Research, Development, and Technology

Turner-Fairbank Highway Research Center

6300 Georgetown Pike

McLean, VA 22101-2296

PUBLICATION NO. FHWA-HRT-23-121 JANUARY 2024

DREDGE (Disaggregate Realistic 

Artificial Data Generator)—Design, 

Development, and Application for 

Crash Safety Analysis, Volume I
FOREWORD

Data-driven safety analysis models help State and local agencies quantify safety data, identify 

high-risk roadway features, and predict the effects of proposed safety measures. However, even a 

model that performs well overall may not accurately represent interactions between variables for 

a specific location or crash because underlying relationships in the real world are unknown. One 

proposed solution is to generate realistic artificial data (RAD) with predetermined safety 

relationships built into them. Since these datasets are known, the RAD can serve as a test bed, 

indicating how well a model reflects underlying cause-and-effect relationships.

This report details a study performed by researchers working under the Federal Highway 

Administration’s Exploratory Advanced Research Program. In the study, the researchers created 

a RAD generation framework on macroscopic and microscopic levels for a diverse selection of 

roadway facility types and developed a web-based tool. The tool’s framework provides users 

with the ability to generate RAD for multiple years at macroscopic segment and microscopic trip 

levels. This report should be of interest to academics and researchers developing crash 

modification factors or functions and statistical models to determine how the models best 

represent real-world relationships. This volume is the first in a series. Volume II in the series is 

FHWA-HRT-23-122. 

Brian P. Cronin, P.E. 

Director, Office of Safety and Operations

Research and Development

Notice

This document is disseminated under the sponsorship of the U.S. Department of Transportation 

(USDOT) in the interest of information exchange. The U.S. Government assumes no liability for 

the use of the information contained in this document.

The U.S. Government does not endorse products or manufacturers. Trademarks or 

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

objective of the document. 

Quality Assurance Statement

The Federal Highway Administration (FHWA) provides high-quality information to serve 

Government, industry, and the public in a manner that promotes public understanding. Standards 

and policies are used to ensure and maximize the quality, objectivity, utility, and integrity of its 

information. FHWA periodically reviews quality issues and adjusts its programs and processes to 

ensure continuous quality improvement. 
TECHNICAL REPORT DOCUMENTATION PAGE

1. Report No.

FHWA-HRT-23-121 

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

4. Title and Subtitle

DREDGE (Disaggregate Realistic Artificial Data Generator)—

Design, Development, and Application for Crash Safety Analysis, 

Volume I

5. Report Date

January 2024

6. Performing Organization Code:

7. Author(s)

John Ivan (ORCID: 0000-0002-8517-4354), Shanshan Zhao 

(ORCID: 0000-0001-5476-6894), Kai Wang 

(ORCID:0000-0003-1452-4000), Oluwaseun Olufowobi 

(ORCID: 0009-0001-1339-9095), Naveen Eluru (ORCID: 0000-

0003-1221-4113), Tanmoy Bhowmik (ORCID:0000-0002-0258-

1692), Lauren Hoover (ORCID: 0009-0004-2431-0860), Md I. 

Jahan (ORCID: 0000-0002-4056-7816), Sudipta Tirtha (ORCID: 

0000-0002-6228-0904), and Mohamad Abdel-Aty (ORCID: 

0000-0002-4838-1573) 

8. Performing Organization Report No.

9. Performing Organization Name and Address

University of Connecticut

Connecticut Transportation Institute

270 Middle Turnpike, Unit 5202

Storrs, CT 06269-5202 

10. Work Unit No.

11. Contract or Grant No.

693JJ31950017 

12. Sponsoring Agency Name and Address

Office of Research and Development

Federal Highway Administration

6300 Georgetown Pike

McLean, VA 22101-2296 

13. Type of Report and Period Covered

Final report: August 2019–January 2024

14. Sponsoring Agency Code

HRSO-2 

15. Supplementary Notes

The contracting officer’s representative was Yusuf Mohamedshah (HRSO-2; ORCID: 0000-0003-0105-5559). 

16. Abstract

Safety analysis primarily focuses on identifying and quantifying the influences of contributing factors to traffic collisions 

and the consequences of these factors. A practice of relying on observed data only allows relative comparisons between 

analysis methods and may not lead to determining how well the methods mimic the true underlying crash-generation 

process, which is often unobserved or known only partially, with varying degrees of certainty. This report describes a 

study, performed by researchers working under the Federal Highway Administration, to help address this data limitation. 

Researchers used two approaches to generate realistic artificial data (RAD): In the macroscopic approach, researchers 

generated RAD at site level by segment or intersection after generating traffic and roadway characteristics by segment or 

intersection for various facility types. Crashes were then generated using known model structures, based on these 

characteristics. The microscopic approach built a high-resolution, disaggregate data-generation process that mimics crash 

occurrences on road facilities at the trip level and accommodates the influence of a full range of crash-contributing 

factors to generate crashes. This crash generation employs data describing the generated trips and involves identifying 

the vehicles involved in the crash, crash location, severity of occupant injuries, and crash type. These generated crashes 

can be aggregated at any spatial or temporal resolution to estimate and evaluate safety models. The researchers thus 

developed a RAD generator as a standalone, customizable software application tool that can prepare multiple realizations 

of RAD. The tool was evaluated using two case studies involving segment crashes and intersection crashes. This volume 

is the first in a series. Volume II in the series is FHWA-HRT-23-122.

17. Key Words

Realistic artificial data, crash analysis, RAD generator

18. Distribution Statement

No restrictions. This document is available to the public through 

the National Technical Information Service, Springfield, VA 

22161. 

https://www.ntis.gov

19. Security Classif. (of this report)

Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of Pages

135 

22. Price

N/A

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

 

 

 

 

 

 
iii

TABLE OF CONTENTS

CHAPTER 1. INTRODUCTION AND MOTIVATION .......................................................... 1

CHAPTER 2. LITERATURE REVIEW .................................................................................... 3

CHAPTER 3. MACROSCOPIC APPROACH FRAMEWORK ........................................... 11

Roadway Data Generation ................................................................................................... 11

Crash Data Generation ......................................................................................................... 13

Data Collection ...................................................................................................................... 14

Roadway Facility Types ....................................................................................................... 15

Roadway Characteristics ...................................................................................................... 16

Crash Types and Severities .................................................................................................. 17

Macroscopic Approach Consolidation and Enhancement ................................................ 17

Roadway Data Generation Inputs ....................................................................................... 19

Crash Data Generation Inputs ............................................................................................ 22

CHAPTER 4. MICROSCOPIC APPROACH FRAMEWORK ............................................ 27

Conceptual Framework ........................................................................................................ 27

Disaggregate Trip Information Generation Module ......................................................... 28

Crash Data Generation Module ........................................................................................ 28

Crash Aggregation Module ............................................................................................... 31

Data Collection ...................................................................................................................... 32

SHRP2 NDS Data ............................................................................................................. 32

CRSS Data ........................................................................................................................ 32

Chicago Trip-Level Data .................................................................................................. 33

Microscopic RAD Module Development ............................................................................ 33

Crash Risk ......................................................................................................................... 34

Crash Location .................................................................................................................. 34

Crash Type ........................................................................................................................ 35

Drivers and Vehicles ......................................................................................................... 35

Crash Severity ................................................................................................................... 36

Microscopic RAD Module Implementation........................................................................ 36

Crash Risk ......................................................................................................................... 36

Crash Location .................................................................................................................. 37

Crash Type ........................................................................................................................ 37

Drivers and Vehicles ......................................................................................................... 37

Crash Severity ................................................................................................................... 38

Validation Check .............................................................................................................. 38

CHAPTER 5. RAD GENERATION TOOL ............................................................................ 39

Macroscopic RAD Datasets .................................................................................................. 39

Microscopic RAD Datasets................................................................................................... 39

CHAPTER 6. CASE STUDY DOCUMENTATION ............................................................... 41

RAD Generation and Validation ......................................................................................... 41

Generation and Validation Context .................................................................................. 41

Data Generation for Case Studies ..................................................................................... 41

Crash Model Estimation ................................................................................................... 42
iv

Application To Rural Two-Lane, Two-Way Undivided Segments .................................. 45

Application To Urban Four-Leg Signalized Intersections ................................................ 64

Validation of Crash Count Prediction ............................................................................... 79

Empirical Analysis for Evaluating Parameter Stability .................................................... 81

Discussion of Findings ...................................................................................................... 87

Driving Simulation Case Study............................................................................................ 88

Introduction ....................................................................................................................... 88

Literature Review.............................................................................................................. 89

Methodology ..................................................................................................................... 91

CHAPTER 7. SUMMARY AND CONCLUSIONS ............................................................... 107

REFERENCES.......................................................................................................................... 109

BIBLIOGRAPHY ..................................................................................................................... 123
v 

LIST OF FIGURES

Figure 1. Equation. Transition probability matrix for discrete Markov chain. ............................. 11

Figure 2. Equation. Transition probability matrix with continuous variable. ............................... 12

Figure 3. Equation. Poisson distribution for crash counts. ........................................................... 13

Figure 4. Equation. Crash prediction model given vector of roadway. ........................................ 14

Figure 5. Equation. Crash prediction model with overdispersion accommodated. ...................... 14

Figure 6. Equation. Crash prediction model with expected crash counts. .................................... 14

Figure 7. Flowchart. RAD generation using a macroscopic approach. ........................................ 18

Figure 8. Equation. Cramer’s V statistic. ...................................................................................... 20

Figure 9. Equation. SPF adjusted by AFs. .................................................................................... 22

Figure 10. Equation. Calibration factor formula for crash count generation. ............................... 26

Figure 11. Flowchart. DREDGE sequential approach I: crash risk to crash type to crash 

severity to crash location. .................................................................................................... 30

Figure 12. Flowchart. DREDGE sequential approach II: crash risk to crash location to 

crash type to crash severity. ................................................................................................ 31

Figure 13. Flowchart. DREDGE generator development for microscopic approach. .................. 34

Figure 14. Chart. Sample variable distribution. ............................................................................ 40

Figure 15. Equation. Poisson regression model. ........................................................................... 42

Figure 16. Equation. NB model. ................................................................................................... 43

Figure 17. Equation. MAD. .......................................................................................................... 45

Figure 18. Equation. MSPE. ......................................................................................................... 45

Figure 19. Equation. Parameter test statistics. .............................................................................. 45

Figure 20. Photo. UConn driving simulator vehicle. .................................................................... 92

Figure 21. Photo. UConn driving simulator control center. ......................................................... 93

Figure 22. Photo. Section of urban roadway used in experiment. ................................................ 96

Figure 23. Photo. Section of rural roadway used in experiment. .................................................. 97

Figure 24. Graph. Conflict severity diagram (Laureshyn and Varhelyi 2020). .......................... 100

Figure 25. Graph. Speed of participants in scenario 1. ............................................................... 103

Figure 26. Graph. Speed of participants in scenario 2. ............................................................... 103

Figure 27. Graph. Speed of participants in scenario 3. ............................................................... 104

Figure 28. Graph. Speed of participants in scenario 4. ............................................................... 104
vi 

LIST OF TABLES

Table 1. Summary of existing literature on RAD generation. ........................................................ 4

Table 2. Initial probability table.................................................................................................... 12

Table 3. Transition probability look-up table. .............................................................................. 13

Table 4. Crash generation parameters for rural, two-lane, two-way, three-leg unsignalized 

intersections. ........................................................................................................................ 24

Table 5. Crash generation parameters for rural, two-lane, two-way, undivided highways. ......... 25

Table 6. Summary of developed models. ...................................................................................... 46

Table 7. Descriptive statistics for continuous variables. ............................................................... 47

Table 8. Descriptive statistics for categorical variables (datasets 1–10). ..................................... 48

Table 9. Poisson regression estimates for dataset 10 (1,000 mi). ................................................. 49

Table 10. NB estimates for dataset 10 (1,000 mi). ....................................................................... 51

Table 11. Random parameters—Poisson regression estimates for dataset 10 (1,000 mi). ........... 53

Table 12. Random parameters—NB regression estimates for dataset 10 (1,000 mi). .................. 55

Table 13. Poisson univariate estimates for dataset 10 (1,000 mi). ............................................... 57

Table 14. Multivariate Poisson lognormal estimates for dataset 10 (1,000 mi). .......................... 59

Table 15. Model prediction using dataset 10 (1,000 mi) for estimation. ...................................... 61

Table 16. Cross validation with dataset 5 (1,000 mi) with parameters estimated using 

dataset 10. ............................................................................................................................ 62

Table 17. Model prediction with dataset 5 (1,000 mi) used for estimation. ................................. 62

Table 18. Cross validation with dataset 10 (1,000 mi) with parameters estimated using 

dataset 5. .............................................................................................................................. 63

Table 19. NB estimates for PDO crashes. ..................................................................................... 65

Table 20. Revised Wald test statistics on NB model parameter estimates (relative to dataset 

10). ....................................................................................................................................... 66

Table 21. Descriptive statistics for continuous variables. ............................................................. 67

Table 22. Descriptive statistics for categorical variables (datasets 1–10). ................................... 68

Table 23. Summary of developed models. .................................................................................... 70

Table 24. Poisson regression model estimation (1,000 intersections). ......................................... 71

Table 25. NB model estimation (1,000 intersections). ................................................................. 73

Table 26. Random parameter Poisson regression (1,000 intersections). ...................................... 75

Table 27. Random parameter—NB regression (1,000 intersections). .......................................... 77

Table 28. Model prediction with dataset 10 (5,000 intersections) used for estimation. ............... 79

Table 29. Cross validation with dataset 5 (5,000 intersections) using dataset 10 model. ............ 80

Table 30. Model prediction with dataset 5 (5,000 intersections) used for estimation. ................. 80

Table 31. Cross validation with dataset 10 (5,000 intersections) using dataset 5 model. ............ 81

Table 32. NB estimates for PDO crashes. ..................................................................................... 82

Table 33. Revised Wald test statistics on NB model parameter estimates (relative to 

dataset 10). .......................................................................................................................... 85

Table 34. Driving simulator participant statistics by scenario. ..................................................... 98

Table 35. Chart. TA values estimated from vehicle speed and distance to collision point 

(Laureshyn and Varhelyi 2020)........................................................................................... 99

Table 36. Example of extracted spreadsheet data. ...................................................................... 102

Table 37. FCW ratios by scenario............................................................................................... 105

Table 38. CMF values by FCW fleet penetration rate. ............................................................... 105
vii 

LIST OF ABBREVIATIONS

AADT annual average daily traffic

AASHTO American Association of State Highway and Transportation Officials

AF adjustment factor

ANN artificial neural network

CMF crash modification factor

CPF cumulative probability function

CRSS Crash Report Sampling System 

CS conflicting speed 

CV connected vehicle

DOCTOR Dutch Traffic Conflict Technique 

DREDGE disaggregate realistic artificial data generator

FCW forward collision warning

FHWA Federal Highway Administration

GIS geographic information system

GOF goodness of fit

GOL generalized ordered logit

HSIS Highway Safety Information System

MAD mean absolute deviation

MDC multiple discrete-continuous 

MMNL mixed multinomial logit

MNL multinomial logit

MPR market penetration rate

MSPE mean-squared prediction error

NB negative binomial

NDS Naturalistic Driving Study

NHTSA National Highway Traffic Safety Administration

OL ordered logit

POLARIS Polychotomous Choice Agent-Based Risk Model for Integrated Travel Demand 

and Network and Operations Simulation

RAD realistic artificial data

SHRP2 Strategic Highway Research Program 2

SPF safety performance function

STCT Swedish Traffic Conflict Technique 

TA time to accident

UConn University of Connecticut 

USTCT U.S. Traffic Conflict Technique

1 

CHAPTER 1. INTRODUCTION AND MOTIVATION

Safety analysis primarily focuses on identifying and quantifying the influences of factors that 

contribute to traffic collisions and the consequences of these factors. A traditional analysis 

paradigm that relies on observed data only allows relative comparisons between analysis 

methods and lacks the ability to show how well the methods mimic the true underlying 

crash-generation process. This process is often unobserved or known only partially, with varying 

degrees of uncertainty. At the same time, existing data sources and the availability of data for 

model calibration and validation pose significant challenges to safety performance and crash 

modification analysis. Most safety performance analysis employs cross-sectional and time series 

datasets. Researchers make assumptions about the data, but whether these assumptions truly 

characterize the safety data that are generated in the real world often remains unknown. 

One possible solution to address this issue is for researchers to artificially generate realistic 

artificial data (RAD) by making assumptions about the underlying crash generation process 

(Bonneson and Ivan 2013). These generated RAD can then provide a context for investigating 

various questions and verifying assorted assumptions related to safety performance and crash 

modification analyses. The idea is that these RAD can potentially be aggregated at any spatial or 

temporal resolution to mimic data from the real world. Furthermore, because researchers 

generating RAD have complete knowledge of the data-generation process, they can compare the 

performance of varied safety analysis methods. 

In this study, the research team proposed two approaches to generate RAD: a macroscopic and a 

microscopic approach. In the macroscopic approach, researchers generated RAD at the site level 

by segment or intersection. Roadway sites that contained traffic and roadway characteristics—

such as annual average daily traffic (AADT), lane width, shoulder width, curvature, and speed 

limit—were generated first, by facility type. The team then generated crashes using known 

model structures (e.g., Poisson and negative binomial (NB) models), based on generated sitelevel characteristics. Meanwhile, the microscopic approach built a high-resolution disaggregate 

data-generation process that mimicked crash occurrences on road facilities at the trip level and 

accommodated the influence of a full range of crash-contributing factors.

The microscopic approach employed detailed information for trips, including start and end time, 

start and end location, characteristics (including solo or group), vehicle used, and precise route. 

The crash generation process involved employing gathered data to identify vehicles involved, 

location, injury severity (including fatal, incapacitating, non-incapacitating, and uninjured), and 

crash type (including head-on, rear-end, and vehicle-pedestrian). The generated crashes were 

further aggregated based on location (intersections versus segments) and time (day, week, 

month, and year).

Next, the research team conducted two types of case studies to supplement RAD generation and 

validate the proposed RAD frameworks. The first case study estimated different types of 

statistical models, using datasets generated by the RAD procedure developed in this study, to 

illustrate how RAD could be used to make comparisons between analysis methods. The second 

case study used driving simulation to evaluate the effects of an advanced forward collision 

warning (FCW) system on crash occurrence.
2 

The three objectives of this study were as follows:

1. Develop frameworks for RAD generation that can be used to evaluate methods used for 

both safety performance and crash modification analysis: The proposed RAD framework 

operates at both the site and trip level and incorporates the influence of a full range of 

crash-contributing factors. RAD frameworks are general enough to generate crash data 

for all roadway facility types, including segments and intersections. The frameworks can 

also generate data for different combinations of inputs, including modeling methods, 

model formulation, input specification, and unobserved heterogeneity. Researchers 

specifically focused on generating RAD to address known knowledge gaps related to 

aggregation issues, statistical and structural methodologies for safety analysis, and crash 

data challenges.

2. Develop DREDGE as a stand-alone software application: The application is customizable 

and can be executed to prepare multiple realizations of RAD. The application embeds 

features so RAD can be distributed efficiently; this embedding ensures that the integrity 

of the RAD process for evaluating methods is maintained. The stand-alone RAD software 

can also be used to study the uncertainty and stochasticity inherent to RAD generation 

and to evaluate model robustness.1

3. Demonstrate proposed RAD tool feasibility and applicability using two types of case 

studies: The first case study estimates different types of statistical models using the RAD 

generated from this study to demonstrate how the RAD can be used to make comparisons 

between analysis methods. The second case study uses driving simulation to evaluate the 

effects of an advanced FCW system on crash occurrence.

1

FHWA. DREDGE (standalone RAD software).
3 

CHAPTER 2. LITERATURE REVIEW

The research team conducted a comprehensive review of previous research efforts on RAD 

approaches across various domains: statistics, econometrics, computer science, ecology, 

medicine, and psychology. In all these disciplines, the primary goal was to assess the ability of 

analysis methods to draw inferences about the underlying assumptions and assertions that 

generated the data. Researchers followed criteria to select studies for this review based on a 

simple core principle of RAD generation. The data generated in the research effort had to be 

based on a framework that was built on research assumptions—as opposed to real observed 

data-based simulation efforts.

The criteria eliminated two major sets of transportation studies that generate simulated data. 

First, several travel demand modeling forecast systems, such as activity-based models and 

synthetic population generators, generate individual-level synthetic data—for example, Eluru, 

Bhat, and Hensher (2008), Kitamura et al.(2000), and Konduri et al.(2016). However, the 

generation is entirely based on models estimated using observed data. Second, artificial data are 

generated in microsimulation frameworks for traffic flow modeling. In these studies, the 

simulated data are generated based on well-calibrated traffic flow models—for example, Asano, 

Iryo, and Kuwahara (2010); Mamun et al. (2020); Ranade, Sadek, and Ivan (2007); and Yu and 

Abdel-Aty (2014). Hence, these studies are also not appropriate for this review.

During the review process, the research team identified 30 research studies (based on the RAD 

generation criteria) that employed artificial data generation in their analyses. These studies 

included transportation (including transportation safety and travel behavior), medical science, 

data science, education, ecological modeling, information analytics, and environmetrics. The 

research team then prepared a summary of the literature review with the objective of developing 

a comprehensive RAD framework. Thus, as opposed to providing a study-by-study summary of 

earlier research, this review provides insight on the important elements of RAD frameworks that 

can be observed from earlier research efforts.

A concise summary of earlier research efforts on RAD generation is presented in table 1. This 

table provides information on study objectives, dataset adopted and study region (if known), 

software and procedures followed for generating RAD, conceptual methods and framework 

employed, contributing field of the study (for example transportation safety), and exact nature of 

the dependent variable (categorical or continuous). For the ease of presentation, the studies 

presented in table 1 are categorized into two groups based on the discipline of the study: first, 

studies related to transportation and second, studies related to other disciplines, including 

statistics, economics, ecology, and computer science. 
4 

Table 1. Summary of existing literature on RAD generation. 

Study

Study 

Objective(s)

Dataset 

Adopted 

(Study 

Region if 

Known)

Software/Procedure for RAD 

Generation Conceptual Methods Field No. of Alt.

Transportation Domain

Bhat (2003)

To propose the 

use of scrambled 

Halton sequence 

for simulation 

estimation.

Simulated 

dataset

Generated data using unordered 

discrete choice models using 

the matrix programming 

language GAUSS (Aptech 

Systems 2023).

Mixed probit and multinomial probit models Travel behavior 3 

Cummings, 

McKnight, and 

Weiss (2003)

To review three 

methods for 

estimating relative 

risks in matchedpair crash data.

Simulated 

and 

observed 

dataset

Generated crash data 

employing Stata® Statistical 

Software with an assumed 

probability of fatality as a 

function of speed and seatbelt 

use (StataCorp 2023).

Mantel–Haenszel stratified methods, doublepair comparison method, conditional Poisson 

regression, 

and Cox proportional hazards regression 

Safety 2 

Salim et al. 

(2007) 

To simulate 

intersection 

environment and 

collision and 

traffic data 

learning.

Simulated

dataset

Vehicles are generated with 

different speed, position, and 

trajectory. 

Ubiquitous intersection awareness 

framework Safety — 

Paez and Scott 

(2007) 

To develop a 

discrete choice 

model that 

incorporates 

elements of social 

influence and 

more conventional 

factors.

Simulated 

dataset

A Monte Carlo simulation was 

designed to explore the 

properties of the econometric 

model proposed. 

Logit probability formulation Travel behavior — 

Bhat et al. 

(2010) 

To propose a 

CML approach to 

estimate 

ordered-response, 

discreet choice 

models with 

flexible, copulabased spatial 

correlation 

structures.

Simulated 

and 

observed 

dataset (San 

Francisco 

Bay area)

Considered three independent 

variables. Drew values from 

univariate normal distribution. 

Assumed fixed coefficients. 

Generated error terms using 

correlation structure. Generated

25 different datasets with 500 

observations. 

Copula-based, spatial ordered-response 

model structure Travel behavior 4 
5 

Study

Study 

Objective(s) 

Dataset 

Adopted 

(Study 

Region if 

Known) 

Software/Procedure for RAD 

Generation Conceptual Methods Field No. of Alt.

Bhat and 

Sidharthan 

(2010) 

To investigate the 

ability of the 

MACML 

estimator to 

recover 

parameters from 

finite samples.

Simulated 

dataset

Considered five independent 

variables. Drew values from 

univariate normal distribution. 

Assumed random coefficients. 

Generated error terms from 

univariate normal distribution 

with 0.5 variance. Generated 20 

datasets with 5,000 

observations. 

Cross-sectional random coefficients model, 

panel interindividual random coefficients 

model, panel intraindividual and 

interindividual random coefficients 

Travel behavior 5 

Pinjari and 

Bhat (2010)

To investigate 

nonworker 

out-of-home 

discretionary 

activity time-use 

and activity 

timing decisions 

on weekdays.

Simulated 

and 

observed 

dataset (San 

Francisco 

Bay area)

Assumed independent variable 

values were uniformly 

distributed. Assumed 

coefficients were nested 

extreme values. Generated data 

for 2,500 hypothetical 

individuals, with an assumption 

that each individual chose the 

value to maximize the total 

random utility.

MDCNEV Travel behavior 3 

Ferdous et al.

(2010) 

To model 

interactions in 

nonwork activity 

decisions across 

household and 

non-household 

members at the 

level of activity 

generation. 

Simulated 

and 

observed 

dataset 

Drew values for independent 

variables from univariate 

normal distribution. Assumed a 

fixed coefficient and used it to 

compute the utility for each 

individual using a linear 

combination. Generated error 

terms using predefined 

correlation structure. Repeated 

the process at least 50 times. 

Multivariate, ordered-response system 

framework Travel behavior 3, 4, 5 

Ye and Lord 

(2011) 

To examine the 

effects of 

underreporting 

crash data.

Simulated 

and 

observed 

dataset

(Texas)

Weighted exogenous sample 

maximum likelihood estimator

method was used for 

appropriately weighting the 

crash outcomes to address the 

underreporting issue in crash 

data. 

MNL, OP, and ML models Safety 5 
6 

Study

Study 

Objective(s) 

Dataset 

Adopted 

(Study 

Region if 

Known) 

Software/Procedure for RAD 

Generation Conceptual Methods Field No. of Alt.

Geedipally, 

Lord, and 

Dhavala (2012) 

To apply an NB, 

generalized linear 

model with 

Lindley mixed 

effects to the 

analysis of traffic 

crash data.

Simulated 

and 

observed 

dataset (road 

segments in 

Indiana, 

Michigan) 

NB NB-Lindley generalized linear model Safety 0 to α

Lord and Kuo 

(2012) 

To examine the 

effects of site 

selection criteria.

Simulated

dataset

Used R software to generate 

sites with crash counts with a 

predefined overall mean for 

different dispersion parameters 

(R Foundation 2021). 

Compared four types of before and after 

studies: Naïve method, control group 

method, EB method based on the method of 

moment, and EB method based on a control 

group. 

Safety — 

Bhat et al. 

(2013) 

To apply the 

MACML 

approach for 

multiple MDCP 

models. 

Simulated 

dataset 

(Michigan) 

Considers five independent 

variables. Draws values from 

univariate normal distribution. 

Generates error terms from 

positive covariance matrix. 

Undertakes data generation 

process 20 times with different 

realizations of coefficient and 

error terms.

MACML Travel behavior 5, 10

Eluru (2013)

To investigate the 

performance of 

ordered and 

unordered injury 

severity response 

frameworks. 

Simulated 

dataset

Considered three independent 

variables. Assumed parameters 

that provide the same aggregate 

shares. Generated 5 realizations 

of the data with 5,000 

observations each for each 

proportional value. Generated a 

total of six aggregate sample 

shares.

MNL, OL, and GOL Safety 4 

Paleti and Bhat 

(2013) 

To compare the 

MSL inference 

and CML 

approaches. 

Simulated 

dataset

Drew independent variables 

from univariate normal 

distribution while coefficients 

were assumed and drawn from 

multivariate normal 

distribution. Considered both 

independent and correlated 

realizations. Generated data at 

least 50 times.

CML and MSL approach Travel behavior 5 
7 

Study

Study 

Objective(s) 

Dataset 

Adopted 

(Study 

Region if 

Known) 

Software/Procedure for RAD 

Generation Conceptual Methods Field No. of Alt.

Wu, Lord and 

Zou (2015) 

To generate CMFs 

using a regression 

model to estimate 

the crash counts 

and compare with 

the actual crash 

distribution 

Simulated 

dataset

Assumed CMF values for lane 

width, curve density, and 

pavement friction and used 

them to generate simulated 

crash counts.

Evaluated the conditions for adopting NB 

regression models for before after studies. Safety 0 to α

Council et al. 

(2017) 2

To use ARD to 

assess the 

performance of 

cross-sectional 

analysis methods.

RAD (rural 

two-lane 

highways, 

Washington) 

Implemented data generation 

by SAS® programs based on an 

assumed model structure for 

AADT and roadway geometry 

factors. 

All available methods of crash count and 

severity analysis Safety 0 to α

Nontransportation Domain

Gamel and 

Vogel (1997)

To compare 

parametric and 

nonparametric 

survival methods.

Simulated

clinical 

dataset

Estimated three parametric 

models using data from breast 

cancer trials. Assumed the 

relationships generated survival 

data in HT Basic 

(Transera 1991). 

Parametric (log-normal) and nonparametric 

test (logrank test, Gray-Tsiatis and 

Laska-Meisner methods) 

Medical science 0 to α

Scott and 

Wilkins (1999)

To evaluate data 

mining 

procedures. 

Artificial 

dataset

Proposed two alternative ways 

of generating artificial data for 

testing data mining approaches.

Self-similarity, classification, and lazy trees Data science 0 to α

Bifulco and 

Bretschneider 

(2001) 

To compare 

methods for 

assessing school 

performance. 

Artificial 

dataset

Generated datasets from loglinear relationships between 

three inputs and two outputs 

and associated parameters. 

Generated datasets without 

bias, with measurement error 

and endogeneity.

Data envelopment analysis and corrected 

ordinary least squares Education — 

Austin et al. 

(2006) 

To evaluate 

statistical methods 

for predicting 

plant species 

distributions. 

Artificial

dataset

Generated data based on two 

plant community theoretical 

models using the computer 

package COMPAS (Minchin 

1987). 

Generalized linear models and generalized 

additive models Ecological modeling 0 to α

2

Council, F., E. Hauer, B. Lan, D. Harwood, and R. Srinivasan. 2017. Use of “Artificial Realistic Data” (ARD) To Assess the Performance of 

Cross-Sectional Analysis Methods in Capturing Causal Relationships Between Individual Roadway Attributes and Safety. Unpublished Report. Washington, DC: 

Federal Highway Administration.
8 

Study

Study 

Objective(s) 

Dataset 

Adopted 

(Study 

Region if 

Known) 

Software/Procedure for RAD 

Generation Conceptual Methods Field No. of Alt.

Bzdusek and 

Christensen 

(2006) 

To compare a new 

variant of PMF 

with other 

receptor modeling 

methods. 

Artificial 

dataset

Generated data set using 

literature source profiles and 

postulated source contributions.

Eigenvalue-based methods and PMF-based 

methods Environmetrics — 

Whiting, Hack, 

and Varley 

(2008) 

To describe a 

method and 

toolset for 

creating realistic, 

synthetic test data.

Realistic 

simulated 

dataset

Generate simulated datasets by 

embedding ground truths. Core 

software and utility software 

were written in JAVA® 

(Arnold, Goslin, and Holmes 

2005). 

Threat stream generator Information analytics — 

Potharst, BenDavid, and 

Van Wezel 

(2009) 

To generate 

monotone 

ordinal datasets.

Observed 

and artificial 

dataset

Used a machine-learning 

algorithm to generate the data.

Algorithms for generating structured and 

unstructured random monotone datasets Data science — 

Zimmermann 

(2012) 

To generate 

diverse datasets 

reflecting realistic 

data 

characteristics.

Artificial 

dataset

Implemented data generator in 

JAVA. Episode mining Data science — 

Devroye, 

Felber, and 

Kohler (2012) 

To estimate a 

density using real 

and artificial data.

Observed 

and artificial 

dataset

Implemented data generator in 

R software. Generated artificial 

data from a regression analysis 

of observed data.

Classical model, finite information model, 

and full information model Data science 0 to α

Hazwani et 

al.(2016) 

To develop an 

automatic 

artificial data 

generator for 

generating 

artificial datasets 

based on real data.

Artificial 

and real 

dataset

Used random permutation 

algorithm to generate different 

sets of artificial data that 

represent realistic data.

Four-phase framework for data generation

Information and 

communication 

technologies 

— 

Dahmen and 

Cook (2019)

To introduce a 

synthetic data 

generation 

method. 

Simulated 

and real 

dataset

SynSys, a machine

learning-based synthetic 

data-generation method.

SynSys, similarity measures, and 

semisupervised learning Medical science — 

Alt. = alternatives; CML = composite marginal likelihood; CMF = crash modification factor; no. = number; EB = empirical Bayes; MACML = maximum 

approximate CML; MDCNEV = multiple discrete-continuous nested extreme value; MDCP = multiple discrete-continuous probit; ML = mixed logit; 

MSL = maximum-simulated likelihood; OP = ordered probit; PMF = Positive matrix factorization. 

*After a concept proposed by Hauer, as presented in Harwood et al. (2003).
9 

Researchers made five important observations based on the information seen in table 1. First, 

earlier research explored RAD applications for wide-ranging topics, including 

statistical/econometric model performance and comparison, travel-demand forecasting, routechoice behavior, and data mining. Second, RAD applications have typically been developed 

using several software packages or platforms, including R, GAUSS, and COMPAS

(R Foundation 2021; Aptech Systems 2023; Minchin 1987). Third, employing RAD datasets, 

researchers considered the performance of several model structures, including ordered logit 

(OL), multinomial logit (MNL), generalized ordered logit (GOL), mixed multinomial logit 

(MMNL) models, probit models (and their cross-sectional and panel variants), multiple discretecontinuous (MDC) frameworks with probit and extreme value formulations, and artificial neural 

networks. Fourth—notably—studies within the transportation domain traditionally adopt RAD 

approaches for econometric models. However, non-transportation domain research typically is 

more focused on machine-learning and data-mining approaches. Fifth, the number of alternatives 

in the RAD variables studied is related to the problem context. The number of alternatives for a 

RAD variable could range from a small number to a very large number. If the RAD variable is a 

binary outcome variable, such as crash/no crash, then the number of alternatives will be two. 

However, if the RAD variable is a continuous value (such as vehicle miles traveled by a 

household), then the number of possible alternatives is, theoretically, infinite. 

The literature review clearly highlights the absence of a single or prevalent software framework 

for developing RAD across various application domains. Several computer programming 

languages, matrix programming languages, and statistical software packages have been 

employed for developing RAD-based frameworks. Thus, RAD framework conceptualization can 

be platform-agnostic. This fact is particularly beneficial in the current context because no 

inherent limitations exist in adopting an open-source software platform for RAD implementation. 

Thus, the framework developed can be widely deployed by Federal Highway Administration 

(FHWA) without any constraints.

The authors of the report noted that the most common experimental set-up for RAD generation 

seen in this review was a statistical or econometric model framework assumed to represent the 

data-generation process of the variable of interest. Within the model framework, independent 

variables (either based on observed data or random realizations) and their impact (defined by 

coefficient values) on the dependent variable are assumed. Using the independent variable 

distribution and coefficient values, a latent propensity (or utility) is computed based on the 

model system. For example, if the dependent variable is a count variable and the model system is 

an NB system, a single propensity is generated with an appropriate overdispersion component. 

However, if the model system represents an unordered discrete outcome model, alternative 

specific utilities are generated. After the model structure is assumed, a random error term is 

added to (or multiplied by) the propensity or utility generated. Subsequently, a choice scenario is 

determined for each record based on the properties of the model. This choice scenario results in 

the formation of the RAD dataset. The process is repeated several times to generate multiple 

copies of the datasets. The choice variable of interest, such as crash occurrence, may possibly be 

a result of several layers of decisions, such as trip, route, and time-of-day decisions. The research 

team’s vision was to use a similar approach to this framework to generate RAD after the 

conceptual framework for the study was finalized. 
10 

Additionally, researchers observed during their literature review that the embedded RAD 

frameworks were consistently single-level frameworks; in other words, the underlying decision 

process consisted of only one layer of decisions. Earlier research in modeling crash occurrence 

related crash occurrence to roadway geometry and traffic volume under prespecified assumptions 

of what variables would influence crash occurrence (for example, AADT and lane width). This

study’s research effort was the first to attempt the development of RAD datasets using a 

multilayered decision process; as such, researchers anticipated the effort would be challenging. 

Hence, the research team was cautious in determining the number of decisions—and thus the 

number of layers—that could be modeled to determine the crash dependent variables in this 

study. As the study commenced in full swing, the research team incorporated multilayered 

complexity within the decision processes to enhance the current state of RAD generation across 

multiple transportation domains.
11 

CHAPTER 3. MACROSCOPIC APPROACH FRAMEWORK

The macroscopic approach has two steps. The first step is to generate roadway data, such as 

AADT, truck percentage, lane width, and shoulder width. The second step is to generate crash 

counts by crash type and severity, based on roadway-level data and statistical and econometric 

methodologies. 

ROADWAY DATA GENERATION 

The basic philosophy for generating roadway data follows the principle of the Markov chain 

(Gagniuc 2017). Given a sequence of random variables X1, X2, X3, ... , Xn (where X represents a 

vector of roadway characteristics), the indices (1, 2, 3, ..., n) represent the walking state in time 

(where the indices represent the contiguous sites for roadway network). The Markov chain 

assumes that the roadway characteristics in the next state depends only on the previous state. A 

higher probability may exist that the values for these roadway characteristics on a given site will 

remain the same as those of the previous site, with lower probabilities that the values will change 

to other possible combinations. 

Overall, a transition probability matrix is required by the Markov chain principle. This matrix 

presents the probabilities for possible roadway characteristics on one site, given the roadway 

characteristics on the previous site. Two types of transition probability matrices are included in 

the Markov chain principle: one matrix is for a discrete Markov chain, and the other matrix is for 

a continuous Markov chain. The discrete Markov chain transition probability matrix is applied to 

the categorical variables in this study—such as lane width, shoulder width, and speed limit—and 

is illustrated in figure 1, which uses numbers that are randomly created to provide a visual 

example.

Figure 1. Equation. Transition probability matrix for discrete Markov chain.

Where: 

I = 1, 2, 3, .... 

n = the ith combination of roadway characteristics, such as 1 = 12-ft lane width, 6-ft shoulder 

width, and 55-mph speed limit while 2 = 11-ft lane width, 4-ft shoulder width, and 45-

mph speed limit. 

Pi = the ith combination of roadway characteristics for the previous site. 

Ci = the ith combination of roadway characteristics for the subject site.
12 

A probability under row Pi and column Cn in the transition probability matrix = the 

probability of roadway characteristics to be the nth combination for the subject site, given 

the ith combination of roadway characteristics for the previous site. 

Note that the probabilities in each row and column must sum to unity in figure 1. 

When a variable is continuous, such as AADT or truck percentage, the transition probability 

matrix is usually represented by a distribution, which can be written as shown in figure 2. 

Figure 2. Equation. Transition probability matrix with continuous variable.

That is, the probability distribution of a roadway characteristic for the subject site follows a 

normal distribution, with a mean equal to the roadway characteristic for the previous site and a 

predefined variance σ2

. 

To illustrate the theoretical process of roadway data generation using the Markov chain, one 

must first generate an initial probability matrix. This initial probability matrix is used to define 

the first site in the RAD, since this site cannot be defined from the transition probability matrix 

in the Markov chain. Table 2 shows a hypothetical probability distribution lookup table, which is 

used to generate all combinations of roadway characteristics for the first roadway site. For 

example, suppose four combinations of roadway characteristics are generated in the RAD, such 

as different combinations of values for lane width, shoulder width, and speed limit. A random 

number between 0 and 1 can be generated and compared to the cumulative probability in table 2

to define the roadway characteristics for the first site. For example, if the random number 

generated is equal to 0.8, then the roadway characteristics following under combination C will be 

set for the first site.

Table 2. Initial probability table.

Roadway Characteristics Initial Probability Cumulative Probability

Combination A 0.25 0.25 

Combination B 0.30 0.55 

Combination C 0.35 0.90 

Combination D 0.10 1.00 

The next step is to generate all remaining sites based on the first site and the transition 

probability matrix using the Markov chain principle. First, generate a transition probability 

matrix from the real data, with the transition probabilities listed in a look-up table (table 3). 

Following this first step, the first site has roadway characteristics of combination C. A random 

number between 0 and 1 can now be generated for the second site, and the random number can 

be compared to the cumulative probability in table 3, where the roadway characteristics for the 

previous site follow combination C to determine the roadway characteristics for the subject site. 

For example, if the random number generated is equal to 0.6, the roadway characteristics for the 

subject site will stay the same as they were. If the random number generated is equal to 0.95, 

then the roadway characteristics for the subject site will change to those values under 

combination D. This process will be repeated until the last site is generated.
13 

Table 3. Transition probability look-up table.

Roadway 

Characteristics for 

Previous Site

Roadway Characteristics 

for Subject Site

Transition

Probability

Cumulative 

Probability

Combination A 

Combination A 0.70 0.70 

Combination B 0.10 0.80 

Combination C 0.10 0.90 

Combination D 0.10 1.00 

Combination B 

Combination A 0.15 0.15 

Combination B 0.75 0.90 

Combination C 0.05 0.95 

Combination D 0.05 1.00 

Combination C 

Combination A 0.10 0.10 

Combination B 0.15 0.25 

Combination C 0.65 0.90 

Combination D 0.10 1.00 

Combination D 

Combination A 0.05 0.05 

Combination B 0.05 0.10 

Combination C 0.05 0.15 

Combination D 0.85 1.00 

CRASH DATA GENERATION

After generating the roadway data, crash counts by crash type and severity are generated using 

the known model structures (for example, Poisson and NB models) and “realistic” relationships 

between crash counts and roadway-level characteristics. Details about how to define the 

“realistic” relationships between crashes and roadway variables are discussed in section “Crash 

Data Generation Inputs” below. 

Suppose a crash prediction model or safety performance function (SPF) exists for the sites 

generated in the roadway data-generation process, and this model or SPF contains the 

coefficients for the relationships between crash counts and all roadway characteristics and other 

model-related parameters. This model or SPF can then be used to estimate the parameters of the 

assumed underlying distribution for each site and then simulate the crash count as a random 

variable. For example, using the two commonly used model frameworks, the Poisson and NB 

models, to illustrate the process. The Poisson distribution for the crash counts can be written as 

shown in figure 3. 

Figure 3. Equation. Poisson distribution for crash counts.

Where: 

Prob[yi|μi] = probability of y crashes occurring at site i.

μi = expected number of crashes at site i. 
14 

Given the vector of roadway site characteristics Xi and the vector of coefficients β in the crash 

prediction model, the expected crashes μi can be predicted as shown in figure 4. 

Figure 4. Equation. Crash prediction model given vector of roadway.

If the overdispersion is accommodated by the crash prediction model, then the NB model is used 

to estimate the vector of coefficients ββ, and the expected crashes μμii can be predicted as shown 

in figure 5. 

Figure 5. Equation. Crash prediction model with overdispersion accommodated.

where:

exp(εi) = error term assumed to follow gamma distribution with mean 1 and variance Error! 

Digit expected.1/σ = k. 

k denotes the overdispersion parameter in the NB model. 

Given the expected crashes μμii, the distribution of crash counts in the NB model can be calculated 

as shown in figure 6. 

Figure 6. Equation. Crash prediction model with expected crash counts.

Where Γ is the common gamma function.

Regardless of which statistical and econometric model is used to estimate the expected crash

counts μi for each site, its probability distribution function can be used to calculate both the 

probabilities and cumulative probabilities of observing 0, 1, 2, and up to yi crashes, respectively, 

for each site. For example, figure 3 and figure 6 illustrate the functions for Poisson and NB, 

respectively. Then, the actual crash counts can be determined for each site using a random 

variable generation procedure similar to the roadway data generation. For example, a random 

number between 0 and 1 can be generated and compared to the cumulative probabilities of 

different crash counts to define the final crash counts.

DATA COLLECTION

The research team collected data from multiple sources to support generating the transition 

probability matrix for the Markov chain principle in the macroscopic approach. Specifically, the 

research team collected the data of all seven States in the current Highway Safety Information 

System (HSIS), including California, Illinois, Maine, Minnesota, North Carolina, Washington, 

and Ohio (FHWA n.d.). Additionally, the research team collected extra data from the Ohio, 
15 

Connecticut, and Florida departments of transportation (DOTs) to supplement the HSIS data 

(Ivan et al. 2021). 

ROADWAY FACILITY TYPES

In this study, the research team attempted to cover RAD generation for as many types of 

roadway facilities as possible, based on the facility categorizations in the current Highway Safety 

Manual (HSM) (American Association of State Highway and Transportation Officials

(AASHTO) 2010). However, the team decided (based on experiences in other safety-related 

studies) that freeway ramp and ramp terminal data would be too challenging to attempt to obtain. 

Therefore, the team mainly focused on all the HSM roadway facility types except these two ramp 

types. However, the team also omitted the following facility types from consideration due to 

too-small sample sizes in their data: urban and suburban three-lane arterials with a center 

two-way, left-turn lane; urban and suburban five-lane arterials with a center two-way, left-turn 

lane; and rural eight-lane freeway segments. Specifically, the research team considered the 

following facility types:

• Rural two-lane, two-way roadways: 

o Two-lane, two-way undivided segments. 

o Three-leg unsignalized intersections with stop control on approaches to minor roads. 

o Four-leg unsignalized intersections with stop control on approaches to minor roads. 

o Four-leg signalized intersections.

• Rural multilane highways:

o Four-lane undivided segments. 

o Four-lane divided segments.

o Three-leg unsignalized intersections with stop control on approaches to minor roads. 

o Four-leg unsignalized intersections with stop control on approaches to minor roads. 

o Four-leg signalized intersections.

• Urban and suburban arterials:

o Two-lane undivided arterials.

o Four-lane undivided arterials.

o Four-lane divided arterials (i.e., including a raised or depressed median). 

o Three-leg unsignalized intersections with stop control on approaches to minor roads. 

o Three-leg signalized intersections. 

o Four-leg unsignalized intersections with stop control on approaches to minor roads. 

o Four-leg signalized intersections.

• Freeway segments:

o Rural four-lane divided freeway segments.

o Rural six-lane divided freeway segments.

o Urban four-lane divided freeway segments.

o Urban six-lane divided freeway segments.

o Urban eight-lane divided freeway segments.

o Urban 10-lane divided freeway segments.
16 

ROADWAY CHARACTERISTICS

Roadway characteristics highly depend on data availability because both the initial probability 

and transition probability in the Markov chain for each characteristic must be calculated from the 

existing data. The research team attempted to include as many roadway characteristics as 

possible in RAD generation, as long as existing data supported the analysis. 

Again, the research team defined roadway characteristics for their RAD generation mainly as 

they were defined in the HSM (AASHTO 2010). The following characteristics were included: 

• Segment-related characteristics:

o AADT.

o Lane width. 

o Left shoulder width. 

o Right shoulder width. 

o Left shoulder type. 

o Right shoulder type. 

o Median width. 

o Median type.

o Horizontal curve-related factors.

o Vertical curve-related factors.

o Centerline rumble strip.

o Two-way left-turn lane.

o Left shoulder rumble strip.

o Right shoulder rumble strip.

o Passing lane.

o Lighting. 

o Roadside characteristics.

o On-street parking.

• Intersection-related characteristics:

o Major road AADT.

o Minor road AADT.

o Number of approaches with exclusive left-turn lanes.

o Number of approaches with exclusive right-turn lanes.

o Intersection lighting.

o Skew angle. 

o Speed limit.

o Left-turn signal phasing. 

o Number of approaches with right-turn-on-red prohibition. 

o Number of lanes to be crossed by a pedestrian.

o Presence of school(s).

o Number of alcohol stores. 

o Number of bus stops. 
17 

CRASH TYPES AND SEVERITIES

In crash data generation, the research team generated crash counts by different crash types and 

severities, as defined in Model Minimum Uniform Crash Criteria (NHTSA 2017). These crash 

counts include the following:

• Crash severity:

o K—Fatal crash.

o A—Incapacitating crash.

o B—Non-incapacitating crash.

o C—Possible crash.

o PDO—Property damage-only crash. 

• Crash type: 

o Single-vehicle crash.

o Fixed-object crash.

o Overturn/rollover crash. 

o Multivehicle crash:

o Angle crash.

o Head-on crash. 

o Rear-end crash. 

o Sideswipe—same-direction crash.

o Sideswipe—Opposite-direction crash.

Macroscopic Approach Consolidation and Enhancement

The overall procedure for generating RAD for the consolidated frameworks for both roadway 

and crash data generation using the macroscopic approach is presented in figure 7; RAD were 

generated by facility type. 
18 

Source: FHWA.

Figure 7. Flowchart. RAD generation using a macroscopic approach.

In terms of a roadway segment-related facility, the research team’s first step was to determine the 

data to be used for generating the initial probability matrix and transition probability matrix in 

the Markov chain principle. The second step was to divide the original data into 0.01-mi units 

with the roadway characteristics to be generated in RAD. Researchers performed this second step 
19 

because 0.01 mi is usually the smallest resolution in most real data, and dividing all sites into the 

same length could help prevent biased results. Next, these two matrices were generated based on 

the 0.01-mi units for all roadway characteristics. After this step, researchers could operate the 

simulation to generate a set of 0.01-mi roadway sites based on the random number criteria 

described in the section “Crash Data Generation.” These contiguous sites could then be 

aggregated into longer roadway segments based on the values of roadway characteristics. These 

steps then culminated with the research team generating the final roadway data in RAD.

At this point, researchers could then generate crash counts by crash type and severity, based on 

the roadway data. Selecting and defining crash prediction models and coefficients for roadway 

characteristics were critical steps in predicting the expected crashes for each site. Two options 

were primarily considered: The first option was using models and parameters in existing research 

and products, such as the HSM (AASHTO 2010). The second option was using coefficients 

estimated by the research team. Researchers predicted that estimating coefficients would be 

challenging. This challenge was due to data limitations, because none of the existing datasets 

contained all the roadway characteristics to be generated in RAD. Thus, the statistical 

significance of the parameters estimated in the models for all crash types and severities could not 

be guaranteed. Therefore, the team selected the first option, which might result in more 

flexibility and variation in the data.

This option was mainly applicable to the segment-related facilities. The Markov chain was not 

directly applicable to intersections because intersections are not contiguous. Instead of using the 

transition probabilities in the Markov chain principle, researchers could simulate the roadway 

characteristics at intersections, based on the intersection’s probability distributions, by 

intersection type in real data. The crash data-generation process was the same as the generation 

process for RAD for segments.

ROADWAY DATA GENERATION INPUTS

During RAD generation, researchers should seek to generate roadway and crash data that are as 

realistic as possible. In terms of roadway data generation, some roadway characteristics are 

highly correlated, including lane width and shoulder width, roadway type, and speed limit. Thus, 

their correlations should be accounted for by the RAD frameworks, so researchers first 

conducted a correlation test for all available roadway characteristics under each facility type. 

This testing allowed researchers to identify variables that were highly correlated in real data and 

so should have been generated together in RAD. To do this testing, Cramer’s V statistic was 

calculated between each pair of roadway characteristics to determine the strength of their 

correlation (Crewson 2006). This statistic can be expressed as shown in figure 8. 
20 

Figure 8. Equation. Cramer’s V statistic.

where: 

χ2 = Chi-square test for correlation. 

N = total number of observations. 

C = number of categories in variable 1. 

R = number of categories in variable 2. 

A Cramer’s V statistic equal to or greater than 0.3 indicated that the two variables were 

moderately or highly correlated and should be generated together. Remaining variables were 

then generated independently. 

Based on the correlation test results, the variables that were generated together in the 

macroscopic approach by facility type were determined to be as follows:

• Rural two-lane, two-way roadways: 

o Two-lane, two-way undivided segments: AADT, right shoulder width, horizontal 

curve. 

o Three-leg unsignalized intersections with stop control on approaches to minor roads: 

Major road AADT, minor road AADT. 

o Four-leg unsignalized intersections with stop control on approaches to minor roads: 

Major road AADT, minor road AADT. 

o Four-leg signalized intersections:

o Major road AADT, minor road AADT.

o Major road AADT, number of approaches with exclusive left-turn lanes.

o Number of approaches with exclusive left-turn lanes, number of approaches with 

exclusive right-turn lanes.

o Intersection lighting, skew angle. 

• Rural multilane highways:

o Four-lane undivided segments: Right shoulder width, right shoulder type, lighting. 

o Four-lane divided segments: Right shoulder width, right shoulder type, speed limit.

o Three-leg unsignalized intersections with stop control on approaches to minor roads: 

o Major road AADT, minor road AADT.

o Major road AADT, intersection lighting.

o Four-leg unsignalized intersections with stop control on approaches to minor roads: 

o Major road AADT, minor road AADT. 

o Major road AADT, intersection lighting.

o Intersection lighting, skew angle. 

o Four-leg signalized intersections:

o Major road AADT, minor road AADT. 

o Major road AADT, number of approaches with exclusive left-turn lanes.

o Major road AADT, intersection lighting.
21 

o Number of approaches with exclusive left-turn lanes, number of approaches with 

exclusive right-turn lanes.

o Intersection lighting, skew angle. 

• Urban and suburban arterials: 

o Two-lane undivided arterials:

o AADT, lighting. 

o Offset to fixed object, on-street parking type, speed limit.

o Four-lane undivided arterials: Offset to fixed object, on-street parking type, speed 

limit.

o Four-lane divided arterials (i.e., including a raised or depressed median):

o Offset to fixed object, on street-parking type, speed limit.

o Median width, lighting. 

o Three-leg unsignalized intersections with stop control on approaches to minor roads: 

Major road AADT, minor road AADT. 

o Three-leg signalized intersections: 

o Major road AADT, minor road AADT. 

o Minor road AADT, speed limit.

o Number of approaches with exclusive left-turn lanes, number of approaches with 

exclusive right-turn lanes.

o Number of approaches with exclusive left-turn lanes, left-turn signal phasing.

o Number of approaches with exclusive left-turn lanes, number of lanes to be 

crossed by a pedestrian.

o Four-leg unsignalized intersections with stop control on approaches to minor roads: 

Major road AADT, minor road AADT. 

o Four-leg signalized intersections:

o Major road AADT, minor road AADT. 

o Number of approaches with exclusive left-turn lanes, number of lanes to be 

crossed by a pedestrian.

o Number of approaches with exclusive left-turn lanes, left-turn signal phasing.

• Freeway segments:

o Rural four-lane divided freeway segments:

o Left shoulder width, right shoulder width. 

o AADT, median width.

o Rural six-lane divided freeway segments:

o Left shoulder width, right shoulder width. 

o Lane width, median width. 

o Urban four-lane divided freeway segments:

o Left shoulder width, right shoulder width. 

o Lane width, speed limit. 

o Left shoulder rumble strip, right shoulder rumble strip, horizontal curve.

o Urban six-lane divided freeway segments:

o Left shoulder width, right shoulder width. 

o Lane width, median width. 

o Left shoulder rumble strip, right shoulder rumble strip, horizontal curve.

o Urban eight-lane divided freeway segments:
22 

o AADT, median width. 

o Left shoulder rumble strip, right shoulder rumble strip, horizontal curve.

o Left shoulder width, right shoulder width. 

o Urban 10-lane divided freeway segments:

o Left shoulder width, right shoulder width. 

o Left shoulder rumble strip, right shoulder rumble strip, horizontal curve.

Researchers then generated both initial probability matrix and transition probability matrix for 

each group of variables and all remaining single variables by each facility type using the 

collected data. The initial probability matrix and transition probability matrix were used as the 

inputs for generating roadway data in the macroscopic approach. Due to the large data size of 

initial probability matrix and transition probability matrix for all facility types, researchers 

decided presenting the matrix values in this report was not feasible. Instead, the matrix tables 

will be included in the source codes accompanying the RAD software developed through this 

project. 

CRASH DATA GENERATION INPUTS

The key to generating crash data for RAD is to estimate the mean (μi) of crashes for each site, 

based on the characteristics generated from the roadway data-generation process. As noted in the 

section on Macroscopic Approach Consolidation and Enhancement, this study uses crash 

prediction models to estimate the mean crashes for each site. The framework for crash prediction 

models is similar to that in the HSM, where the crash mean is first predicted using a base 

condition SPF and then adjusted by multiple adjustment factors (AFs), which are expressed as 

figure 9. 

Figure 9. Equation. SPF adjusted by AFs.

where: 

Y = number of years of crashes to be generated. 

SPFBASE = predicted crashes using the base condition SPF, with segment length and AADT 

used as predictors for segment-related facilities and AADT on both major and minor 

approaches used as predictors for intersection-related facilities. 

AFs = adjustment factors related to all roadway characteristics generated from the roadway 

data-generation process. 

Researchers collected the coefficients of base condition SPF and all AFs for each facility type. 

Most of the existing literature the research team looked at estimated SPFs by injury severity 

group instead of by individual severity level, especially for severe injury crashes, such as K and 

A crashes (Wang et. al 2022; Garber and Rivera 2010). Thus, researchers collected both the SPF 

coefficients and AFs by three injury severity groups, namely PDO, B + C, and K + A. Further, to 

account for crash randomness and crash generation variation, researchers generated a uniform 

distribution for each of the SPF coefficients and AFs, based on all the AF values. The collected 

coefficients and AFs and generated distributions are not presented in this report to hide the 

underlying parameters used in generating the data. 
23 

For each parameter, if multiple sources provided values, researchers could directly determine the 

lower and upper limits of the uniform distribution based on all the values. However, if 

researchers found that only one source included a parameter estimate, they calculated the lower 

limit of the uniform distribution as 10 percent below that parameter estimate. Similarly, they 

calculated the upper limit as 10 percent above that parameter estimate. A random value could 

then be selected for each of the SPF coefficients and AFs from the generated uniform 

distributions for each site, to be used in predicting the mean of crashes by each injury severity 

group. 

Table 4 shows an example of uniform distributions generated for the SPF coefficients and AFs 

for the intersection facility, and table 5 shows an example of uniform distributions generated for 

the SPF coefficients and AFs for the segment facility. The numbers and letters used in these 

tables are fabricated to demonstrate the overall procedure for generating the uniform 

distributions.
24 

Table 4. Crash generation parameters for rural, two-lane, two-way, three-leg unsignalized intersections.

Variables 

Example Generated Parameters and AFs Uniform Distributions for Parameters and AFs

PDO B+C K+A PDO B+C K+A 

Parameters for Base Condition SPFs

Major AADT A1, A2 B1, B2, B3 C1, C2, C3 U(A1, A2) U(B1, B3) U(C1, C3)

Minor AADT D1 E1, E2 F1, F2, F3 U(0.9*D1, 1.1*D1) U(E1, E2) U(F1, F3) 

Parameters for AFs 

Number of 

approaches with 

exclusive left-turn 

lanes: 

1 

2 

G1, G2

G3, G4

H1, H2

H3, H4

NA

NA

U(G1, G2)

U(G3, G4)

U(H1, H2)

U(H3, H4)

U(1, 1) 

U(1, 1) 

Overdispersion for NB 

k J1, J2 K1, K2 L1, L2 U(J1, J2) U(K1, K2) U(L1, L2) 
25 

Table 5. Crash generation parameters for rural, two-lane, two-way, undivided highways.

Variables 

Example Generated Parameters and AFs Uniform Distributions for Parameters and AFs

PDO B+C K+A PDO B+C K+A 

Parameters for Base Condition SPFs

Intercept A1, A2 B1, B2, B3 C1, C2, C3 U(A1, A2) U(B1, B3) U(C1, C3)

AADT D1 E1, E2 F1, F2, F3 U(0.9*D1, 1.1*D1) U(E1, E2) U(F1, F3) 

Parameters for AFs

≤9-ft lane width G1, G2 H1, H2 J1, J2 U(G1, G2) U(H1, H2) U(J1, J2) 

10-ft lane width G3, G4 H3, H4 J3, J4 U(G3, G4) U(H3, H4) U(J3, J4) 

11-ft lane width G5, G6 H5, H6 J5, J6 U(G5, G6) U(H5, H6) U(J5, J6) 

≥12-ft lane width G7, G8 H7, H8 J7, J8 U(G7, G8) U(H7, H8) U(J7, J8) 

Right Shoulder Width 

2 ft K1, K2 L1, L2 M1, M2 U(K1, K2) U(L1, L2) U(M1, M2) 

4 ft K3, K4 L3, L4 M3, M4 U(K3, K4) U(L3, L4) 

U(M3, 

M4) — — 

6 ft K5, K6 L5, L6 M5, M6 U(K5, K6) U(L5, L6) 

U(M5, 

M6) — — 

≥8 ft K7, K8 L7, L8 M7, M8 U(K7, K8) U(L7, L8) 

U(M7, 

M8) — — 

Overdispersion for NB: k = β/Length

β N1

N2

O1

O2

P1

P2

P3

U(N1, N2) U(O1, O2) U(P1, P3) 

—No data. 

*Variables are insignificant at a 90-percent confidence level.
26 

The mean crashes by the three injury severity groups were calculated using the uniform 

distributions generated in table 4 and table 5, respectively. Then, the crash counts were further 

simulated using two commonly used crash prediction modeling frameworks: the Poisson and NB 

models (figure 2 and figure 4). Next, a uniform distribution between 0.6 and 0.8 will be used to 

generate a weight for NB crash counts for each site, and the final crash counts for each site will 

be calculated by incorporating the weighted Poisson crash counts. Researchers assigned a greater 

weight to the NB model because it could account for overdispersion, which is a common issue in 

crash data. Mixing the Poisson and NB models in the crash generation process may further help 

account for crash generation variations for the RAD.

After researchers generated crashes for each of the three severity groups, they further 

disaggregated crash counts for the B+C group into individual B and C crash counts by randomly 

selecting and multiplying a proportion of B crashes over B and C in total from a pooled sample. 

They created a pooled sample using the real crash data for the specific facility type collected in 

this study. They applied the same procedure to K and A crashes and generated crash counts by 

crash type by each crash severity using a similar approach.

Although crashes could then be generated by both injury severity and crash type for each facility 

type, the generated crash counts still might not be feasible and realistic, because the research 

team collected the parameters for both the base condition SPFs and the AFs used in crash 

generation from multiple sources. The selected parameters are not presented in this report to hide 

the underlying parameters we used in generating the data to preserve the value of the RAD 

generated for testing data analysis approaches. Other literature that was not considered might use 

a different modeling formula, which could suggest a different range of values for the predicted 

mean of crashes.

For example, if the literature estimates crash prediction models using “Number of Years*365” as 

the offset, the estimated intercept will be relatively small, and the predicted mean of crashes in 

the RAD will be extremely low, leading to a zero crash count in the final crash generation. 

However, if the literature estimates crash prediction models without offset, the estimated 

intercept will be relatively large, and the predicted mean of crashes in the RAD will be extremely 

high, leading to unexpectedly high crash counts in the final crash generation. Therefore, the 

original uniform distributions generated for the intercepts needs to be calibrated to make the 

generated crash counts realistic. To accomplish this, crash counts are first generated by the three 

injury severity groups using the initial parameters for each facility type, then the crash counts by 

the injury severity group are also calculated using the actual crash data collected in this study, 

based on the same data size as the generated crash counts. For example, if the crash counts are 

generated from N intersections or M-mile segments, the crash counts are also calculated from N

intersections or M-mile segments from the collected crash data. Next, a calibration factor is 

calculated (figure 10) for each of the injury severity groups for each facility type, and the 

calibration factor is added to the original uniform distribution for the intercept to calibrate the 

crash generation framework.

Figure 10. Equation. Calibration factor formula for crash count generation. 
27 

CHAPTER 4. MICROSCOPIC APPROACH FRAMEWORK

This section will outline the research team’s development of a high-resolution, disaggregate 

realistic artificial data generator (DREDGE). The conceptualization of the disaggregate 

DREDGE framework, the data compiled to build the DREDGE, and the implementation 

procedures are described. In this research, the authors proposed a general framework of RAD 

generation embedded with heterogeneous causal structures that generated crash data by 

considering crash occurrence as a trip-level event impacted by trip-level factors, demographic 

characteristics, roadway facilities, and vehicle attributes.

The proposed framework was general enough to generate crashes for all roadway facility types, 

including segments and junctions. Additionally, the framework can generate data for different 

combinations of inputs, including modeling methods, model formulation, input specification, and 

unobserved heterogeneity. This chapter presents the motivation for this approach, along with 

conceptual framework details, a description of the data, and implementation details of the 

microscopic DREDGE framework.

CONCEPTUAL FRAMEWORK

Safety analysis primarily focuses on identifying and quantifying the influences of factors that 

contribute to traffic collisions and the consequences of these factors. The authors of this report 

proposed and built a high-resolution, disaggregate data-generation process that mimics crash 

occurrences on transportation facilities at the trip level and accommodates the influence of a full 

range of crash-contributing factors. The proposed DREDGE recognizes that crashes result from 

travel decisions made by people. Hence, it is important to examine crash occurrence as a 

trip-level decision to mimic the true crash-generation process.

Considering such disaggregate treatments will allow the crash process incorporated in the 

DREDGE to resemble the true process more closely. For example, if data are simulated at the 

segment level, driver-related factors can be included either in an aggregate form or not at all. In 

contrast, all contributing factors—including trip-related factors—can be realistically 

incorporated if the data are simulated at the trip level. Researchers will consider trip-level 

attributes for DREDGE framework development that include driver and other occupant 

characteristics, vehicle characteristics, and roadway attributes.

The first step in the proposed framework was to evaluate the crash risk for each trip by 

employing trip-level travel information. Then, researchers generated detailed crash 

characteristics for trips identified for crash involvement. The DREDGE framework employs a 

suite of models to process trips with crashes, including crash type, crash severity, and crash 

location—as well as driver and vehicle characteristics.

Researchers developed the DREDGE framework by employing multiple datasets, including 

Strategic Highway Research Program 2 (SHRP2) Naturalistic Driving Study (NDS) data and 

Crash Report Sampling System (CRSS) data (Virginia Tech Transportation Institute 2020; 

NHTSA n.d.). The framework is organized around three specific modules: a) disaggregate trip 

information generation, b) crash data generation, and c) crash data aggregation. This report 

presents detailed documentation about each module in the next three sections.
28 

Disaggregate Trip Information Generation Module

The paradigm for regional travel demand modeling has undergone a transformation from an 

aggregate zonal level statistical framework (such as a four-step or trip-based model) to a 

disaggregate individual-level framework (tour level and/or activity-based models) (Kamel et al. 

2019; Pinjari et al. 2008). The disaggregate frameworks accommodating various influences 

provide a representation of any individual’s travel in continuous time and space. These 

influences include sociodemographic characteristics (such as income, age, household structure, 

education, and car ownership), employment characteristics (such as employment industry and 

location), transportation network characteristics (such as access to travel mode and travel time by 

mode), and built environmental measures (such as population density, land-use mix, and public 

transit density).

From these travel patterns, researchers retrieved high-resolution information for trips were

retrieved, including trip start and end time, trip start and end location, trip characteristics (such as 

alone or group trip), vehicle used for trip, and precise route considered. In consultation with 

FHWA research staff, the research team recognized that developing a standalone, activity-based 

model system was beyond the scope of this study. Hence, the research team identified 

Polychotomous Choice Agent-Based Risk Model for Integrated Travel Demand and Network 

and Operations Simulation (POLARIS), a travel-demand modeling tool developed by Argonne 

National Laboratory, for use in the study in place of developing a new standalone modeling 

system. POLARIS is a well-established, activity-based travel demand modeling tool. It is highperformance, open-source, and has an agent-based modeling framework that includes traffic flow 

simulation, activity-based demand simulation, model building, and geographic information 

system (GIS) tools (Auld et al. 2016).

To use POLARIS data in the study, the research team reached out to Argonne National Lab 

personnel, who agreed to provide calibrated outputs from POLARIS. (For more information on 

POLARIS, please see Auld et al.’s (2016) article titled “POLARIS: Agent-Based Modeling 

Framework Development and Implementation for Integrated Travel Demand and Network and 

Operations Simulations.”)

Crash Data Generation Module

The objective of the crash data generation module is to generate crashes on the transportation 

system. The framework utilizes the detailed trip information from the trip information module 

and the disaggregate trip information generation to generate crashes. This process involves 

identifying the vehicles involved in the crash, crash location, injury severity of the occupants 

(such as fatal, incapacitating, non-incapacitating, and no injury), and crash type (such as head-on, 

rear-end, and vehicle-pedestrian). The framework employed for crash generation is described in 

the following paragraphs. 

In the first step of the framework, the research team classified all the trips on the transportation 

system into two categories: No Crash and Crash. In urban regions, trips in a typical day amount 

to several million, and their data are likely to take up large amounts of storage space, with 

high-resolution details on routing characteristics with GIS coordinates. The team’s proposed 

classification process allowed for a reduced number of trips to be used for crash data generation. 
29 

Given the relatively small proportion of crash-involved trips, this classification approach 

provides an elegant solution to computational and data burdens. The classification problem was 

modeled using a binary logit model. Specifically, the research team estimated the proposed 

binary model using SHRP2 NDS data (Virginia Tech Transportation Institute 2020). This NDS 

data provided the study with a large sample of trips and an associated indicator for trips involved 

in crashes. Further, using the NDS data allowed the research team to incorporate trip-level 

details, including average speed, travel distance, time of day, and vehicle type, into crash 

probability (Virginia Tech Transportation Institute 2020).

The second step of the framework took place after the research team identified and tagged the 

trips with crashes. During the second step, the team processed crash-tagged trips to determine 

detailed characteristics of the crashes, including crash type, crash location, and injury severity. 

Notably, crash type and crash severity have fixed and well-defined alternatives, but crash 

location alternatives are more complicated. Thus, depending on when crash location is examined, 

alternative structures for crash variable generation become possible. For example, one sequence 

can be as follows: Researchers estimate a trip-level model for the crash-tagged trips to identify 

the crash type, such as head-on, rear-end, or vehicle-pedestrian. Researchers then create a 

subsequent model for crash severity with this crash type and develop a crash location model that 

is conditional on crash type and severity. Notably, more information is available in the latter 

models in the sequence (i.e., additional independent variables can be included in the model 

estimation). For instance, if crash severity follows a crash type model, crash type can be included 

as an independent variable in the model. 

The resulting crash location model can have crash type and crash severity as independent 

variables. A visual representation of the proposed structure for crash variable generation is 

presented in figure 11. 
30 

Source: FHWA.

. . . = other options not listed. 

Figure 11. Flowchart. DREDGE sequential approach I: crash risk to crash type to crash 

severity to crash location. 

Alternatively, the sequence of the variables can be altered to crash location, followed by crash 

type and crash severity. In this sequence, crash location model estimation will be based on 

trip-level characteristics, and crash type and crash severity variables will have access to location 

variables in the model. Figure 12 provides a potential model structure. As is evident, the two 

sequences described in figure 11 and figure 12 might result in different behavioral frameworks. 

Safety literature has placed significant emphasis on the role of crash location in modeling 

severity and crash type.

Hence, the research team chose the sequence listed in figure 12 for this study. The significant 

data processing and modeling efforts involved did not allow for testing the two sequences. 

Finally, the research team also generated driver and vehicle information (including the attributes 
31 

of other drivers and vehicles for multivehicle crashes) involved in crashes, based on the driver 

and vehicle characteristics of the crash trips.

Source: FHWA.

. . . = other options not listed. 

Figure 12. Flowchart. DREDGE sequential approach II: crash risk to crash location to

crash type to crash severity. 

The final step of the crash data generation framework involved determining the appropriate 

model framework. Given that crash type and crash location were categorical variables, an MNL 

model framework was appropriate. For the severity variable—given the inherent ordered nature 

of the variable—a GOL model structure was employed. 

Crash Aggregation Module 

The crash data generation module outputs are crash data—including crash type, crash severity, 

and crash location—and time and number of vehicle occupants involved in a crash for a typical 
32 

day of the year. However, for crash datasets, it might be necessary to aggregate data by facility 

type both at temporal resolution (such as crashes on a segment or intersection in a 6-mo period or 

multiple years) and spatial resolution (such as crashes in a zone or county). To generate 

aggregate data, the research team implemented framework to run a typical day multiple times 

with different random seeds to ensure duplication of the same crashes did not occur between 

runs. 

DATA COLLECTION

The microscopic DREDGE generator was developed employing three different datasets: SHRP2 

NDS data from Virginia Tech Transportation Institute, CRSS data from NHTSA, and Chicago 

trip-level data from Argonne National Laboratory (Virginia Tech Transportation Institute 2020; 

NHTSA n.d.; Auld et al. 2016).

SHRP2 NDS Data

To develop models for crash risk and crash location, researchers used SHRP2 NDS data, which 

had been collected during the NDS with cameras and sensors placed in participants’ cars that

tracked their driving over an extended period (Virginia Tech Transportation Institute 2020). For 

the current study, researchers requested five different data components from the NDS: 

time-series data, a trip summary table, an event table, driver demographics, and roadway 

characteristics data. Time-series data provided trip-related information, (e.g., speed, acceleration, 

deceleration, location, lighting, and airbag deployment information) for every tenth of a second. 

The SHRP2 trip table provided trip details about all 5.4 million trips, including start time, end 

time, day of week, facility locations, facility speeds (mean/maximum speed), mean/maximum 

acceleration, mean/maximum deceleration, and headways versus distance traveled. The SHRP2 

event table provided information on 1,951 crashes, including event severity, crash severity, 

traffic condition, and weather condition. Driver demographics included driver age, gender, 

income, and educational level. Finally, the roadway characteristics data provided information on 

corresponding road segments, including AADT, number of lanes, and roadway classification 

(arterial, major/minor collector, freeway) (Virginia Tech Transportation Institute 2020). These

variables were included in analyzing crash risk, crash location, crash type, and crash severity

appropriately for each analysis variables. 

For their model development, the research team selected 1 million trips that did not result in 

crashes and 1,951 trips that resulted in crashes. The 1 million trips were randomly selected from 

a full sample of 5,512,900 trips (Hankey, Perez, and McClafferty 2016). For the 1,951 trips 

during which crashes occurred, 814 crashes were categorized as low-risk tire strikes and 

removed from the list of trips with crashes. This removal left 1,137 trips with crashes for the 

final model development.

CRSS Data

The CRSS was used by the research team to develop models for crash type, drivers and vehicles, 

and crash severity. The CRSS database contains a nationally representative, weighted and 

stratified sample of road crashes collected and compiled from about 60 jurisdictions across the 

50 States and District of Columbia. The database includes information from reports compiled by 
33 

police officers for roadway crashes involving at least one motor vehicle and resulted in property 

damage, injury, or death (NHTSA n.d.).

The research team obtained these CRSS databases from the U.S. Department of Transportation 

and NHTSA’s National Center for Statistics and Analysis. NHTSA collected the crash data over 

the last 50 years from 1970 to the present. The data included information on a multitude of 

factors regarding crash situation and events, including driver characteristics (e.g., driver age, 

gender), vehicle characteristics (e.g., vehicle type, vehicle age, vehicle model), roadway design 

and operational attributes (e.g., speed limit, number of lanes), environmental factors (e.g., 

weather, lighting condition), and crash characteristics (e.g., hour, day, location, crash type, crash 

severity) (NHTSA n.d.).

This study focused on the most recent 4 yr of CRSS data to develop crash type, driver, vehicle, 

and crash severity modules. Between 2016 and 2019, the CRSS crash database recorded a record 

204,332 crashes involving 362,596 motor vehicles and 512,312 people (NHTSA n.d.). To 

prepare the final CRSS dataset for module development, the research team removed records with 

missing information on essential attributes. Thus, the final CRSS dataset consisted of about 

113,983 crashes involving 211,311 motor vehicles and 298,382 people. 

Chicago Trip-Level Data

The research team used the Chicago trip-level data as an input when it implemented DREDGE. 

The team sourced the trip-level data from an activity-travel realization for an urban region, 

generated from the POLARIS model for the Chicago region. The data contained 2,256,502 trips, 

with information on trip data (start time, duration, etc.), driver demographics (age, education, 

household size, number of vehicles, number of workers, etc.), and roadway segments (segment 

length, AADT, number of lanes, and roadway type: major/minor arterial, freeway, collector or 

local, etc.) (Auld et al. 2016).

MICROSCOPIC RAD MODULE DEVELOPMENT

The research team’s proposed microscopic DREDGE framework for implementing the 

sequential approach presented in figure 12. This framework has five stages/modules of crash data 

generation, as illustrated in figure 13. Stage 1 (the crash risk module) evaluates a series of trips 

using a binary logit model to classify each trip as crash or no crash. In stage 2 (the crash location 

module), the location of each “crash” trip is determined using an MNL model. Stage 3 (the crash 

type module) is where the type of each crash is determined using an MNL model. In stage 4 (the 

driver and vehicle module), data on the driver(s) and vehicle(s) associated with each crash are 

generated using a probability distribution table. In stage 5 (the crash severity module), the 

severity of the crash is generated for each driver involved using an OL model. Each of these 

stages/modules is implemented sequentially in DREDGE using Python® programming language 

(Van and Drake 1995; Python Software Foundation 2023). In the following five sections, this 

report presents a detailed description of each of the five stages/modules involved in the 

microscopic approach framework.
34 

Source: FHWA.

Figure 13. Flowchart. DREDGE generator development for microscopic approach.

Crash Risk

The first stage of the DREDGE generator is the crash risk module. The goal of this module is to 

evaluate each trip and determine stochastically if a crash will occur during the trip. The research 

team used the SHRP2 NDS dataset to develop the crash risk model. The dataset included 

1,137 trips that resulted in crashes and 1 million trips that did not (Virginia Tech Transportation 

Institute 2020). For this model, the research team removed any trips missing relevant trip or 

driver information. These removals resulted in 1,004 trips resulting in a crash and 774,873 trips 

that did not result in a crash. The data were then further filtered because trips with crashes 

accounted for only 0.13 percent of the total trips, making them difficult to model. Therefore, the 

trips not resulting in a crash were under sampled, as researchers randomly selected 10 percent to 

be used for analysis. The final dataset the research team used for model development contained 

78,336 trips; 1,004 resulted in crashes, and 77,332 did not. 

The research team used a binary logit model for modeling crash risk. The development of the 

model was based on removing statistically insignificant variables in a systematic manner. Since 

different datasets were used for modeling and implementation (SHRP2 NDS data were used for 

modeling and Chicago data were used for implementation), only those variables that were 

present in both datasets were considered when developing the model. Four variables were 

common in both datasets, including driver age and income and trip start time and length. Among 

these variables, driver age was found to have a significant impact on the trip crash risk. Drivers 

less than 30 years old (with teenage drivers being the most likely) and greater than 74 years old 

were found to be more likely to be involved in crashes than other drivers. 

Finally, as the model was developed with under-sampled, noncrash trips, the constant in the 

binary logit model will need to be modified for applying the crash risk model for the full 

population. Researchers calibrated the constant to match the true population crash shares.

Crash Location 

The second stage of the DREDGE framework is the crash location module. The goal of this 

module is to evaluate the sequence of roadway segments traversed during a trip and determine 

stochastically where a crash will occur. The SHRP2 NDS dataset provided 1,004 crash records 

and was used to develop the crash location model (Virginia Tech Transportation Institute 2020). 
35 

For this module, any crashes that occurred at an intersection or crashes without a defined 

location were removed. However, there was missing information for multiple attributes for these 

857 crashes. As a result, the research team imputed the missing information in alignment with 

existing distributions observed in the data. 

For modeling the crash segments, the research team considered a sampling of alternative 

segment approaches to avoid computational complexity for large trips spanning several 

segments. The sampling process included the crash segment alternative and 29 additional 

segments randomly sampled from the trip segments. The research team’s development of the 

model was based on removing statistically insignificant variables in a systematic manner. The 

results indicated that longer segments tended to have a higher risk of crash occurrence. 

Additionally, for each trip segment with more lanes, roadways with higher AADTs and collector 

roadways tend to have lower risks of crash occurrence.

Crash Type

The goal of the third stage of the DREDGE generator, the crash type module, is to generate the 

type of crash that will occur based on trip and roadway variables. The research team used the 

CRSS dataset to develop this crash type module (NHTSA n.d.). In the CRSS dataset, researchers 

randomly selected 25,000 crashes (out of a total of 113,983 crashes) to use for developing the 

crash type module. The alternatives researchers considered for crash type were rear-end crash, 

head-on crash, angular crash, sideswipe crash, crash with fixed objects, crash with nonfixed 

objects, and nonmotorized crash. Since different datasets were used for modeling and 

implementation, the research team considered only variables that were present in both datasets 

when developing the module. 

Rear-end crashes are used as the base alternative for this module, with angular crashes and 

crashes with fixed and nonfixed objects having a higher probability of occurrence, and head-on 

crashes, sideswipe crashes, and nonmotorized crashes having a lower probability of occurrence. 

Also, as the number of lanes increases, the probability of any crash, except for a rear-end crash, 

decreases. Crashes on freeways have a higher likelihood of sideswipe crashes and a lower 

probability of head-on crashes, angular crashes, crashes with nonfixed objects, and nonmotorized 

crashes. On weekdays, the probability of rear-end crashes increases, and the probability of 

head-on crashes and crashes with fixed and nonfixed objects decreases. During the morning peak 

(7 a.m. to 10 a.m.), the probability of crashes with fixed and nonfixed objects and nonmotorized 

crashes decreases. During the evening peak (4 p.m. to 7 p.m.), the probability of any crash other 

than a rear-end crash decreases.

Drivers and Vehicles

The goal of the fourth stage of the DREDGE generator, the drivers and vehicles module, is to 

generate data for drivers and vehicles involved in crashes. Illinois crashes from the CRSS dataset 

are used for this module (NHTSA n.d.). From this data, the research team developed a 

probability distribution of different driver demographics (such as age, gender, and seatbelt use) 

and vehicle characteristics (such as type and age). The research team then used this distribution 

to generate driver and vehicle information for the generated crashes.
36 

Crash Severity

The goal of the fifth stage of the DREDGE framework, the crash severity module, is to generate 

the severity of the crash for each driver based on trip data, roadway information, driver 

demographics, vehicle information, and crash type. Of the 25,000 crashes that were used in the 

crash type module, driver information was available for 24,351 crashes, resulting in 

42,039 drivers that researchers used in developing the crash severity module. 

Researchers used an OL model to model crash severity. For the crash severity module the 

alternatives were PDO, minor, major, and severe. Since researchers used different datasets for 

modeling and implementation (CRSS and Chicago trips, respectively), they only considered 

those variables that were present in both datasets when developing the model (NHTSA n.d.; 

Auld et al. 2016). According to this model, drivers under 25 years old are less likely to 

experience a high-severity crash. Crashes that occur on freeways or involve more than a few 

lanes are more likely to result in high severity. Crashes that occur on weekdays or during peak 

hours are likely to be less severe. Using rear-end crashes, crashes with nonfixed objects, and 

nonmotorized crashes as a base, sideswipe crashes are less likely to result in severe crashes; 

meanwhile, head-on crashes, angular crashes, and crashes with fixed objects are more likely to 

result in severe crashes. Using automobiles, motorcycles, and buses as a base, drivers in utility 

vehicles and trucks are less likely to sustain severe injuries.

MICROSCOPIC RAD MODULE IMPLEMENTATION

The research team employed Monte Carlo simulation for each DREDGE module’s 

implementation (Spence 1983). Typically, the simulation process involves generating the 

cumulative probability function (CPF) for all alternatives using the module-specific model. 

Then, the chosen alternative is identified by generating a uniform random number between 0 and 

1 and comparing it with the CPF. Across different modules, different CPF formulae are 

employed. The rest of the process remains stable across all modules. The implemented routines 

are validated and checked to ensure the model outcomes follow expected distributions. A 

detailed documentation on the implementation of microscopic DREDGE modules and validation 

checks implemented are presented in the following six sections.

Crash Risk

The research team implemented the binary logit model in the DREDGE framework after 

developing the crash risk module. For the first step of the DREDGE framework, the Chicago 

data’s 2,256,502 trips were used to simulate a day of traffic. Each trip was evaluated using the 

binary logit model for crash risk, and the probability of a crash was calculated. A random 

number between 0 and 1 was then generated. If the random number was less than the probability, 

then that trip was classified as resulting in a crash and added to the list of crashes for that day. 

This process repeats 365 times using the same set of trips to simulate a full year of traffic. Since 

a random number is generated for each trip, different trips will result in a crash each day; 

however, those with a higher probability of a crash are more likely to be selected. When using 

DREDGE to generate a full year of data, an average of 563 crashes per day occurred. This

average equates to approximately 0.025 percent of the trips resulting in a crash. This average is 
37 

comparable to the NDS data that were used for model development, where 0.021 percent of the 

trips resulted in a crash (Virginia Tech Transportation Institute 2020).

Crash Location 

After developing the crash location model, the research team processed it in the DREDGE tool to 

determine the likely crash locations. A sequence of roadway segments is provided for each trip 

that occurs in the Chicago dataset. Each segment in this sequence is considered using the MNL

model developed for crash location. This model calculates the probability of a crash for each 

segment. These probabilities are then combined to create a cumulative probability table. A 

random number is then generated from a uniform distribution from 0 to 1 and used to select the 

roadway segment. The random number is compared to each of the segments in the cumulative 

distribution table, and the segment with a probability greater than the generated number is the 

one where the crash occurred. The roadway information for this segment is then appended to the 

trip data to be returned to the user at the end.

Crash Type

The third step of the DREDGE tool determines the type of crash for each trip where a crash is 

determined to have occurred. Trip and roadway information are used to calculate the probability 

of each type of crash using the MNL model. Since the same data are used for each day, a 

weekday variable is not in the original dataset. Instead, this variable is generated based on the 

specific day being evaluated. The generator will determine a random day of the week for the first 

day of generation and then assign each following day accordingly. Similar to the crash location 

module, this model calculates the probability of each crash type. These probabilities are then 

combined to create a cumulative probability table. A random number is then generated from a 

uniform distribution from 0 to 1, which is used to select the crash type. The random number is 

compared to the cumulative distribution table, and a crash type with a probability greater than the 

generated number is assigned. The crash type is then appended to the trip data, to be returned to 

the user at the end.

Drivers and Vehicles

The first step in generating driver and vehicle information is determining the number of vehicles 

involved in the crash. This step is partially based on the crash type generated in the crash type 

module. If the crash type was defined as crash with fixed objects, crash with nonfixed objects, or 

nonmotorized crash, then it was considered a single vehicle crash. Otherwise, the number of cars 

was generated as 2 or 3. The probabilities calculated from the Illinois dataset for multivehicle 

crashes were an 88.1-percent likelihood of two vehicles being involved and an 11.9-percent 

likelihood of three vehicles being involved. Researchers generated the number using a 

cumulative probability table. Data was generated for each driver and vehicle involved in a crash 

once the number of vehicles involved was determined. Researchers assigned the first driver to 

have the same age as the age of the primary driver in the trip data; subsequently, all other 

information was generated using probability distributions observed from Illinois CRSS data 

(NHTSA n.d.).
38 

Crash Severity

The final step of the DREDGE generator determines the severity of the crash for each trip where 

a crash occurred. The trip, roadway, and generated crash information are used to calculate the 

probability of each crash severity using the OL model. Using the OL model, the probability for 

each crash severity across each driver is calculated. These probabilities are then combined to 

create a cumulative probability table. A random number is then generated from a uniform 

distribution from 0 to 1, which is used to select the crash severity. The random number is 

compared to all the crash severities in the cumulative distribution table, and the severity with a 

probability greater than the generated number is selected as the assigned crash severity for the 

driver. The crash severity is then appended to the driver data to be returned to the user at the end. 

The highest severity for all the drivers involved in a crash is then appended to the crash data to 

be returned to the user at the end. 

Validation Check

The research team used the DREDGE generator to generate a full year of data to test its 

accuracy. Data on crash type, driver and vehicle characteristics, and crash severity were 

generated and compared to the CRSS dataset (see also appendix A, which is in the second 

volume of this publication). The RAD was similar to the CRSS dataset. The biggest differences 

were rear-end crashes (a slight underestimation) and crashes with fixed and nonfixed objects (a 

slight overestimation). Additionally, differences were found in the number of vehicles and driver 

age distribution. These differences could be partially attributed to inputs from preceding 

modules. Crash severity results were well-aligned with the input data. 
39 

CHAPTER 5. RAD GENERATION TOOL

The team developed an aggregated RAD software. 3 This software used Python and incorporated

both the macroscopic approach and microscopic approach into a single platform (Python 

Software Foundation 2023). The team compiled all scripts and required inputs into a single 

executable file, which could be used directly by the end user to operate the RAD tool, with no 

need to install any software on their local computer.

MACROSCOPIC RAD DATASETS

The research team implemented the macroscopic approach to provide the user with capabilities 

to generate RAD for 22 facility types, with 10 intersection facilities and 12 segment facilities. 

For each facility type, the software produced two data files: the roadway file (containing total 

crash counts by severity and roadway geometric characteristics) and the crash file (detailed crash 

severity and crash type information). The two files are cross-linked, and the corresponding ID 

column can be used to join the two files together when needed. 

Researchers can use roadway data generated by the RAD software to compare the crash 

prediction accuracy among different statistical and econometric methodologies by five injury 

severity levels: K, A, B, C, and PDO crashes.4

 Additionally, the software can be used to 

investigate the underlying relationships between crash and/or crash severity and traffic and 

roadway characteristics (such as AADT, lane width, shoulder width, speed limit, and curvature). 

The crash data generated by the RAD software can be used to further incorporate crash types into 

the analysis. The crash data includes a total of 10 crash types: angle, front-to-front, front-to-rear, 

sideswipe from the opposite direction, sideswipe from the same direction, other multivehicle, 

fixed object, nonfixed object, overturn/rollover, and other single-vehicle.

MICROSCOPIC RAD DATASETS

In terms of the microscopic approach, researchers implemented the disaggregate realistic 

artificial data generator (DREDGE) generator to produce three data files, as follows: 

• Crash file: Contains information on crash details, such as location, type, and severity . 

• Driver file: Contains information on each driver involved in a crash and their individual 

injury severity . 

• Vehicle file: Contains information on each vehicle involved in a crash. 

The three files are cross-linked, and columns from one dataset can be readily merged into the 

other two files as needed. The user can specify the number of years of crash data to be produced 

by the DREDGE generator, as well as the number of instances of data for that number of years. 

For example, a user can specify that they want two sets of 3 yr of crash data. When the 

3

FHWA. DREDGE (standalone RAD software). 4

FHWA. DREDGE (standalone RAD software).
40 

DREDGE generator is run, it will then produce two different crash files, two different driver 

files, and two different vehicle files, each containing three years of data.

The crash dataset produced by the DREDGE generator can be used in a variety of ways. To 

analyze the crash data produced, it can be aggregated by facility type, such as by crashes on a 

segment in a 6-mo period or multiple years. It can also be aggregated spatially, such as by 

crashes in a zone or county. Additionally, multiple variables can be used for analysis. A selection 

of these variables (and their distributions) are shown in figure 14. A user can analyze the data for 

roadway characteristics, such as for number of lanes, type of roadway, or AADT. A user can also 

analyze the data by crash characteristics, such as time of crash, type of crash, or severity of 

crash.

Source: FHWA.

Figure 14. Chart. Sample variable distribution.

The crash databases generated in this study can be employed for future use by transportation 

agencies to compare frequency models, severity models, crash type, and various other 

dimensions by facility type. The DREDGE generator the researchers developed can serve as a 

universal benchmarking system for alternative model frameworks in safety literature.
41 

CHAPTER 6. CASE STUDY DOCUMENTATION

RAD GENERATION AND VALIDATION

Generation and Validation Context 

Crash prediction models are essential tools in improving highway safety because they can 

identify both the expected frequency of crash occurrences and their contributing factors, which 

can subsequently be addressed by safety countermeasures. Using information from vehicle 

crashes, these prediction models predict both the frequency of crash occurrence and the degree of 

crash severity. Although the quantity of crash data has generally expanded over the years, the 

quality of crash data has not necessarily improved in tandem with methodological advancements 

in crash analysis. Prior studies tended to concentrate on the modeling component of the overall 

crash prediction process by developing cutting-edge modeling approaches that offered better fits

to the observed data, and the acquired data were implicitly assumed to be suitable representations 

of reality (Bonneson and Ivan 2013). In developing DREDGE, the research team hoped to 

mitigate this situation. The purpose of the case studies was to demonstrate the usefulness of the 

macroscopic DREDGE tool’s applications and establish its capability to reliably generate 

realistic data. Researchers completed the following analyses to support this purpose:

1. Generated a collection of datasets of varying sizes for two different facility types using 

the RAD tool. 

2. Estimated crash prediction models using each dataset and evaluated their predictive 

performance using mean absolute deviation (MAD) and mean-squared prediction error 

(MSPE) to select the best model.

3. Examined the estimated model parameters for stability from one dataset to another for 

each facility type.

The research team intended for these analyses to demonstrate how the RAD tool can serve as a 

testbed and help determine if any statistical models developed using the RAD can capture 

underlying relationships between independent variables and resultant crashes. Additionally, the 

team intended for these analyses to help guide and improve the practical application of statistical 

methods that influence highway safety policy, eventually leading to more effective safety 

countermeasures to reduce highway-related injuries and fatalities. The team also sought to 

demonstrate whether the data generated by the tool reliably represented real-world data and 

determine the tool’s consistency from one random generation session to another.

Data Generation for Case Studies

The team used datasets produced by the macroscopic RAD tool for two different facility types as 

an illustration to accomplish the purpose of the case studies. The tool was activated to generate 

10 different datasets for each facility, as follows: 

• Two sets of segments with 150, 300, 500, 750, and 1,000 mi each, with varying random 

seed values . 
42 

• Two sets of signalized intersections with 1,000, 2,000, 3,000, 4,000, and 

5,000 intersections each, with varying random seed values.

Crash Model Estimation

A wide variety of methods have been historically applied to address the data and methodological 

issues associated with crash-frequency data. Articles by Lord and Mannering (2010) and 

Abdulhafedh (2017) provide detailed reviews of the key issues associated with crash-frequency 

data and discuss strengths and weaknesses of various methodological approaches that researchers 

have used to address these issues. The research team considered several different statistical 

approaches for estimating models with the RAD, as described in the next few sections.

Poisson Regression Model

Using standard ordinary least-squares regression (which assumes a continuous dependent 

variable) is not appropriate for crash prediction because the crash frequency observations are 

non-negative integers (Lord and Mannering 2010). In a Poisson regression model, the probability 

of roadway entity (segment, intersection, etc.) I having yi crashes per some period (where yi is a 

non-negative integer) is given as shown in figure 15. 

Figure 15. Equation. Poisson regression model. 

where: 

P(yi) = probability of roadway entity I having yi crashes per period. 

λi = Poisson parameter for roadway entity I, which is equal to roadway entity i’s expected 

number of crashes per period E[yi]. 

Poisson regression models are estimated by specifying the Poisson parameter λi (the expected 

number of crashes per period) as a function of explanatory variables, the most common 

functional form being λi = EXP(βXi), where Xi is a vector of explanatory variables, and β is a 

vector of estimable parameters

However, researchers have often found that crash data exhibit characteristics that make the 

application of the simple Poisson regression problematic (Washington, Karlaftis, and Mannering 

2010). Poisson models cannot handle overdispersion and underdispersion, and they can be 

adversely affected by low sample means and produce biased results in small samples. These 

models have been applied to crash-frequency data by Jones, Janssen, and Mannering (1991); 

Miaou (1994), Greibe (2003), Al-Jabri (2015), Santamarina and Perez (2021), Khattak et 

al. (2021), and Shahzad et al. (2021).

NB Regression (Poisson-Gamma) Model

Some researchers have employed the use of NB as an alternative to Poisson regression 

(Abbas 2004; Amoros, Martin, and Laumon 2003; Hirst, Mountain, and Maher 2004; Lord and 
43 

Bonneson 2007; Miaou and Lord 2003). The NB (or Poisson-gamma) model is an extension of 

the Poisson model that was developed to overcome possible overdispersion in the data. The 

NB/Poisson-gamma model assumes that the Poisson parameter follows a gamma probability 

distribution governed by an additional parameter that is estimated. The NB distribution offers a 

simple way to accommodate the overdispersion, especially since the optimization function has a 

closed form, and the mathematics to manipulate the relationship between the mean and the 

variance structures is relatively simple. The NB model is derived by rewriting the Poisson 

parameter for each observation I, as shown in figure 16. 

Figure 16. Equation. NB model.

Where EXP(εεi) EXP(ᶓ)is a gamma-distributed error term with a mean of 1 and a variance of α.

The addition of this term allows the variance to differ from the mean as VAR[yi] = E[yi][1+ 

αE[yi]] = E[yi]+αE[yi] (Lord and Miranda-Moreno 2008; Miaou and Lord 2003). 

The Poisson-gamma/NB model is likely the most frequently used model in crash-frequency 

modeling. However, the model does have its limitations, most notably its inability to handle 

underdispersed data and dispersion-parameter-estimation problems when the data are 

characterized by low sample-mean values and small sample sizes (Washington, Karlaftis, and 

Mannering 2010).

Random Parameter Model 

The random parameter model can be viewed as an extension of the random effects model. The 

latter is designed to address the correlations in crash data arising from spatial considerations 

(data from the same geographic region may share unobserved effects), temporal considerations 

(such as in panel data, where data collected from the same observational unit over successive 

periods may share unobserved effects), or a combination of the two. To account for such 

correlations, the following models can be considered (Lord and Mannering 2010): 

• Random effects models where common, unobserved effects are assumed to be distributed 

over spatial and/or temporal units according to some distribution, and shared, unobserved 

effects are assumed to be uncorrelated with explanatory variables.

• Fixed effects models where common, unobserved effects are accounted for by indicator 

variables, and shared unobserved effects are assumed to be correlated with independent 

variables.

Random parameter models can be viewed as extensions of random effects models; however, 

rather than effectively only influencing the intercept of the model, random parameter models 

allow estimated parameter values to vary across all the observations in the dataset. Random 

parameters attempt to account for unobserved heterogeneity from one roadway site to another 

(Milton, Shankar, and Mannering 2008). Some researchers have applied this approach to crash 

frequency data. For example, Anastasopoulos and Mannering (2009) explored the use of random 
44 

parameter count models as a methodological alternative for analyzing crash frequencies. Their 

findings showed that ignoring the possibility of random parameters when estimating count data 

models can result in different marginal effects and subsequent inferences relating to the 

magnitude of the effect of factors affecting accident frequencies. This result can also be observed 

in studies from Bhat (2001) and Eluru, Bhat, and Hensher (2008). 

To allow for such random parameters in count data models, estimable parameters can be written 

as , βi = β + φi where φi is a randomly distributed term—for example, a normally distributed term 

with mean zero and variance σ2. With this equation, the Poisson parameter becomes 

λi|φi = EXP(βiXi) in the Poisson model and λi|φi = EXP(βiXi+εi) in the NB/Poisson-gamma, with 

the corresponding probabilities for Poisson or NB now P(yi|φi). Each observation has its own 

parameters; the final model will often provide a statistical fit that is significantly better than a 

model with traditional fixed parameters. However, random parameter models are complex to 

estimate, and they are not guaranteed to improve predictive capability (Washington, Karlaftis, 

and Mannering 2010). 

Poisson Lognormal Models 

Some researchers have proposed using the Poisson-lognormal model as an alternative to the 

NB/Poisson-gamma model for modeling crash data (Aguero-Valverde and Jovanis 2008; Lord 

and Miranda-Moreno 2008). The Poisson-lognormal model is similar to the NB/Poisson-gamma 

model; however, the EXP(εi) term used to compute the Poisson parameter is lognormal rather 

than gamma-distributed. The Poisson-lognormal model addresses limitations of the NB model

because it is more flexible in handling overdispersion. 

Multivariate Poisson Lognormal Models

Researchers have found this method necessary when instead of total crash counts, one wishes to 

model mutually exclusive categories of crash counts, such as counts by severity or type of 

collision (Aguero-Valverde and Jovanis 2008; N’Guessan and Langrand 2005; N’Guessan 

2010). Modeling the counts of mutually exclusive crash types, as opposed to total crashes, with 

independent count models can be statistically invalid because the categorical counts are not 

strictly independent of one another. That is, the counts of crashes resulting in fatalities cannot 

increase or decrease without affecting the counts of crashes resulting in injuries and no injuries. 

Bivariate/multivariate models are used to resolve this problem, as they explicitly consider the 

correlations among the severity levels (Lord and Mannering 2010). The multivariate Poisson 

Lognormal approach calculates a covariance matrix that captures additional heterogeneity that 

other models do not. 

Validating Crash Count Predictions 

An objective assessment of the predictive performance of a particular model can be made only 

through the evaluation of several goodness of fit (GOF) criteria. GOF measures used to conduct 

external model validation include MAD and MSPE (Washington, Karlaftis, and Mannering 

2010). The model building effort in this case study used the first dataset at each total distance 

increment for model estimation and the other dataset for cross validation. This process was 
45 

reversed, and the data used for cross validation were then used for estimation, and the validation 

dataset was also used for estimation.

The GOF measures are calculated using the equations shown in figure 17 and figure 18. 

Figure 17. Equation. MAD.

Figure 18. Equation. MSPE.

where:

Ymodel = predicted crash frequency. 

Yobserved = observed crash frequency. 

n = sample size. 

Evaluating Parameter Stability

The parameter estimates from the model estimation are used to check if the stochasticity 

embedded in the RAD generation process will be consistent for different random seeds used to 

generate data with the tool. To achieve this goal, researchers examined parameter estimates from 

the models for each dataset using revised Wald test statistics created by Hoover, Bhowmik, 

Yasmin, and Eluru (2022), as shown in figure 19. 

Figure 19. Equation. Parameter test statistics.

Where SE denotes the standard error for the corresponding sample. 

If the parameter test statistic computed was higher than the 90 percent t-statistic, the result would 

indicate significant difference across the parameters. The research team employed this test 

statistic (figure 19) to compute revised t-statistics for all the parameters across all samples.

Application To Rural Two-Lane, Two-Way Undivided Segments

Generation of Datasets

Crash prediction models were estimated using each of the alternative approaches, as detailed in 

table 6. 
46 

Table 6. Summary of developed models. 

ID Facility Datasets Models

Severity 

Level

No. of 

Models

1 Two-lane, 

two-way, 

undivided 

segment

Two sets each 

of 150, 300, 

500, 750, and 

1,000 mi 

Poisson regression K, A, B, C, 

PDO

5×2 = 10

NB regression K, A, B, C, 

PDO

5×2 = 10

Poisson regression—

random parameters

K, A, B, C, 

PDO

5×2 = 10

NB—random 

parameters

K, A, B, C, 

PDO

5×2 = 10

Univariate Poisson 

lognormal

K, A, B, C, 

PDO

5×2 = 10

Multivariate Poisson 

lognormal

K, A, B, C, 

PDO

5×2 = 10

The descriptive statistics of the datasets generated by the tool are shown in table 7 and table 8. 

Notably, the same size dataset (i.e., the two sets of 150 mi) resulted in different numbers of 

observations, as the datasets were randomly generated. The data contained horizontal curve data, 

roadway data, and crash data. The list of variables included is summarized in table 7 and table 8. 

Model Estimation 

Researchers estimated crash prediction models using the parameters given in table 6 of this 

report. A total of 60 models were developed for rural two-lane undivided segments for each 

severity level: K, A, B, C, and O. For brevity, only the model parameter estimates and model fit 

statistics for the 1,000-mi dataset models are included in table 9, table 10, table 11, table 12, 

table 13, and table 14. Parameter estimates and model fit statistics for all models are provided in 

appendix B, which is in the second volume of this publication.
47 

Table 7. Descriptive statistics for continuous variables.

Continuous 

Variables 

Dataset 1 

(n = 1,351): 

150 mi

Dataset 2 

(n = 2,742): 

300 mi

Dataset 3 

(n = 4,690): 

500 mi

Dataset 4 

(n = 4,140): 

750 mi

Dataset 5 

(n = 8,667): 

1,000 mi

Dataset 6 

(n = 1,361): 

150 mi

Dataset 7 

(n = 2,229): 

300 mi

Dataset 8 

(n = 4,270): 

500 mi

Dataset 9 

(n = 3,749): 

750 mi

Dataset 10 

(n = 7,050): 

1,000 mi

Crash counts PDO 0.938a, 

1.752b

0.856, 

1.615 

0.853, 

1.684 

1.231, 

2.52 

0.950, 

1.80 

0.870, 

1.751 

0.988, 

1.997 

0.923, 

1.841 

1.432, 

2.751 

1.087, 

2.13 

Crash counts K 0.005, 

0.094 

0.003, 

0.060 

0.005, 

0.092 

0.007, 

0.150 

0.004, 

0.088 

0.004, 

0.085 

0.008 

0.232 

0.0014, 

0.037 

0.008, 

0.230 

0.0079, 

0.27 

Crash counts A 0.168, 

0.565 

0.137, 

0.477 

0.143, 

0.504 

0.223, 

0.765 

0.168, 

0.583 

0.153, 

0.561 

0.156 

0.536 

0.165, 

0.629 

0.247, 

0.855 

0.167, 

0.63 

Crash counts B 0.041, 

0.247 

0.041, 

0.241 

0.047, 

0.278 

0.060, 

0.374 

0.057, 

0.323 

0.036, 

0.234 

0.052 

0.306 

0.054, 

0.336 

0.073, 

0.386 

0.064, 

0.370 

Crash counts C 0.157, 

0.454 

0.165, 

0.493 

0.158, 

0.486 

0.282, 

0.908 

0.193, 

0.552 

0.196, 

0.527 

0.213 

0.613 

0.186, 

0.618 

0.397, 

0.810 

0.219, 

0.641 

Pavement roughness 104, 

34.04 

104.5, 

34.656 

105.3, 

40.64 

107.6, 

40.30 

105.9, 

39.800 

104.7, 

33.54 

104.9 

34.96 

105.1, 

39.29 

108.5, 

40.83 

106.7, 

40.7 

Pavement condition 40.170, 

3.090 

39.85, 

3.077 

23.0, 

3.714 

39.37, 

3.791 

39.33, 

3.730 

40.28, 

3.173 

39.63 

3.18 

39.59, 

3.76 

39.4, 

3.728 

39.24, 

3.70 

Average 

super-elevation

0.769, 

2.603 

0.769, 

2.586 

−8.00, 

2.576 

−8.183, 

11.94 

−19.61, 

9.240 

0.802, 

2.60 

0.771 

2.608 

0.737, 

2.59 

−9.14, 

12.14 

−19.07, 

9.74 

Curvature degree 4.320, 

8.435 

5.217, 

10.354 

5.21, 

9.30 

0.00, 

4.750 

5.52, 

10.91 

5.155, 

10.17 

4.624 

8.563 

5.062, 

8.943 

3.122, 

2.970 

5.367, 

10.27 

Arc angle 36.53, 

20.62 

35.16, 

21.904 

10.0, 

22.08 

43.29, 

12.42 

36, 

22.670 

36.06, 

22.72 

40.75 

18.904 

37.79, 

19.92 

45.33, 

5.557 

37.41, 

20.93 

Log (radius) 7.686, 

0.904 

7.189, 

0.943 

7.17, 

0.830 

8.281, 

0.842 

7.153, 

0.920 

7.143, 

0.931 

7.694 

1.094 

7.463, 

1.051 

8.657, 

0.304 

7.397, 

1.034 

Log segment length −2.710, 

1.066 

−2.72, 

1.058 

−2.75, 

1.056 

−2.383, 

1.22 

−2.63, 

1.101 

−2.69, 

1.053 

−2.58 

1.126 

−2.69, 

1.087 

−2.29, 

1.249 

−2.576, 

1.150 

Vertical approach 0.225, 

0.828 

0.293, 

0.910 

0.355, 

0.895 

0.504, 

1.094 

0.440, 

0.960 

0.232, 

0.850 

0.316 

0.950 

0.361, 

0.913 

0.548, 

1.134 

0.459, 

0.990 

Vertical leaving 0.239, 

0.873 

0.213, 

0.822 

0.221, 

0.831 

0.340, 

1.017 

0.26, 

0.930 

0.252, 

0.905 

0.250 

0.874 

0.236, 

0.848 

0.369, 

1.055 

0.277, 

0.936 

Log (AADT) 7.735, 

0.570 

7.394, 

0.845 

7.628, 

0.832 

7.381, 

0.851 

7.56, 

0.804 

7.739, 

0.575 

7.411 

0.858 

7.616, 

0.835 

7.357, 

0.891 

7.542, 

0.820 

Grade 0.353, 

1.075 

0.402, 

1.090 

0.462, 

1.080 

0.671, 

1.30 

0.56, 

1.170 

0.370, 

1.05 

0.446 

1.145 

0.474, 

1.104 

0.728, 

1.35 

0.594, 

1.200 a

mean. b

standard deviation.
48 

Table 8. Descriptive statistics for categorical variables (datasets 1–10).

Variable Dataset 1 Dataset 2 Dataset 3 Dataset 4 Dataset 5 Dataset 6 Dataset 7 Dataset 8 Dataset 9 Dataset 10

Shoulder Width

0 ft 42 80 280 208 465 16 69 260 194 413

2 ft 454 845 939 1134 2029 20 697 860 1044 121

4 ft 186 589 1196 1083 2070 444 467 1081 970 2876

6 ft 387 462 620 593 1387 171 379 570 532 1824

8 ft 282 755 1655 1122 2116 705 617 1499 1005 1816

Speed Limit

25 mph 95 92 100 78 203 94 109 104 51 157

30 mph 26 34 75 73 159 19 41 60 57 129

35 mph 26 35 55 81 220 33 35 54 75 129

40 mph 54 138 272 229 472 50 101 260 200 393

45 mph 85 133 311 258 486 86 122 253 224 418

50 mph 31 96 151 258 486 86 122 253 224 418

55 mph 1037 2205 3726 3303 6339 1043 1747 3399 3025 5604

Lane Width

9 ft 0 3 37 107 247 0 3 37 103 203

10 ft 50 183 354 311 382 50 180 327 267 347

11 ft 269 664 11075 919 1829 264 558 912 835 1651

12 ft 1032 1881 3224 2803 5609 1042 1488 2992 2540 4849

Lighting

Present 77 201 353 344 530 72 172 316 328 530

Not 

present 1274 2530 4337 3796 7537 1284 2057 3954 3417 6520
49 

Table 9. Poisson regression estimates for dataset 10 (1,000 mi). 

Variables

K A B C PDO 

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter

Estimate

Std.

Error

Intercept −25.066* 4.175 −7.434 0.678 −4.478 1.004 −3.609 0.560 −2.099 0.250

Ln (Segment 

Length) 1.292 0.178 0.984 0.032 1.023 0.051 1.000 0.028 1.059 0.013

Ln (AADT) 1.389 0.303 0.813 0.050 0.533 0.075 0.469 0.040 0.536 0.018

Shoulder Width

8 ft Base Level

<2 ft −0.580* 1.059 0.317 0.140 0.656 0.214 0.324 0.129 0.231 0.062

≥2 ft < 4 ft −16.357* 1672.361 0.736 0.180 −0.817 0.589 −0.020 0.227 0.349 0.087

≥4 ft < 6 ft −0.080 0.397 0.260 0.076 0.369 0.126 0.359 0.066 0.340 0.030

≥6 ft < 8 ft −0.874 0.583 0.174 0.088 0.395 0.140 0.003 0.079 0.263 0.034

Lane Width

>12 ft Base Level

≤9 ft 3.028 0.567 0.279 0.160 0.011 0.249 0.246 0.142 0.145 0.064

9.5 ft−10.5 ft 0.411 0.593 0.129 0.091 −0.376 0.149 0.179 0.076 0.188 0.034

11 ft−11.5 ft −12.372* 1520.181 0.478 0.229 −0.341 0.393 0.343 0.194 0.223 0.089

Speed Limit

45 mph Base Level

25 mph 2.826* 1.088 0.791 0.181 1.089 0.323 0.365 0.157 0.483 0.068

30 mph −15.228* 1864.827 0.387 0.199 1.239 0.303 0.560 0.150 0.081 0.078

35 mph 0.773* 1.250 0.642 0.184 0.696 0.329 0.038 0.167 −0.359 0.088

40 mph 1.575* 1.096 −0.676 0.214 −0.479 0.374 −0.677 0.163 −1.044 0.084

50 mph −14.297* 1595.923 0.321 0.217 −0.256 0.505 −0.356 0.219 −0.151 0.092

55 mph −0.123 1.072 −0.217 0.136 0.246 0.249 −0.356 0.106 −0.258 0.049

Roadside Hazard Rating (Zegeer et al. 1988) 

3 Base Level

4 −1.069 0.547 0.045 0.084 0.030 0.138 0.146 0.072 0.069 0.032

5 −1.395 0.678 0.115 0.103 0.279 0.164 0.233 0.089 0.180 0.040

6 0.054 0.689 −0.025 0.134 0.430 0.204 0.161 0.116 0.159 0.052

7 −0.715 1.066 0.237 0.171 0.558 0.279 0.110 0.159 0.251 0.068

Pavement 

Condition 0.249 0.068 0.021 0.012 −0.019 0.018 0.008 0.011 −0.002 0.005

Pavement 

Roughness 0.014 0.008 0.002 0.001 −0.002 0.002 0.001 0.001 0.000 0.001
50 

Variables

K A B C PDO 

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter

Estimate

Std.

Error

Presence of 

Lighting −1.631 1.220 −0.180 0.164 −0.491 0.288 −0.237 0.152 −0.417 0.070

Presence of 

Horizontal 

Curve

−14.516* 1042.264 −1.408 0.411 −14.329* 262.136 −1.225 0.336 0.345 0.073

Vertical Curve 

Approaching 

Slope

−3.043* 5.011 0.173 0.157 −0.359 0.289 0.158 0.106 −0.053 0.053

Vertical Curve 

Leaving Slope −2.276* 5.020 −0.104 0.114 0.163 0.324 0.332 0.107 0.077 0.051

Grade 2.775* 5.118 −0.058 0.175 −0.061 0.374 −0.336 0.140 −0.005 0.065

Curvature 

Degree −0.007 0.024 0.006 0.003 0.002 0.005 0.000 0.003 0.002 0.001

Fit Statistics

AIC 443.88 N/A 5693.6 N/A 3018.8 N/A 6641.1 N/A 15812 N/A

BIC 635.97 N/A 5885.67 N/A 3210.9 N/A 6833.24 N/A 16004.4 N/A

—No data. 

*Variables are insignificant at a 90-percent confidence level.

N/A = not applicable; std. error = standard error.
51 

Table 10. NB estimates for dataset 10 (1,000 mi).

Variables

K A B C PDO 

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Intercept −18.145* 6.912 −5.918 0.948 −4.412 1.829 −3.124 0.766 −2.002 0.389

Ln (Segment 

Length) 0.976 0.339 1.006 0.046 1.200 0.097 1.049 0.039 1.058 0.020

Ln (AADT) 1.206 0.512 0.847 0.064 0.204 0.108 0.438 0.049 0.452 0.025

Shoulder Width

8 ft Base Level

<2 ft 0.005 1.406 0.322 0.210 −0.515 0.631 0.837 0.164 0.240 0.094

≥2 ft < 4 ft −1.063 1.226 0.538 0.119 1.019 0.243 0.528 0.103 0.292 0.051

≥4 ft < 6 ft 1.001 0.828 0.189 0.124 0.558 0.259 0.558 0.104 0.258 0.051

≥6 ft < 8 ft 0.338 0.910 0.115 0.144 0.482 0.296 0.367 0.122 0.246 0.059

Lane Width

>12 ft Base Level

≤9 ft −49.933* 38132 0.116 0.222 0.514 0.444 0.157 0.183 0.256 0.095

9.5 ft−10.5 ft 0.920 0.818 0.027 0.123 0.051 0.247 0.126 0.100 0.091 0.052

11 ft−11.5 ft −46.906* 65065 −0.065* 0.308 0.120 0.675 0.167 0.255 0.505 0.130

Speed Limit

45 mph Base Level

25 mph 3.088* 1.793 0.303 0.313 −0.222 0.660 0.549 0.252 0.041 0.131

30 mph −0.097* 3.564 0.353 0.300 1.175 0.552 0.817 0.237 0.041 0.136

35 mph 2.391* 1.939 −0.001 0.332 1.096 0.529 0.717 0.241 −0.295 0.144

40 mph −46.975* 4453 −0.920 0.275 −0.604 0.498 −0.553 0.230 −1.110 0.120

50 mph −49.941* 6182.0 −0.552 0.341 −0.248 0.612 0.211 0.258 0.004 0.120

55 mph 0.634 1.399 −0.396 0.166 −0.546 0.349 −0.101 0.153 −0.330 0.071

Roadside Hazard Rating (Zegeer et al 1988) 

3 Base Level

4 1.392 1.159 0.122 0.120 −0.146 0.238 −0.092 0.100 0.015 0.049

5 0.485 1.391 0.083 0.152 0.073 0.301 0.093 0.125 0.051 0.064

6 0.867 1.585 0.019 0.191 0.313 0.371 0.205 0.156 0.251 0.080

7 2.325 1.695 −0.078 0.261 −0.083 0.536 0.348 0.206 0.180 0.110

Pavement 

Condition 0.081 0.125 −0.014 0.018 0.040 0.038 −0.006 0.015 0.014 0.008

Pavement 

Roughness −0.002 0.012 0.003 0.002 0.002 0.004 0.001 0.002 0.000 0.001
52 

Variables

K A B C PDO 

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Presence of 

Lighting 0.688 1.347 −0.404 0.238 −0.240 0.449 −0.722 0.211 −0.531 0.102

Presence of 

Horizontal 

Curve

−42.583* 45853 −2.856 1.032 −1.361 1.065 −1.429 0.462 0.186 0.113

Vertical Curve 

Approaching 

Slope

0.163 1.167 0.528 0.244 −0.008 0.528 0.125 0.157 −0.011 0.075

Vertical Curve 

Leaving Slope 1.851 2.132 0.107 0.151 −0.243 0.409 0.079 0.135 −0.050 0.064

Grade −1.984 2.366 −0.546 0.274 0.033 0.587 −0.140 0.189 0.060 0.089

Curvature 

Degree 0.105 0.038 −0.016 0.013 −0.011 0.024 0.002 0.009 0.000 0.005

Fit Statistics

AIC 370.24 N/A 6386.8 N/A 3011.2 N/A 7191.9 N/A 17318 N/A

BIC 439.225 N/A 3843.97 N/A 1686.52 N/A 4457.75 N/A 9481.62 N/A

Overdispersion 0.0139 — 1.182 — 4.329 — `0.567 — 0.375 — 

—No data.

*Variables are insignificant at a 90-percent confidence level.
53 

Table 11. Random parameters—Poisson regression estimates for dataset 10 (1,000 mi).

Variables

K A B C PDO 

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Intercept −17.604* 10.500 −6.423 0.966 −5.983 2.342 −3.449 0.777 −2.262 0.385

Ln (Segment 

Length) 0.855 0.617 0.988 0.049 1.111 0.123 1.040 0.040 1.053 0.020

Ln (AADT) 0.862 1.023 0.830 0.067 0.251 0.144 0.438 0.051 0.468 0.026

Shoulder Width

8 ft Base Level

<2 ft 0.265 2.649 0.204 0.224 −0.460 0.810 0.805 0.168 0.241 0.092

≥2 ft < 4 ft −1.248 2.812 0.497 0.120 1.234 0.310 0.530 0.105 0.305 0.049

≥4 ft < 6 ft 0.685 1.488 0.172 0.124 0.522 0.325 0.552 0.105 0.272 0.050

≥6 ft < 8 ft 0.059 1.608 0.068 0.145 0.066 0.399 0.355 0.124 0.253 0.057

Lane Width

>12 ft Base Level

≤9 ft 0.265 2.649 0.204 0.224 −0.460 0.810 0.805 0.168 0.241 0.092

9.5 ft−10.5 ft −1.248 2.812 0.497 0.120 1.234 0.310 0.530 0.105 0.305 0.049

11 ft−11.5 ft 0.685 1.488 0.172 0.124 0.522 0.325 0.552 0.105 0.272 0.050

Speed Limit

45 mph Base Level

25 mph 1.650* 4.138 0.296 0.333 −0.412 0.828 0.541 0.258 0.017 0.128

30 mph 1.833* 4.425 0.517 0.303 0.646 0.689 0.770 0.244 0.027 0.132

35 mph 3.153 3.556 −0.071 0.351 −0.180 0.823 0.596 0.251 −0.377 0.142

40 mph −3.141* 20.861 −0.855 0.280 −1.315 0.629 −0.557 0.234 −1.106 0.117

50 mph −30.351* 609.50 −0.391 0.332 −1.131 0.838 0.222 0.262 −0.009 0.116

55 mph −0.242 2.618 −0.342 0.168 −1.005 0.398 −0.123 0.154 −0.339 0.068

Roadside Hazard Rating (Zegeer et al 1988) 

3 Base Level

4 2.522* 1.945 0.053 0.122 −0.417 0.307 −0.090 0.101 0.030 0.048

5 1.539* 2.566 0.062 0.156 −0.401 0.397 0.094 0.128 0.049 0.063

6 1.953* 2.526 −0.014 0.193 −0.023 0.445 0.211 0.156 0.278 0.077

7 2.678* 2.636 −0.113 0.267 −0.223 0.647 0.381 0.211 0.206 0.107

Pavement Condition 0.037 0.165 −0.009 0.018 0.014 0.047 −0.003 0.015 0.016 0.008

Pavement 

Roughness 0.006 0.021 0.003 0.002 0.005 0.005 0.001 0.002 0.000 0.001

Presence of Lighting 0.128 2.363 −0.420 0.244 −0.330 0.553 −0.754 0.217 −0.534 0.100
54 

Variables

K A B C PDO 

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Presence of 

Horizontal Curve −27.666* 2949.6 −2.918 1.050 −1.583 1.350 −1.479 0.470 0.175 0.111

Vertical Curve 

Approaching Slope 0.286 2.322 0.505 0.236 −0.165 0.593 0.133 0.159 −0.006 0.075

Vertical Curve 

Leaving Slope 2.126 5.974 0.145 0.152 −0.340 0.483 0.089 0.138 −0.033 0.064

Grade −2.241 6.215 −0.547 0.267 0.275 0.665 −0.159 0.193 0.038 0.089

Curvature Degree −0.007 0.024 0.006 0.003 0.002 0.005 0.000 0.003 0.002 0.001

SD (Curvature 

Degree) 1.027 0.052 1.097 0.022 1.2011 0.107 0.838 0.152 0.584 0.029 

Fit Statistics

AIC 443.88 N/A 5693.6 N/A 3018.8 N/A 6641.1 N/A 15812 N/A

BIC 635.97 N/A 5885.67 N/A 3210.9 N/A 6833.24 N/A 16004.4 N/A

—No data.

*Variables are insignificant at a 90-percent confidence level.
55 

Table 12. Random parameters—NB regression estimates for dataset 10 (1,000 mi).

Variables

K A B C PDO 

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Intercept −25.066 4.175 2.522 0.678 −3.181 1.004 −2.594 0.560 −1.875 0.250

Ln (Segment 

Length) 1.323 0.178 1.539 0.032 1.318 0.051 1.000 0.028 1.0946 0.013

Ln (AADT) 0.367 0.303 1.953 0.050 0.133 0.075 0.469 0.040 0.090 0.018

Standard Deviation 

(Length) 0.263 0.184 2.678 2.968 0.197 1.560 0.432 0.371 0.520 0.149

Standard Deviation 

(AADT) 0.012 0.219 0.037 1.149 0.134 0.793 0.021 0.387 0.116 0.324

Shoulder Width

8 ft Base Level

<2 ft −0.580* 1.059 0.317 0.140 0.656 0.214 0.324 0.129 0.231 0.062

≥2 ft < 4 ft −16.357* 1672.361 −0.364 0.180 −0.817 0.589 −0.020 0.227 0.349 0.087

≥4 ft < 6 ft −0.080* 0.397 0.260 0.076 0.369 0.126 0.359 0.066 0.340 0.030

≥6 ft < 8 ft −0.874* 0.583 0.174 0.088 0.395 0.140 0.003 0.079 0.263 0.034

Lane Width

>12 ft Base Level

≤9 ft 3.028 0.567 0.279 0.160 0.011 0.249 0.246 0.142 0.145 0.064

9.5 ft−10.5 ft 0.411 0.593 0.129 0.091 −0.376 0.149 0.179 0.076 0.188 0.034

11 ft−11.5 ft −12.372* 1520.181 0.478 0.229 −0.341 0.393 0.343 0.194 0.223 0.089

Speed Limit

45 mph Base Level

25 mph 2.826* 1.088 0.791 0.181 1.089 0.323 0.365 0.157 0.483 0.068

30 mph −15.228* 1864.827 0.387 0.199 1.239 0.303 0.560 0.150 0.081 0.078

35 mph 0.773* 1.250 0.642 0.184 0.696 0.329 0.038 0.167 −0.359 0.088

40 mph 1.575* 1.096 −0.676 0.214 −0.479 0.374 −0.677 0.163 −1.044 0.084

50 mph −14.297* 1595.923 0.321 0.217 −0.256 0.505 −0.356 0.219 −0.151 0.092

55 mph −0.123* 1.072 −0.217 0.136 0.246 0.249 −0.356 0.106 −0.258 0.049

Roadside Hazard Rating (Zegeer et al 1988) 

3 Base Level

4 −1.690* 0.547 0.045 0.084 0.030 0.138 0.146 0.072 0.069 0.032

5 −1.395 0.678 0.115 0.103 0.279 0.164 0.233 0.089 0.180 0.040

6 0.054 0.689 −0.025 0.134 0.430 0.204 0.161 0.116 0.159 0.052

7 −0.715 1.066 0.237 0.171 0.558 0.279 0.110 0.159 0.251 0.068
56 

Variables

K A B C PDO 

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Pavement Condition 0.249 0.068 0.021 0.012 −0.019 0.018 0.008 0.011 −0.002 0.005

Pavement 

Roughness 0.014 0.208 0.002 0.371 −0.297 0.284 0.387 0.248 0.452 0.732

Presence of Lighting −1.631 1.220 −0.180 0.164 −0.491 0.288 −0.237 0.152 −0.417 0.070

Presence of 

Horizontal Curve −14.516* 1042.264 −1.408 0.411 −14.329 262.136 −1.225 0.336 0.345 0.073

Vertical Curve 

Approaching Slope −3.043 5.011 0.173 0.157 −0.359 0.289 0.158 0.106 −0.053 0.053

Vertical Curve 

Leaving Slope −2.276 5.020 −0.104 0.114 0.163 0.324 0.332 0.107 0.077 0.051

Grade 2.775 5.118 −0.058 0.175 −0.061 0.374 −0.336 0.140 −0.005 0.065

Curvature Degree −0.007 0.024 0.006 0.003 0.002 0.005 0.000 0.003 0.002 0.001

Fit Statistics

AIC 443.88 N/A 5693.6 N/A 3018.8 N/A 6641.1 N/A 15812 N/A

BIC 635.97 N/A 5885.67 N/A 3210.9 N/A 6833.24 N/A 16004.4 N/A

—No data.

*Variables are insignificant at a 90-percent confidence level.
57 

Table 13. Poisson univariate estimates for dataset 10 (1,000 mi). 

Variables

K A B C PDO 

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Intercept −25.066* 4.175 −7.434 0.678 −4.478 1.004 −3.609 0.560 −2.099 0.250

Ln (Segment Length) 1.292 0.178 0.984 0.032 1.023 0.051 1.000 0.028 1.059 0.013

Ln (AADT) 1.389 0.303 0.813 0.050 0.533 0.075 0.469 0.040 0.536 0.018

Shoulder Width

8 ft Base Level

<2 ft −0.580* 1.059 0.317 0.140 0.656 0.214 0.324 0.129 0.231 0.062

≥2 ft < 4 ft −16.357* 1672.361 0.736 0.180 −0.817 0.589 −0.020 0.227 0.349 0.087

≥4 ft < 6 ft −0.080 0.397 0.260 0.076 0.369 0.126 0.359 0.066 0.340 0.030

≥6 ft < 8 ft −0.874 0.583 0.174 0.088 0.395 0.140 0.003 0.079 0.263 0.034

Lane Width

>12 ft Base Level

≤9 ft 3.028* 0.567 0.279 0.160 0.011 0.249 0.246 0.142 0.145 0.064

9.5 ft−10.5 ft 0.411 0.593 0.129 0.091 −0.376 0.149 0.179 0.076 0.188 0.034

11 ft−11.5 ft −12.372* 1520.181 0.478 0.229 −0.341 0.393 0.343 0.194 0.223 0.089

Speed Limit

45 mph Base Level

25 mph 2.826* 1.088 0.791 0.181 1.089 0.323 0.365 0.157 0.483 0.068

30 mph −15.228* 1864.827 0.387 0.199 1.239 0.303 0.560 0.150 0.081 0.078

35 mph 0.773 1.250 0.642 0.184 0.696 0.329 0.038 0.167 −0.359 0.088

40 mph 1.575 1.096 −0.676 0.214 −0.479 0.374 −0.677 0.163 −1.044 0.084

50 mph −14.297* 1595.923 0.321 0.217 −0.256 0.505 −0.356 0.219 −0.151 0.092

55 mph −0.123 1.072 −0.217 0.136 0.246 0.249 −0.356 0.106 −0.258 0.049

Roadside Hazard Rating (Zegeer et al 1988) 

3 Base Level

4 −1.069 0.547 0.045 0.084 0.030 0.138 0.146 0.072 0.069 0.032

5 −1.395 0.678 0.115 0.103 0.279 0.164 0.233 0.089 0.180 0.040

6 0.054 0.689 −0.025 0.134 0.430 0.204 0.161 0.116 0.159 0.052

7 −0.715 1.066 0.237 0.171 0.558 0.279 0.110 0.159 0.251 0.068

Pavement Condition 0.249 0.068 0.021 0.012 −0.019 0.018 0.008 0.011 −0.002 0.005

Pavement Roughness 0.014 0.008 0.002 0.001 −0.002 0.002 0.001 0.001 0.000 0.001

Presence of Lighting −1.631 1.220 −0.180 0.164 −0.491 0.288 −0.237 0.152 −0.417 0.070

Presence of Horizontal 

Curve −14.516* 1042.264 −1.408 0.411 −14.329 262.136 −1.225 0.336 0.345 0.073
58 

Variables

K A B C PDO 

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Vertical Curve 

Approaching Slope −3.043* 5.011 0.173 0.157 −0.359 0.289 0.158 0.106 −0.053 0.053

Vertical Curve Leaving 

Slope −2.276* 5.020 −0.104 0.114 0.163 0.324 0.332 0.107 0.077 0.051

Grade 2.775* 5.118 −0.058 0.175 −0.061 0.374 −0.336 0.140 −0.005 0.065

Curvature Degree −0.007 0.024 0.006 0.003 0.002 0.005 0.000 0.003 0.002 0.001

Fit Statistics

AIC 443.88 N/A 5693.6 N/A 3018.8 N/A 6641.1 N/A 15812 N/A

BIC 635.97 N/A 5885.67 N/A 3210.9 N/A 6833.24 N/A 16004.4 N/A

—No data.

*Variables are insignificant at a 90-percent confidence level.
59 

Table 14. Multivariate Poisson lognormal estimates for dataset 10 (1,000 mi). 

Variables 

K A B C PDO 

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Intercept −25.066* 4.175 −7.434 0.678 −4.478 1.004 −3.609 0.560 −2.099 0.250

Ln (Segment Length) 1.292 0.178 0.984 0.032 1.023 0.051 1.000 0.028 1.059 0.013

Ln (AADT) 1.389 0.303 0.813 0.050 0.533 0.075 0.469 0.040 0.536 0.018

Shoulder Width

8 ft Base Level

<2 ft −0.580 1.059 0.317 0.140 0.656 0.214 0.324 0.129 0.231 0.062

≥2 ft < 4 ft −16.357* 1672.361 0.736 0.180 −0.817 0.589 −0.020 0.227 0.349 0.087

≥4 ft < 6 ft −0.080 0.397 0.260 0.076 0.369 0.126 0.359 0.066 0.340 0.030

≥6 ft < 8 ft −0.874 0.583 0.174 0.088 0.395 0.140 0.003 0.079 0.263 0.034

Lane Width

>12 ft Base Level

≤9 ft 3.028* 0.567 0.279 0.160 0.011 0.249 0.246 0.142 0.145 0.064

9.5 ft−10.5 ft 0.411 0.593 0.129 0.091 −0.376 0.149 0.179 0.076 0.188 0.034

11 ft−11.5 ft −12.372* 1520.181 0.478 0.229 −0.341 0.393 0.343 0.194 0.223 0.089

Speed Limit

45 mph Base Level

25 mph 2.826 1.088 0.791 0.181 1.089 0.323 0.365 0.157 0.483 0.068

30 mph −15.228* 1864.827 0.387 0.199 1.239 0.303 0.560 0.150 0.081 0.078

35 mph 0.773 1.250 0.642 0.184 0.696 0.329 0.038 0.167 −0.359 0.088

40 mph 1.575 1.096 −0.676 0.214 −0.479 0.374 −0.677 0.163 −1.044 0.084

50 mph −14.297* 1595.923 0.321 0.217 −0.256 0.505 −0.356 0.219 −0.151 0.092

55 mph −0.123 1.072 −0.217 0.136 0.246 0.249 −0.356 0.106 −0.258 0.049

Roadside Hazard Rating (Zegeer et al 1988) 

3 Base Level

4 −1.069 0.547 0.045 0.084 0.030 0.138 0.146 0.072 0.069 0.032

5 −1.395 0.678 0.115 0.103 0.279 0.164 0.233 0.089 0.180 0.040

6 0.054 0.689 −0.025 0.134 0.430 0.204 0.161 0.116 0.159 0.052

7 −0.715 1.066 0.237 0.171 0.558 0.279 0.110 0.159 0.251 0.068

Pavement Condition 0.249 0.068 0.021 0.012 −0.019 0.018 0.008 0.011 −0.002 0.005

Pavement Roughness 0.014 0.008 0.002 0.001 −0.002 0.002 0.001 0.001 0.000 0.001

Presence of Lighting −1.631 1.220 −0.180 0.164 −0.491 0.288 −0.237 0.152 −0.417 0.070

Presence of Horizontal 

Curve −14.516* 1042.264 −1.408 0.411 −14.329 262.136 −1.225 0.336 0.345 0.073
60 

Variables 

K A B C PDO 

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Parameter 

Estimate

Std.

Error

Vertical Curve 

Approaching Slope −3.043* 5.011 0.173 0.157 −0.359 0.289 0.158 0.106 −0.053 0.053

Vertical Curve Leaving 

Slope −2.276* 5.020 −0.104 0.114 0.163 0.324 0.332 0.107 0.077 0.051

Grade 2.775* 5.118 −0.058 0.175 −0.061 0.374 −0.336 0.140 −0.005 0.065

Curvature Degree −0.007 0.024 0.006 0.003 0.002 0.005 0.000 0.003 0.002 0.001

Variance-Covariance Matrix

K 0.218 0.625 0.248 0.138 0.086

A — 1.011 0.526 0.491 0.516

B — — 1.600 0.733 0.715

C — — — 0.501 0.477

PDO — — — — 0.531

—No data.

*Variables are insignificant at a 90-percent confidence level.
61 

Crash Validation 

The model prediction of dataset 10 used for model estimation is shown in table 15. Dataset 5, 

which had the same mileage, was used for cross-validation. 

Table 15. Model prediction using dataset 10 (1,000 mi) for estimation. 

Models Metric K A B C PDO

Poisson regression MAD 11.15 2.74 3.71 2.48 1.73

MSPE 202.4 9.25 15.5 7.49 5.17

NB regression MAD 14.92 2.74 3.69 2.49 1.73

MSPE 451.31 9.23 15.36 7.50 5.18

Poisson regression—random

parameters

MAD 36.91 3.26 5.30 2.80 1.78

MSPE 450.7 12.29 30.74 9.00 7.57

NB—random parameters MAD 24.5 3.16 6.10 2.61 1.73

MSPE 331.3 11.5 40.04 8.03 5.20

Univariate Poisson lognormal MAD 29.1 2.83 4.38 2.53 1.88

MSPE 135.32 9.81 31.5 8.36 6.28

Multivariate Poisson lognormal MAD 26.92 3.72 3.98 3.13 1.34

MSPE 472.31 10.50 15.57 7.90 5.70

From the results, Poisson regression had a lower MSPE and MAD for K crashes. Poisson 

regression had the lowest MAD and NB had the lowest MSPE for A crashes, which means that it 

performed better. Univariate Poisson lognormal had the lowest MAD for B crashes; meanwhile, 

Poisson regression had the lowest MSPE value. Poisson regression performed better for C 

crashes for MAD while NB did better for MSPE. Finally, Poisson did better for MAD for PDO 

crashes while univariate Poisson lognormal did better for MSPE.

The cross-validation results are shown in table 16, table 17, and table 18. 
62 

Table 16. Cross validation with dataset 5 (1,000 mi) with parameters estimated using 

dataset 10. 

Models Metric K A B C PDO

Poisson regression MAD 11.41 2.74 3.73 2.52 1.85

MSPE 210.24 9.34 15.82 7.73 6.21

NB regression MAD 15.20 2.73 3.74 2.53 1.85

MSPE 466.20 9.29 15.83 7.75 6.24

Poisson regression—random 

parameters

MAD 39.01 3.27 5.36 2.83 1.99

MSPE 593.8 12.70 31.36 9.48 6.86

NB—random parameters MAD 25.20 3.15 6.21 2.64 1.86

MSPE 379.1 11.8 41.48 8.36 6.26

Univariate Poisson lognormal MAD 14.92 2.74 3.69 2.49 1.73

MSPE 351.31 9.23 15.36 7.50 5.18

Multivariate Poisson 

lognormal

MAD 15.92 2.92 4.68 3.51 2.73

MSPE 462.31 10.25 15.56 7.54 5.27

Table 17. Model prediction with dataset 5 (1,000 mi) used for estimation.

Models Metric K A B C PDO

Poisson regression MAD 9.81 2.83 4.38 2.53 1.88

MSPE 135.32 9.81 31.5 8.36 6.28

NB regression MAD 14.03 2.86 5.05 2.51 1.88

MSPE 533.60 10.04 66.41 7.74 6.30

Poisson regression—random 

parameters

MAD 24.01 3.26 5.30 2.79 1.88

MSPE 233.2 12.29 30.74 8.98 5.182

NB—random parameters MAD 24.59 3.14 6.11 2.60 1.739

MSPE 301.3 11.4 40.08 8.02 5.203

Univariate Poisson lognormal MAD 10.1 2.75 2.52 2.01 1.12

MSPE 254.7 8.32 12.30 6.90 10.04

Multivariate Poisson 

lognormal

MAD 14.38 3.61 2.67 2.91 2.01 

MSPE 441.5 9.19 12.99 6.98 10.24
63 

Table 18. Cross validation with dataset 10 (1,000 mi) with parameters estimated using 

dataset 5. 

Models Metric K A B C PDO

Poisson regression MAD 13.54 2.74 4.64 2.59 1.77

MSPE 263.07 9.274 33.51 8.14 5.34

NB regression MAD 25.19 2.80 5.25 2.58 1.78

MSPE 335.4 9.65 66.39 8.060 5.401

Poisson regression—random 

parameters

MAD 27.63 3.41 5.31 2.78 2.01

MSPE 2671.7 13.12 31.13 8.96 6.28

NB—random parameters MAD 22.86 3.31 9.05 2.55 1.88

MSPE 303.1 12.59 310.34 7.84 6.31

Univariate Poisson lognormal MAD 10.2 2.77 2.98 2.70 1.12

MSPE 254.7 8.24 13.67 6.79 10.14

Multivariate Poisson lognormal MAD 15.38 2.36 2.77 2.93 2.31

MSPE 267.5 9.19 12.89 6.89 10.24

Cross validation was used to assess the predictive performance of the models and to judge how 

they perform outside the sample to a new dataset also known as test data. Without cross 

validation, there is only information on how each model performs relative to the in-sample data. 

The range of numbers from the cross-validation table shows that the models are robust enough to 

be relied on which suggests that the data from the tool can be trusted to produce convincing 

result each time various models are run on them.

Empirical Analysis for Evaluating Parameter Stability

Parameter estimates were examined to determine consistency using the revised Wald statistics. 

The analysis was conducted for all approaches, but only the results for the NB approach with 

PDO crashes are shown here, as an example. Dataset 10 (1,000 mi) was used as the population 

benchmark to evaluate if each parameter in any model is statistically different from the 

corresponding one in that dataset. If the parameter test statistic is greater than the 90 percent 

t-statistic, this result indicates a significant difference between the datasets.

In contrast, if the parameter test statistics are below the 90-percent t-statistic, no significant 

difference exists between the datasets, and the stochasticity in the RAD tool can be assumed to 

be consistent across different generations with different random seeds. The research team used 

dataset 10 because it had the largest number of observations and was believed to produce the 

most convincing parameter estimates. The combined NB estimates of PDO crashes for the 10 

datasets are displayed in table 19, along with their standard errors for illustrative purposes. The 

updated Wald test statistics for the estimated model parameter values are shown in table 20. For 

example, the test statistics for segment length across the datasets were lower than the 90-percent 

confidence value of 1.65, which could mean that there is no significant difference between the 

estimated parameters in the corresponding dataset and dataset 10.
64 

Application To Urban Four-Leg Signalized Intersections

RAD Generation

The descriptive statistics of the datasets generated by the tool are shown in table 21. The data in 

table 22 contains traffic, geometric, and intersection feature characteristics. The values for each 

of the continuous variables fall within a reasonable range, which shows that the dataset generated 

from the tool, even though randomly generated, will be within expected ranges. The same 

outcome was noticed for the categorical variables.
65 

Table 19. NB estimates for PDO crashes.

Parameter Dataset 1 Dataset 2 Dataset 3 Dataset 4 Dataset 5 Dataset 6 Dataset 7 Dataset 8 Dataset 9 Dataset 10

Intercept −0.62a

(0.860b 

−3.870, 

(0.685)

−1.856, 

(0.369)

−2.001, 

(0.389)

−1.965, 

0.291

−1.000*, 

(1.176)

−3.278, 

(0.671)

−1.917, 

(0.448)

−2.438, 

(0.385)

−2.223, 

(0.321)

Ln (Segment 

Length)

0.963, 

(0.036)

1.054, 

(0.034)

1.023, 

(0.021)

1.058, 

(0.019)

1.049, 

(0.017)

0.965, 

(0.050)

1.050, 

(0.031)

1.074, 

(0.025)

1.048, 

(0.021)

1.064, 

(0.017)

Ln (AADT) 0.392, 

(0.065)

0.627, 

(0.045)

0.548, 

(0.027)

0.452, 

(0.025)

0.474, 

(0.023)

0.415, 

(0.089)

0.533, 

(0.044)

0.464, 

(0.032)

0.530, 

(0.043)

0.546, 

(0.088)

Shoulder Width

8-ft Base Level

<2 ft 0.167, 

(0.329)

0.610, 

(0.155)

0.381, 

(0.076)

0.240, 

(0.093)

0.349, 

(0.071)

0.818, 

(0.365)

0.334, 

(0.159)

0.266, 

(0.093)

0.189, 

(0.095)

0.246, 

(0.076)

≥2 ft < 4 ft 0.361, 

(0.092)

0.497, 

(0.077)

0.360, 

(0.051)

0.292, 

(0.050)

0.383, 

(0.043)

0.781, 

(0.373)

0.471, 

(0.076)

0.249, 

(0.061)

0.341, 

(0.051)

0.329, 

(0.118)

≥4 ft < 6 ft 0.144, 

(0.083)

0.446, 

(0.083)

0.277, 

(0.048)

0.257, 

(0.051)

0.243, 

(0.042)

0.361, 

(0.095)

0.410, 

(0.083)

0.218, 

(0.058)

0.307, 

(0.051)

0.351, 

(0.094)

≥6 ft < 8 ft 0.168, 

(0.096)

0.479, 

(0.089)

0.137, 

(0.058)

0.246, 

(0.059)

0.217, 

(0.048)

0.117*, 

(0.136)

0.332, 

(0.088)

0.126, 

(0.070)

0.177, 

(0.060)

0.287, 

(0.044)

Lane Width

12-ft Base Level

≤9 ft — 0.932, 

(1.071)

0.326, 

(0.196)

0.5047, 

(0.129)

0.415, 

(0.112) — 0.599*, (1.088)

0.566, 

(0.232)

0.403, 

(0.130)

0.235, 

(0.117)

9.5 ft−10.5 ft 0.210*, 

(0.208)

0.252, 

(0.143)

0.125, 

(0.089)

0.256, 

(0.094)

0.289, 

(0.081)

−0.151, 

(0.285)

0.255, 

(0.142)

0.184, 

(0.111)

0.268, 

(0.093)

0.152, 

(0.084)

11 ft−11.5 ft 0.008*

(0.101)

0.116, 

(0.086)

0.183, 

(0.051)

0.091, 

(0.052)

0.141, 

(0.043)

0.047*, 

(0.142)

0.374, 

(0.083)

0.102*, 

(0.064)

0.152, 

(0.051)

0.210, 

(0.043)

Speed Limit

45-mph Base Level

25 mph 0.475, 

(0.062)

0.140, 

(0.188)

0.439, 

(0.116)

0.041, 

(0.030)

0.606, 

(0.102)

0.291, 

(0.125)

0.429, 

(0.174)

0.519*, 

(0.140)

0.244, 

(0.139)

0.483, 

(0.098)

30 mph 0.171*, 

(0.242)

0.428, 

(0.244)

0.210, 

(0.136)

0.041, 

(0.035)

0.349, 

(0.110)

0.313, 

(0.173)

0.385*, 

(0.241)

−0.05*, 

(0.184)

0.234, 

(0.139)

0.035*, 

(0.111)

35 mph −0.84*, 

(0.271)

−0.701, 

(0.306)

−0.271, 

(0.174)

−0.295, 

(0.143)

0.069*, 

(0.109)

0.268*, 

(0.343)

−0.045*, 

(0.253)

−0.567, 

0.237

−0.222, 

(0.150)

−0.336, 

(0.112)

40 mph −0.956, 

(0.252)

−0.824, 

(0.197)

−0.822, 

(0.114)

−1.109, 

(0.119)

−0.655, 

(0.099)

−0.879, 

(0.319)

−0.992, 

(0.216)

−0.91*, 

0.144, 

−0.969, 

(0.123)

−1.021, 

(0.103)

50 mph 0.120*, 

(0.233)

0.095, 

(0.172)

−0.163, 

(0.122)

0.003*, 

(0.119)

−0.087, 

(0.124)

−0.425, 

(0.367)

−0.036, 

(0.178)

0.117*, 

0.139

−0.057, 

(0.130)

−0.115, 

(0.116)
66 

55 mph −0.28*, 

(0.132)

−0.332, 

(0.118)

−0.254, 

(0.072)

−0.329, 

(0.071)

−0.011, 

(0.067)

−0.196, 

(0.180)

−0.066, 

(0.115)

−0.226, 

(0.089)

−0.125, 

(0.074)

−0.270, 

(0.064)

Presence of 

Lighting

−0.341, 

(0.184)

−0.603, 

(0.151)

−0.523, 

(0.151)

−0.531, 

(0.101)

−0.412, 

(0.085)

−0.300, 

(0.259)

−0.334, 

(0.026)

−0.658, 

(0.122)

−0.306, 

(0.098)

−0.431, 

(0.085)

Presence of 

Horizontal 

Curve

0.549*, 

(0.239)

0.294*

(0.205)

0.078*

(0.116)

0.186

(0.112)

0.264

(0.088)

0.269*

(0.244)

0.407

(0.036)

0.652

(0.111)

0.168*

(0.112)

0.365

(0.085)

Overdispersio

n 

0.257 0.463 0.344 0.199 0.375 0.725 0.271 0.447 0.216 0.307

—No data.

*Variables insignificant at a 90-percent confidence level. a

Parameter estimate. b

Standard error. 

Table 20. Revised Wald test statistics on NB model parameter estimates (relative to dataset 10).

Parameter Dataset 1 Dataset 2 Dataset 3 Dataset 4 Dataset 5 Dataset 6 Dataset 7 Dataset 8 Dataset 9

Ln (Segment Length) 2.536 0.263 1.517 0.235 0.623 1.874 0.395 0.330 0.592

Ln (AADT) 1.407 0.819 0.022 1.027 0.792 1.046 0.132 0.875 0.163

Shoulder Width

0 ft 0.233 2.108 1.256 0.049 0.990 1.534 0.499 0.166 0.468

2 ft 0.213 1.192 0.241 0.288 0.429 1.155 1.011 0.602 0.093

4 ft 1.650 0.757 0.701 0.878 1.048 0.074 0.470 1.204 0.411

6 ft 1.126 1.933 2.060 0.557 1.075 1.189 0.457 1.947 1.478

Lane Width

9 ft 2.008 0.646 0.398 1.544 1.111 2.008 0.332 1.273 0.960

10 ft 0.258 0.602 0.220 0.825 1.174 1.019 0.624 0.229 0.925

11 ft 0.199 0.977 0.404 1.763 1.134 1.098 1.754 1.400 0.869

Speed Limit

25 mph 0.042 1.617 0.289 2.714 0.869 0.782 0.270 0.210 1.405

30 mph 0.510 1.466 0.996 0.034 2.009 0.123 1.319 0.437 1.118

35 mph 1.718 1.120 0.314 0.225 2.591 1.673 1.051 0.881 0.608

40 mph 0.238 0.886 1.295 0.559 2.561 0.423 0.121 0.593 0.324

50 mph 0.902 1.012 0.285 0.710 0.164 0.805 0.371 1.281 0.332

55 mph 0.068 0.461 0.166 0.617 3.033 0.387 1.550 0.401 1.482

Presence of Lighting 0.907 0.992 0.530 0.757 0.158 0.480 1.091 1.526 0.963

Presence of 

Horizontal Curve 0.725 0.319 1.995 1.273 0.825 0.371 0.455 2.052 1.401
67 

Table 21. Descriptive statistics for continuous variables. 

Continuous

Variables Statistic 

Dataset

1 

Dataset

2 

Dataset

3 

Dataset

4 

Dataset

5 

Dataset

6 

Dataset

7 

Dataset

8 

Dataset

9 

Dataset

10

Crash Counts PDO Mean 52.17a 48.0 48.43 47.94 48.03 49.03 47.51 48.94 40.070 50.76

Crash Counts PDO . Std. Dev. 69.974b 62.397 59.587 62.614 65.780 63.277 63.49 69.625 64.699 68.167

Crash Counts K Mean 0.066 0.074 0.06 0.071 0.0742 0.072 0.073 0.083 0.070 0.072

Crash Counts K Std. Dev. 0.437 0.446 0.427 0.451 0.503 0.463 0.440 0.576 0.448 0.503

Crash Counts A Mean 1.138 1.048 1.057 1.078 1.108 1.09 1.085 1.1 1.122 1.112

Crash Counts A Std. Dev. 2.129 1.769 1.698 1.72 1.926 1.887 1.979 1.945 1.965 1.933

Crash Counts B. Mean 4.098 4.142 4.249 4.432 4.354 4.038 4.062 4.216 4.178 4.34

Crash Counts B. Std. Dev. 8.345 8.561 9.578 11.732 10.366 7.153 8.462 8.537 8.450 9.621

Crash Counts C Mean 10.9 11.41 11.42 11.79 11.31 10.94 11.08 11.22 11.66 11.26

Crash Counts C Std. Dev. 15.755 20.053 17.871 20.982 18.151 18.19 16.94 18.00 20.36 17.479

Ln (AADT Major) Mean 9.565 9.547 9.55 9.557 9.557 9.563 9.54 9.541 9.570 9.566

Ln (AADT Major) Std. Dev. 0.565 0.551 0.554 0.5620 0.5612 0.551 0.557 0.584 0.572 0.569

Ln (AADT Minor) Mean 8.458 8.43 8.435 8.443 8.441 8.470 8.431 8.428 8.443 8.451

Ln (AADT Minor) Std. Dev. 0.812 40.798 0.812 0.815 0.818 0.815 0.79 0.817 0.821 0.819

a

Parameter estimate. b

Standard error
68 

Table 22. Descriptive statistics for categorical variables (datasets 1–10). 

Variables Dataset 1 Dataset 2 Dataset 3 Dataset 4 Dataset 5 Dataset 6 Dataset 7 Dataset 8 Dataset 9 Dataset 10

LTLs

1 370 729 1147 1479 1871 411 733 1145 1445 1805

2 232 505 727 998 1225 238 488 740 998 1250

3 127 256 379 520 622 121 273 407 531 667

4 271 510 747 1003 1282 230 506 708 1026 1278

RTLs

1 874 1730 2575 3450 4313 859 1721 2649 3436 4334

2 95 180 303 380 487 106 182 246 396 462

3 22 64 84 105 118 22 65 64 108 138

4 9 26 38 65 82 13 32 41 60 66

Speed Limit (mph)

25 175 319 507 660 826 167 322 517 656 882

30 216 431 627 790 1016 180 414 549 810 965

35 281 600 841 1167 1399 287 586 867 1157 1395

40 199 406 656 869 1139 230 433 696 897 1111

45 100 205 300 430 495 107 204 309 397 530

50 20 32 50 72 99 21 33 51 69 96

55 7 4 14 6 19 6 3 8 10 13

65 2 3 5 6 7 2 5 3 4 8

LTLs Permitted

1 715 1432 2212 2936 3595 760 1424 2221 2883 3617

2 180 373 535 715 920 152 388 508 752 928

3 43 86 116 172 258 51 91 146 182 221

4 62 109 137 177 227 37 97 126 183 234

LTLs Protected

1 934 1818 2744 3691 4644 919 1833 2765 3662 4628

2 47 134 182 234 253 64 126 179 243 282

3 5 19 25 34 41 6 17 26 44 40

4 14 29 49 41 62 11 24 30 51 50

LTLs Mix

1 716 1475 2175 2832 3604 743 1465 2174 2868 3510

2 145 261 445 618 738 133 272 433 604 805
69 

Variables Dataset 1 Dataset 2 Dataset 3 Dataset 4 Dataset 5 Dataset 6 Dataset 7 Dataset 8 Dataset 9 Dataset 10

3 50 112 143 187 257 49 92 154 181 271

4 89 152 237 363 401 75 171 239 347 414

RTOR

1 909 1799 2711 3601 4475 892 1780 2680 3553 4628

2 31 79 101 158 183 42 82 94 165 282

3 35 67 119 148 219 43 93 140 168 40

4 25 55 69 93 123 23 45 86 114 50

Lighting

No 145 245 382 526 617 144 267 408 551 637

Yes 855 1755 2618 3474 4383 856 1733 2592 3449 4363

Presence of School Within 1,000 ft of Intersection

No 874 1745 2965 3503 4393 893 1751 2610 3538 4384

Yes 126 255 35 487 607 107 249 390 462 616

Number of Bus Stops Within 1,000 ft of Intersection

0 823 1696 2543 3356 4246 845 1697 2540 3392 4226

1–2 45 83 108 147 165 38 71 103 129 185

≥3 132 221 349 497 589 117 232 357 479 539

Number of Alcohol Sale Establishments Within 1,000 ft of Intersection

0 930 1846 2750 3673 4625 917 1857 2772 3684 4634

1–8 70 154 250 327 375 83 143 228 316 366

≥9 — — — — — 0 — — — — 

Maximum Crossing Lanes

1 1 1 3 1 2 0 1 3 3 3 

2 189 369 626 798 1006 230 396 594 793 973

3 179 314 441 644 864 158 306 463 652 812

4 234 490 714 958 1179 246 478 712 955 1169

5 252 572 823 1060 1297 229 547 806 1026 1341

6 115 198 291 410 489 107 210 330 433 540

7 30 56 102 129 163 28 62 92 138 162

LTL = left-turn lane; RTL = right-turn lane; RTOR = right turn on red.
70 

Crash Prediction Model Estimation

Researchers estimated crash prediction models as described in table 23 section of this report. A 

total of 60 models were developed and validated for each severity level: K, A, B, C, and O. 

Table 23 summarizes the crash prediction model created using the dataset from the RAD tool. 

Estimated model parameters and fit statistics are provided here for only the 1,000-intersection 

model for brevity. The results for all the models are provided in appendix C through appendix M, 

which are in the second volume of this publication. 

Table 23. Summary of developed models.

ID Facility Datasets Models

Severity 

Level

No. of 

Models

1 Urban fourleg signalized 

intersections

Two sets each of 

1,000, 2,000, 

3,000, 4,000, 

and 5,000 

intersections

Poisson regression K, A, B, C, 

PDO

5×2 = 10

NB regression K, A, B, C, 

PDO

5×2 = 10

Poisson regression—random 

parameters

K, A, B, C, 

PDO

5×2 = 10

NB—random parameters K, A, B, C, 

PDO

5×2 = 10

Univariate Poisson 

lognormal

K, A, B, C, 

PDO

5×2 = 10

Multivariate Poisson 

lognormal

K, A, B, C, 

PDO

5×2 = 10

The statistical software R: A Language and Environment for Statistical Computing 

(version 12.0) was used to estimate the model parameters (R Foundation 2021). Table 24, table 

25, table 26, and table 27 summarize the model parameter estimates and their associated statistics 

under the Poisson, NB, random parameter, and univariate and multivariate Poisson lognormal 

models, respectively. An examination of the tables indicates that the model parameter estimates 

are significant at the 95-percent confidence level; however, the estimates that are followed by 

asterisks in these tables were not significant at a 95-percent confidence level. 
71 

Table 24. Poisson regression model estimation (1,000 intersections).

Variables 

K A B C PDO

Parameter

Estimate 

Std.

Error 

Parameter

Estimate 

Std.

Error 

Parameter

Estimate 

Std.

Error 

Parameter

Estimate 

Std.

Error 

Parameter

Estimate 

Std.

Error 

Intercept −29.068* 9426.618 −13.673 1.189 −6.419 0.376 −6.788 0.294 −9.225 0.184 

Ln (AADT 

Major) 0.778 0.296 0.956 0.072 0.673 0.035 0.748 0.022 0.919 0.011 

Ln (AADT 

Minor) 0.282 0.176 0.518 0.047 0.361 0.023 0.323 0.014 0.484 0.007 

Left-Turn Lane

1 Base Level

2 −1.164 0.645 −0.540 0.141 −0.495 0.069 −0.403 0.042 −0.345 0.019 

3 −2.139 1.272 −0.579 0.209 −0.543 0.112 −0.562 0.066 −0.134 0.030 

4 0.232 1.355 −1.037 0.283 −0.641 0.142 −0.356 0.084 −0.368 0.040 

Right-Turn Lane

1 Base Level

2 −0.708 0.529 −0.167 0.108 −0.328 0.061 −0.279 0.037 −0.100 0.016 

3 0.009 0.744 0.050 0.174 −0.721 0.136 −0.524 0.078 0.203 0.025 

4 0.878 0.757 −0.490 0.340 −0.642 0.189 0.572 0.074 −0.661 0.057 

Speed Limit

35 mph Base Level

25 −0.208 0.610 −0.230 0.098 0.311 0.048 0.273 0.029 0.099 0.014 

30 0.974* 0.415 −0.165 0.093 0.105 0.049 −0.010 0.029 0.007 0.013 

40 1.206* 0.404 0.197 0.084 0.263 0.047 0.001 0.029 0.063 0.013 

45 1.458* 0.450 −0.027 0.115 −0.191 0.070 0.037 0.037 0.216 0.016 

50 0.332* 1.070 0.410 0.165 0.767 0.086 0.011 0.071 −0.633 0.039 

55 −13.996* 3185.047 0.063 0.463 0.735 0.175 −0.059 0.148 −0.321 0.076 

65 −16.579* 6618.226 1.455 0.264 0.658 0.191 −0.102 0.177 −0.361 0.081 

Lighting 

Not Present Base Level

Present −0.364 0.315 −0.228 0.082 −0.115 0.044 −0.347 0.025 −0.330 0.012 

RTOR

1 Base Level

2 −15.958* 1347.927 0.190 0.174 0.164 0.089 0.023 0.059 −0.054 0.029 

3 −0.738 1.019 −0.207 0.194 0.245 0.086 0.180 0.050 0.053 0.024 

4 −16.383* 1505.027 −0.227 0.231 −0.262 0.114 −0.213 0.071 −0.222 0.033 

Maximum No. Crossing Lanes

1 Base Level

2 15.660* 9426.618 0.407 1.004 −1.345 0.205 −0.427 0.220 0.494 0.159 

3 16.853* 9426.618 0.384 1.007 −1.336 0.209 −0.324 0.221 0.540 0.159 
72 

Variables 

K A B C PDO

Parameter

Estimate 

Std.

Error 

Parameter

Estimate 

Std.

Error 

Parameter

Estimate 

Std.

Error 

Parameter

Estimate 

Std.

Error 

Parameter

Estimate 

Std.

Error 

4 16.052* 9426.618 0.328 1.004 −1.623 0.206 −0.271 0.220 0.641 0.159 

5 16.731* 9426.618 0.191 1.007 −1.416 0.209 −0.164 0.221 0.472 0.159 

6 16.526* 9426.618 0.330 1.010 −1.419 0.213 −0.298 0.223 0.453 0.160 

7 15.758* 9426.618 0.163 1.026 −2.054 0.248 −0.325 0.228 0.339 0.161 

LTLs Permitted

1 Base Level

2 0.284 0.745 0.730 0.147 −0.110 0.079 0.276 0.045 0.002 0.021 

3 −15.817* 1146.280 0.337 0.256 0.349 0.129 0.041 0.082 −0.380 0.038 

4 −0.408 1.377 0.623 0.312 0.159 0.155 −0.096 0.091 −0.012 0.043 

LTLs Mix

1 Base Level

2 0.000 0.745 0.060 0.162 −0.077 0.081 −0.214 0.049 −0.152 0.022 

3 −1.393 1.623 0.224 0.267 −0.153 0.139 −0.174 0.084 −0.173 0.037 

4 −0.355 1.351 0.812 0.287 0.061 0.147 −0.507 0.090 −0.194 0.041 

LTLs Protected

1 Base Level

2 −1.923 1.295 0.048 0.236 −0.123 0.125 −0.114 0.070 −0.251 0.033 

3 −16.255* 3477.698 0.720 0.621 −0.364 0.395 −0.409 0.225 −0.575 0.110 

4 −16.865* 2270.057 0.852 0.377 −0.466 0.236 −0.454 0.129 −0.722 0.066 

Bus 

0 Base Level

1–2 −0.285 0.547 0.052 0.137 0.032 0.073 0.142 0.041 −0.092 0.020 

≥3 0.493 0.330 0.058 0.089 −0.236 0.052 −0.054 0.029 0.006 0.013 

Alcohol 

0 Base Level

1–8 −0.210 0.609 0.246 0.116 −0.477 0.078 −0.170 0.042 −0.017 0.019 

School 

Not Present Base Level

Present −0.153 0.412 −0.093 0.095 −0.119 0.050 −0.349 0.034 −0.055 0.014 

Fit Statistics

AIC 527.48 N/A 2879.3 N/A 9162.4 N/A 13433 N/A 29547 N/A

BIC 718.879 N/A 3070.68 N/A 9353.81 N/A 13624.59 N/A 29738.76 N/A

—No data.

*Variables insignificant at a 90-percent confidence level. 
73 

Table 25. NB model estimation (1,000 intersections).

Variables 

K A B C PDO

Parameter 

Estimate 

Std.

Error 

Parameter

Estimate 

Std.

Error 

Parameter

Estimate 

Std.

Error 

Parameter 

Estimate 

Std.

Error 

Parameter 

Estimate 

Std.

Error 

Intercept −52.286* 6710.0 −13.408 1.523 −6.687 1.722 −6.839 1.125 −9.720 0.759 

Ln (AADT 

Major) 0.999 0.427 0.963 0.093 0.659 0.100 0.723 0.064 0.991 0.042 

Ln (AADT 

Minor) 0.178 0.267 0.473 0.061 0.443 0.069 0.362 0.044 0.460 0.029 

Left-Turn Lane

1 Base Level

2 −1.177 0.940 −0.519 0.190 −0.364 0.214 −0.445 0.138 −0.371 0.091 

3 −1.997 1.569 −0.497 0.287 −0.580 0.326 −0.676 0.208 −0.171 0.136 

4 0.646 1.954 −0.866 0.373 −0.491 0.412 −0.571 0.264 −0.471 0.173 

Right-Turn 

Lane

1 Base Level

2 −0.610 0.750 −0.167 0.149 −0.302 0.174 −0.314 0.111 −0.142 0.072 

3 0.422 1.120 0.030 0.266 −0.442 0.353 −0.326 0.222 0.357 0.143 

4 1.263 1.686 −0.545 0.475 −0.611 0.547 0.316 0.333 −0.440 0.226 

Speed Limit (mph)

35 Base Level

25 −0.422* 0.743 −0.162 0.131 0.321 0.152 0.154 0.097 0.084 0.064 

30 0.800 0.584 −0.117 0.124 0.077 0.144 −0.068 0.092 0.037 0.060 

40 1.041 0.559 0.136 0.119 0.340 0.145 0.007 0.094 0.055 0.062 

45 0.832 0.752 0.025 0.157 −0.302 0.191 −0.085 0.120 0.136 0.078 

50 0.587 1.373 0.448 0.271 0.345 0.359 −0.109 0.235 −0.188 0.154 

55 −43.061* 2565.3 0.173 0.545 0.633 0.594 −0.119 0.397 −0.385 0.261 

65 −58.519* 47453.2 1.430 0.614 0.612 1.046 −0.191 0.689 −0.152 0.458 

Lighting 

Not Present Base Level

Present 0.145 0.537 −0.181 0.117 −0.401 0.140 −0.376 0.090 −0.346 0.060 

RTOR

1 Base Level

2 −54.061* 12077.2 0.151 0.242 0.121 0.288 0.087 0.185 −0.074 0.122 

3 −0.588 1.331 −0.118 0.244 0.224 0.272 0.153 0.173 0.041 0.115 

4 −54.394* 1357.1 −0.185 0.296 −0.077 0.327 −0.055 0.210 0.041 0.136 
74 

Variables 

K A B C PDO

Parameter 

Estimate 

Std.

Error 

Parameter

Estimate 

Std.

Error 

Parameter

Estimate 

Std.

Error 

Parameter 

Estimate 

Std.

Error 

Parameter 

Estimate 

Std.

Error 

Maximum No. Crossing Lanes

1 Base Level

2 37.701* 671000 0.499 1.264 −1.535 1.456 −0.406 0.959 0.510 0.652 

3 38.491* 671000 0.388 1.268 −1.547 1.461 −0.315 0.962 0.449 0.654 

4 38.019* 664.000 0.269 1.264 −1.681 1.456 −0.234 0.959 0.600 0.652 

5 38.271* 671000 0.213 1.268 −1.490 1.460 −0.151 0.962 0.448 0.654 

6 38.020* 67000 0.317 1.272 −1.386 1.465 −0.283 0.965 0.440 0.656 

7 37.828* 67000 0.016 1.295 −1.578 1.485 −0.380 0.978 0.588 0.663 

LTLs Permitted

1 Base Level

2 0.367 1.012 0.663 0.196 −0.181 0.222 0.284 0.141 0.020 0.093 

3 −54.576* 1024.8 0.239 0.343 0.416 0.387 0.098 0.250 −0.226 0.163 

4 −0.640 2.006 0.494 0.402 −0.201 0.441 0.073 0.282 0.023 0.185 

LTLs Mix

1 Base Level

2 −0.094 1.038 −0.015 0.211 −0.102 0.230 −0.182 0.148 −0.026 0.097 

3 −1.172 2.167 0.216 0.346 −0.169 0.389 −0.157 0.250 −0.013 0.162 

4 −1.191 1.948 0.586 0.380 −0.079 0.423 −0.260 0.272 −0.014 0.178 

LTLs Protected

1 Base Level

2 −2.032 1.684 −0.017 0.304 −0.195 0.332 0.053 0.211 −0.133 0.139 

3 −57.193* 3001.9 0.521 0.750 −0.500 0.842 −0.293 0.521 −0.497 0.334 

4 −58.888* 17935.6 0.696 0.506 −0.005 0.581 0.225 0.368 −0.567 0.247 

Bus 

0 Base Level

1–2 −0.309 0.990 0.104 0.193 −0.051 0.240 0.256 0.152 0.189 0.101 

≥3 0.260 0.611 0.092 0.122 −0.108 0.150 0.031 0.095 −0.031 0.063 

Alcohol 

0 Base Level

1–8 0.307 0.803 0.252 0.161 −0.431 0.202 −0.106 0.126 −0.028 0.083 

School 

Not Present Base Level

Present 0.065 0.615 −0.117 0.132 −0.046 0.153 −0.226 0.099 −0.021 0.064 

Fit Statistics

AIC 446.93 N/A 2666.8 N/A 4579.6 N/A 6536.7 N/A 9018.6 N/A

BIC 643.23 N/A 2863.10 N/A 4775.91 N/A 6732.98 N/A 9214.88 N/A

Overdispersio

n 

0.0769 — 1.718 — 0.486 — 1.157 — 2.52 — 

—No data.

*Variables insignificant at a 90-percent confidence level.
75 

Table 26. Random parameter Poisson regression (1,000 intersections). 

Variables

K A B C PDO

Parameter 

Estimate

Std.

Error

Parameter

Estimate

Std.

Error

Parameter

Estimate

Std.

Error

Parameter

Estimate

Std.

Error

Parameter

Estimate

Std.

Error

Intercept −42.122* 6.385 −7.907 1.048 −2.428 1.570 −3.551 0.860 −1.760 0.449 

Ln (AADT 

Major) 1.071 0.323 1.007 0.055 1.134 0.101 1.008 0.049 1.022 0.026 

Ln (AADT 

Minor) 0.762 0.470 0.879 0.079 0.260 0.117 0.452 0.063 0.531 0.033 

Left-Turn Lane

1 Base Level

2 0.585 0.550 0.449 0.110 0.456 0.189 0.466 0.107 0.274 0.046 

3 0.253 0.547 0.103 0.111 0.438 0.185 0.328 0.104 0.292 0.043 

4 −0.741 0.781 −0.004 0.130 −0.015 0.232 0.187 0.122 0.151 0.051 

Right-Turn Lane

1 Base Level

2 0.026 0.092 0.002 0.015 0.041 0.030 −0.012 0.013 0.012 0.006 

3 0.002 0.009 0.003 0.023 0.589 0.298 0.369 0.759 0.379 0.349 

4 0.281 1.065 −0.234 0.197 −0.299 0.371 −0.686 0.191 −0.507 0.088 

Speed Limit

35 mph Base Level

25 −0.422* 0.743 −0.162 0.131 0.321 0.152 0.154 0.097 0.084 0.064 

30 0.800* 0.584 −0.117 0.124 0.077 0.144 −0.068 0.092 0.037 0.060 

40 1.041* 0.559 0.136 0.119 0.340 0.145 0.007 0.094 0.055 0.062 

45 0.832* 0.752 0.025 0.157 −0.302 0.191 −0.085 0.120 0.136 0.078 

50 0.587* 1.373 0.448 0.271 0.345 0.359 −0.109 0.235 −0.188 0.154 

55 −43.061* 2565.3 0.173 0.545 0.633 0.594 −0.119 0.397 −0.385 0.261 

65 −58.519* 47453.2 1.430 0.614 0.612 1.046 −0.191 0.689 −0.152 0.458 

Lighting 

Not Present Base Level

Present 0.288 0.537 −0.317 0.107 −0.210 0.131 −0.268 0.080 −0.185 0.051 

RTOR

1 Base Level

2 0.037 0.165 −0.009 0.018 0.014 0.047 −0.003 0.015 0.016 0.008 

3 0.006 0.021 0.003 0.002 0.005 0.005 0.001 0.002 0.000 0.001 

4 0.128 2.363 −0.420 0.244 −0.330 0.553 −0.754 0.217 −0.534 0.100 

Maximum No. Crossing Lanes

1 Base Level

2 20.453 1.037 0.243 0.126 0.400 0.234 −0.164 0.112 0.164 0.051 

3 22.119 2.250 0.223 0.163 0.253 0.305 0.155 0.133 0.203 0.066 

4 20.509 2.606 0.597 0.200 0.239 0.417 −0.056 0.186 0.464 0.084 
76 

Variables

K A B C PDO

Parameter 

Estimate

Std.

Error

Parameter

Estimate

Std.

Error

Parameter

Estimate

Std.

Error

Parameter

Estimate

Std.

Error

Parameter

Estimate

Std.

Error

5 20.073 0.158 −0.016 0.015 −0.013 0.028 −0.009 0.014 0.009 0.006 

6 20.017 0.017 −0.001 0.002 0.000 0.003 0.004 0.001 −0.001 0.001 

7 22.600 1.683 −0.526 0.219 −0.351 0.377 −0.288 0.171 −0.287 0.084 

LTLs Permitted

1 Base Level

2 0.892 1.323 0.093 0.178 0.341 0.299 −0.204 0.162 0.242 0.071 

3 0.684 0.582 0.182 0.095 0.237 0.175 0.094 0.086 0.141 0.040 

4 0.259 1.662 0.279 0.220 0.186 0.440 −0.128 0.213 0.343 0.095 

LTLs Mix

1 Base Level

2 0.376 0.682 0.267 0.094 0.122 0.166 0.232 0.089 0.207 0.040 

3 1.849 0.720 0.275 0.122 0.609 0.197 0.151 0.117 0.309 0.050 

4 1.050 1.230 0.245 0.169 0.405 0.282 0.638 0.142 0.377 0.067 

LTLs Protected 

1 Base Level

2 0.130 0.202 0.015 0.017 −0.012 0.028 −0.019 0.014 0.005 0.007 

3 0.005 0.016 0.002 0.002 −0.006 0.003 0.002 0.002 0.002 0.001 

4 2.760 1.848 0.071 0.204 −0.022 0.352 −0.139 0.191 −0.642 0.098 

Bus 

0 Base Level

1–2 −0.679 1.421 0.067 0.202 0.213 0.238 0.120 0.147 0.073 0.093 

≥3 −1.488 0.919 0.084 0.132 0.061 0.146 0.140 0.089 −0.061 0.056 

Alcohol 

0 Base Level

1–8 2.419 0.550 0.884 0.122 −0.031 0.169 −0.054 0.104 −0.074 0.066 

School 

Not Present Base Level

Present 1.093 0.574 0.432 0.118 −0.073 0.151 0.083 0.092 −0.097 0.058 

Fit Statistics

AIC 446.97 N/A 2589.9 N/A 4680.3 N/A 6346.2 N/A 8563.1 N/A

BIC 638.375 N/A 2781.29 N/A 4871.75 N/A 6537.60 N/A 8,754.53 N/A

—No data.

*Variables insignificant at a 90-percent confidence level.
77 

Table 27. Random parameter—NB regression (1,000 intersections).

Variables 

K A B C PDO

Parameter 

Estimate

Std.

Error

Parameter

Estimate

Std.

Error

Parameter

Estimate

Parameter 

Estimate

Std.

Error

Parameter

Estimate

Std.

Error

Parameter

Estimate

Intercept 45.292* 7.476 −4.946 2.538 −4.689 4.723 −1.963 1.874 −1.003 1.176 

Ln AADT 

Major 1.196 0.677 0.856 0.601 1.523 0.232 1.030 0.084 0.966 0.050 

Ln AADT 

Minor 0.329 1.325 0.683 0.199 0.421 0.360 0.419 0.147 0.415 0.090 

Left-Turn Lane

1 Base Level

2 0.537 0.693 0.122 0.129 0.373 0.221 0.320 0.114 0.288 0.059 

3 −0.491 0.957 −0.020 0.155 −0.096 0.279 0.197 0.134 0.167 0.071 

4 0.911 0.707 0.475 0.130 0.475 0.226 0.500 0.117 0.298 0.063 

Right-Turn Lane 

1 Base Level

2 1.392 1.159 0.122 0.120 −0.146 0.238 −0.092 0.100 0.015 0.049 

3 0.485 1.391 0.083 0.152 0.073 0.301 0.093 0.125 0.051 0.064 

4 0.867 1.585 0.019 0.191 0.313 0.371 0.205 0.156 0.251 0.080 

Speed Limit

35 mph Base Level

25 3.088* 1.793 0.303 0.313 −0.222 0.660 0.549 0.252 0.041 0.131 

30 −0.097* 3.564 0.353 0.300 1.175 0.552 0.817 0.237 0.041 0.136 

40 2.391* 1.939 −0.001 0.332 1.096 0.529 0.717 0.241 −0.295 0.144 

45 −2.975* 445322 −0.920 0.275 −0.604 0.498 −0.553 0.230 −1.110 0.120 

50 −2.941* 618229 −0.552 0.341 −0.248 0.612 0.211 0.258 0.004 0.120 

55 0.634* 1.399 −0.396 0.166 −0.546 0.349 −0.101 0.153 −0.330 0.071 

65 0.288* 0.537 −0.317 0.107 −0.210 0.131 −0.268 0.080 −0.185 0.051 

Lighting 

Not Present Base Level

Present −1.923 1.295 0.048 0.236 −0.123 0.125 −0.114 0.070 −0.251 0.033 

RTOR

1 Base Level

2 7.258* 17237 1.068 0.589 0.747 1.238 0.896 0.480 −0.013 0.374 

3 −0.928 14010 −0.386 0.869 0.807 1.332 −0.061 0.656 0.268 0.343 

4 6.742* 12284 −1.169 0.730 0.730 1.051 −1.625 0.774 −0.879 0.319 

Maximum No. Crossing Lanes

1 Base Level

2 22.158* 1098.5 1.219 1.723 −0.001 0.840 0.130 0.568 0.487 0.435 

3 21.745* 1098.5 1.143 1.426 0.068 0.841 0.166 0.569 0.722 0.352 
78 

Variables 

K A B C PDO

Parameter 

Estimate

Std.

Error

Parameter

Estimate

Std.

Error

Parameter

Estimate

Parameter 

Estimate

Std.

Error

Parameter

Estimate

Std.

Error

Parameter

Estimate

4 22.356* 1088.5 1.216 1.625 −0.069 0.840 0.047 0.568 1.146 0.598 

5 22.316* 1088.5 1.181 1.129 −0.086 0.841 0.124 0.569 1.403 0.360 

6 22.162* 1088.5 1.115 1.128 0.083 0.843 0.034 0.570 1.774 0.369 

7 22.422* 1088.5 1.123 1.132 −0.087 0.848 −0.078 0.573 1.935 0.334 

LTLs Permitted

1 Base Level

2 −1.164 0.645 −0.540 0.141 −0.495 0.069 −0.403 0.042 −0.345 0.019 

3 −2.139 1.272 −0.579 0.209 −0.543 0.112 −0.562 0.066 −0.134 0.030 

4 0.232 1.355 −1.037 0.283 −0.641 0.142 −0.356 0.084 −0.368 0.040 

LTLs Mix

1 Base Level

2 0.000 0.745 0.060 0.162 −0.077 0.081 −0.214 0.049 −0.152 0.022 

3 −1.393 1.623 0.224 0.267 −0.153 0.139 −0.174 0.084 −0.173 0.037 

4 −0.355 1.351 0.812 0.287 0.061 0.147 −0.507 0.090 −0.194 0.041 

LTLs Protected 

1 Base Level

2 −0.337 8413.9 −0.794 0.383 0.038 0.784 −1.140 0.372 −1.039 0.180 

3 −1.311 84015 −0.874 0.342 −0.347 0.747 −0.463 0.281 −0.194 0.133 

4 −0.279 8926.8 −0.055 0.347 0.058 0.863 −0.460 0.326 −0.141 0.151 

Bus 

0 Base Level

1–2 0.957 0.406 1.146 0.068 1.200 0.127 1.011 0.060 1.006 0.026 

≥3 0.490 0.546 0.636 0.085 0.414 0.150 0.519 0.076 0.612 0.034 

Alcohol 

0 Base Level

1−8 0.425 0.294 0.506 0.056 −0.016 0.075 0.013 0.049 −0.085 0.028 

School 

Not Present Base Level

Present 0.511 0.257 0.482 0.050 0.018 0.064 0.096 0.043 0.029 0.024 

Fit Statistics

AIC 2018.3 N/A 12901 N/A 23319 N/A 32515 N/A 42322 N/A

BIC 2279.01 N/A 13161.73 N/A 23579.51 N/A 32776.17 N/A 42583.07 N/A

Overdispersion 0.0557 — 2.026 — 0.5428 — 1.1794 — 3.907 — 

—No data.

*Variables insignificant at a 90-percent confidence level. 
79 

Validation of Crash Count Prediction

The model prediction of dataset 10 used for model estimation is shown in table 28; meanwhile, 

dataset 5, which had the same number of intersections, was used for cross validation.

Table 28. Model prediction with dataset 10 (5,000 intersections) used for estimation.

Models Metric K A B C PDO

Poisson Regression MAD 3.24 1.38 3.16 9.13 44.56

MSPE 11.94 4.68 113.18 402.25 6236.01

NB Regression MAD 3.25 1.37 3.14 9.23 44.58

MSPE 13.59 6.67 113.30 402.24 6236.42

Poisson regression—random 

parameters

MAD 3.35 2.37 4.14 9.34 43.88

MSPE 14.59 4.80 131.30 522.74 5036.42

NB—random parameters MAD 3.32 1.73 3.15 9.23 44.68

MSPE 15.60 4.76 113.27 420.24 5236.42

Univariate Poisson 

lognormal

MAD 3.20 1.38 3.12 9.21 44.56

MSPE 12.84 4.80 112.18 402.25 5106.01

Multivariate Poisson 

lognormal

MAD 4.25 1.67 3.24 9.32 45.58

MSPE 15.59 5.67 123.30 502.24 5236.42

In this case, univariate Poisson lognormal performed better in the prediction of MSPE for K 

crashes while Poisson regression did better for MAD for K crashes. NB had the lowest MAD and 

MSPE for A crashes, which shows that it performed better. Univariate Poisson lognormal had the 

lowest MAD for B crashes while Poisson regression had the lowest for MSPE. Poisson 

regression performed better for C crashes for MAD while NB did better for MSPE. Finally, 

Poisson did better for MAD for PDO crashes while univariate Poisson lognormal did better for 

MSPE. The cross-validation results are shown in table 29, table 30, and table 31. Cross 

validation was used to assess the predictive performance of the models and judge how they 

performed outside the sample in a new dataset, also known as test data. The cross-validation 

values for the different models fall within the same range, which could mean the models are 

robust enough to be relied on. 
80 

Table 29. Cross validation with dataset 5 (5,000 intersections) using dataset 10 model. 

Models Metric K A B C PDO

Poisson regression MAD 3.25 1.39 3.15 9.08 47.28

MSPE 12.15 4.74 98.66 378.05 6800.66

NB regression MAD 3.27 1.38 3.23 7.13 47.27

MSPE 14.21 4.72 98.68 98.69 6800.94

Poisson regression—random 

parameters

MAD 3.35 2.73 3.14 7.23 63.87

MSPE 15.90 5.60 131.30 522.74 5036.42

NB—random parameters MAD 3.19 1.40 3.20 3.32 48.27

MSPE 13.21 3.72 97.86 99.79 6832.49

Univariate Poisson 

lognormal

MAD 4.20 2.83 2.42 9.32 45.56

MSPE 10.84 3.79 102.82 502.25 5216.12

Multivariate Poisson 

lognormal

MAD 3.30 1.38 3.13 3.13 46.27

MSPE 15.21 5.72 99.70 377.69 5800.94

Table 30. Model prediction with dataset 5 (5,000 intersections) used for estimation.

Models Metric K A B C PDO

Poisson regression MAD 3.31 1.37 3.13 9.07 53.99

MSPE 12.58 4.66 98.43 377.75 7489.81

NB regression MAD 3.33 1.37 3.12 9.07 47.24

MSPE 14.91 4.67 98.49 377.67 6796.74

Poisson regression—random 

parameters

MAD 3.25 3.01 2.01 10.98 49.6

MSPE 15.79 5.89 96.91 383.82 7116.87

NB—random parameters MAD 4.14 1.45 2.51 10.95 48.85

MSPE 16.01 4.75 98.31 393.32 6920.38

Univariate Poisson lognormal MAD 3.1 2.14 2.91 10.23 51.99

MSPE 13.92 4.58 95.83 370.86 6873.26

Multivariate Poisson lognormal MAD 5.05 2.41 2.85 9.5 54.54

MSPE 15.47 4.93 97.2 376.7 7092.44
81 

Table 31. Cross validation with dataset 10 (5,000 intersections) using dataset 5 model.

Models Metric K A B C PDO

Poisson regression MAD 3.3 1.38 1.38 9.07 53.98

MSPE 12.68 36.22 54.09 103.12 7000.66

NB regression MAD 3.33 1.38 3.12 9.07 47.26

MSPE 14.49 6.19 103 388.34 6879.1

Poisson regression—random 

parameters

MAD 4.78 2.79 2.59 9.95 53.4

MSPE 12.4 7.71 99.22 159.65 6608.34

NB—random parameters MAD 3.32 1.88 1.51 8.43 48.51

MSPE 12.59 10 99.41 120.17 6634.81

Univariate Poisson lognormal MAD 4.75 1.56 1.16 8.97 50.16

MSPE 12.8 9.93 99.29 232.12 6759.22

Multivariate Poisson 

lognormal

MAD 3.36 2.68 1.87 8.78 51.07

MSPE 13.92 8.67 65.5 169.32 6772.57

Empirical Analysis for Evaluating Parameter Stability

Dataset 10 (5,000 intersections) was used as the population benchmark to evaluate if each 

parameter in any model is statistically different from the corresponding one in that dataset. The 

combined NB estimates of PDO crashes for the 10 datasets are displayed in table 32 with their 

standard errors for illustrative purposes. Table 33 shows the updated Wald test statistics for the 

estimated model parameter values (Hoover, Bhowmik, Yasmin, and Eluru 2022).
82 

Table 32. NB estimates for PDO crashes.

Variables Dataset 1 Dataset 2 Dataset 3 Dataset 4 Dataset 5 Dataset 6 Dataset 7 Dataset 8 Dataset 9 Dataset 10

Intercept −9.719a, 

0.758b

−10.514,0.

648

−9.438, 

0.364

−9.829, 

0.550

−9.945, 

0.442

−9.107, 

0.339

−10.705, 

0.733

−9.367, 

0.486

−9.918, 

0.364

−9.858, 

0.361

Ln (AADT 

Major)

0.991,

0.042

0.854, 

0.025

0.881, 

0.020

0.867, 

0.016

0.834, 

0.016

0.847, 

0.036

0.888, 

0.024

0.813, 

0.026

0.868, 

0.017

0.855, 

0.015

Ln (AADT 

Minor) 

0.459,

0.028

0.576, 

0.017

0.543, 

0.013

0.575, 

0.011

0.561, 

0.011

0.545, 

0.024

0.539, 

0.017

0.602, 

0.018

0.563, 

0.012

0.575, 

0.011

LTLs 

1 Base Level

2 −0.370, 

0.091

−0.603, 

0.053

−0.564, 

0.044

−0.582, 

0.037

−0.700, 

0.035

−0.652, 

0.076

−0.741, 

0.056

−0.569, 

0.062

−0.679, 

0.039

−0.686, 

0.034

3 −0.171, 

0.136

−0.669, 

0.077

−0.495, 

0.061

−0.553, 

0.053

−0.652, 

0.050

−0.590, 

0.109

−0.763, 

0.076

−0.645, 

0.085

−0.732, 

0.055

−0.690, 

0.049

4 −0.471, 

0.173

−0.748, 

0.097

−0.739, 

0.081

−0.632, 

0.070

−0.808, 

0.064

−0.697, 

0.140

−0.952, 

0.100

−0.798, 

0.111

−0.832, 

0.072

−0.844, 

0.063

RTLs 

1 Base Level

2 −0.142, 

0.072

−0.083, 

0.042

−0.102, 

0.033

−0.108, 

0.029

−0.039, 

0.026

−0.120, 

0.059

−0.032, 

0.042

−0.012, 

0.050

−0.084, 

0.029

−0.103, 

0.027

3 0.356, 

0.143

−0.102, 

0.069

−0.126, 

0.060

−0.186, 

0.054

−0.123, 

0.052

−0.287, 

0.128

−0.037, 

0.067

−0.080, 

0.095

−0.107, 

0.054

−0.186, 

0.047

4 −0.440, 

0.226

−0.423, 

0.107

−0.146, 

0.088

−0.280, 

0.067

−0.257, 

0.063

−0.286, 

0.161

−0.196, 

0.098

−0.285, 

0.118

−0.242, 

0.073

−0.269, 

0.070

Speed Limit (mph)

35 Base Level

25 0.083, 

0.064

0.117, 

0.037

0.038, 

0.030

0.0107, 

0.0260

0.011, 

0.024

−0.077, 

0.055

−0.005, 

0.037

−0.018, 

0.041

0.005, 

0.027

−0.020, 

0.023

30 0.036, 

0.060

0.052, 

0.034

0.079, 

0.028

−0.027, 

0.024

0.043, 

0.023

0.032, 

0.054

0.010, 

0.035

−0.027, 

0.041

−0.022, 

0.025

−0.013, 

0.023

40 0.055, 

0.062

0.048, 

0.035

0.045, 

0.028

0.006, 

0.023

−0.002, 

0.022

0.010, 

0.050

0.046, 

0.034

−0.027, 

0.038

−0.029, 

0.025

−0.027, 

0.022

45 0.136, 

0.078

0.004, 

0.044

−0.013, 

0.036

−0.021, 

0.030

−0.017, 

0.029

−0.065, 

0.064

−0.069, 

0.044

0.058, 

0.049

0.009, 

0.032

−0.033, 

0.028

50 −0.187, 

0.154

0.042, 

0.098

0.019, 

0.078

0.128, 

0.064

0.029, 

0.057

−0.199, 

0.131

−0.080, 

0.098

0.082, 

0.106

−0.065, 

0.068

0.064, 

0.058

55 −0.384, 

0.261

−0.270, 

0.280

−0.004, 

0.146

−0.212, 

0.216

0.012, 

0.124

0.125, 

0.246

−0.283, 

0.363

−0.292, 

0.272

−0.314, 

0.177

−0.261, 

0.151
83 

Variables Dataset 1 Dataset 2 Dataset 3 Dataset 4 Dataset 5 Dataset 6 Dataset 7 Dataset 8 Dataset 9 Dataset 10

65 −0.151, 

0.458

−0.368, 

0.320

−0.113, 

0.249

−0.409, 

0.222

0.215, 

0.202

0.033, 

0.406

0.111, 

0.241

−0.510, 

0.451

0.419, 

0.265

−0.024, 

0.192

LTLs Permitted

1 Base Level

2 0.019, 

0.093

0.144, 

0.053

0.047, 

0.044

0.078, 

0.038

0.161, 

0.035

0.080, 

0.079

0.206, 

0.055

0.065, 

0.062

0.089, 

0.039

0.109, 

0.034

3 −0.225, 

0.163

0.137, 

0.096

0.042, 

0.078

0.032, 

0.067

0.093, 

0.060

0.131, 

0.134

0.309, 

0.094

0.181, 

0.105

0.153, 

0.068

0.146, 

0.061

4 0.023, 

0.185

0.116, 

0.106

−0.004, 

0.090

0.023, 

0.078

0.037, 

0.071

−0.009, 

0.163

0.262, 

0.110

0.094, 

0.125

0.077, 

0.080

0.153, 

0.070

LTLs Protected

1 Base Level

2 −0.133, 

0.139

−0.324, 

0.073

−0.316, 

0.059

−0.302, 

0.052

−0.179, 

0.050

−0.215, 

0.107

−0.221, 

0.076

−0.337, 

0.084

−0.264, 

0.054

−0.256, 

0.048

3 −0.496, 

0.334

−0.398, 

0.154

−0.352, 

0.128

−0.443, 

0.112

−0.328, 

0.104

−0.302, 

0.260

−0.481, 

0.164

−0.195, 

0.175

−0.350, 

0.105

−0.453, 

0.104

4 −0.567, 

0.247

−0.535, 

0.144

−0.409, 

0.111

−0.583, 

0.109

−0.502, 

0.095

−0.310, 

0.216

−0.226, 

0.149

−0.963, 

0.178

−0.505, 

0.105

−0.402, 

0.101

LTLs Mix

1 Base Level

2 −0.026, 

0.097

−0.107, 

0.058

−0.131, 

0.047

−0.158, 

0.040

−0.091, 

0.037

−0.103, 

0.081

−0.003, 

0.059

−0.052, 

0.064

−0.099, 

0.041

−0.060, 

0.037

3 −0.013, 

0.162

−0.115, 

0.092

−0.227, 

0.078

−0.246, 

0.067

−0.119, 

0.062

−0.157, 

0.136

0.050, 

0.096

−0.225, 

0.106

−0.081, 

0.069

−0.116, 

0.060

4 −0.014, 

0.178

−0.138, 

0.103

−0.202, 

0.085

−0.298, 

0.073

−0.135, 

0.067

−0.253, 

0.149

0.017, 

0.105

−0.147, 

0.116

−0.169, 

0.075

−0.170, 

0.067

RTOR

1 Base Level

2 −0.073, 

0.122

−0.180, 

0.063

−0.137, 

0.055

0.071, 

0.043

−0.016, 

0.041

−0.109, 

0.089

−0.141, 

0.061

−0.162, 

0.078

−0.057, 

0.044

−0.113, 

0.038

3 0.040, 

0.115

−0.117, 

0.068

−0.085, 

0.051

−0.069, 

0.045

−0.051, 

0.038

−0.190, 

0.089

−0.056, 

0.057

−0.079, 

0.064

−0.041, 

0.044

−0.063, 

0.041

4 0.041, 

0.136

−0.222, 

0.075

−0.145, 

0.066

0.013, 

0.056

−0.026, 

0.051

−0.203, 

0.119

−0.035, 

0.080

−0.013, 

0.082

−0.068, 

0.052

−0.150, 

0.052

Lighting

No Base Level

Yes −0.346, 

0.060

−0.213, 

0.036

−0.211, 

0.029

−0.267, 

0.024

−0.297, 

0.023

−0.185, 

0.051

−0.216, 

0.035

−0.247, 

0.039

−0.280, 

0.025

−0.278, 

0.023
84 

Variables Dataset 1 Dataset 2 Dataset 3 Dataset 4 Dataset 5 Dataset 6 Dataset 7 Dataset 8 Dataset 9 Dataset 10

Presence of School Within 1,000 ft of Intersection

No Base Level

Yes −0.021, 

0.064

0.028, 

0.036

0.057, 

0.090

0.022, 

0.025

0.017, 

0.024

−0.097, 

0.058

0.017, 

0.037

0.005, 

0.040

−0.085, 

0.028

0.029, 

0.024

Number of Bus Stops Within 1,000 ft of Intersection

0 Base Level

1–2 0.189, 

0.101

−0.096, 

0.061

−0.010, 

0.053

−0.080, 

0.044

0.030, 

0.043

0.073, 

0.093

−0.040, 

0.065

0.035, 

0.074

−0.028, 

0.050

0.006, 

0.041

≥3 −0.031, 

0.063

−0.040, 

0.039

0.036, 

0.030

−0.027, 

0.025

0.003, 

0.024

−0.061, 

0.056

0.015, 

0.038

0.019, 

0.042

0.001, 

0.027

−0.015, 

0.024

Number of Alcohol Sale Establishments Within 1,000 ft of Intersection

0 Base Level

1–8 −0.027, 

0.064

−0.024, 

0.046

−0.092, 

0.036

−0.005, 

0.030

−0.006, 

0.029

−0.074, 

0.066

−0.025, 

0.047

−0.053, 

0.051

0.007, 

0.032

−0.010, 

0.030

Maximum Crossing Lanes

1 Base Level

2 0.510, 

0.652

1.068, 

0.609

−0.042, 

0.323

0.268, 

0.527

0.840, 

0.419

0.291, 

0.067

1.272, 

0.704

0.082, 

0.418

0.540, 

0.331

0.487, 

0.332

3 0.448, 

0.654

1.209, 

0.610

0.208, 

0.324

0.479, 

0.527

1.062, 

0.420

0.575, 

0.053

1.435, 

0.704

0.368, 

0.420

0.782, 

0.332

0.722, 

0.332

4 0.600, 

0.652

1.621, 

0.609

0.619, 

0.323

0.889, 

0.527

1.497, 

0.419

0.875, 

0.065

1.862, 

0.703

0.730, 

0.418

1.188, 

0.331

1.146, 

0.332

5 0.447, 

0.654

1.878, 

0.610

0.839, 

0.324

1.126, 

0.527

1.742, 

0.420

1.241, 

0.077

2.054, 

0.704

1.005, 

0.419

1.473, 

0.331

1.403, 

0.332

6 0.439, 

0.656

2.240, 

0.610

1.288, 

0.325

1.523, 

0.528

2.101, 

0.420

1.354, 

0.116

2.527, 

0.705

1.375, 

0.421

1.815, 

0.332

1.774, 

0.332

7 0.587, 

0.663

2.349, 

0.613

1.306, 

0.327

1.624, 

0.529

2.261, 

0.421

0.291, 

0.067

2.596, 

0.707

1.548, 

0.425

2.011, 

0.334

1.935, 

0.334

*Variables insignificant at a 90-percent confidence level. a

Parameter estimate. b

Standard error.
85 

Table 33. Revised Wald test statistics on NB model parameter estimates (relative to dataset 10). 

Variables Dataset 1 Dataset 2 Dataset 3 Dataset 4 Dataset 5 Dataset 6 Dataset 7 Dataset 8 Dataset 9

Ln (AADT 

Major) 0.165 0.029 1.030 0.043 0.940 0.195 1.143 1.388 0.555

Ln (AADT 

Minor) 3.051 0.065 1.910 0.561 0.962 1.117 1.853 1.274 0.787

LTLs

1 Base Level

2 3.251 1.303 2.170 2.011 0.290 0.400 0.840 1.644 0.134

3 3.602 0.231 2.500 1.910 0.546 0.841 0.811 0.466 0.567

4 2.018 0.829 1.018 2.232 0.395 0.955 0.915 0.358 0.126

RTLs

1 Base Level

2 0.501 0.415 0.041 0.137 1.709 0.250 1.432 1.611 0.494

3 3.594 0.994 0.779 0.012 0.903 0.746 1.801 1.002 1.100

4 0.721 1.205 1.108 0.113 0.128 0.097 0.614 0.110 0.273

Speed Limit

35 mph Base Level

25 mph 1.521 3.127 1.527 0.878 0.922 0.958 0.351 0.033 0.704

30 mph 0.772 1.590 2.525 0.421 1.743 0.767 0.561 0.301 0.258

40 mph 1.252 1.827 2.027 1.024 0.792 0.670 1.780 0.003 0.075

45 mph 2.041 0.718 0.446 0.288 0.409 0.460 0.688 1.610 0.997

50 mph 1.525 0.189 0.462 0.753 0.430 1.830 1.272 0.150 1.438

55 mph 0.411 0.029 1.217 0.184 1.389 1.334 0.058 0.100 0.227

65 mph 0.258 0.921 0.282 1.312 0.856 0.126 0.436 0.992 1.353

LTLs Permitted

1 Base Level

2 0.899 0.557 1.107 0.600 1.058 0.162 1.488 0.629 0.393

3 2.131 0.077 1.046 1.250 0.619 0.888 1.449 0.286 0.073

4 0.658 0.288 1.378 1.230 1.159 2.389 0.830 0.410 0.711

LTLs Protected

1 Base Level

2 0.835 0.787 0.787 0.659 1.112 0.175 0.383 0.844 0.115

3 0.124 0.300 0.613 0.063 0.853 0.597 0.141 1.268 0.702

4 0.617 0.755 0.047 1.215 0.722 3.466 0.980 2.734 0.700

LTLs Mix

1 Base Level

2 0.323 0.682 1.206 1.808 0.589 0.690 0.829 0.109 0.705
86 

Variables Dataset 1 Dataset 2 Dataset 3 Dataset 4 Dataset 5 Dataset 6 Dataset 7 Dataset 8 Dataset 9

3 0.595 0.002 1.135 1.446 0.038 0.853 1.466 0.902 0.381

4 0.819 0.264 0.298 1.300 0.377 0.358 1.506 0.170 0.014

RTOR

1 Base Level

2 0.309 0.905 0.359 3.205 1.725 0.044 0.383 0.560 0.957

3 0.845 0.686 0.338 0.111 0.201 1.306 0.091 0.218 0.354

4 1.312 0.787 0.068 2.147 1.722 0.403 1.209 1.423 1.117

Lighting

No Base Level

Yes 1.067 1.495 1.803 0.295 0.595 1.671 1.479 0.664 0.055

Presence of School Within 1,000 ft of Intersection

No Base Level

Yes 0.731 0.015 0.305 0.172 0.354 1.231 0.267 0.516 3.142

Number of Bus Stops Within 1,000 ft of Intersection

0 Base Level

1–2 1.687 1.389 0.234 1.428 0.409 0.964 0.602 0.346 0.520

≥3 0.242 0.549 1.328 0.343 0.528 0.848 0.666 0.699 0.447

Number of Alcohol Sale Establishments Within 1,000 ft of Intersection

0 Base Level

1–8 0.205 0.264 1.778 0.108 0.102 1.329 0.720 0.727 0.385

Maximum Crossing Lanes

1 Base Level

2 0.032 0.837 1.142 0.351 0.660 0.580 1.009 0.758 0.114

3 0.372 0.702 1.108 0.389 0.635 0.438 0.916 0.661 0.128

4 0.746 0.684 1.139 0.413 0.657 0.802 0.920 0.781 0.088

5 1.303 0.684 1.218 0.443 0.633 0.477 0.836 0.745 0.149

6 1.815 0.671 1.047 0.401 0.611 1.192 0.966 0.743 0.088

7 1.814 0.593 1.344 0.495 0.606 5.402 0.845 0.714 0.161

Overall Percent 71.4 92.8 71.42 78.57 92.85 71.4 100 92.85 92.85
87 

From table 33, the parameter for major and minor AADT presents a t-statistic lower than the 

90-percent confidence value of 1.65 for all instances except for dataset 1 for the minor AADT, 

perhaps because the parameter was barely significant in the prediction model itself. A lower 

confidence value of 1.65 for the AADTs indicates no significant differences between the AADTs 

across the datasets. The parameter for “left-turn lane” presents a range higher than the 90-percent 

confidence value of 1.65, which is also not surprising, given the variable was only marginally 

significant. The parameter for “right-turn lane” for all models has a range lower than the 

confidence value of 1.65. The speed limit parameter, with levels from 25 to 65 mph, had a range 

lower than the confidence value of 1.65, except for 45 mph in dataset 1. Overall, out of the 14 

variables included in the models, 71.4 percent of the variable values in the dataset 1 fell into the 

range for the tested confidence value. For dataset 2, 92.8 percent of the variables fell within the 

range of significant value. The exception was “25-mph speed limit,” which presented a value 

higher than the 90-percent confidence value of 1.65. This conjecture applies to the other datasets 

used for this case study. 

Overall, most variables had test statistics lower than the confidence value of 1.65, which 

indicated there were no significant differences between this dataset and the corresponding dataset 

(10) that was used for testing the stability of this study’s data.

Discussion of Findings

Segments

The case study generated various crash model predictions, using 10 different datasets for rural, 

two-lane, undivided segment roads. The significant variables are intuitive relative to other 

studies. Less variables were significant when the sample size was small. Poisson Regression had 

a lower MSPE and MAD for K crashes. Poisson regression had the lowest MAD and NB had the 

lowest MSPE for A crashes, which meant that NB performed better. Univariate Poisson 

lognormal had the lowest MAD for B crashes while Poisson regression had the lowest for MSPE. 

Poisson regression performed better for C crashes for MAD while NB did better for MSPE. 

Finally, Poisson did better for MAD for PDO crashes while univariate Poisson lognormal did 

better for MSPE.

The research team used cross validation to assess the predictive performance of the models and 

judge how they performed outside the sample to a new dataset also known as test data. Without 

cross validation, there is only information on how the model performs relative to in-sample data. 

Ideally, it is desirable to see the model prediction accuracy on new data. The results from the 

cross-validation exercise showed that the dataset was robust enough to be relied on. Finally, the 

t-statistic estimates showed that the differences among the parameter value across the dataset are 

within a statistically acceptable level. The test statistics across the datasets for the AADT 

parameter were also lower than the 90-percent confidence value of 1.65 indicating that the 

variation across the different datasets is within a statistically acceptable level.

Intersections

The case study presents the crash prediction models for urban four-leg signalized intersections. 

Univariate Poisson lognormal performed better in the prediction of MSPE for K crashes while 
88 

Poisson regression did better for MAD for K crashes. NB had the lowest MAD and MSPE for A 

crashes which meant that it performed better. Univariate Poisson lognormal had the lowest MAD 

for B crashes while Poisson regression had the lowest for MSPE. Poisson regression performed 

better for C crashes for MAD while NB did better for MSPE. Finally, Poisson did better for 

MAD for PDO crashes while univariate Poisson lognormal did better for MSPE.

The results from the cross-validation exercise show that the dataset was robust enough to be 

considered reliable. Finally, the t-statistic estimates show that the differences among the 

parameter values across the dataset are within a statistically acceptable level.

Summary and Conclusions

The purpose of this case study was to demonstrate the usefulness of the RAD tool, how to apply 

it, and establish that it could be relied upon to generate realistic data. Hence, the team developed 

various prediction models for different severity levels. Revised Wald Test Statistics were 

conducted to check if the variation across the different datasets is within a statistically acceptable 

level using dataset 10 as the benchmark. The result clearly highlights the stability in various 

parameter estimates across the datasets. The resulting stability found across the datasets indicates 

that the parameter estimates using RAD will be consistent, regardless of the miles of segmentrelated data generated using the tool. With this knowledge, the dataset from the RAD tool can be 

used for other possible purposes, such as estimating other possible prediction models, comparing 

the performance of varied safety analysis methods, and helping to determine an adequate sample 

size to get convincing results, especially for low realizations of fatal crashes. RAD can also be 

helpful in generating large datasets with consistent conditions in cases where it is not possible to 

go back many years due to changes in roadway characteristics or drivers. This makes it easier to 

estimate models for unusual and rare events, such as minor crashes.

The results from this case study may have significant implications on highway safety research in 

the development of information that can be used to make roads safer and crashes less severe. 

Since the data-generation process in the RAD tool is completely known, its use will allow 

objective evaluation and validation of various safety analysis methods used to verify various 

assumptions related to safety performance. In turn, these capabilities may help transportation 

agencies produce effective countermeasures to prevent and address crashes.

DRIVING SIMULATION CASE STUDY

Introduction

One of the most significant socioeconomic issues facing the world today is the frequency of 

traffic crashes. According to a recent report by NHTSA, 38,824 people in the United States died 

in car crashes in 2020 (National Center for Statistics and Analysis 2021). Rear-end collisions 

accounted for almost one-third of these crashes. Driver inattention was a major causal factor in 

about 91 percent of rear-end crashes. This inattention may prevent a driver from detecting an 

object ahead and can be caused by distraction, fatigue, or atmospheric conditions (such as fog or 

sun glare). Several strategies exist to aid drivers in avoiding collisions (National Transportation 

Safety Board 2002). These strategies differ in terms of degree of intervention—from alerts that 

suggest subtle speed adjustments to automatic emergency braking.
89 

With the emergence of these strategies in the future, vehicles with these various safety systems 

installed are likely to have fewer total crashes. One such strategy is the FCW system. This 

system is designed to help drivers reduce the severity of collisions or avoid them, especially rearend crashes, with visual, auditory, or tactile warnings of possibly impending collisions (Kusano 

and Gabler 2012). Additionally, a 2002 Daimler-Chrysler study found that 60 percent of rear-end 

collisions could likely be avoided if drivers had 0.5 additional seconds of warning, and 

90 percent of rear-end collisions could likely be avoided if drivers had an extra second of 

warning (NHTSA 2002). This research places FCW high on the list of solutions that can 

contribute significantly to reducing crash numbers and severity. 

The objective of this case study was to quantify the observed number of serious conflicts for 

drivers in vehicles equipped with an FCW system versus those in vehicles without these systems 

using driving simulator experiments. For this case study, driving simulation scenarios were 

generated that exposed participating drivers to road and traffic conditions deliberately designed 

to test their responses to unexpected forward obstacles with and without warnings from an FCW 

system. This case study differs from a previous study by Lee et al. (2002), which focused on the 

value of an FCW system in a scenario where the driver was intentionally distracted as part of the 

experiment and the hazard was a stopped lead vehicle. This case study explores the value of the 

FCW system for drivers who may or may not be distracted; the hazard may be a vehicle that 

stops or crosses the travel path suddenly or a nonmotorized user crossing the street unexpectedly. 

Literature Review

Studies on FCW Systems

FCW systems have been estimated to potentially reduce front to rear-end crash rates by 

27 percent and front to rear injury crash rates by 20 percent (Cicchino 2017). Although these 

systems do not prevent every crash, they have been proven to significantly mitigate the 

likelihood of a crash. Lee et al. (2002) presented users of a driving simulator with a scenario 

involving a stopped lead vehicle in which the driver was intentionally distracted and found that 

the FCW system reduced the number of collisions for that scenario by 80.7 percent. Teoh (2021) 

estimated the effectiveness of FCW systems using detailed data from exposure measures 

extracted from video footage and concluded that FCW systems were associated with a 

statistically significant 22-percent reduction in the rate of police-reportable crashes per vehicle 

miles traveled.

A common human factor discussion surrounding FCW concerns the criteria that determine the 

guidance on warning timing. McLaughlin, Hankey, and Dingus (2008) noted that early warning 

has great potential to prevent crashes. Abe and Richardson (2006) found that early alarm timing 

may improve driver’s trust compared with late alarm timing. In addition, other studies 

investigated how various driving situations and hazard warning experience can influence driver 

responses to warning and hazardous situations. Incorrect warnings and failures of the system to 

produce warnings that drivers find useful and understandable may diminish drivers’ reliance on 

and compliance with warnings (Glassco and Cohen 2001; Lee and Lee 2007; Najim and Smith 

2004; Reinmueller and Steinhauser 2019).
90 

A considerable amount of literature describes the possible safety benefits of these advanced 

technologies. Most of the literature the research team reviewed focused on the market 

penetration rate (MPR) of these systems. For example, Xiao et al. (2021) carried out a 

meta-analysis evaluating crash reduction by penetration rate. The results indicated that the 

number of conflicts is exponentially reduced as MPR goes up; safety is enhanced by 4 percent 

with an MPR of 10 percent and by 43 percent with 90 percent MPR. Furthermore, Papadoulis, 

Quddus, and Imprialou (2019) evaluated the safety impact of connected vehicles (CVs) using 

traffic simulation. The results showed that CVs bring about compelling benefits to road safety, as 

traffic conflicts significantly reduce even at relatively low market penetration rates. Specifically, 

estimated traffic conflicts were reduced from 12 to 47 percent, 50 to 80 percent, 82 to 92 percent, 

and 90 to 94 percent for 25-percent, 50-percent, 75-percent, and 100-percent CV penetration 

rates, respectively.

Based on 37 precrash scenarios developed by Najm et al. (2010), Jermakian (2011) estimated the 

maximum potential for U.S. crash reductions for four crash avoidance technologies: side view 

assistance, FCW, lane departure warning, and adaptive headlights. Jermakian (2011) estimated 

that FCW holds the greatest potential for preventing crashes of any severity, up to 1.2 million 

crashes per year in the United States, or 20 percent of the annual 5.8 million police-reported 

crashes. Kusano and Gabler (2012) examined the safety benefits of FCW systems in rear-end 

collisions simulating scenarios from real world rear-end crash data extracted from the National 

Automotive Sampling System/Crashworthiness Data System. Their study indicated a dramatic 

reduction in serious and fatal injuries when using these safety features.

Studies on Crashes and Serious Conflicts

Surrogate safety assessment is an alternative method of assessing safety that relies on the 

analysis of safety-critical events known as traffic conflicts. Definitions of traffic conflict vary; 

the first mention of the term was by Klebelsberg (1964), as cited by Tarko (2018), who defined 

traffic conflicts as dangerous traffic interactions. Laureshyn and Varhelyi (2020) define a traffic 

conflict as “an observable situation in which two or more road users approach each other in 

space and time to such an extent that there is a risk of collision if their movements remain 

unchanged.” The proximity of road users to each other can be physically measured in temporal 

and/or spatial dimensions, and thresholds can be used to identify conflicts. Perkins and Harris 

(1968) used traffic conflicts to define situations necessitating evasive actions, such as braking. 

According to this definition, conflicts and crashes are of a similar nature but for the presence and 

success of an evasive action (Zheng, Ismail, and Meng 2014). 

Various conflict indicators have been developed to measure the severity of an interaction by 

quantifying the spatial and temporal proximity of two or more road users. A comprehensive 

summary of the different indicators is provided in Brown (1994) and Tarko (2009). Typical 

measures of conflict severity include conflicting speed and severity index (Autey, Sayed, and 

Zaki 2012; Essa and Sayed 2020). The time-to-collision indicator used in this study has been 

widely used to measure the severity of traffic conflicts. Time to collision is defined as “...the 

time that remains until a collision between two vehicles would have occurred if the collision 

course and speed difference are maintained” (Laureshyn and Varhelyi 2020). The main 

advantage of conflict indicators is their ability to capture the severity of an interaction in an 

objective and quantitative way. Various studies have used traffic conflicts together with crashes 
91 

for safety analysis and have shown that conflicts can adequately serve as a proxy for crashes 

(Laureshyn and Varhelyi (2020) ; Charly and Mathew 2019; Peesapati, Hunter, and Rodgers 

2013; Xie et al. 2016). 

Although previous studies have contributed to examining the effectiveness of FCW systems, 

none of these studies focused on estimating the relationship between crashes and serious conflict 

with the aim of providing information for crash modification factors for the FCW systems. In 

this case study, near crashes, also known as serious conflicts, were used as surrogates for crashes 

to represent quantitative safety. This substitution was made because researchers expected crashes 

to be extremely rare, given the number of drivers to be observed in the experiment, and because 

identifying crashes in the driving simulation environment would be challenging. 

Methodology 

This section outlines the general description of methods used for the entire experiment and 

describes the participants used for the experiment, followed by the description of the Connecticut 

Transportation Safety Research Center driving simulator used for the main experiments. The 

driving scenarios used for the experiment are also described in detail. Finally, there is a 

discussion regarding the traffic conflict technique the research team used to evaluate traffic 

conflicts.

Participants

A total of 142 participants—64 females, 77 males, and 1 person who declined to identify as 

either male or female—were recruited for the study. Simulator sickness prevented nine people 

from completing the experiment. Of the 133 participants who completed the experiment, 58 

(41 percent) were between the ages of 18 and 29, 55 (41 percent) were between the ages of 30 

and 64, and 20 (15 percent) were 65 or older. Each participant had a valid driver’s license and 

was in good physical health. They were recruited via flyers distributed on college bulletin 

boards, in accordance with Institutional Review Board regulations. The college bulletin boards 

they were posted to belonged to University of Connecticut (UConn). Flyers were also distributed 

at senior and community centers to increase the participation of senior drivers and area residents 

not associated with UConn. 

Equipment

The experiment made use of a driving simulator, made by Technology Company A, which had a 

full cab and a segmented screen. Additional equipment included a series of cameras, four 

projectors (rear, front, right-side, and left-side projections), and various screens to provide a 

high-fidelity virtual environment. All the equipment was owned and maintained by the 

Connecticut Transportation Safety Research Center located at UConn. Some features of the 

driving simulator include the Internet Scene Assembler ®, which is used in the modification of 

the virtual environment, and the SimObserver®, which is integrated with the virtual environment 

and used for data and video synchronization, video capture, and after-action review (Realtime 

Technologies 2023). Figure 20 shows the UConn driving simulator. 
92 

Source: FHWA. 

Figure 20. Photo. UConn driving simulator vehicle.

In figure 21, the computer on the left side was used to run the simulator and edit the model. The 

computer on the right side was used to extract data and video clips of the main driving 

experiment. 
93 

Source: FHWA.

Figure 21. Photo. UConn driving simulator control center.

The software programs used in the driving simulator were SimCreator®, SimObserver®, 

SimCreator DX®, and Data Distillery® (Realtime Technologies 2023). Additionally, audio 

software and hardware were used to simulate engine sounds, tire sounds, and vehicle noises for 

the participants.

SimCreator® is used for graphical simulation and as a modeling system. Different components 

are connected with each other to make a model. Each component can either be a group of 

different components or a C/C++ code component. Once a model has been developed, it can be 

simulated, and the result plotted (Realtime Technologies 2023). 

SimObserver® is a data collection system designed to capture data for after-action review that is 

controlled by the SimObserver computer. It procures the following output files (Realtime 

Technologies 2023): 

• Video file (MPEG-2 format): The format of the file is “.mpg file.” It contains a video of 

the entire experiment. The video is used to determine the length of road over which 

various tasks are performed.

• Log file: The format of this file is “.log file.” It includes system messages, errors, and 

warnings. 
94 

• Event file: The format of this file is “.vt file.” It contains start and stop times and labels 

all events logged during video capture. 

• DAT file: The format of this file is “.dat file” (in the form of a video on DVD). It is the 

main data file used to measure driving performance. Data are categorized into different

columns. Some of the variables in the DAT file are longitudinal acceleration, lateral 

acceleration, throttle, and headway distance.

SimCreatorDX® is the graphical user interface used to create, monitor, and control scenarios. It 

allows the researcher to observe the current state of the SimCreator simulation through graphical 

displays and detailed data view and is intended to be the primary tool used for development, 

tuning, and creation of experimental scenarios. After launching SimCreatorDX, the user has the 

option of either developer or experimental mode. In developer mode, the purpose is to design 

scenarios to fit the purpose of the study. Experimental mode allows for a quick interface to run 

multiple participants for any studies (Realtime Technologies 2023).

Data Distillery® is a data review and reduction software package with a main purpose of 

improving the efficiency of data reduction. This tool is also used for compiling the captured 

video and data from SimObserver. Data Distillery provides fine details of the collected data to 

understand the nature of the behavior or system being observed. The log file, video file, and data 

file are all displayed on the same screen. It also can be utilized to find the position of the vehicle 

(Realtime Technologies 2023).

Procedure and Experimental Design

Each participant was given an informed consent document to read and sign upon arrival on the 

day of the study. This consent included permission to record video. Each participant then 

completed a questionnaire that included demographic questions, such as their sex, age, 

race/ethnicity, and years of driving experience. Researchers then asked participants to complete 

an approximately 5-min driving simulator training sequence to acclimatize them to driving in the 

simulator. If participants requested more time, an additional 5-min training was provided. 

After the training sequence, each participant was placed into one of two groups based on the 

answers to the questionnaire. This allowed researchers to balance the distribution of age, sex, and 

years of driving experience within each group. The race/ethnicity distribution of each group was 

also monitored to keep those factors as balanced as possible. Both groups completed a defined 

driving scenario that included a mix of city and highway driving. The scenario took about 15 min 

to complete. Participants were asked to drive the course as if they were driving a real car on a 

real road. They were told that the purpose of the experiment was to observe their reactions to the 

scenario they drove. At some point during the scenario, a stimulus was introduced that required 

the driver to execute a sudden, unexpected braking maneuver. For example, a parked vehicle 

would suddenly pull in front of the vehicle without warning, or another vehicle, animal, or 

pedestrian would cross the vehicle path. 

Half the participants drove the course with a FCW system programmed into the simulation, 

giving the participant an audible and visual warning on the dashboard about the imminent 

collision. The other half did not have the FCW system and received no warning. Before the 
95 

driving began, participants with the FCW system were informed that the vehicle they were 

driving provided an audible sound and/or flashing pedestrian on the dashboard to warn of 

imminent forward collision. 

After they completed their turns in the driving simulator, participants were debriefed by a 

researcher and given a copy of the UCONN debriefing form for the case study, titled “Estimating 

a Crash Modification Factor for Forward Collision Warning Systems in the Vehicle Fleet.” The 

debriefing form included contact info for the principal investigator and student researcher and 

listed the study sponsor as FHWA. Additionally, it included the following statement about the 

study: 

The purpose of this study is to find out how much of a reduction in accidents is possible 

when vehicles are equipped with an automatic FCW system. Participants were placed in 

one of two groups: one that included the automatic forward collision avoidance system 

and one that did not. At one point during the experiment, you were faced with a sudden, 

unexpected obstacle that required you to stop. We noted the vehicle speed and the 

distance from the obstacle when you took evasive action and will use them to calculate 

the severity of the resulting conflict. Once we are finished running experiments, we will 

compare the conflict severities of the drivers who had the FCW system with those who 

did not to estimate the safety value of having this system installed.

Additionally, the researcher gave a verbal description of the study and its aims. As the 

experiment did not involve deception, this debriefing was only informational. 

Driving Scenarios

Driving scenarios were created for both urban and rural settings. Researchers programmed 

pedestrians walking along sidewalks into each scenario to help the scenes be as realistic as 

possible. Participants were asked to drive as they normally would on their way to work or 

college but to always stay in the right lane. These instructions allowed for the unexpected event 

to occur in the right lane. (Although it would be most ideal to run this experiment with no 

same-way lane restrictions on drivers, programming the driving simulator to accommodate this 

change was not possible due to constraints in the technology available. Due to its uniqueness, 

this case study is still extremely valuable for the insights it can provide.)

The urban setting consisted of an undivided roadway section with two lanes of traffic in both 

directions. The roadway included all the typical features of an urban road, such as speed limit 

signs and traffic signals, and had paved curbs on both sides of the road. The roadside features 

included restaurants and gas stations placed along the route with high-rise buildings. Most 

sections of the road were straight, but there were a few right-angled turns, as shown in figure 22. 
96 

Source: FHWA.

Figure 22. Photo. Section of urban roadway used in experiment.

The rural scenario consisted of a two-lane undivided roadway section with traffic simulated into 

the experiment to make it feel like real-world driving. The rural setting was a mix of countryside 

and residential areas, as shown in figure 23. Most sections of the road were straight, with a few 

horizontal curves. 
97 

Source: FHWA.

Figure 23. Photo. Section of rural roadway used in experiment. 

During the experiment, participants were asked to follow the designated route as a recorded 

message that was broadcast through speakers in the car provided directions. Four scenarios were 

modeled, as follows:

• Scenario 1: Urban setting plus entering car. A parked car suddenly pulls over into the 

path of the driver.

• Scenario 2: Urban setting plus pedestrian. A person suddenly crosses the road into the 

path of the driver.

• Scenario 3: Rural setting plus animal. A deer crosses into the path of the driver.

• Scenario 4: Rural setting plus pedestrian. A small child crosses the road unexpectedly.

The demographics of participants in each scenario are demonstrated in table 34. Care was taken 

to ensure a balance for the age categories by scenario.
98 

Table 34. Driving simulator participant statistics by scenario.

Scenario

Male 

18−29

Male 

30–64 

Male 

65+ 

Female

18−29

Female

30−64

Female

65+ Total 

Urban plus entering car 4a, 4b 4, 5 1, 1 2, 3 3, 1 1, 1 15,15 

Urban plus pedestrian 4 ,4 4, 3 0, 1 4, 3 5, 5 2, 3 19, 19 

Rural plus animal 2, 2 2, 3 3, 2 4, 5 4, 3 2, 2 17, 17 

Rural plus pedestrian 4, 4 3, 3 1, 0 4, 4 2, 3 1, 1 15, 15 

Total 14, 14 13, 14 5, 4 14, 15 14, 12 6, 7 66, 66 

a

With FCW.

b

Without FCW.

Traffic Conflict Methodology

To consistently use conflict techniques, many countries have formed their own standards and 

published manuals or handbooks to guide field observations. Some examples include Swedish 

Traffic Conflict Technique (STCT), U.S. Traffic Conflict Technique (USTCT), Dutch Traffic 

Conflict Technique (DOCTOR), and the German Traffic Conflict Technique. Both the STCT and 

USTCT use the time to accident (TA) and conflicting speed (CS) values as a conflict severity 

indicator; meanwhile, DOCTOR is more concerned with driver error and incorporates it into a 

safety continuum (Zheng, Ismail, and Meng 2014). 

The STCT is used in this effort to evaluate traffic conflicts. Traffic conflicts are typically 

categorized based on two indicators: the TA value and the CS. The TA is the time between when 

a particular road user successfully performs an evasive action and when the collision would have 

occurred. The CS is the speed of the roadway user at the point of taking the evasive action. The 

TA and CS values are used to define the conflict severity. The CS affects the outcome of the 

collision (the resulting pedestrian injury depends on the speed), and a higher speed requires 

longer stopping time and distance. Thus, a higher CS indicates a more severe conflict for the 

same TA (Laureshyn and Varhelyi 2020). Directly estimating the TA in live traffic can be 

challenging, so TA can instead be estimated from the vehicle speed and the distance to the 

collision point using table 35. The conflict severity is then determined from the TA and the CS, 

according to the graph in figure 24.
99 

Table 35. Chart. TA values estimated from vehicle speed and distance to collision point (Laureshyn and Varhelyi 2020).

Speed Distance, m

km/h m/s 0.5 1 2 3 4 5 6 7 8 9 10 15 20 25 30 35 40 45 50 55

5 1.4 0.4 0.7 1.4 2.2 2.9 3.6 4.3 5.0 5.8 6.5 7.2 — — — — — — — — —

10 2,8 0.2 0.4 0.7 1.1 1.4 1.8 2.2 2.5 2.9 3.2 3.6 5.4 7.2 9.0 — — — — — —

15 4.2 0.1 0.2 0.5 0.7 1.0 1.2 1.4 1.7 1.9 2.2 2.4 3.6 4.8 6.0 7.2 8.4 9.6 — — —

20 5.6 0.1 0.2 0.4 0.5 0.7 0.9 1.1 1.3 1.4 1.6 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 9.0 9.9 

25 6.9 0.1 0.1 0.3 0.4 0.6 0.7 0.9 1.0 1.2 1.3 1.4 2.2 2.9 3.6 4.3 5.0 5.8 6.5 7.2 7.9 

30 8.3 0.1 0.1 0.2 0.4 0.5 0.6 0.7 0.8 1.0 1.1 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 6.6 

35 9.7 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.5 2.1 2.6 3.1 3.6 4.1 4.6 5.1 5.7 

40 11.1 0.0 0.1 0.2 0.3 0.4 0.5 0.5 0.6 0.7 0.8 0.9 1.4 1.8 2.3 2.7 3.2 3.6 4.1 4.5 5.0 

45 12.5 — 0.1 0.2 0.2 0.3 0.4 0.5 0.6 0.6 0.7 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 

50 13.9 — 0.1 0.1 0.2 0.3 0.4 0.4 0.5 0.6 0.6 0.7 1.1 1.4 1.8 2.2 2.5 2.9 3.2 3.6 4.0 

55 15.3 — 0.1 0.1 0.2 0.3 0.3 0.4 0.5 0.5 0.6 0.7 1.0 1.3 1.6 2.0 2.3 2.6 2.9 3.3 3.6 

60 16.7 — 0.1 0.1 0.2 0.2 0.3 0.4 0.4 0.5 05 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 3.3 

65 18.1 — 0.1 0.1 0,2 0.2 0.3 0.3 0.4 0.4 0.5 0.6 0.8 1.1 1.4 1.7 1.9 2.2 2.5 2.8 3.0 

70 19.4 — 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.8 1.0 1.3 1.5 1.8 2.1 2.3 2.6 2.8 

75 20.8 — 0.0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.7 1.0 1.2 1.4 1.7 1.9 2.2 2.4 2.6 

80 22.2 — 0.0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.7 0.9 1.1 1.4 1.6 1.8 2.0 2.3 2.5 

85 23.6 — 0.0 0.1 0.1 0.2 0.2 0.3 0.3 0.3 0.4 0.4 0.6 0.8 1.1 1.3 1.5 1.7 1.9 2.1 2.3 

90 25.0 — 0.0 0.1 0.1 0.2 0.2 0.2 0.3 0.3 0.4 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 

95 26.4 — 0.0 0.1 0.1 0.2 0.2 0.2 0.3 0.3 0.3 0.4 0.6 0.8 0.9 1.1 1.3 1.5 1.7 1.9 2.1 

100 27.8 — 0.0 0.1 0.1 0.1 0.2 0.2 0.3 0.3 0.3 0.4 0.5 0.7 0.9 1.1 1.3 1.4 1.6 1.8 2.0 

© 2018 Lund University. 

—No data.
100 

© 2018 Lund University. 

Figure 24. Graph. Conflict severity diagram (Laureshyn and Varhelyi 2020). 

Conflicts with a severity level above 26 are ranked as serious. For instance, a vehicle 

approaching at 30 mph (48.3 km/hr) with a TA of 2.8 sec would pass the potential conflict point 

with too much time remaining for the interaction to be classified as a serious conflict. However, 

a 45 mph (72.4 km/hr.) approach speed would just exceed that level. These severity levels can be 

used to understand serious conflicts to estimate the crash modification factor (CMF) for FCW 

systems. A CMF is the ratio of the number of crashes expected with a countermeasure to the 

number of crashes expected without it. In other words, when implementing a countermeasure, 

multiplying the existing expected number of crashes by the CMF gives an estimate of the 

expected number of crashes after implementation of the countermeasure. CMFs with a value less 

than one indicate an expected decrease in crashes; meanwhile, a value greater than one indicates 

an expected increase. The FCW system can be said to be a countermeasure installed in vehicles 

to help mitigate the occurrence of crashes in vehicles that have it installed. A CMF for this 

“countermeasure” would have to also account for the penetration rate of the FCW system in the 

vehicle fleet. Estimating a CMF value for FCW systems would be helpful for adjusting the 

expected number of crashes in future conditions, as the proportion of vehicles with such systems 

increases.

The steps to estimate the CMF are as follows:

• Estimate a ratio to account for the safety effect of having a FCW system (RFCW):

 RFCW = nFCW∕ no, where nFCW is the number of serious conflicts in vehicles with the FCW, and no 

is the number of serious conflicts in vehicles without the FCW. 
101 

• Define the proportion of the vehicle fleet with the FCW system (PFCW): Five different 

proportions of the vehicle fleet from 10 to 50 percent in increments of 10 percent were 

investigated. 

• Calculate the estimated CMF for the FCW system: CMFFCW = 1+ PFCW(RFCW – 1) Error! 

Digit expected.. 

Data Collection and Analysis 

TA and CS are the two variables required to calculate conflict severity. Velocity is the output of 

the speed in the simulator and reflects the speed with which a participant drove. Lane position 

can be defined as the position of the vehicle measured from the center of the road, in meters and 

was used to represent the lateral control of the vehicle. A positive number indicates a vehicle on 

the right side of the center line. A negative number indicates a vehicle on the left. . Data 

collected on these two dependent variables were at a frequency of 60 Hz through the 

SimObserver® proprietary software of the driving simulator (Realtime Technologies 2023). 

The lateral position where the event occurred was collected for both the scenarios with and 

without the FCW system. Table 36 shows an example of the resulting extracted data from the

simulator. The X and Y axes indicate the lateral position of each participant at the time the event 

was introduced into the experiment. The speed is the speed the participant was driving before 

carrying out the evasive action of either hitting the brakes or swerving. 
102 

Table 36. Example of extracted spreadsheet data.

Participants X Y Speed (m/s)

1 (without FCW) FHWA01 3995.18 -6895.13 17.56 

FHWA03 2681.54 -6895.13 17.56 

FHWA 20 2694.1 -6895.31 18.89 

FHWA 28 2673.54 -6896.02 17.61 

FHWA 30 2672.73 -6895.13 15.95 

FHWA 40 2711.58 -6895.38 28.23 

FHWA 42 2674.42 -6895.55 14.129 

FHWA45 2667.56 -6897.24 17.57 

FHWA 51 2677.33 -6895.81 15.192 

FHWA 52 2686.56 -6897.07 16.96 

FHWA 95 2665.24 -6895.48 18.04 

FHWA125 2681.99 -6895.33 28.23 

FHWA126 2705.15 -6895.73 14.129 

FHWA127 2669.16 -6895.67 15.129 

FHWA128 2672.73 -6895.13 15.5 

1 (with FCW) FHWA 09 2680.06 -6895.81 13.95 

FHWA 10 2678.98 -6895.1 13.75 

FHWA 15 2668.53 -6895.37 15.07 

FHWA 29 2676.24 -6895.5 15.63 

FHWA 31 2671.76 -6895.96 13.44 

FHWA 34 2687 -6895.84 15.12 

FHWA 68 2681.99 -6895.33 15.38 

FHWA 71 2669.16 -6895.67 10.6 

FHWA 74 2665.45 -6895.66 12.01 

FHWA 105 2667.92 -6895.83 16.61 

FHWA 112 2705.15 -6895.73 13.29 

FHWA 122 2660.42 -6897.24 12.47 

FHWA 123 2667.45 -6896.09 15.07 

FHWA 124 3995.18 -6895.13 15.63 

FHWA129 2711.58 -6895.38 15.63 

Thirty participants were in scenario 1, 38 in scenario 2, 34 in scenario 3, and 30 in scenario 4. 

The numbers of participants in each scenario was balanced, as noted in table 34. Figure 25 shows 

the speed of the participants with and without the FCW. When scenario 1 (a parked car suddenly 

drives into the path of the participant) occurred, the speed was lower with participants with the 

FCW—possibly because the warning sound made drivers immediately aware of the upcoming 

event. 
103 

Source: FHWA.

Figure 25. Graph. Speed of participants in scenario 1.

Figure 26 shows the speed of the participants with and without FCW when scenario 2 (a 

pedestrian crossing into the path of the participant) occurred. It shows that the speed was lower 

for participants with FCW. This outcome may be explained by the fact that the warning sound 

made drivers immediately aware of the upcoming event. 

Source: FHWA.

Figure 26. Graph. Speed of participants in scenario 2.

Figure 27 and figure 28 show a similar trend, as the speed remains greater for participants 

without the FCW.
104 

Source: FHWA.

Figure 27. Graph. Speed of participants in scenario 3.

 Source: FHWA.

Figure 28. Graph. Speed of participants in scenario 4.

Calculation of the Effectiveness of FCW

The safety effect of having FCW was calculated by the ratio of serious conflict (nFCW) in the 

experiment with FCW to the ratio of serious conflict without FCW, as discussed in the traffic 

conflict methodology section. Table 37 shows the raw factor for the different scenarios 

conducted in the experiment. The raw factor was also calculated for all cases with and without 

FCW to see if there were substantial differences between individual scenarios and overall cases.
105 

Table 37. FCW ratios by scenario.

Scenario nFCW nO RFCW

All cases 24 42 0.57

Urban plus entering 

car 

6 9 0.67

Urban plus pedestrian 6 13 0.46

Rural plus animal 7 10 0.70

Rural plus pedestrian 5 10 0.50

The CMF was calculated based on the various penetration rates of the system and is shown in 

table 38. 

Table 38. CMF values by FCW fleet penetration rate.

Scenario PFCW= 0.1 PFCW= 0.2 PFCW= 0.3 PFCW= 0.4 PFCW= 0.5 

All cases 0.95 0.91 0.87 0.82 0.78

Urban plus entering 

car 

0.96 0.93 0.89 0.86 0.83

Urban plus pedestrian 0.94 0.89 0.83 0.78 0.73

Rural plus animal 0.97 0.94 0.91 0.88 0.85

Rural plus pedestrian 0.95 0.90 0.85 0.80 0.75

Considering all simulation scenarios, when the penetration rate is 10 percent, the CMF is 0.95, 

which translates to a 5-percent reduction in crashes. The CMF increases to 22 percent for a 

penetration rate of 50 percent. For scenario 1, when the FCW penetration rate is 10 percent, a 

4-percent reduction in crashes is expected with a 17-percent reduction at a 50-percent penetration 

rate. In the case of scenario 2, a CMF of 0.94 for a 10-percent penetration rate means a 6-percent 

reduction in crashes, with a 27-percent reduction expected when the penetration rate of FCW is 

at 50-percent. Scenario 3 had a CMF of 0.97, which translates to a 3-percent reduction in crashes 

when the penetration rate is at 10 percent and 0.85, a 15-percent reduction when the penetration 

rate is at 50 percent. Scenario 4 had a CMF of 0.95, a 5-percent reduction in crashes when the 

penetration rate is at 10 percent and a CMF of 0.75 and a 25-percent reduction in crashes when 

the penetration is at 50 percent. Notably, as the overall penetration rate increases, the number of 

crashes decreases.

Driving Simulator Case Study Findings 

The team examined the effectiveness of FCW to provide information for crash modification 

factors for the FCW systems for five different FCW system market penetration rates for the 

vehicle fleet from 10 to 50 percent in increments of 10 percent. Four scenarios were modeled in 

urban and rural settings in which participants were exposed to conditions deliberately designed 

to test their responses to unexpected events when they got warnings from FCW systems or when 

there was no system. To achieve this, two variables, lane position and speed, were used to 

measure conflict severity. This X was used as a proxy for crashes because researchers expected 

crashes to be extremely rare, given the relatively small number of drivers observed in the 

experiment. The conflict severity was determined from the TA and CS values of the road users. 

A total of 133 participants completed the experiment.
106 

The results suggest that FCW systems have the potential to reduce incidences of crashes in 

vehicles that have them installed, and reduction in incidences of crashes will occur as market 

penetration rate goes up. The overall CMF values for all the scenarios conducted in the 

experiment ranged from 0.78 to 0.95, which is approximately a 10 to 22 percent reduction in 

crashes, with an R value of 0.57. The CMF values for scenario 1 and 2 ranged from 0.83 to 0.96 

and 0.73 to 0.94, which is approximately a 4 to 17 percent and 6 to 27 percent reduction in 

crashes, respectively. R values were 0.67 and 0.46. In the rural scenario, the CMF values for 

scenario 3 and 4 were from 0.85 to 0.97 and 0.75 to 0.95, with R values of 0.70 and 0.50, 

respectively. These data indicate that incidences of crashes were reduced by 3 to 15 percent for 

rural plus animal and 5 to 10 percent for rural plus pedestrian. The data also reveals that FCW 

enhances driver responses over a range of velocities, strongly suggesting that FCW can enhance 

driver responses in scenarios that may lead to collisions.

This study shows that FCW systems have enormous benefits, even though some unanswered 

questions remain. Specifically, although the warning enhanced driver’s awareness of potential 

conflict in the simulator, warnings might be perceived as a nuisance in an actual driving situation 

when they are too sensitive or too early. Further data collection might be required to assess 

drivers’ responses to nuisance alarms generated by early warning or if dynamic adjustment of the 

warning is feasible. Also, a longer-term follow-up study might be helpful in understanding 

changes in driver attitudes and validating benefits observed in this study. The use of naturalistic 

studies may provide more insight into driver behavior during routine trips. Data from such 

potential studies can help validate the benefits of FCW and assist researchers in more completely 

understanding driver attitudes toward FCW systems.
107 

CHAPTER 7. SUMMARY AND CONCLUSIONS

This study created a system for generating RAD that can be used for safety analysis.

First, the team developed two independent RAD frameworks that can be adopted by other safety 

researchers interested in generating RAD for crashes of interest.

The macroscopic framework randomly generates a dataset of segments or intersections for a 

specified facility type (e.g., urban or rural two-lane, four-lane segment, or three-leg or four-leg 

signal or stop-controlled intersection), length of roadway or number of intersections, and number 

of years of crashes. The data are generated in two parts. First, physical and traffic characteristics 

of the segment or intersection are randomly generated according to distributions and sequential 

patterns (in the case of segments) observed in real datasets. Second, crashes are generated 

randomly using SPFs synthesized from distributions of parameter and CMF values found in a 

literature search of crash prediction models. Randomness is added to the resulting crash counts 

using a random mixture of several probability distributions to avoid producing a dataset with an 

easily discoverable common probability distribution. 

The microscopic framework takes an alternative approach, generating vehicle trips for a 

specified region, each with an origin, destination, travel route, purpose, and time of day, and then 

the likelihood of each trip resulting in a crash, along with where on the route the crash occurs, is 

used to generate crashes. A rich trip dataset available from Argonne National Lab and the 

SHRP2 NDS was used to develop the model for predicting the likelihood of any trip resulting in 

a crash (Auld et al. 2016; Virginia Tech Transportation Institute 2020). The result is a database 

of crashes generated in a manner that attempts to replicate how crashes occur.

Second, the team developed an open-source software application that can be used to generate 

RAD for different combinations of inputs, including facility type, period of time, and geographic 

extent. A user desiring to generate a dataset can choose to use either the macroscopic or the 

microscopic approach. For the macroscopic approach, a user-friendly interface permits 

specification of facility type, miles of roadway segments or number of intersections, number of 

years of crashes, and a random number seed. The tool then generates a roadway file containing 

the roadway characteristics for each segment or intersection and total crash counts by injury 

severity and a crash file containing detailed crash severity and crash type information. For the 

microscopic approach, the interface has fewer parameters, requesting only the number of years 

of crashes. This tool generates annual crash data for the study region, providing crash, person, 

and vehicle files.

Third, the team conducted two case studies to demonstrate the feasibility and applicability of the 

software. They generated multiple datasets of various sizes using the resulting RAD tool for rural 

two-lane highways and urban four-leg signalized intersections. SPFs were estimated using 

statistical estimation software and then used to predict crashes on a different one of the RAD 

datasets. A variety of distributional assumptions and modeling approaches were considered for 

estimating the SPFs. The SPFs included a full complement of predictor variables. All variables 

used for generating the RAD were considered in each SPF. Statistical tests were used to compare 

the estimated parameter values from one model to another to evaluate dataset consistency. 
108 

Researchers found that, in general, the parameter values do not vary significantly from one 

dataset to another, even from the smallest to the largest datasets or among the modeling 

approaches. The resulting SPFs also demonstrated predictive transferability from one dataset to 

another. 

Using driving simulator studies, the team explored the potential safety benefits of FCW systems. 

Participants were recruited to drive in one of four simulated scenarios in urban and rural areas. 

During the scenarios, an unexpected emergency stop was required due to either a pedestrian, 

animal, or other vehicle suddenly entering the vehicle’s path. Half the drivers in each scenario 

were given an FCW system, which gave them an audible and/or visible warning about the 

hazard. Participants with the FCW system had a much higher TA than those without, and thus a 

much-reduced rate of serious conflicts and collisions with the hazard. The resulting reduced risk 

of conflict was used to estimate CMFs overall and for each scenario for a range of possible 

market penetration rates for vehicles with FCW systems.

In conclusion, the resulting RAD generation tool was proven to be reliable for generating 

datasets for testing new crash prediction approaches. The tool provides flexibility for generating 

datasets of any size desired, including many road facility types and crashes of different types and 

severity. The microscopic tool includes variables describing demographic and vehicle 

characteristics that can be used to investigate approaches for crash severity modeling. The 

macroscopic tool can also be used to generate crashes over an extended period of time, which is 

typically not possible with real crash data, in that road characteristics cannot be assumed to be 

constant. Having this tool available will permit further investigation of approaches for predicting 

crashes in scenarios where crashes are much too scarce to estimate models.
109 

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