Ci] can be further

“calibrated” to existing local conditions by means of the EB procedure, which is described in

section 6 of this report. It is important to recognize that for projects in which existing accident

histories can be used, weights are assigned to the outputs from the accident prediction algorithm

and the site-specific accident history within the EB procedure. Thus, a greater degree of

importance can be assigned to site-specific accident histories than to the algorithm-generated

prediction.

For alternatives primarily on new alignment, the EB procedure does not apply because the

site-specific accident history of the old alignment is not necessarily representative of conditions on

the new alignment. Thus, such projects should be analyzed without the EB procedure, but using

the applicable calibration factors, Cr and Ci. If the analyst is concerned that safety conditions in a

particular local area may differ substantially from the conditions represented by the “global”

calibration factors, a special calibration study for a local area (possibly with a reduced sample

size of roadway segments and/or intersections) can be performed using the procedures presented

in this appendix. This might be particularly suitable for smaller areas with distinct populations or

climate conditions.
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APPENDIX D

DEFINITIONS OF ROADSIDE HAZARD RATINGS USED WITH THE

ACCIDENT PREDICTION ALGORITHM

The accident prediction algorithm uses a roadside hazard rating system developed by Zegeer,

et al. to characterize the accident potential for roadside designs found on two-lane highways.(6)

Roadside hazard is ranked on a seven-point categorical scale from 1 (best) to 7 (worst). The

seven categories of roadside hazard rating are defined as follows:

Rating = 1

• Wide clear zones greater than or equal to 9 m (30 ft) from the pavement edgeline.

• Sideslope flatter than 1:4.

• Recoverable.

Rating = 2

• Clear zone between 6 and 7.5 m (20 and 25 ft) from pavement edgeline.

• Sideslope about 1:4.

• Recoverable.

Rating = 3

• Clear zone about 3 m (10 ft) from pavement edgeline.

• Sideslope about 1:3 or 1:4.

• Rough roadside surface.

• Marginally recoverable.

Rating = 4

• Clear zone between 1.5 and 3 m (5 to 10 ft) from pavement edgeline.

• Sideslope about 1:3 or 1:4.

• May have guardrail (1.5 to 2 m [5 to 6.5 ft] from pavement edgeline).

• May have exposed trees, poles, or other objects (about 3 m or 10 ft from pavement

edgeline).
190

• Marginally forgiving, but increased chance of a reportable roadside collision.

Rating = 5

• Clear zone between 1.5 and 3 m (5 to 10 ft) from pavement edgeline.

• Sideslope about 1:3.

• May have guardrail (0 to 1.5 m [0 to 5 ft] from pavement edgeline).

• May have rigid obstacles or embankment within 2 to 3 m (6.5 to 10 ft) of pavement edgeline.

• Virtually non-recoverable.

Rating = 6

• Clear zone less than or equal to 1.5 m (5 ft).

• Sideslope about 1:2.

• No guardrail.

• Exposed rigid obstacles within 0 to 2 m (0 to 6.5 ft) of the pavement edgeline.

• Non-recoverable.

Rating = 7

• Clear zone less than or equal to 1.5 m (5 ft).

• Sideslope 1:2 or steeper.

• Cliff or vertical rock cut.

• No guardrail.

• Non-recoverable with high likelihood of severe injuries from roadside collision.

Figures 8 through 14 present photographs illustrating the seven roadside hazard rating

categories.
191

Figure 8. Typical Roadway with Roadside Hazard Rating Equal to 1.

Figure 9. Typical Roadway with Roadside Hazard Rating Equal to 2.
192

Figure 10. Typical Roadway with Roadside Hazard Rating Equal to 3.

 

Figure 11. Typical Roadway with Roadside Hazard Rating Equal to 4.
193

Figure 12. Typical Roadway with Roadside Hazard Rating Equal to 5.

Figure 13. Typical Roadway with Roadside Hazard Rating Equal to 6.
194

 

Figure 14. Typical Roadway with Roadside Hazard Rating Equal to 7.