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This is to certify that the

dissertation entitled

EVALUATION OF SAFETY AT FREEWAY INTERCHANGES

presented by

Nakmoon Sung

has been accepted towards fulfillment
of the requirements for

Ph.D. Civil Engineering

degree in

 

 

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11m Wm.“

EVALUATION OF SAFETY AT FREEWAY INTERCHANGES
By

Nakmoon Sung

A DISSERTATION
Submitted to
Michigan State University
In partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY

Department of Civil and Environmental Engineering

2000

ABSTRACT
EVALUATION OF SAFETY AT FREEWAY INTERCHANGES
By

Nakmoon Sung

This research focused on several issues that arise when the Negative Binomial
distribution rather than the Poisson distribution(which have been the commonly accepted

assmnption in analyzing traffic accidents), is found to better fit the accident data.

On the basis of the Negative Binomial distribution, the framework of the rate
quality control method was redefined as a basis for the identification of hazardous sites.
This produced conceptually more reasonable results than the existing approaches such as

the Poisson distribution based rate quality control method, or the Bayes approach.

However, it is sometimes not efficient for traffic engineers to apply this approach
since the parameters of the Negative Binomial distribution can not be easily estimated.
Therefore, a Normal approximation method to overcome this issue was developed. The
Normal approximation method identified the same hazardous sites from a list of two

common interchange types found on several freeways in Michigan.

Although the rate quality control method based on the Negative Binomial
distribution is an effective technique for the identification of hazardous sites, it has two

limitations. First, the selection of reference sites is a matter of judgement.

Second, a sufficient number of reference sites with similar characteristics are not always
available to assure statistical accuracy. As an alternative, a prediction model method was
developed. This method produced results similar to those from the rate quality control
method. By using the prediction model method, the conceptual and practical problems
associated with the identification of hazardous sites can be reduced. The Generalized

Linear Model concept was used to calibrate the accident prediction models.

To Haein and J inmo

ACKNOWLEDGEMENTS

I would like to express my sincere appreciation to my major professor, Dr. William C.

Taylor for his invaluable wisdom and kindness during the journey of this research.

I would like to express my gratitude to the other members of the committee, Dr.
Thomas L. Maleck, Dr. Virginia P. Sisiopiku and Dr. Vincent Melfi for their suggestions

and interest in this work.

A particular note of thanks is extended to Dr. Robert Maki and Mr. Robert Rios
of the Michigan Department of Transportation for helping me use several databases, and
Mr. Oleg V. Makhnin, Mr. Linyuan Li and Mr. Jan Hannig of the Statistical Consultant
Service(SCS) of the Department of Statistics and Probability for advice in understanding

various statistical concepts.

Finally, I would like to express my thanks to my wife, Byungcheon, for her love

and patience, and my parents for their encouragement through the years.

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................. ix
LIST OF FIGURES ........................................................................................................... xi
CHAPTER 1
INTRODUCTION ............................................................................................................... l
1.1 Background and problem identification ......................................................................... 1
1.2 Proposed research objectives ......................................................................................... 4
1.3 Structure of this dissertation .......................................................................................... 5
CHAPTER 2
THE PROBABILITY DISTRIBUTION OF TRAFFIC CRASHES AT FREEWAY
INTERCHANGES ............................................................................................................... 6
2.1 General ........................................................................................................................... 6
2.2 Concept of the Poisson distribution and Negative Binomial distribution ...................... 8
2.3 Phenomena of over-dispersion over time ...................................................................... 9
2.3.1 Analysis by crash types ......................................................................................... 9
2.3.2 Analysis by annual crash frequency per interchange
(Diamond interchanges) ....................................................................................... 13
2.3.3 The results ........................................................................................................... 16
CHAPTER 3
A TRAFFIC CRASH PREDICTION MODEL FOR FREEWAY INTERCHANGES17
3.1 General ......................................................................................................................... 17
3.2 Dependent variable description .................................................................................... 20
3.2.1 Classification by interchange type ...................................................................... 21
3.2.2 Crash data summary ............................................................................................ 25
3.3 Independent variable description ................................................................................. 33
3.4 Preliminary analyses of variables ................................................................................ 36
3.4.1 Correlation analysis ............................................................................................ 36
3.4.2 Analysis of variance (ANOVA) .......................................................................... 38
3.5 Model structure ............................................................................................................ 42
3.6 Model calibration and analysis .................................................................................... 49
3.6.1 Link functions for the Generalized linear model (GLIM) approach ................... 50
3.6.2 Assessing the goodness of fit of the model ......................................................... 52
3.6.3 The procedures used in parameter calibration .................................................... 55
3.6.4 Results of the model calibration ......................................................................... 58
3.6.5 Pearson Residuals ............................................................................................... 61
3.7 A comparison of model calibration results according to the error structure
(Normal versus Negative Binomial assumption) ......................................................... 63
3.8 Sensitivity analysis of the accident prediction model .................................................. 65

vi

CHAPTER 4
THE DEVELOPMENT OF A METHOD TO IDENTIFY HAZARDOUS SITES

BASED ON THE NEGATIVE BINOMIAL DISTRIBUTION ........................................ 68
4.1 General ......................................................................................................................... 68
4.2 A review of the statistical methods for identifying hazardous sites ............................ 69
4.2.1 The rate quality control method .......................................................................... 69
4.2.2 The Bayes approach ............................................................................................ 74
4.2.3 The problems resulting from the use of the Poisson distribution ....................... 77
4.3 The identification of hazardous sites based on the Negative Binomial distribution....79
4.3.1 Concept ............................................................................................................... 79
4.3.2 Estimation of the parameters .............................................................................. 81
4.3.3 Application and validation of the Negative Binomial approach ......................... 87
CHAPTER 5
A SIMPLIFIED APPROACH FOR OVER-DISPERSION
(NORMAL APPROXIMATION METHOD) ................................................................... 93
5.1 General ......................................................................................................................... 93
5.2 Concept ........................................................................................................................ 93
5.3 Application and validation of the Normal approximation method .............................. 95
5.3.1 Estimation of parameters .................................................................................... 96
5.3.2 Validation of the Normal approximation approach ............................................ 96
5.4 An examination of the assumptions ........................................................................... 100
5.4.1 Goodness of fit of the Normal distribution ....................................................... 100
5.4.2 Large values of u, ............................................................................................. 101
CHAPTER 6
IDENTIFICATION OF HAZARDOUS SITES USING A TRAFFIC CRASH
PREDICTION MODEL (PREDICTION MODEL METHOD) ...................................... 104
6.1 The limitation of the rate quality control method
(or upper control limit method) .................................................................................. 104
6.2 The concept of the prediction model method ............................................................ 107
6.3 Application and validation of the prediction model method ..................................... 111
6.3.1 Illustration of the prediction model method ...................................................... l 1 1
6.3.2 Validation of the prediction model method ...................................................... 1 14
6.4 Evaluation of Michigan freeway interchanges on the basis of the prediction
model ......................................................................................................................... 1 19
CHAPTER 7
SUMMARY AND CONCLUSIONS .............................................................................. 121
7.1 Summary .................................................................................................................... 121
7.1.1 Traffic crash distribution ................................................................................... 121
7.1.2 Traffic crash prediction model .......................................................................... 122
7.1.3 The rate quality control method based on the Negative Binomial
distribution ........................................................................................................ 123
7.1.4 The Normal approximation method .................................................................. 124
7.1.5 The prediction model method ........................................................................... 125

vii

7.2 Conclusions ................................................................................................................ 126

BIBLIOGRAPHY ............................................................................................................ 127
APPENDICES ................................................................................................................. 133
A. The results of accident prediction model calibration .......................................... 134
B. The results of evaluating of freeway interchanges .............................................. 145

viii

LIST OF TABLES

Table 2.1 The correlation and residual values according to the distribution ..................... 16
Table 3.1 Interchange classification .................................................................................. 24
Table 3.2 Summary of crashes per interchange (1996 ~ 1998) ......................................... 25
Table 3.3 Injury percent by interchange type (1996 ~ 1998) ............................................. 29
Table 3.4 Summary of mainline and ramp crashes (1996 ~ 1998) .................................... 31
Table 3.5 Summary of the crash types ( 1996 ~ 1998) ....................................................... 32
Table 3.6 Classification of independent variables ............................................................. 35
Table 3.7 Correlations between independent variables ..................................................... 37
Table 3.8 ANOVA for traffic efi‘ects ................................................................................. 39
Table 3.9 ANOVA for geometric effects ........................................................................... 41
Table 3.10 Cross tabulation of crashes by mainline volume and ramp volume ................ 44
Table 3.11 The results of crash prediction model calibration (Interchange type 11) ........ 60
Table 3.12 A comparison of model calibration results according to error structure
(Normal and Negative Binomial assumption) .................................................. 64
Table 3.13 Sensitivity analysis (effect of main geometric variables) ................................ 67
Table 4.1 The estimation of parameters ............................................................................. 87

Table 4.2 A comparison of hazardous sites according to the methods
(Diamond interchanges) ..................................................................................... 91

Table 4.3 A comparison of hazardous sites according to the methods

(Par Clo A or B 4 Q Interchanges) .................................................................... 92
Table 5.1 Estimation of parameters ................................................................................... 96
Table 5.2 A comparison with the results using the Negative Binomial distribution ......... 98
Table 6.1 A comparison of results ................................................................................... 117

ix

Table 6.2 The abnormal sites by the prediction model method ....................................... 120

Table A.1 The results of accident prediction model calibration
(Interchange type 11) ....................................................................................... 135

Table A2 The results of accident prediction model calibration
(Interchange type 12) ....................................................................................... 136

Table A.3 The results of accident prediction model calibration
(Interchange type 13) ....................................................................................... 137

Table A.4 The results of accident prediction model calibration
(Interchange type 14) ....................................................................................... 138

Table A5 The results of accident prediction model calibration
(Interchange type 21) ....................................................................................... 139

Table A6 The results of accident prediction model calibration
(Interchange type 31) ....................................................................................... 140

Table A.7 The results of accident prediction model calibration
(Interchange type 33) ....................................................................................... 141

Table A.8 The results of accident prediction model calibration
(Interchange type 35) ....................................................................................... 142

Table A9 The results of accident prediction model calibration
(Interchange type 41) ....................................................................................... 143

Table A.10 The results of accident prediction model calibration
(Interchange type 51) ....................................................................................... 144

Table B] The results of evaluating of freeway interchanges
by the prediction model method ..................................................................... 146

LIST OF FIGURES

Figure 2.1 Observed and theory variances by crash types (Poisson distribution) ............. l I

Figure 2.2 Observed and theory variances by crash types
(Negative Binomial distribution) ...................................................................... 12

Figure 2.3 Observed and theory variances of annual crash frequency
(Poisson distribution) ........................................................................................ 14

Figure 2.4 Observed and theory variances by annual crash frequency

(Negative Binomial distribution) ...................................................................... 15
Figure 3.1 Boundary of the interchange ............................................................................ 23
Figure 3.2 Total crashes and injury crash percent ............................................................. 28
Figure 3.3 Mainline traffic volume and crashes ................................................................ 45
Figure 3.4 Ramp traffic volume and crashes ..................................................................... 46
Figure 3.5 The process to calibrate coefficients & k parameter ........................................ 57
Figure 3.6 Pearson Residuals and E(x) .............................................................................. 62
Figure 4.1 The concept of rate quality control method under the Poisson distribution ..... 73
Figure 4.2 Identification of hazardous sites by Bayes approach ....................................... 76

Figure 4.3 A comparison of the Negative Binomial and Poisson assumption in applying
rate quality control method .............................................................................. 78

Figure 5.1 A comparison of the results of the Negative Binomial and Normal

approximation approaches ............................................................................... 99
Figure 5.2 The difference between the true and approximate upper control limits ......... 103
Figure 6.1 Basic geometric classification tree for reference sites ................................... 106
Figure 6.2 Concept of prediction model method ............................................................. 108
Figure 6.3 An example of application of prediction model method (95 %) .................... l 13

xi

Figure 6.4 A comparison of results of the rate quality control method and prediction
model method ................................................................................................. 1 18

Chapter 1

INTRODUCTION

1.1 Background and problem identification

In response to limited budgets, it has become very important to ensure that funding
available for road improvements is efficiently utilized. A typical safety program includes
identification, diagnosis, and remediation of hazardous locations, and hence the success
of the safety program can be enhanced by efficiently identifying hazardous locations.

A hazardous location is defined as a site where the observed number of crashes is larger
than a specific norm (a record of crashes at locations with similar characteristics). That is,
a site is deemed hazardous if its crash history over a given period exceeds a
predetermined level which is based on the concept of confidence levels within the context

of classical statistics (Witkowski 1988).

The observed number of crashes over a specific period at a specific site can
usually be obtained from a database related to traffic crashes. However, several
difficulties arise in determining a base for comparing this number to an expected number
of crashes at reference sites that are defined as sites with similar geometric and traffic
characteristics. Hauer (92) recognized that the identification of hazardous sites using
reference sites causes conceptual and practical problem in nature. The main conceptual
problem is that of choosing suitable reference sites, which is a matter of judgement. The

practical problem is that if very similar sites are chosen to reduce the variations caused

fiom the conceptual difficulties, the number of reference sites will usually be too small to
allow for an accurate estimate of the hazard at a given site. These same questions were
also raised by Mahalel (1982), Hauer and Persaud (1987), and Mountain and Fawaz

(1989).

There are 397 interchanges along the four main Interstates (I-69, I-75, I-94 and

I-96) in Michigan. In order to define reference sites for the evaluation of a given
interchange in Michigan, the interchanges were first classified according to their
geometry; such as interchange type, the number of ramps, shoulder width, the number of
lanes, ramp length et al., and second according to traffic conditions. However, with this
level of stratification, it was not possible to obtain enough reference sites to guarantee a
significant level of accuracy for each type of interchange. To overcome these difficulties,
a method using a crash prediction model to identify hazardous sites was examined in this

study.

The basic concept of the prediction model method is that the expected value of
crashes at the reference sites E(6) can be obtained by developing a crash prediction
model rather than on the basis of reference sites. A specific site is deemed to be
hazardous if the probability of the number of observed crashes occurring at the site is
smaller than some predetermined values (i.e.,0.05). That is, a hazardous location is one in
which the deviation from the expected crash frequency E(G) is large. The prediction

model method is a technique to identify hazardous sites, based on an expected value

which is calculated by accident prediction models. Thus, if this method is to be accurate,

it is important to develop the traffic crash prediction models under appropriate rationale.

There are generally two kinds of crash prediction models which differ according
to the assumption of the error structures. One is the conventional linear regression model
with a constant normal error structure, the other is a regression model with a non normal
and heterogeneous error structure (i.e., Poisson and Negative Binomial distribution). In
this research, we have examined the error structures of crash occurrences in various
respects on the basis of the observed data, and verified that crashes on freeway
interchanges follow the Negative Binomial distribution rather than a Normal or Poisson
distribution. Accordingly, the model parameters should be calibrated under the

assumption of the Negative Binomial error structure.

The classical rate quality control method has been used by many transportation
agencies to identify hazardous sites since it was first proposed in the transportation field
in 1956 (Stokes and Mutabazi 1996). This method uses a statistical test to determine
whether the crash rate of a site is abnormally high, compared with that of reference sites.
Therefore, if the crashes follow the Negative Binomial distribution, the rate quality
control method should be reexamined because it is based on the assumption that the
probability of traffic crash occurrences can be approximated by the Poisson distribution

(Norden et a1. 1956, Morin 1967, and Stokes and Mutabazi 1996).

1.2 Proposed research objectives

The four major objectives of this research are:

1) to verify that the fieeway traffic crashes follow the Negative Binomial distribution
rather than the Poisson distribution,

2) to develop crash prediction models for freeway interchanges using the Negative
Binomial distribution,

3) to provide a new framework for the rate quality control method for identifying
hazardous sites on the basis of the Negative Binomial distribution, and

4) to propose a method for the identification of hazardous sites using a traffic crash

model calibrated on the basis of the Negative Binomial distribution.

Even though there are several objectives for this research, each is based on the
assumption that the error structure follows a Negative Binomial distribution. First, this
research will describe how traffic engineers can apply the rate quality control method
based on the Negative Binomial distribution. However, there are interchanges where this
method can not be applied because an insufficient number of reference sites are available
to allow for an accurate evaluation. To solve this kind of problem, a method for
identifying hazardous sites using a crash prediction model is proposed. The prediction
model method can be used to evaluate a freeway interchange without reference sites, and

to determine the sites in need of remedial actions.

1.3 Structure of this dissertation

The background and objective of this research have been discussed in the first chapter.
The issues related to the distribution of crash occurrences are analyzed in chapter 2.

In chapter 3, the effort is focused on parameter calibration of the crash prediction models
for freeway interchanges based on the Negative Binomial error structure. This chapter
includes the description of independent variables, such as traffic and geometric features,
the model structures, methods to converge nonlinear regression models, and measures of
model accuracy. In addition, sensitivity analyses of the models is discussed in this

chapter.

Chapter 4 presents the problem resulting from applying the Poisson error
assumption in the existing rate quality control method, and develops a new framework for
the rate quality control method on the basis of the Negative Binomial error assumption.
Chapter 5 focuses on how this rate quality control method based on the Negative
Binomial distribution can be simplified through a Normal approximation for the purpose
of user convenience. This chapter also demonstrates that the Normal approximation

method produces the same results as the proposed rate quality control method.

Chapter 6 describes how the prediction model method can be used as an
alternative for the identification of hazardous sites when the number of reference sites is
insufficient to allow for significant results. Based on the prediction model method, about
200 interchanges along Michigan freeways are evaluated. A summary and conclusions

occupy the last chapter of this dissertation.

Chapter 2

THE PROBABILITY DISTRIBUTION OF TRAFFIC CRASHES AT FREEWAY

IN TERCHANGES

2.1 General

The most appropriate distribution of crash occurrences is a fundamental question that
ofien arises in the traffic safety field. For example, the Poisson distribution frequently
appears in articles identifying hazardous locations using control limit charts, because of
its simplicity caused from the assumption that the variance is the same as the mean
(Norden et al 1956, Hauer 1996). It has also been recognized that the Poisson distribution
provides a better fit to traffic crash data than the Normal distribution (Miaou et al 1992,

Jovanis and Chang 1993).

However, in studying the injury severity to the front seat occupants of vehicles in
crashes, Hutchinson and Mayne (1977) realized that there appeared to be more variability
of different severity levels occurring in different years than would be expected on the
hypothesis of the Poisson distribution. When there is greater variability than expected by
Poisson' law, we call this phenomenon over-dispersion. Issues related to this over-
dispersion are also implicit in the works of earlier researchers (Benneson and McCoy

1997, Vogt and Bared 1999).

Consequently, two distributions (Poisson and Negative Binomial) have been
assumed for traffic crash occurrences. However, no researcher has yet provided a full
discussion of the issue, even though the assumption of the probability distribution for
crash occurrence is very important for the identification of hazardous sites necessary for

highway safety programs and for the calibration of crash prediction models.

For example, with the rate quality control method, a site is identified as
hazardous if its observed crash rate exceeds the upper control limit, which is the mean
crash rate of reference sites plus a multiple of the standard deviation of the site crash rates
(Stokes and Mutabazi 1996). Herein, the standard deviation is equal to the square root of
the mean for a Poisson distribution and the square root of the (mean + mean 2 /k) for the

Negative Binomial distribution, respectively (Rice 1997).

Three distributions have generally been assumed for the calibration of traffic
crash prediction models (i.e., constant normal, Poisson and Negative Binomial).
However, recently there is an implicit agreement between traffic engineers that the
Poisson or Negative Binomial distributions are more desirable assumptions than the
constant normal distribution. Crash prediction models with a heterogeneous error
structure such as the Poisson or Negative Binomial distribution, are generally calibrated
using weighted least squares (Seber and Wild 1989). In weighted least square regression,
data points are weighted by the reciprocal of their variances. Thus, in calibrating traffic
crash models, the assumption of error structures is a very critical issue in determining the

accuracy of coefficients. Because of the importance of the distribution of crash

occurrences, the year to year variability in the number of crashes is examined and

discussed in this chapter.

2.2 Concept of the Poisson distribution and Negative Binomial distribution

The Poisson distribution is often the first option considered for random counts; it has the
property that the mean of the distribution is equal to the variance (Rice 1997) and the

following fiequency function:

 

X
p(X = x) = exM-m'xm) (2.1)
x.
where,
m = mean

However, when the variance of the counts is substantially larger than the mean,
consideration is given to the Negative Binomial distribution, which is a discrete

distribution with the following frequency function (Rice 1997):

 

_k x
_ 1r F(k +x) m
f(x/m,k)—[l+ k] x!F(k) (”Hka (2.2)
where,
m = mean

k 2 negative binomial parameter

2.3 Phenomena of over-dispersion over time

In examing the freeway interchange crash data over time, there appeared to be more
variability than would be expected under the hypothesis of the Poisson distribution. The
large variability could be expected because there are many factors to cause the annual
crash frequency to vary, including maintenance activities, the weather and traffic

changes.

The Negative Binomial distribution might be considered as a model for the
situation in which the rate varies over time or space(Rice 1997). The Negative Binomial
distribution has been assumed to explain various physical phenomena; the distribution of
insect counts if the insect hatch from the depositions of larvae(Rice 1997). Thus, it is not

unique to apply the Negative Binomial distribution in analyzing discrete random counts.

Two kinds of data sets are utilized to test the over-dispersion. One is the number
of crashes classified by type, the other is the number of crashes per interchange per year
across 84 interchanges. Analyses of the over-dispersion were performed for the crashes

during the 5 year-period 1994-1998.

2.3.1 Analysis by crash types

To test over-dispersion of the crashes which occurred in freeway interchanges, crash
frequencies of each of 24 types of crashes were obtained separately for each of 5 years
from 1994 to 1998. The variance and the mean annual number of crashes were calculated

on the basis of the crashes that occurred over the 5 years.

To test whether the crash occurrences follow the Poisson distribution, the
observed variances of the annual number of crashes were plotted against the annual mean
value. Therefore, there are 24 points corresponding to the 24 types of crashes. In Figure
2.1, the solid line is the variance that would be expected on the hypothesis of the Poisson
distribution. Ifthe Poisson distribution is a good fit, the observed variances should lie
along the solid line. However, the figure shows that there is larger variability than would

be expected under the Poisson distribution.

There is a much larger variability in the most common types of crashes (rear end,
sideswipe) than for the less common types of crashes (backing, fixed object). This

phenomenon was discussed in previous research (Hutchinson and Mayne 1977).

Noting that the Negative Binomial distribution is an alternative to reflect the
phenomenon of over-dispersion, the maximum likelihood estimate of k was determined
to be about 71 by fitting the data to the Negative Binomial distribution. In Figure 2.2, the
solid line is the variance that would be expected on the hypothesis of the Negative
Binomial distribution. This figure shows that the Negative Binomial distribution fits the

data much better than the Poisson distribution shown in Figure 2.1.

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2.3.2 Analysis by annual crash frequency per interchange (Diamond interchanges)
To see how widely this relationship applies, a similar approach was used for Diamond

interchanges, which is the most common type of freeway interchange in Michigan.

The variance and the mean annual number of crashes were calculated from the
total number of crashes that occurred on the same 84 interchanges from 1994 through
1998. The observed variances in the annual numbers of crashes were also plotted against
the mean annual numbers, with a data point corresponding to each of the 84 interchanges.
In Figure 2.3, the solid line is the variance that would be expected on the hypothesis of
the Poisson distribution, and we see that there is also greater variability than expected by
the Poisson distribution, as in the previous case. When the data were fit to the Negative
Binomial distribution, it was found that the maximum likelihood estimate for k is about
21. Figure 2.4 shows that the Negative Binomial distribution fits the data much better

than the Poisson distribution.

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2.3.3 The results

For theoretical support of these results, correlation coefficients and squared residuals
were calculated for the data in Figure 2.1 through Figure 2.4. As shown in Table 2.1,
the correlation coefficients between the observed and the expected variances increased
from 0.91 to 0.97 and from 0.84 to 0.90 in the analysis of 24 crash types and annual total
crashes, respectively, when the Negative Binomial distribution was assumed. Squared
residuals were calculated using the observed variances and expected variances. The
residuals were reduced by more than 80 % when the Negative Binomial distribution was

assumed as shown in Table 2.1.

Thus, we can conclude that the Negative Binomial distribution is a more

reasonable assumption for the distribution of freeway interchange crashes than the

Poisson distribution.

Table 2.1 The correlation and residual values according to the distribution

 

 

 

Poisson Negative Binomial
Correlation coefficient Correlation coefficient Squared
Residual
Accident type 0.91 0.97 87%U
Annual crash
frequency 0.84 0.90 84%U

 

 

 

 

 

 

l6

Chapter 3

A TRAFFIC CRASH PREDICTION MODEL FOR FREEWAY INTERCHANGES

3.1 General

There have been several studies which purpose was to develop crash prediction models
using the relationship between traffic crashes and various independent variables.

In all such studies, the first issue is selection of the independent variables. Using
characteristics of a county, Maleck (1980) and Tarko et a1 (1996) developed models for
predicting the expected annual crashes for a county. Independent variables in these
models consist of a subset of the following factors: the number of licensed drivers, the
number of registered vehicles, population, median family income, road mileage, and

percentage of state roads over all ones.

Mcguigan (1981), Maher and Summersgill (1996), Persaud and Nguyen (1998),
Rodriguez and Sayed (1999), Bonneson and McCoy (1997), Lau and May (1988), and
Belanger (1994) developed crash prediction models for signalized or unsignalized
intersections. These models include one or more of the following independent variables;
major road traffic volume, minor road traffic volume, pedestrian volume and
channelization on the main road. The main road traffic and minor road traffic have been

found to be the most significant variables.

17

Hauer and Griffith (1994), Vogt and Bared (1999), Seder and Livneh (1981), and
Moutain et a1 (1996) developed crash prediction models for road sections using only the
traffic volume. In addition, Hauer and Persaud (1987) used traffic volume and train
volume for crash models of rail-highway grade crossings, and Miaou et a1 (1992)
modeled truck crashes using geometric characteristics and truck ADT. A few researchers
modeled the effects of independent variables on traffic crashes on freeways. Kim (1989)
used interchange types, traffic volume, population and the number of ramps to develop a
crash prediction model for freeway interchanges. All of these models would be classified
as macroscopic models because they use average daily traffic (ADT), rather than the

traffic volume at the time of the crash.

Persaud and Dzbik (1993) developed a microscopic model to estimate crashes on
freeway sections. Microscopic models relate crash occurrences to the specific flow at the
time of the crash rather than to the average daily traffic (ADT). Hence a freeway with
intense flow during rush hour periods would have a higher crash potential than a freeway

with the same ADT, but with flow more evenly distributed during the day.
As noted above, traffic volume is considered the main contributing factor in
predicting traffic crashes in most of the models, with additional geometric variables

chosen based on the objective of modeling.

The second issue in the development of an accident prediction model is how to

calibrate the model parameters, which usually depend on the error structure. There are

18

two approaches that are often used when calibrating model parameters. One is a
conventional linear regression approach, with its assumption of a normally distributed
and homogeneous error structure. The linear regression approach has been recognized to
be lacking the distribution properties to adequately describe the discrete, nonnegative,
and sporadic traflic crash events with a low mean value (Mahalel 1986, Miaou and
Lum1993). Before the Poisson approach was introduced, most models were developed on
the basis of multi linear regression, with the assumption of a normal distribution. For
example, McGuigan (1981), Kim (1989), and Lau and May (1988) used the normal error

structure to calibrate their crash prediction models.

The other approach is the use of a regression model, with a non -normal and
heterogeneous error structure. These include the Poisson, Negative Binomial and Gamma
distributions. It has been generally recognized that crash frequencies better fit a model
using the assumption of a Poisson distribution rather than a Normal distribution. For
example, Miaou et a1. (1992, 1994) proposed the Poisson model to develop the
relationship between truck crashes and geometric design. J ovanis and Chang (1993) also

used the Poisson model to relate crashes to mileage and environmental variables.

However, the Poisson model also has its weakness. For example, the Poisson
model assumes that the variance is the same as the expected number, and hence it can not
reflect the phenomenon of "over-dispersion" which ofien occurs in traffic crashes. In

order to overcome this problem, Persaud and Nguyen (1998), and Rodriguez and Sayed

19

(1999) have proposed regression models with the Negative Binomial error structure to

predict signalized intersection crashes.

The phenomenon of over-dispersion on freeway crashes has been verified and
discussed in chapter 2. In this chapter, a crash prediction model for freeway interchanges

will be developed under the assumption of a Negative Binomial error structure.

3.2 Dependent variable description

The focus on freeway interchange crashes requires a working definition of the boundary
of an interchange. In this study, the interchange is composed of ramps and mainlines. The
ramps include on- ramps and off-ramps, and the mainlines are defined as the section
within 500 feet from the beginning of the off- ramp to 500 feet from the end of the on-
ramp as shown in Figure 3.1. This definition is the same as that of the Michigan DOT
interchange inventory file. The crashes on cross roads are not included in this study
because of the practical barrier that traffic volume for the cross road is not available, and
the engineering intuition that the crashes on the cross road may have very different

characteristics (i.e., low severity, high percentage of angle crashes).

May (1964) found that there is little to be gained by using a study period longer
than three years. Subsequently, many previous researchers have used three years of crash
data in developing crash prediction models (Miaou and Lum 1993, Bonneson and Macoy

1993, Persaud and Nguyen 1998). Noting that data older than three years may not reflect

20

the current situations, the number of crashes that occurred in the past 3 years (1996

through 1998) are used as the dependent variable for this study.

The accident rate will not be used as the dependent variable since accurate
volume data for each element of the interchange is not available. The original source of
the crash data is the "Official Michigan Traffic Accident Report' (form UD-lO). The

crash data are summarized in section 3.2.2

3.2.1 Classification by Interchange type

A lack of homogeneity refers to the understanding that different relationships may hold
between variables on the basis of the values of various characteristics (i.e., geometry,
control, traffic, and so on). In many cases, tree structures which are easily understood
and interpreted, are built describing the main factors and interactions between factors
(Lau and May 1988). However, the tree structures can be used only in the case of large
samples, and hence this method may be inadequate in developing crash prediction models

for freeway interchanges, even though it is a conceptually powerful and systematic tool.

In this study, a total of 199 interchanges are grouped into 10 categories as shown
in Table 3.1. We can not classify the interchanges more specifically because of the
limitation of sample sizes, even though the Michigan interchange inspection file includes
22 categories of interchanges. In the approach to grouping interchanges, the independent
variables (i.e., traffic volume, ramp length, et a1) were explicitly excluded from the

features which were used in the classification of interchange types.

21

As shown in Table 3.1, the number of type 11 and type 31 interchanges is relatively

large compared with those of other types.

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23

Table 3.1 Interchange classification

 

 

 

 

 

 

 

 

 

 

 

 

 

CLASSIFICATION INTERCHANGE TYPE SAMPLE
SIZE
Type 11 . Diamond 34
l. DIAMOND
INTERCHANGE Type 12 . Tight Diamond 19
. Modified Tight Diamond
Type 13 . Partial Diamond 24
. Partial Tight Diamond
Type 14 . Split Diamond 14
. Modified Diamond
Type 21 o Trumpet — A
2. T-INTERCHANGES . Trumpet _ B 9
. Partial Clover A
. Partial Clover B
Type 31 . Partial Clover A 4 Quadrant 41
. Partial Clover B 4 Quadrant
3. CLOVER LEAFS
. Partial Clover AB
Type 33 . Partial Clover AB 4 Quadrant 21
. Clover
Type 35 . Clover with CD 8
. Full Directional
. Partial Directional
4' DIRECTIONAL Type 41 . Directional Y 21
. Partial Directional Y
5- OTHERS Type 51 . Others 8
TOTAL
199

 

 

 

 

24

 

3.2.2 Crash data summary

3.2.2.1 Summary of crashes per interchange

The summary statistics describing the crashes that have occurred over 3 years in each
interchange are provided in Table 3.2. As listed in the table, an average of 126 crashes
occurred in each interchange, 28 % of which were injury crashes. The average number of

crashes is highest in Directional interchanges, and lowest in T-interchanges.

Table 3.2 Summary of crashes per interchange (1996~1998)

 

 

 

 

 

 

 

 

 

 

 

 

 

Interchange type Total crashes Injury crashes
(include fatal crashes)
Max Min Average Max Min Average

Type 11 321 24 132 93 6 39

Diam‘md Type 12 492 42 123 156 6 33
Type 13 252 18 120 84 3 33

Type 14 393 24 99 135 3 27

T-interchange Type 21 156 21 75 69 6 24
Type 31 402 33 135 99 6 33

ample“ Type 33 237 24 84 54 3 21
Type 35 405 51 168 138 12 48

Directional Type 41 408 21 186 111 3 54
Others Type 51 408 21 180 45 6 21
Total 492 18 126 156 3 36

 

 

 

 

 

 

 

 

 

 

25

3.2.2.2 Summary of injury data

Figure 3.2 shows the relationship between total crashes and injury crash percentage. As
shown in the figure, the smaller the total number of crashes, the greater the scatter of
injury crash percentage. Therefore, total crashes are a more reliable dependent variable
than injury crashes, because there is always implicit variability in injury crashes. In the
case of the interchanges with a small number of crashes, this variability may

inappropriately model the effects of the independent variables on crashes.

Table 3.3 contains summary statistics of injury crashes that occurred in the past 3
years. It is not surprising that the percent of injury crashes is relatively high for T-
interchanges and Directional interchanges (30.8 % and 29.2 % respectively), considering
that the vehicle operating speeds on these types of interchanges are high compared with

those on other types of interchanges.

The coefficient of variation V(x) is a stable measure of the variability of a random

variable x, which is defined as (Harr 1996):

V(x) = 52—8 x 100 (%)

The higher the coefficient of variation V(x), the greater will be the scatter. As a rule of
thumb, coefficients of variation below 15 % are thought to be low, between 15 and 30 %

moderate, and greater than 30 °/o high (Harr 1996).

26

As shown in the last row of the Table 3.3, the coefficient of variation of injury
percent across the interchange types is 10. 8 %, which is low. This implies that
interchange types are related to the number of crashes, but not the severity of the crashes.
Thus, for this study, the total number of crashes is used as the dependent variable for the

development of traffic crash prediction models.

27

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28

Table 3.3 Injury percent by interchange type (1996~1998)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Interchange type Total crashes Injury crashes Injury (°/o)
Type 11 4479 1272 28.4
Diamm‘d Type 12 2211 600 27.1
Type 13 2886 822 28.5
Type 14 1380 393 28.5
T-interchange Type 21 681 210 30.8
Type 31 5388 1380 25.6
Clmr'leaf Type 33 1779 453 25.5
Type 35 1347 381 28.3
Directional Type 41 4074 1188 29.2
Others Type 51 699 177 25.3
Total 24924 6876 27.6
V(x) - - 10.8

 

 

 

 

 

29

 

3.2.2.3 Summary of mainline and ramp crashes

Table 3.4 presents a statistical summary of mainline and ramp crashes that occurred from
1996 to 1998. Ramp accidents are about 4300 of the total 25000 crashes, or about 17 %.
There is a large variability in the percent of ramp crashes across the interchange types, as
shown in the table. That is, the coefficient of variation is 344 %, which is extremely high.
This implies that different explanatory variables are needed when developing crash

prediction models by interchange type.

Table 3.5 presents data on the crash type according to the interchange type. In
our sample sites, rear end crashes account for 39.7 % of total crashes. Rear end crashes
are especially high in Type 11(Diamond) and Type 35 (Clover leaf) interchanges, and
low in Type 33(Partial Clover AB or Partial Clover AB 4 Q). Fixed object and sideswipe
crashes are 20.9 % and 14.1 %, respectively, as shown in the table. The coefficients of
variance of a special type of crash percent across interchange types range from 53 % to
172 %, which are high. Accordingly, one recognizes that the different types of

interchanges are associated with different types of crashes.

It is very important to analyze crash type by interchange type because the crash
type provides clues for treatment of a hazardous site. For example, if there were a high
percent of sideswipe crashes in an interchange, traffic engineers would analyze in detail
the merge section to find the solution. If there were many rear end crashes at an
interchange, one possibility is that the ramp length is too short to accelerate to freeway

speeds before vehicles enter the mainline. If there are many overturn crashes at an

30

interchange, one possibility is that there may be an imbalance between the radius of the

ramp curve and the exit speed limit onto the ramp. Thus it is valuable to classify crashes

according to the crash type.

Table 3.4 Summary of mainline and ramp crashes (1996~1998)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Interchange type Total Mainline Ramp
crashes Crashes % Crashes %
Type 11 4479 3780 84.4 699 15.6
Diamm‘l Type 12 2211 1872 84.7 339 15.3
Type 13 2886 2634 91.3 252 8.7
Type 14 1380 1329 96.3 51 3.7
T-interchange Type 21 681 486 71.4 195 28.6
Type 31 5388 4323 80.2 1065 19.8
Clover‘leaf Type 33 1779 1470 82.6 309 17.4
Type 35 1347 960 71.3 387 28.7
Directional Type 41 4074 3135 77.0 939 23.0
Others Type 51 699 642 91.8 57 8.2
Total 24924 20631 82.8 4293 17.2
V(X) - - - - 344

 

31

 

Table 3.5 Summary of the crash types (1996~1998)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Total Rear end Fixed obj Sideswipe Others
Interchange type crashes (overturn)
# °/o # % # % # %

Type 1 1 4479 2247 50.2 908 20.3 463 10.3 861 19.2
Diamond .

Type 12 2211 910 41.2 450 20.3 360 16.3 492 22.2

Type 13 2886 963 33.4 615 21.3 467 16.2 842 29.2

Type 14 1380 604 43.7 321 23.3 102 7.4 353 25.6

T- Type 2] 681 205 30.0 184 27.1 88 12.9 205 30.0

interchange

Type 31 5388 2122 39.4 1275 23.7 816 15.1 1175 21.8
Clover'leaf Type 33 1779 400 22.5 434 24.4 374 21.0 571 32.1

Type 35 1347 694) 51.5 234 17.4 130 9.7 289 21.5
Directional Type 41 4074 1527 37.5 625 15.3 590 14.5 1332 32.7
Others Type 51 699 233 33.3 153 21.9 117 16.8 196 28.0
Total 24924 9905 39.7 5199 20.9 3509 14.1 6314 25.3
V(x) 172 53 118 85

 

 

 

 

 

 

 

 

 

 

 

32

 

3.3 Independent variable description

Independent variables used for this study consist of traffic data and geometric data. The
traffic data are:

1) Mainline traffic volume,

2) Ramp traffic volume, and

3) Truck traffic volume and truck percent.

Average daily traffic (ADT) on mainlines of freeways has been shown to be an
important contributing factor in predicting interchange traffic crashes. The Michigan
Department of Transportation (MDOT) maintains about 100 permanent traffic recorders
located at various sites throughout the state. The traffic volume data at these counter

locations are used to estimate the ADT on all highway segments each year.

Ramp ADT is also considered to be an important independent variable for model
development. The ramp ADT are traffic volumes on every on and off ramp (including
loop) within the Ramp Counting program jurisdiction (Detroit Metropolitan area, Flint,
Lansing, Grand Rapids, Jackson, etc). Any missing ramp data is estimated by reviewing
previous years' traffic volumes and adjacent ramps. This adjustment implies an
assumption that if traffic exits a freeway, it will return through the same intersection,

going the opposite way.

33

Truck percent was also included, based on engineering intuition that truck ADT
and mainline ADT, or truck ADT and ramp ADT may have the same mechanistic origin,

which causes multicollinearity in crash prediction models.

Geometric data were obtained from the sufficiency rating file(1994) and freeway
interchange inventory fi1e(1997), which are maintained by the Michigan DOT. Table 3.6
presents all variables that are intuitively thought to effect crash frequency, and are
possible to obtain. An analysis of variance (ANOVA) of all independent variables was
performed to determine which variables have a significant effect on the dependent
variable (i.e., crash frequency). The results of this preliminary analysis are discussed in

detail in section 3.4.

34

Table 3.6 Classification of independent variables

 

Independent variables

 

Variable type 1 Variable type 2

 

Traffic 0 Mainline traffic(ADT)

effects a On ramp traffic(ADT)

o On and Off ramp
traffic(ADT)

- Truck traffic(Truck ADT)

0 Truck percent (°/o)

 

 

Geometric o Interchange length (miles) 0 Number of lanes

effects 0 Average spread - ramp 0 Number of on ramps
length (miles) 0 Total number of ramps
0 Average loop- ramp 0 Shoulder width(feet)
length (miles) 0 Lighting condition

 

 

 

35

 

3.4 Preliminary analyses of variables

3.4.1 Correlation analysis

There is an implicit assumption in statistical model development that the independent
variables are mutually independent. It is generally accepted that multicollinearity exists
when a linear combination of independent variables is highly correlated, and that it is
difficult to identify independent variable effects on the dependent variable (Neter et a1.
1992, Sever and Wild 1989). Therefore, explanatory variables with low collinearity

should be selected in the process of modeling.

To evaluate the mutual independence between variables, a correlation table was
produced. As shown in Table 3.7, some of the independent variables are identified as
relatively highly correlated. For example, the correlation between the ramp traffic volume
and the interchange size, and the correlation between the mainline traffic volume and
shoulder width are 0.454 and - 0.411 respectively. Those are not high enough to be
excluded in the first stage of model developments. However, these variables are carefully

dealt with in the detailed process of modeling.

36

 

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3.4.2 Analysis of variance (ANOVA )

Analysis of variance (ANOVA) techniques are a useful tool for analyzing the statistical
relationship between a dependent variable and independent variables. In fact, these may
be considered as a special case of linear regression. However, ANOVA models allow
analyses of statistical relations from a different perspective than with linear regression,
and therefore are widely used. In this section the ANOVA is used for the preliminary
analyses of the relationship between the independent variables and a dependent variable.
The independent variables are categorized into several groups before the ANOVA
models are applied (i.e., for mainline ADT, 1: under 10000, 2:10000~15000, 3:

15000~20000, 4: over 20000).

The next step is to carry out a test whether or not the category means uj are equal.

The hypothesis for this test is the following (N eter et a1. 1992)

H03 “1:112:19 ------- 21%

H1 : Not all llj are equal

Here, H0 implies that all of the probability distributions have the same mean, and thus
there are no factor effects. Alternative H1 implies that the means are not equal, and hence
that there are factor effects. The F - test statistic and p-value are used as a decision rule for

this test, and statistical package SPSS (9.0 version) is used to investigate the ANOVA.

38

3.4.2.1 ANOVA for traffic effects

.When a: 0.05, F(0.95; 3, 195) is equal to 2.65. For mainline ADT from Table 3.8, the
F- test statisitic=l 7.578>2.65. Thus we conclude Hl- that the mean crash frequency is not
the same for the different mainline ADT categories. Similarly, ANOVA of ramp ADT
and truck percent result in the same interpretion as that of mainline ADT. However, for
truck ADT, the F-test statistic 0.244 is less than the critical value of 3.04, and hence we
conclude Ho -that the mean crash frequencies are the same for different truck ADT. The
large p-value of the test in this table provides strong evidence that the sample data are in
accord with equal mean frequencies for the different truck ADT. Mainline ADT, ramp
ADT, and truck percent are thus expected to be contributing factors in the crash

prediction models

Table 3.8 ANOVA for traffic effects

 

 

 

 

 

 

 

 

 

 

Source of variance d.o.f Mean F-test P-value
square
Statistic Critical
Value
(a=0.05)
Mainline ADT Hypothesis 3 12887 17.578 2.65 0.000
Error 195 733
Ramp ADT Hypothesis 2 28635 45.134 3.04 0.000
Error 196 634
Truck ADT Hypothesis 2 225 0.244 3.04 0.784
Error 196 924
Truck percent Hypothesis 2 10434 12.722 3.04 0.000
Error 196 820

 

 

 

 

 

 

 

 

 

39

3.4.2.2 ANOVA for geometric effects

Table 3.9 presents the results of ANOVA for geometric effects. For the variables of
interchange size and average spread ramp length, the F-test statistics are 6.760 and 3.901,
respectively, which exceed the critical value of 3.04. This implies that the mean accidents
are not the same for the different length of interchange, or the different length of spread
ramps. However, for average loop ramp length, the F-test statistic 0.146 is very small,
compared to the critical value of 3.11, and hence we conclude Ho - that the mean crashes
are the same for the different length of loop ramps. The small P-value of the test in this

table provides strong evidence of this conclusion.

On the other hand, the number of lanes and shoulder width are expected to be
important independent variables for the prediction models based on F-test statistics that
exceed critical values at a 0.05. However, for lighting, the F ~test statistic (1.953) is less
than the critical value of 3.04, and hence we can not conclude that mean accident
frequencies are not the same for the different lighting conditions. In addition, the F- test
statistic for the number of on-off ramps is 1.818, which is close to the critical value of

1.93.

Thus, the number of on and off ramps, the number of lanes, shoulder width,
interchange length and average spread ramp length are expected to be contributing
factors. However, there are no factor effects caused by lighting condition and average

loop ramp length, and thus no further analyses which include these variables is required.

40

Table 3.9

AN OVA for geometric effects

 

A. Variable type 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Source of variance d.o.f Mean F -test P-value
square
Statistic Critical
value
(0L=0.05)
Interchange Hypothesis 2 5860 6.760 3.04 0.001
length Error 196 866
Average Hypothesis 2 1 1 5 0. 146 3.11 0.703
10"" ramp Error 83 782
length
Average Hypothesis 2 3 565 3 .901 3 .04 0.021
Spread ”ml” Error 193 9112
length
B. Variable type 2

Number of on Hypothesis 9 1608 1.818 1.93 0.067
and “f “‘me Error 189 884
Number of Hypothesis 4 2206 2.477 2.42 0.046
lanes Error 194 890
Shoulder Hypothesis 1 17458 20.950 3.89 0.000
Width Error 197 833
Lighting Hypothesis 2 l 703 1.953 3.04 0.144

Error 196 872

 

 

 

 

 

 

 

41

 

3.5 Model structure

Model structure is another issue in building an accident prediction model. However it is
very difficult to choose the form of model equations because modeling remains, partly at
least, an art (McCullagh and Nelder 1989). There are, however, some principles related to

model structures which are summarized as follows.

( McCullagh and Nelder 1989):
O A good model is one that fits the observed data very well.

0 Simplicity is a desirable feature of any model; we should not include parameters that

we do not need.
0 Models should make sense intuitively.

0 If main effects are found from several studies bearing on the same phenomenon, the

main effects should usually be included whether significant or not.

The above principles were used in the process of choosing model structures for
this study. There are a few research papers on freeway interchanges, as mentioned in
section 3.1. But these may not be appropriate guides for this study, since the models are
based on a normally distributed and homogeneous error structure. For this reason, the
findings from these studies related to traffic crash estimation at intersections have been
reviewed based on the engineering intuition that the crash patterns at interchanges would

be similar to those at intersections.

Several studies (Maher and Summersgill 1996, Persaud and Nguyen 1998,

Bonneson and McCoy 1997, Vogt and Bared 1999) found that nonlinear relation is

42

mainly proposed, and traffic volume belongs in the main effect group among the various
variables.

To confirm the model structure, the cross tabulation between crash frequency and
traffic volume were produced as shown in Table 3.10. This approach was performed in a
similar manner by Bonneson and McCoy (1993), and Hauer et al.(l988). In Table 3.10,
the traffic ranges were selected such that the same traffic ranges are located in each row,
or each column, in order to obtain equal weight in calculating the average number of
crashes per interchange. Therefore, 52 interchanges with traffic volumes that exceed

these ranges were excluded in building the table.

The cells give the average number of crashes that have occurred for 3 years at
interchanges with mainline volume and ramp volume given in the left-most column and
the upper row. The brief examination of the row and column summaries indicates a
positive relation between crashes and both mainline volume and ramp volume as shown
in Figure 3.3 and Figure 3.4. However, the rate of increase may be different, depending

on the traffic volume.

For example, while crashes are always increasing over all ranges of mainline
ADT, the increase is very small between mainline ADT 10000~15000 and 15000
~20000, compared with other ranges of mainline ADT. This implies that the increase of
crashes with mainline ADT is nonlinear, and the increase can be captured by a function

such as V B, where V is mainline ADT and B is a coefficient larger than 0.0.

43

Table 3.10 Cross tabulation of crashes by mainline volume and ramp volume

 

 

 

 

 

 

 

Ramp volume 5000 5000 5000 5000 Summary
~ ~ ~ ~ Row
Mainline volume - 15000 15000 15000 15000
5000 50“ 88 66 62 66
10000 9039/18” 1233/14 132/2 186/3 2454/37
10000 55 100 108 148 98
15000 721/13 2091/21 1624/15 1038/7 5474/56
15000 57 122 90 133 103
20000 454/8 1095/9 270/3 1065/8 2884/28
20000 116 170 178 175 159
25000 815/7 851/5 1420/8 1049/6 4135/26
Summary 63 108 123 139 102
Column
2893/46 5270/49 3446/28 3338/24 14947/147

 

 

 

 

 

 

1): Average number of crashes per interchange

2): Total crashes

3): The number of interchanges

44

 

8

{3‘

Crashes

8

 

'8‘

 

 

 

 

5000~10000 10000~15000 15000~20000 20000-25000
ADT/lane

Figure 3.3 Mainline traffic volume and crashes

45

 

200

150

Crashes
8

01
O

 

 

/

 

5000~15000 15000~25000 25000~35000 35000~45000
Ramp ADT

Figure 3.4 Ramp traffic volume and crashes

46

 

We can also determine from Table 3.10 that there is a nonlinear relationship
between crash frequency and traffic volume. For example, in the first column, crash
frequencies increase sharply from 57 to 116 when the mainline volumes are changed
from 15000~20000 to 20000 ~25000, whereas the crash frequencies increase only
slightly (from 50 to 57) when the mainline volumes are changed from 5000~10000 to
15000~20000. These combinations can be found in other cells in Table 3.10, which is
conceptually consistent with the nonlinear product of flows to power formulation as

follows:

15(67): AleBl xV232 (3.1)

where,
E(6) : Expected number of crashes

Vl :Mainline volume
V2 :Ramp volume
A,B1,B2 :Parameters

In principle, one should seek a model structure that best fits each interchange
type. However, in this case, the model structure would be based on too small of a sample
size to allow for finesse. Therefore, we regard equation (3.1) as the basic model structure

describing the main effects of traffic variables on the interchange crash frequency.

47

The range of geometric variables is also an issue in choosing the appropriate
model structure. The previous research found that the expected number of crashes can be
represented by a product of geometric variables raised to various powers (Mountain et a1.
1996), or by an exponential applied to a linear fimction of the geometric variables (Vogt

and Bared 1999, Mahel and Summersgill 1996).

The effect of the range of possible geometric variables can not be evaluated
efficiently, and hence, iterative tests of the model structures were performed. The results
showed that a product of variables raised to various powers is appropriate for variables of
type 1 (such as the size of interchanges), whereas an exponential applied to a linear

function is appropriate for variables of type 2 (such as the number of on and off ramps).

On the basis of the literature review, the principles of model structures, and the
results of the analyses, the general model structure for this study was finally determined

to be of the following form:

13(19): AxViBi ijCj xepo(Ck ka) (3.2)

where,
E(19) : Expected number of crashes
Vi :Traffic variables

G j : Geometric variables(type 1)

GR : Geometric variables(type 2)
A,Bi,Cj,Ck :Parameters

48

3.6 Model calibration and analysis

In section 3.4, the results of a preliminary analysis used to determine which variables
have a significant effect on crash occurrences were discussed. The basic model structure
that has been proposed in section 3.5 includes the independent variables that are
significant as a result of the AN OVA. However, a variable can be insignificant when we
put the variable into a nonlinear model structure stratified by interchange type, even
though it has been evaluated to be significant in the preliminary analysis, because the
preliminary ANOVA was performed independent of the interchange type. This issue is

related to the simplicity of the model.

Simplicity is a desirable feature of any model as noted by McCullagh and Nelder
(1989). This means that we should not include insignificant parameters in a model, noting
that not only does a simple model enable the researchers to think about their data, but the
model that involves only the correct variables gives better predictions than one that
includes unnecessary variables. In this stage, the irrelevant terms from the general model
structure are excluded, and the models are calibrated through checks on the fit of a model

to the data, for example by residuals and other statistics.

A nonlinear regression model was proposed in the preceding section, and it has
been verified that the crash occurrences follow a Negative Binomial distribution in
chapter 2. Therefore, it is necessary to calibrate the coefficients of the crash prediction
models and the Negative Binomial distribution parameter k simultaneously. There are

two methods used to calibrate nonlinear regression models with a heterogeneous error

49

structure (such as the Negative Binomial distribution): transformation of the model and

generalized linear models (GLIM).

However, the transformation of models causes a change of scale in the data (Sever
and Wild 1987, and McCullagh and Nelder 1989), which results in a violation of the
Negative Binomial error assumption. Therefore, the analyses that follow are performed
on the original scale of the data. This feature is a characteristic of generalized linear
models (McCullagh and Nelder 1989). Previous researchers have suggested that the
generalized linear models can be a technique to overcome the shortcomings of the
conventional normally distributed error assumption in describing random, discrete and
non-negative events which often occur in the traffic crash field (Rodriguez and Sayed

1999).

3.6.] Link functions for the Generalized Linear Model (GLIM ) approach
Recognizing that traffic crashes follow the Negative Binomial distribution as mentioned
in chapter 2, the GLIM approach is utilized for model calibration. The GLIM approach
used herein is based on the work of McCullagh and Nelder (1989), and Lawless (1987).
The generalized linear modeling technique introduces a link function n that relates the
linear equation to the expected value of an observation. This link function equates the

non- linear relationship to a linear one.

On the other hand, there is a specific link function that is associated with the error

structure of a distribution. This is defined as the natural link function. For example,

50

natural link functions can be described for Normal, Poisson and Negative Binomial

distributions as follows (McCullagh and Nelder 1989):

Normal : 77 = E(6)

Poisson : 77 = ln[E(6)]

15(6) 1

Negative Binomial : 77 =
K + E((9)

In order to describe the use of the Poisson link function, equation (3.2) in section 3.5 can

be changed into a linear predictor as follows:

n=mwwfl
= lniA >< VrBi X jSj xexPZw/r X CH]

=lnA+Bi1nVi+CjGj +Z(Ck ka)

Now, this is a linear predictive equation after applying the Poisson link function.
However, our models for crash occurrence are based on the Negative Binomial
distribution, and it is much harder to calculate a linear predictor from the natural link
function for the Negative Binomial distribution. In fact, it is not algebraically possible to
derive the linear predictor using the natural link function for the Negative Binomial

distribution (Bonneson and Macoy 1997). Therefore, the Poisson link function is utilized

51

instead, recognizing that the use of a natural link function is not a requirement for the

GLIM approach (McCullagh and Nelder 1989).

In order to calibrate the prediction model, a dispersion parameter (Dp) will be

utilized. That is, if Dp is greater than 1.0, then the data has a greater dispersion than is

explained by the Poisson error assumption, and further analysis using the Negative
Binomial error structure is required. In this case, the parameters are estimated in the
iterative process using the maximum likelihood method. The model calibration

procedures are explained in section 3.6.3.

3.6.2 Assessing the goodness of fit of the model
This section describes a basis of measuring the model significance. To make

understanding easier, the following notations are used:

yi : the observed number of crashes at a site i
E(0)i: the expected number of crashes at a site i
E(0): the average expected number of crashes
Var(yi): estimated variance in crashes at a site i

11: sample size

p: the number of parameters

Several measures are used to assess the model fit and the significance of the

model parameters, based on the studies of McCullagh and Nelder (1989), and Bonneson

52

and McCoy (1997). One such measure is the generalized Pearson x2 statistic, which is

calculated as:

" (y- —-E<<9>-)2
P 2 = 1 1
earson I ; var(y,-)

 

where var(yi) is estimated from the variance equation of the Negative Binomial
distribution which has been shown in equation (2.2). McCullagh and Nelder( 1 989)
indicate that the generalized Pearson x2 statistic has the exact x2 distribution for a

Normal linear model, while asymptotic results are available for other distributions. The
asymptotic results may not be relevant to statistics calculated from a small sample size.
Therefore it sometimes can not be used as an absolute measure for assessing the fit of a

model.

A second measure of model fit is the Dispersion parameter (Dp), which can be

calculated as:

Pearson [2
n — P

 

Dispersion parameter(DP) =

As shown in the above formula, Dp can be obtained by dividing the Pearson x2 by n - p.

McCullagh and Nelder(l989) indicated that it is a useful measure for assessing the fit of a

53

model. A Dp value near 1.0 means that the error assumption of the model is equivalent to

that found in observed data. If Dp is greater than 1.0, then the observed data has greater

dispersion than is assumed in the model. This concept will be utilized in estimating the
" k parameter " in the Negative Binomial distribution and the coefficients of the accident

prediction models. This will be described in detail in the following section.

The third measure of model fit is the coefficient of determination R2, which can

be calculated as:

where

SSE = 2110509), —y,.]2

l

SST = ib, - 177(6)] 2

This measure is commonly used for the fit of a linear regression model based on the

normally distributed error assumption. Nevertheless, this statistic can still be useful in
assessing the model fit, recognizing the findings that the coefficient of determination R2

is still efficient in assessing a model calibrated under a non normal error structure

(Kvalseth 1985).

54

The fourth measure of model fit is the Pearson Residual, which can be calculated

as:

5(9); ‘y,’

V var(y,' )

Pearson Residual (PRi) 2

As shown in this formula, this is defined as the difference between the predicted and
observed data divided by the standard deviation. The Pearson Residual will be discussed

again in section 3.6.5.

In addition to these measures, the standard error and t-value are used for assessing
the significance of variable coefficients. The t-value is the ratio between the variable
coefficient and its standard error. The detailed descriptions of these statistics are not
presented here since the concepts are commonly applied in measuring the fit of linear

regression models.

3.6.3 The procedures used in parameter calibration
The calibration of model parameters was performed based on the works of Lawless

( 1987). The calibration for this research is a multi-step process as shown in Figure 3.5.

55

First, the model parameters are estimated based on the Poisson errorstructure that
the variance equals to the expected value. Using the expected number being calculated in
the first step, the second step is to estimate the "k" parameter. If I/k is not greater than
0.0, then there is no over-dispersion in the observed data and the procedure stops. If 1/k is
greater than 0.0, then a third step is to calculate new model coefficients under the
Negative Binomial error structure using the k from the second step. In this step, the

maximum likelihood estimates of the model coefficients are obtained by iterative
weighted least squares. The final step is to calculate the Dispersion parameter (Dp). If Dp
does not equal 1.0, the k parameter is increased (or decreased) and then a feedback loop
is performed to the third step. The analysis is repeated in an iterative manner until the

Dispersion parameter (Dp) converges to 1.0.

Models with Negative Binomial errors can not be calibrated using conventional
statistical packages (i.e., SPSS, SYSTAT), and thus a statistical package for Generalized
Linear Interactive Modeling (GLIM), which is specially designed to calibrate models
with special types of errors (i.e., Negative Binomial, Poisson and Gamma), was used.
Rodriguez and Sayed (1999) used a similar process in calibrating the traffic crash

prediction models for urban unsignalized intersections.

56

I

 

1. Poisson Model Calibration
: Variance = Mean

 

 

 

I

 

2. Residual analysis
2

i (y,- - #i) z

[’1'

k )

i=1 #i (1 +

 

 

 

I

 

Initial model coefficients

:B1.BZ.B3.B4 .......

 

 

 

Initial 1/k

 

 

 

 

NO
.°

 

 

 

YES F

 

3. NB Model Calibration
: Maximum likelihood estimation
by iterative weighted least squares

 

 

New model coefficient

:01, 02, B3, [34

 

 

 

 

I

 

4. Dispersion parameter (Dp)
Calculation

_ pearsonz 2
n ‘ P

Dp

 

 

 

 

 

 

I

Converge ?

(1)1351)

  
 
 

 

  

YES

Figure 3.5 The process to calibrate coefficients & k parameter

 
 

 

 

2
n . -— .
‘ pearsonz2 = Z£J/'—#—;l)—
i=1 i
- 1+ _-
#,( k
Adjusted 1/k

 

57

 

 

 

 

 

 

3.6.4 Results of the model calibration
On the basis of the procedures for assessing the model fit explained in the previous
sections, the crash prediction models have been calibrated. The logarithmic link function

has the following basic form, as mentioned in section 3.6.1:

1n[E(6)] =lnA+Biani+CjGj+Z(Ck ka) (3.3)

This equation can be rewritten in a more useful form as:

,3. CH
E(6)=AxVi szj 1 xepo(Ckak) (3.4)
where,

E (9) : Expected number of crashes
Vi :Traffic variables

G j :Geometric variables

G k :Geometric variables

A,Bi,C j‘C k : Parameters

The model calibration process starts with individual models according to the interchange
types that have been classified in section 3.2. Table 3.11 presents several statistics
relating to the calibrated crash prediction model for interchange type 11. In determining
the significance of the variable coefficients, the 95 percent confidence level is used with a
few exceptions. In the second row of the table, the statistic for the constant terms does not

have any meaning since the logarithm results in a change of scale.

58

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The table indicates that several variables have a significant effect on the
frequency of interchange crashes. These variables are mainline traffic, ramp traffic, truck
percent, interchange size, spread ramp length, and shoulder width. However, the number
of lanes and the number of total ramps are not included in this model because the effect
of these variables is not significant. The calibrated coefficients can be applied to the
equation (3.4) that is the basic model structure, in order to predict the number of traffic
crashes that would be expected for 3 years in interchange type 1 1. The resulting model

can be written as follow:

E(6) = 3.448 V,"401 V20'186 V30'620 010738 exp(—1.267 G2 — 0.156 G5)

where,
Vl : Mainline traffic volume per lane

: Ramp traffic volume

w< N<

: Truck percent

: Interchange length

~

: Average spread - ramp length
: Shoulder width

000
N

LII

A k parameter of 8.05 is found to yield a dispersion parameter of 1.0. The Pearson
x2 is 28.84, and the degrees of freedom are 27(n-p-1=34-6-1). This statistic is less than x2
0 05, 27 = 40.11, and hence the hypothesis that the model fits the data can not be rejected. It
implies that the model is consistent with the observed data.

Several statistics associated with the calibrated crash prediction models for other

interchange types are included as Appendix 1.

59

Table 3.11 The results of crash prediction model calibration

(Interchange type 1

1)

 

 

 

 

 

 

 

 

 

 

 

 

Coefficient Variable definition Unit Estimate Std t -
error statistic
A Constant - 3.448
Log(A) (1.238) (0.67) (1.85)
B1 V1: Mainline traffic volume per lane (ADT/1000) 1-401 0-30 4-66
B2 V2 : Ramp traffic volume (ADT/1000) 0.186 0.12 1.55
0
B3 V3 :1.ka percent (/o) 0.620 0.19 3.26
C] G. : Interchange length (Mile) 0-733 0-15 4-92
C2 G2 : Average spread- ramp length (Mile) '1'267 0'97 '1 '31
C3 G3 : The number of lanes -
C4 G4 : The number of total ramps -
(Feet) -0.156 0.12 -l.30
C5 G5 2 Shoulder Width
Model statistic
[)p Dispersion parameter 1.0
X2 Pearson chi -square 28.84 (12 0.95, 27 = 40.11)
R2 Coefficient of determination 0.60
K Negative Binomial parameter 8.0 5

 

 

 

60

 

3.6.5 Pearson Residuals

A useful subjective measure of the model fit is the Pearson Residuals(PR),which are
normalized residuals in the context that Pearson Residuals are the difference between the
predicted and observed data divided by the standard deviation as described in section
3.6.2. One can visually assess the goodness of model fit by plotting the Pearson Residuals
versus the estimates of the expected number of crashes. A good model will have the

Pearson Residuals centered around 0.0.

Pearson Residuals are plotted against the expected frequency for the 199
interchanges in Figure 3.6. As shown in the figure, Pearson Residuals are centered
around 0.0 for the entire range of expected frequency, which indicates that the calibrated
models fit the observed data well. The advantage of the Negative Binomial error

assumption in crash model development will be examined again in section 3.7.

61

 

 

 

 

 

0 100 200 300 400 500
E(x)

Figure 3.6 Pearson Residuals and E(x)

62

3.7 A comparison of model calibration results according to the error structure
(Normal versus Negative Binomial assumption)

In section 3.1, it was noted that there are two error structures used in calibrating traffic
crash prediction models. One is a normally distributed and homogeneous error structure,
the other is a non- normal and heterogeneous error structure. Recently, Poisson or
Negative Binomial error structures are most often assmned in modeling traffic crashes.
Nevertheless, Iiteratures reviewed did not contain a full description of the advantages
relating to this approach. In order to examine the advantages of the Negative Binomial
error assumption, the results of model calibrations for interchange type 11 and type 12
are compared in Table 3.12. The parameter estimates and their standard errors are very

sensitive to the error structures assumed.

An extremely useful relative measure of the scatter of a random variable is its
coefficient of variation V(x) (Harr 1996). This implies that the coefficient of variation is
a measure of the reliability of the calibrated model coefficients. There were large
reductions in the coefficient of variation, as shown in the table, when the models were
calibrated using the Negative Binomial distribution instead of the Normal distribution.
This reduction in the coefficient of variation occurs in all coefficients that have been
calibrated as shown in the table, with a maximum reduction of 80 % and an average
reduction of 30 %. These results support the hypothesis that the Negative Binomial
distribution is a desirable assumption in calibrating crash prediction models relating to

freeway interchanges.

63

(Normal and Negative Binomial assumption)

Table 3.12 A comparison of model calibration results according to error structure

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Typell Normal Negative Binomial Reduction
of V(x)
Parameter Estimate Std.err V(x) Estimate Std.err V(x)
A B B/Axl 0 C D / x
() () ( 0) () () (DC100) (%)
(°/o)
B1 1.118 0.37 33 1.401 0.30 21 36
B2 0.108 0.15 142 0.186 0.12 64 55
B3 0.425 0.24 55 0.620 0.19 30 45
G1 0.558 0.20 36 0.738 0.15 20 44
G2 -0.600 1.13 189 -l .267 0.97 77 50
G5 -0.297 0.26 87 -0.156 0.12 77 11
Type 12
B1 1.003 0.26 26 0.946 0.24 25 4
G1 0.570 0.22 39 0.933 0.36 39 0
G2 -0.705 1.18 167 -3.842 1.31 34 80
Max:
80
Min :
0
Avg :
36

 

 

64

 

3.8 Sensitivity analysis of the crash prediction model

There are two objectives associated with a sensitivity analysis: One is to examine the
possibility that the crash prediction model violates conceptual rules. For example, if a
model were designed such that its predicted crashes would decrease with an increase in
ramp volume, the model should be rejected because it violates a conceptual rule. The
other objective is to determine the effects of individual variables on the crash frequency

at freeway interchanges.

The sensitivity analysis is performed for the major geometric variables, but not
for the traffic variables because it is possible to change the geometry, but changing traffic
is difficult. During the sensitivity analysis of a specific variable, other design parameters
are assumed to be a constant. For this analysis, an experimental matrix was established,
which includes 3 experiments (A: 0.1 mile shorter than mean, B: mean, C: 0.1 mile
longer than mean) for interchange length, 3 experiments (A: 0.1 mile longer than mean,
B: mean, C: 0.1 mile shorter than mean) for spread -ramp length, and 2 experiments (A:

12 feet and B: 10 feet) for shoulder width.

Table 3.13 illustrates the results of the sensitivity analyses. In the sensitivity
analysis of interchange size, when the interchange size is increased by 0.1 mile, traffic
accidents increase in all interchange types which use this variable as a model component.

The average increase is 14 %.

65

In the sensitivity analysis of the spread- ramp length, traffic crashes increase by
an average of 26 % when the spread- ramp length is decreased by 0.1 mile. The traffic
crashes increase most rapidly for interchange type 12 (Tight diamond interchanges),
which increases by 47 %. The crash frequency is very sensitive to shoulder width for both
interchange types that include this variable, and especially for type 41(Directional
interchanges). In the sensitivity analyses, no violation of conceptual rules of traffic

crashes were found.

66

Table 3.13 Sensitivity analysis(effect of main geometric variables)

 

 

 

 

 

Parameter Interchange Experiment Experiment Experiment Effects
type (A) (B) (C)
Interchange Length 0.534 mile 0.634 mile 0.734 mile 0,1 Mile (T)
Length
Type 11 0.629 0.714 0.796 1.12
Type 12 0.557 0.654 0.749 1.16
Type 13 0.599 0.689 0.777 1.14
Type 14 0.438 0.549 0.666 1.23
Type 31 0.819 0.865 0.906 1.05
Type 33 0.549 0.647 0.744 1.16
Mean 0.599 0.686 0.773 1.14
Spread- Length 0.33 mile 0.23 mile 0.13 mile 0.1 mile (I)
ramp
length Type 11 0.658 0.747 0.848 1.14
Type 12 0.281 0.413 0.607 1.47
Type 14 0.472 0.592 0.744 1.26
Type 33 0.438 0.563 0.723 1.28
Mean 0.462 0.579 0.730 1.26
Shoulder Width 12 11 10 ft 2.0 feet(I«)
width
Type 11 0.154 0.211 1.37
Type 41 0.057 0.093 1.63
Mean 0.106 0.152 1.50

 

 

 

 

 

 

67

 

Chapter 4

THE DEVELOPMENT OF A METHOD TO IDENTIFY HAZARDOUS SITES

BASED ON THE NEGATIVE BINOMIAL DISTRIBUTION

4.1 General

The traditional rate quality control method is based on the assumption that the probability
of crash occurrences follow a Poisson distribution(Zegeer and Deen 1977) in which the
mean and the variance are equal. The normal approximation to the Poisson provides a
control chart without tedious interpolation from the table of the Poisson
distribution(Orlansky and Jacobs 1956), and this chart has been commonly used for the

identification of hazardous locations.

In chapter 2, the fact that the variance of crash occurrences at freeway
interchanges is substantially larger than the mean was discussed, based on the observed
data. This over-dispersion can be better explained by using the Negative Binomial
distribution. A control chart constructed under the assumption of the Poisson
distribution, can not reflect the phenomenon of over-dispersion in identifying hazardous

locations.

The purpose of this chapter is to describe a technique to overcome the limitation

of the rate quality control method based on the Poisson distribution. One statistician

68

(Rice 1997) suggests that the Negative Binomial distribution might be considered as a
model for situations in which the rate varies over time and over space.
Thus, in this chapter, the rate quality control method will be developed under the

assumption of the Negative Binomial distribution.

4.2 A review of the statistical methods identifying hazardous sites

4.2.1 The rate quality control method

The rate quality control method is one of the most common methods used to identify
hazardous sites. This method was originally developed as a means to control the quality
of industrial production(Norden et a1. 1 956). This approach uses a statistical test to
determine whether the traffic accident rate for a particular location is abnormally high
compared with the rate of reference sites with similar properties. The statistical test is
based on the assumption that traffic crashes are rare, hence the probability of their

occurrences follows a Poisson distribution(Zegeer and Deen 1977).

There have been changes in the original equations based on a comparison of the
errors between real values and estimated values obtained from the rate quality control
method formula. The following is a brief description of changes of the rate quality

control method.

The rate quality control method was proposed as a way to analyze crash data on

highway sections in 1957 using the following formula:

69

 

UCL = ,1 + 2.576\/2 + 0'8” + 1 (4.1)
V, V

 

i 2Vi
LCL = )1 — 2.576 1 + 0829 - 1 (4.2)
VI. Vi 2V].
where

UCL : upper control limit
LCL : lower control limit

)1 : average accident rate of reference sites(Z N, /Z Vi)
N ,- : the number of accidents at site i

V,- :the number of vehicles at site i

A decade later, it was recommended that the correction term(0.829/V 1) be
eliminated to improve the validity of the equations(Morin 1967). Thus, the following
equations are currently in use to calculate the upper and lower limits for the rate quality

control method.

UCL=I1+z i+L (4.3)
VI. 2V,-

LCLzli—z i+——1— (4.4)
VI. 2V1.

where

z : predetermined significance level

70

With the rate quality control method, a site is identified as hazardous if its
observed accident rate exceeds the mean accident rate of similar sites plus a multiple of
the standard deviation of the site accident rate, which is called the critical accident rate.
The critical accident rates can be calculated for each site by applying the following

equation:

RCi=2+z i+—1—— (4.5)
Vi 2V4

where

RCI- : Critical rate for site i

In the above equation, the first two terms result from the Normal approximation to
the Poisson distribution, the third term is a correction factor necessary because only
integer values are possible for the observed number of accidents. The coefficient of the
second term describes a probability factor determined by the level of statistical
Significance desired for RC.. The FHWA, however, proposes the following equation for

calculating the critical accident rate(Stokes and Mutabazi 1996).

Vi 2V1.

where
RC ,- :Critical rate for Site i

71

Equation(4.6) is different from equation(4.5) in that the Sign of last term is
negative, and the difference results from whether a probability should be included or
excluded if the rate is equal to the critical rate. The method of identifying hazardous sites
by the rate quality control method can be visually explained by Figure 4.1, where filled

stars correspond to the hazardous sites chosen under the rate quality control method.

grassed“. 523m 05 .32.: .552: .8289 5:35 8a.. me 3355 2:. — .v ohswwm

 

 

 

 

 

 

 

EVE—85mm“ .8325 .3 :8: .9555 L253

 

 

 

_ ea

73

4.2.2 The Bayes approach

Higle and Witkowski(l988) introduced and illustrated how Bayesian theory can be used
to identify the hazardous intersections from among many signalized intersections.
Hauer(1986) then proposed the application of Emperical Bayesian(EB) theory in traffic
safety problems, based on Robbin's work(Robbin l977,1979,1980) and this method has
subsequently been used by many researchers (Maher and Summersgill 1996, Persaud
1993, Belanger 1994). Both the Bayesian approach and the EB method which are

described above, have the following assumptions(Higle and Witkowski 1988).

Assumption 1: At a given site, when the average accident rate of reference sites(k) is
known, the count of accidents(N) obeys the Poisson probability law with expected

value( 1V1).

N.
exp( —/1Vi)(/1Vi) '
N.

I

(4n

 

MNzNflHQz

where,
,1 :average accident rate of reference Sites( 2 Ni /2 Vi)

2V1. :the expected value at site i

Assumption 2: The accident rate of reference sites(which the given site belongs to) can

be described by a Gamma probability density function such as:

__,B_(_z_ 01—1 :62
f(/1) - Ha) /l. e (4.8)

where,
f(/i) : gamma probability density function of reference sites

a, ,6 : parameters

74

Equation (4.8) is also denoted as the probability density function of the prior distribution
in terms of Bayes theory. Here, parameters 01 and [3 can be estimated through the method

of moment estimates(MME) or the maximum likelihood estimates(MLE).

Under the assumption that 01 and B were calibrated, if the observed data at a given
site i are N and V,, the probability density function of the posterior distribution can be

described as:

N.
(,B+V,)0[+ ’ a+N.-1 —(fl+V.),t

,1 I e I
T(a+Ni)

 

f(/1/Ni,V1-)= (4.9)

Equation (4.9) is the posterior distribution using Bayesian theory in its original meaning,
and we can evaluate the hazard of a given site using this equation. That is, the probability

that Bayes accident rate at site i, M, exceeds an average accident rate of reference sites,

Aavg:

P(2,. 2 2mg) = 1 — Pu, < 2mg)

 

4avg a+N-
+V. ’ N.—1 — V2
:1. [w ') 71““ e('6+’) d2 (4.10)
0 l'(a+Ni)

Figure 4.2 shows a graphical representation of the equation (4.10)

75

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”CK

 

 

 

 

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9...“ 5:: 82 a z :2: 9.526...—

 

76

4.2.3 The problems resulting from the use of the Poisson distribution

The rate quality control method as used in practice identifies a site as hazardous if its
observed accident rate over a given period exceeds its critical accident rate, which is the
average accident rate over reference sites plus a multiple of the standard deviation of the
accident rate of the site over the same period. This rate quality control method is based on
the Poisson distribution as mentioned is section 4.1. However, the Negative Binomial
distribution fits the freeway interchange crash data much better than the Poisson

distribution. Thus, this identification of hazardous sites may not be valid.

Recognizing that the variance of the Poisson distribution equals the mean,
whereas the variance of the Negative Binomial distribution is (mean+mean 2/ k), the
existing approach under the Poisson assumption will identify more sites as hazardous
than would be expected under the Negative Binomial assumption. For example, in Figure
4.3, the solid line is the upper control limit chart based on the Poisson distribution while
the dotted line is the upper control limit chart based on the Negative Binomial
distribution. The stars correspond the hazardous sites which are chosen under the Poisson

or Negative Binomial distribution.

77

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78

4.3 The identification of hazardous sites based on the Negative Binomial

distribution.

4.3.1 Concept

In equation (4. 1) and (4.3), the true value of the upper control limit for the Normal
approximation to the Poisson distribution on traffic crash frequency can be computed

from the following equation:

 

U—l -m y

P: 2: )3" (4.11)
y:

where,

P : predetermined probability level
U : true upper control limit

m : expected value

In equation(4.l 1),
m=lfi,

Ni

 

where,

N,- = the accident fi'equency at site i

V,- = the number of vehicles at site i

79

Under the condition that the average accident rate of reference sites, A, is known, the true
control limit for a site i can be calculated by selecting the value that meets the

predetermined probability levels (i.e., 0.9, 0.95) from equation (4.11).

The above concept can be utilized for the Negative Binomial distribution. That is,
when crash occurrences follow the Negative Binomial distribution, the formula for the

true upper control limit is as follows:

 

U—l -k y

1):: (1+3) F<k+y)[ ’" ] (4.12)
yzo k y!F(k) m+k

where,

U : the true upper control limit
m :mean

k :parameter

From equation (4.12), we can compute the true upper control limit under a desired
probability level based on the Negative Binomial distribution for a given site. However,
for the Negative Binomial distribution, the estimates of parameters (m ,k) are not as
simple as those of the Poisson distribution. The method for these estimates will be

described at length in section 4.3.2.

80

4.3.2 Estimation of the parameters

4.3.2.1 Derivation of maximum likelihood equation

There are two methods for estimating the parameters of the Negative Binomial
distribution: one is a moment method, the other is a maximum likelihood method (Rice
1997). As mentioned in section (4.1), over-dispersion occurs over time, which implies
that each site has its own distribution, and hence in fact, there are many distributions in a
sample. Thus, for this research, the maximum likelihood method is used, recognizing that
the parameter k can not be determined with an acceptable efficiency by a moment

method for multiple distributions(Bliss 195 3).

In order to estimate the parameters, two equations were derived from the
maximum likelihood function, based on the Lawless' work(1987). For simplicity,

equation (2.2) is transformed as equation (4.13), and thus the parameter k equals l/a.

 

 

-1 y a
f(y/a,m)=r(y+a )[ am ][ 1 J (4.13)

y!F(a_1) 1+am 1+am

The likelihood function is:

 

 

 

L< amrfi “kw—”i am VI 1 j (414)
y,, ’ i=1 [‘(a‘l) 1+am 1+am '
—1
-1 J’,‘ a
n P -+a am V.
L(yisaam09Vi)= H (y! ) 0 I —-—l__ (4.15)
I=1 [‘(a‘l) 1+am0Vi 1+amOVi

81

Noting that for any c > O,F(y + c) / F(c) = c(c +1)(c + 2) ------ (c + y — 1),

-I
M:l[l+lj{l+2j ......... (1+yi —1]
ma“) a a a a

Now, we can write the log likelihood fimction, log L(yi, a, mo, Vi) as

—1

y- a
’7 F(y-+1/a) am V. ' 1
l(yi,a,mO,Vi)=zlog ' [ 0 I J ——
i

i=1 F(1/ a) 1+ amOV 1+ amOVi

 

,V‘ a
" F .+1/a am V. ’
=2 log—(I):’—-——)+log ——0—'- +log —1—— (4.16)
~ (l/a) 1+am0Vi 1+am0Vi

In equation (4.16)

The first term,

2'21. _._1“<yi+“a>=,. 1(1+_aj(1+2a)(1+3a]..... 1.0,-..
i=1 g F(1/a) ga a a a a

=log1+log(1+ a)+log(1+ 2a) - - - log(1 + (yi — 1)a)— yl- loga

 

 

le

= Zlog(1+ aj) — yi loga (4.17)
j=O

82

The second term,

’7 amOVi y"
Zlog —1———7 = yi log(amOVi)—yi log(1+ amOVi)

= yi loga + yi log mO +yi log VI. -yl. log(1+ amOVi) (4.18)
The third term,

" 1 a 1
Zlog —— = ——log(l + amOVi ) (4.19)
(1

Thus, from equations (4.17),(4.18)and(4.19) the log likelihood function can be

summarized as follow :

log(1+ aj) + yi log mO — y,- log(1+ amO Vi) — llog(1+ amO Vi) (4.20)
a

83

The simplest way to obtain (m0 ,a) is to maximize [(yi ,a,mO‘Vl-) with respect to (m0,a),

and thus, we need to set the partial derivatives of the l(y,- , (1, m0 Vi) equal to zero.

That is,

51(yi,a,mO‘V,-) _

 

0 4.21
6m0 ( >

 

61(yi,a,m0‘Vi) _ 0

 

 

 

4.22
6a ( )
01(yi,a,m0.Vl-):i fi_ yiaVi _ Vi
(3m0 i=1 m0 1+am0Vi 1+amOVi
zfi yi’moVi (423)
i=1 m0(1+amOVi)

 

 

61(y.,a,m V.) n yi—1 ' .m V. m V.
’ 0" =Z[Z[ j ']—————y' 0' +L210g(1+amOVi)-l_—Ol

6a i=1 1+am0Vi a a 1+am0Vi

 

 

n y:'“1 - V- - _1
=Z[ ( J.]+;12—10g(1+am0Vi)-m0'(y'+a 1 (4.24)

l+ amOVi

84

From equation (4.23),

n

m0 = L?" (4.25)
Vi

=1

I

From equation (4.24),

mOVi(yi + a_1) _

 

 

.Vi-l .
Z[ J j + —l3log(1 + amOVi) — 0 (4.26)
j=0 1+aj a 1+am0Vi

The maximum likelihood estimate of " m0" can be easily calculated using
equation (4.25), but that of " a " is not simply obtained because it is not a closed form.

Therefore, a numerical approach is used to solve equation (4.26).

85

4.3.2.2 Parameter estimation

It is not easy to verify that one method is superior to another in the absence of perfect
information, which can not be obtained naturally in the traffic safety field. Therefore, in
this section the results of the Negative Binomial approach are compared with those of the
existing two methods that are commonly used in the traffic safety field: the rate quality
control method and the Bayes identification method. Because the two methods have
already been reviewed at length in section 4.2 and 4.3, the parameters are estimated and

the results obtained without expanding on them here.

There is a limitation in choosing the sample sites to examine the Negative
Binomial approach because sites with similar geometry should be used for reference
sites. Therefore, two data sets are selected for this study. The first data set includes 16
diamond interchanges with similar geometric properties (i.e., Diamond type, 6 lanes, 10
ft shoulder width, 4 ramps). The second data set includes 14 partial clover A or B 4
Quadrant interchanges which have similar geometric characteristics (i.e., 6 lanes, 10 ft
shoulder width, 6 ramps). It is not possible to get a data set with exactly the same
geometric conditions in practice, and the more classification variables used, the smaller

the sample size.

With the above data sets, the new approach has been tested and compared with
the results of the existing two methods. The parameters for Bayes approach have been
estimated through the method of moment estimates (MME), and the parameters for the

rate quality control method based on the Negative Binomial distribution have been

86

estimated through the procedures described in the preceding sections. The estimates of
the parameters are summarized in Table 4.1. Note that the k parameter for Diamond
interchanges is larger than that of Partial Clover 4 Quadrant interchanges. It implies that

the variance of Diamond interchanges is less than that of Partial Clover 4 Quadrant

 

 

 

 

 

interchanges.
Table 4.1 The estimation of parameters
Method Parameters Estimates
Diamond Par clo A or B 4 Q

Negative Binomial m0 0.0010 0.00117

a 0.105 0.095

k(=1/a) 9.52 10.52
Poisson A, 0.0010 0.001 17

Bayes approach a 6.51 10.98
5 6160.40 10194.61

 

 

 

 

 

 

4.3.3 Application and validation of the Negative Binomial approach

In order to choose hazardous sites based on the estimates of the parameters, the

following scenarios are developed.

Scenario 1: a site i is hazardous if the observed accident rate (N (N i) exceeds the

upper control limit which is a function of the average accident rate of reference sites and

87

a desired probability level. This scenario will be used to test the rate quality control

method based on the Negative Binomial distribution.

Scenario 2: a site i is hazardous if the observed accident rate (N (N i) exceeds the

upper control limit which is a function of the average accident rate of reference sites and
a desired probability level. This scenario will be used to test the rate quality control

method based on the Poisson distribution.

Scenario 3: a site i is hazardous if the probability that its Bayes accident rate
exceeds the average accident rate of reference sites is greater than a predetermined

probability level.

For this study, a 95 % probability level is applied for all scenarios. Table 4.2
presents the identification of hazardous sites for Diamond interchanges. An asterisk (*)
corresponds to the sites that have been identified as hazardous on the basis of the above
scenarios. For example, under the existing methods (rate quality control method based
on the Poisson distribution, and Bayes approach), 7 sites (i.e., sites: 1, 2, 4, 7, 8, 9, 11)
out of 16 are identified as hazardous, whereas under the new method (rate quality
control method based on the Negative Binomial distribution), 2 sites(i.e., sites: 2 and 4)

are identified as hazardous.

In each of the these scenarios (1, 2 and 3), a 95 % probability level was used,

which implies that there is approximately 1 abnormal or hazardous site out of 20

88

random sites in the statistical sense. Therefore, 7 sites out of 16 is an unreasonably high
percentage in the context of the 95 % probability level. Thus, the new method is clearly
more conceptually persuasive in identifying hazardous locations than the existing

methods.

For Par Clo A or B 4 Q interchanges, 4(i.e., sites: 7, 12, 13, 14) of 14 sites were
chosen with the existing methods, whereas only 1 site was identified as hazardous when
assuming the Negative Binomial error as shown in Table 4.3. Thus similar conclusions

can also be reached.

The disagreement between the existing methods and the new method is probably
best explained in the context of the underlying assumptions. The existing methods are
both based on the widely accepted assumption that crashes occur according to the
Poisson distribution, whereas the new method is based on the assumption that the

occurrence of accidents follows a Negative Binomial distribution.

The upper control limits, which are functions of the average accident rate of the
reference sites and the variance in the accident rate at the given site, were lower when a
Poisson distribution instead of a Negative Binomial distribution is assumed. This causes
the procedure to identify more hazardous sites than are expected, which was shown in

Figure 4.3.

89

Chapter 2 verified and discussed at length that the variance of crashes is
substantially larger than the mean, and hence the Negative Binomial distribution is an
appropriate assumption for the occurrence of crashes at freeway interchanges. Thus, the
rate quality control method based on the Negative Binomial distribution would be an
effective measure for the identification of hazardous sites, in these cases where the

variance of accidents exceeds the mean (over-dispersion).

90

Table 4.2 A comparison of hazardous sites according to the methods

(Diamond interchanges)

 

Diamond interchanges

 

 

 

 

 

 

Site(i) Accidents Vehicles Observed rate Upper control limit Bayesian
( Ni) ( vi) (NiNi) Negative Poisson APPmaCh
Binomial
1 213 150529 0.00142 0.00153 0.00114 * 1.000 *
2 322 166103 0.00194 0.00154 * 0.00114 * 1.000 *
3 137 123131 0.00111 0.00154 0.00116 0.875
4 196 128354 0.00153 0.00153 * 0.00115 * 1.000 *
5 193 240902 0.00080 0.00153 0.00111 0.001
6 194 227992 0.00085 0.00153 0.00112 0.010
7 247 171559 0.00144 0.00153 0.00113 * 1.000 *
8 164 137889 0.00119 0.00154 0.00115 * 0.981 *
9 207 161545 0.00128 0.00154 0.00114 * 1.000 *
10 160 183034 0.00087 0.00154 0.00113 0.039
11 242 179213 0.00135 0.00153 0.00113 * 1.000 *
12 102 207928 0.00049 0.00153 0.00112 0.000
13 1 1 1 203816 0.00054 0.00153 0.00112 0.000
14 158 207045 0.00076 0.00153 0.00112 0.000
15 161 217719 0.00074 0.00153 0.00112 0.000
16 121 209255 0.00058 0.00153 0.00112 0.000

 

 

 

 

 

 

 

91

 

Table 4.3 A comparison of hazardous sites according to the methods

 

 

 

 

 

 

 

(Par Clo A or B 4 Q)
Par Clo A or B 4Q
Sile(i) Accidents Vehicles Observed rate Upper control limit Bayes
(N) (v,) (N, Ni) Negative Poisson Approx“
Binomial
1 39 39434 0.00099 0.00183 0.00147 0.127
2 45 65947 0.00068 0.00179 0.00140 0.000
3 53 86205 0.00061 0.00177 0.00137 0.000
4 62 94429 0.00066 0.00178 0.00136 0.000
5 127 100303 0.00127 0.00177 0.00135 0.764
6 120 103676 0.00116 0.00177 0.00135 0.407
7 157 108898 0.00144 0.00177 0.00135 * 0.990 *
8 111 110946 0.00100 0.00177 0.00134 0.041
9 103 112020 0.00092 0.00177 0.00134 0.005
10 1 17 127060 0.00092 0.00176 0.00133 0.003
11 131 133995 0.00098 0.00177 0.00133 0.016
12 226 169887 0.00133 0.00176 0.00131 * 0.959 *
13 286 202817 0.00141 0.00176 0.00130 * 0.998 *
14 403 235275 0.00171 0.00170 * 0.00129 * 1.000 *

 

 

 

 

 

 

 

92

 

Chapter 5

A SIMPLIFIED APPROACH FOR OVER- DISPERSION (NORMAL

APPROXIMATION METHOD)

5.1 General

The rate quality control method based on the Negative Binomial distribution was
discussed in the preceding chapter, and it was found that this method produces reasonable
results in the statistical sense. However it is not easy for traffic engineers to apply this
technique in the safety field because the parameters can not be estimated as simply as
those of the Poisson distribution. This chapter provides a simple approach for the
identification of hazardous sites when the Negative Binomial distribution should be

assumed because of the phenomenon of over-dispersion.

5.2 Concept

The Negative Binomial approach can be simplified using the Normal approximation as:
N,- ~N(.U,'ad.ui) (5-1)

where,
#i = 4Vi

 

Ni

N I. = the number of accidents at site i
V, = the number of vehicles at site i

d/JI- = variance (d 2 1)

93

In equation (5.1), the variance is larger than the mean, which is conceptually consistent
with the error structure of the Negative Binomial distribution.
On the other hand, under ideal conditions, the Poisson distribution can be
approximated by the Normal distribution for large values of in, because the probability

mass function of the Poisson distribution becomes more symmetric and bell-shaped as 11,

increases (Rice 1997). Let N, be a sequence of Poisson random variables with the

corresponding parameters. Then, E(Ni)= Var(Ni)= iii. If we wish to approximate the

Poisson distribution by a Normal distribution, the Normal distribution should have the
same mean and variance as the Poisson, and hence the random variables can be

standardized by letting,

X. z—N"_#i

' (a?

then, E(Xi) = 0, Var(Xi) =1.

(5.2)

That is,Xi ~ N(O,1)

However, the assumption is :
E(Xi) = 0, Var(Xi) = d. That is,

X, ~ N(O, d), (5.3)

94

Therefore, the random variables with over - dispersion can be standardized as follow :

 

N,- —,ul.
Z]. = (5.4)
dfli
Then E(Zi) = 0, Var(ZI-) = 1.That is,
Zi ~ N( 0,1) (5.5)

Thus, Z, can be applied to the identification of hazardous sites based on the traffic crash

data with over—dispersion.

5. 3 Application and validation of the Normal approximation method

The Normal approximation method is an alternative to solving the difficulties associated
with the estimation of parameters in applying the rate quality control method based on the
Negative Binomial distribution. Therefore, the results should be analogous to those of
this rate quality control method. The validity of the Normal approximation method was
tested using the two data sets (16 Diamond interchanges and 14 partial clover A, or B 4

Quadrant interchanges) which were used in chapter 4.

95

5.3.1 Estimation of parameters

In order to apply the Normal approximation method for over-dispersion, two parameters
need to be estimated. One is the average accident rate, 2». The other is a parameter
relating to the over-dispersion, (1. Here, the average accident rate is the same as that of the
Poisson or Negative Binomial distributions described in chapter 4, whereas d is the
variance of random variables that have been standardized by formula (5. 2). Table 5.1
shows the results of the estimated parameters. The "d" estimates are 30.39 and 11.96 for

Diamond, and Par Clo A or B 4 Q interchanges, respectively.

Table 5.1 Estimation of parameters

 

 

 

Method Parameters Estimates
Diamond Par Clo A or B 4 Q
Normal approximation )c 0.0010 0.001 17
method d 30.39 1 1.96

 

 

 

 

 

 

5.3.2 Validation of the Normal approximation approach

Based on the estimates of parameters in Table 5.1, the Normal approximation method

was tested for the 2 data sets. Table 5.2 shows that the use of the Normal approximation
method produced similar results with those of the approach using the Negative Binomial
distribution. In this table, asterisks (*) correspond to the sites that have been identified as

hazardous on the basis of the two approaches.

96

For example, under a 95 °/o probability level, the Normal approximation method identifies
2 sites as hazardous, whereas the rate quality control method under the Negative
Binomial distribution identifies 3 sites as hazardous out of 30 sites. That is, site 4 is not
identified as a hazardous site by the Normal approximation method, whereas it is
hazardous based on the Negative Binomial approach. However, this difference is not
substantial with the probabilities being 0.94 and 0.95 respectively, when we use each
approach.

Figure 5. 1 presents a comparison of the probabilities with which a site is
identified as hazardous by the two methods. As shown in the figure, results of both
methods are consistent, even though there are a few sites that disagree slightly. It is
expected that the differences would be reduced even further with larger data sets.

Thus, for the identification of hazardous sites, the Normal approximation method
can be used as an alternative to solving the difficulties associated with the estimation of
the parameters for the rate quality control method. There is only a slight loss of accuracy

as discussed here.

97

Table 5.2 A comparison with the results using the Negative Binomial distribution

 

 

 

 

 

 

 

 

 

Site Probability
( i ) Interchange type lli Xi Zi Approximation Negative
Binomial
1 Diamond 151 5.03 0.91 0.82 0.91
2 Diamond 167 12.02 2.18 0.99* 0.99*
3 Diamond 124 1.20 0.22 0.58 0.72
4 Diamond 129 5.91 1.53 0.94 0.95*
5 Diamond 242 -3.14 -O.57 0.38 0.36
6 Diamond 229 -2.31 -0.42 0.33 0.42
7 Diamond 172 5.69 1.03 0.85 0.92
8 Diamond 138 2.17 0.39 0.65 0.79
9 Diamond 162 3.52 0.64 0.74 0.85
10 Diamond 184 -1.75 -0.32 0.38 0.45
1 1 Diamond 180 4.63 0.84 0.80 0.90
12 Diamond 209 -7.39 -1.34 0.09 0.05
13 Diamond 205 -6.55 -1.19 0.12 0.09
14 Diamond 208 -3.46 -0.63 0.26 0.31
15 Diamond 219 -3.90 -0.71 0.24 0.28
16 Diamond 210 -6.15 -1.12 0.13 0.11
1 Par Clo A 4 Q 46 -1.06 -0.31 0.39 0.43
2 Par Clo A 4 Q 77 -3.67 -1.06 0.14 0.12
3 Par Clo B 4 Q 101 -4.77 -l .38 0.08 0.07
4 Par Clo A 4 Q 111 -4.62 -l.34 0.09 0.09
5 Par Clo A 4 Q 117 0.88 0.25 0.61 0.77
6 Par Clo B 4 Q 121 -0.13 -0.04 0.48 0.59
7 Par Clo B 4 Q 128 2.61 0.76 0.78 0.83
8 Par Clo A 4 Q 130 -1.66 -0.48 0.32 0.55
9 Par Clo A 4 Q 131 -2.46 -0.71 0.34 0.33
10 Par Clo B 4 Q 149 -2.61 -0.75 0.33 0.33
11 Par Clo A 4 Q 157 -2.07 -0.60 0.38 0.39
12 Par Clo A 4 Q 199 1.92 0.55 0.71 0.75
13 Par Clo A 4 Q 237 3.15 0.91 0.82 0.81
14 Par Clo B 4 Q 276 7.68 2.22 0.99* 0.95*

 

 

 

 

 

 

 

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5.4 An examination of the assumptions

In the preceding sections, the following assumptions were made: 1) random variables X,
which are standardized follow a Normal distribution, and 2) the expected values u, should
be large enough to be approximated by the Normal distribution. Therefore, these

assumptions need to be examined.

5.4.1 Goodness of fit of the Normal distribution

In section 5.2, we assumed that X, follows a Normal distribution without any verification.
In order to enhance the credibility of this method, we need to test the goodness of fit of
the random variable X, to a Normal distribution. The Chi-square test was used to conduct
this test after partitioning a Normal distribution into eight intervals of equal probability

(Neter et al. 1992). Thus, if Ho holds (that is, X, is Normally distributed), then X2 follows
an approximate x2 distribution with n-p-1=8-2-1=5 degrees of freedom.
For (1:005, we require x2(0.95; 5)=11.07.

Hence, the decision rule is as follows:

If X2 S 11.07, conclude H0

If X2 > 11.07, conclude H1

The analysis of Diamond interchanges and Par-Clo A, or B 4 Q interchanges, which are

the same data sets as used in the previous section, found X2 values of 6.00 and 5.43,

respectively. Thus, a Normal distribution is a reasonable assumption for Xi.

100

5.4.2 Large values of 11,

In section 5.2, we assumed that 11, should be large enough to be approximated by the
Normal distribution. This approximation method can not be used, if it, is a small number.
Accordingly, the effects of various values of u, were tested to determine the limits of the

approximation method. The analysis focuses on the calculation of the difference between

the true and approximate upper control limits over the values of 11,, as computed from the

following equations.

 

U—l “/1 y
p_—_ Z 9 "1 (5-6)
1
y=0 y
Ua = p + kfi (5.7)
where,

P : predetermined probability level
U : true upper control limit
Ua : approximated upper control limit
k : standard normal variate corresponding to the predetermined probability level

Figure 5.2 shows the difference between the true and approximate upper control
limits for a range of expected frequencies (pi) from 0 to 60 crashes using the 95

probability level (k=1.645). Note that the difference is very large when the expected
values are less than 5, then the curve flattens for expected values in the range from 5 to

15.

101

The difference is very small when the expected values are larger than 40. For this

research, pi is large enough to be approximated, recognizing that the minimum value of

u, is 151 and 46, for Diamond and Par Clo A or B 4 Q data sets, respectively as shown in

Table 5.2.

102

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103

Chapter 6

IDENTIFICATION OF HAZARDOUS SITES USING A TRAFFIC CRASH

PREDICTION MODEL (PREDICTION MODEL METHOD)

6.1 The limitation of the rate quality control method (or upper control limit)

As mentioned before, a rate quality control method is commonly used for identification
of hazardous sites. In order to overcome the problem caused by over-dispersion, the rate
quality control method based on the Negative Binomial distribution rather than the
Poisson distribution has been examined and proposed as an alternative. Nevertheless, in
identifying the hazardous sites using reference sites we recognize that there are still

limitations as defined by others (Elvik 1988, Mountain and Fawaz 1989, Hauer 1992):

First, the selection of reference sites is a matter of judgement, and hence the same
site can be evaluated differently, depending on the researchers. Second, the number of
reference sites will likely not be large enough to permit the accurate identification of

hazardous sites in practice.

For example, suppose that the objective was to evaluate the safety of all
interchanges in Michigan using the rate quality method. The first step is to classify all
interchanges to find the reference sites with similar properties. Figure 6.1 presents
classification trees considering only the basic contributing factors to traffic crashes, and

1056 groups ( = 22x2x2x3x4) are produced, even though other contributing factors (i.e.,

104

ramp length, interchange size, et al) are not considered in grouping the interchanges. This
implies that we need reference sites for 1056 groups to evaluate all the interchanges using
the rate quality control method described in chapter 4. Thus, it is not possible to identify
hazardous interchanges using reference sites, recognizing that there are a total of only
397 interchanges along the four main freeways (I-69, I-75, I-94, I-96) in Michigan. There

are many interchange types for which a sizeable number of reference sites does not exist.

An alternative to the use of reference sites would be to use data from other states
for the evaluation of freeway interchanges in Michigan. However, this approach causes
several linked difficulties. For example, the definition of traffic crashes is different across
states (i.e., total damage of $400 in Michigan, $500 in New Mexico and $1000 in
Wisconsin: Michigan, New Mexico and Wisconsin traffic crash facts (1998 )), and
interchange crashes are sensitive to weather conditions (i.e., in Michigan, winter crashes
are approximately 15 % higher than in other seasons). In addition, it is not easy to obtain
well defined geometric and traffic data from other states. Thus it is obviously not a good
approach to use data from other states for the identification of hazardous sites. In this
chapter, a method to search for hazardous sites using an accident prediction model is

examined.

lOS

 

 

T1: 22

 

 

 

 

 

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T2: 2
A3: 2
(A4: 3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Classification of geometg
T1: Interchange types

- 22 interchange types T5: 4
T2: Area

- Urban or rural
T3: Shoulder width
-12 ft or 10 ft
T4: The number of lanes
- 4 lanes
- 6 lanes
- 8 lanes
T5: The number of ramps
- 2 ramps
- 4 ramps
- 6 ramps
- 8 ramps

 

 

 

 

 

 

 

 

Figure 6.1 Basic geometric classification tree for reference sites

106

 

6.2 The concept of the prediction model method

In section 6.1, several limitations to the use of the rate quality control method for the
identification of hazardous interchanges was discussed. The conceptual problem can be
solved by diminishing the scope of individual judgement through logical procedures,
whereas the practical problem can be treated by estimating the effects of special
contributing factors to traffic crashes at a given site through an analysis of relevant traits
at other sites. Previous researchers (J orgensen 1972, Flak and Barbaresso 1982) have
recommended that hazardous sites be estimated by the difference between the observed
accident frequency (B) of a site and the expected frequency (A) as predicted by an
accident prediction model as shown in Figure 6.2. McGuian (1981) noted that this
difference represents the size of the potential crash reduction when a safety improvement
project is implemented at the site. These ideas can be updated to solve both the

conceptual problem and the practical problem which have been identified.

Suppose that the goal is to estimate the hazardness of site i using a statistical
concept like the rate quality control method. In order to evaluate site i using the rate
quality control method, reference sites with similar properties should be selected, and the
accident rate of the site i compared with that of the reference sites. However, in the strict
sense, there are no reference sites which exactly reflect the site i. Thus, the idea of the
prediction model method is that the value of E(O) obtained from the crash prediction
model can be used instead of the average crashes of the reference sites to which the site i
belongs. Using this approach, the reference sites match exactly the traits of the site i

(these are imaginary reference sites as denoted by Hauer (1992).

107

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This approach is similar to the rate quality control method in the sense that both
use the mean and standard deviation for identification of hazardous sites. However, the
difference is that the mean is the expected value 13(9), based on a calibrated model for the
prediction model method, whereas the mean is the average of the reference sites for the
rate quality control method. This is why "E(B)" instead of "m" is used in formula 6.].
Therefore, the calibration of the crash prediction model based on the correct error

structure is extremely important to the identification of hazardous sites.

It has already been shown that the desirable assumption for freeway crash models
is the Negative Binomial rather than the Normal or Poisson error structure. In order to
illustrate the prediction model method for the identification of hazardous sites, the

Negative Binomial distribution function is again mentioned as equation (6.1).

 

P=Ui1 [NEE-11k m”) E(B) x (61)
,20 k x!r(k) E(6)+ k '
where,

U : the true upper control limit
E( 6 ) : expected values

k : parameter

109

In equation (6.1), E(G) would be obtained from the crash prediction model and the
parameter k would be estimated in the process of calibrating coefficients of the crash
prediction model, which were discussed in detail in chapter 3. From equation (6.1),
therefore, the upper control limit for identification of hazardous sites at a desired

probability level can be computed.

The variance of the Negative Binomial distribution is E(0)+E(9)2/k, as discussed

in chapter 2 and chapter 3, and hence the upper control limit will increase sharply with

E(G) as shown by the thick dotted line in Figure 6.2.

However, if an accident prediction model is developed under a constant normal
error structure, the upper control limits would be a constant distance from the accident
prediction line as shown by the thin dotted line in the Figure. This approach is similar to

that of previous research (J orgensen 1972, Flak and Barbaresso 1982).

110

6. 3 Application and validation of the prediction model method

6. 3.1 Illustration of the prediction model method

Suppose that the goal is to estimate the safety of a special site using a crash prediction
model that has been calibrated under the Negative Binomial error structure.

Again, k can be estimated by the parameter calibration procedure described in chapter 3,
and E(G) can be computed from the crash prediction model using several independent
variables of the site. Thus, the true upper control limit 'U' can be found from equation

(6. 1) for a given site under the desired probability level.

For example, consider site 1 in Table 6.1. Using the crash prediction model

developed in chapter 3, the expected value at site 1,E (0) is

1.401 0.186 V 0.620 0.738

= 3.448 V1 V2 3 G1 exp(—1.267 G2 — 0.156 G5) (6.2)

= 141 .6 accidents/3years

The standard deviation at site 1

 

= JE(9)+ 13(9)2 /k

= 51.3 accidents/ 3years

111

The parameter k was determined to be equal to 8.05 in chapter 3. In equation (6.1), (6.2)
and (6.3), the upper control limit ' U ' is 233 crashes for 3 years under the 95 percent

probability level as follows:

 

U-l -8.05 x
(1+141.6) I"(8.05+x)[ 141.6 ] (6.3)

x!F(8.05) 141.6 + 8.05

 

However, there were only 213 crashes over 3 years at the given site. Thus this site is not
identified as hazardous under the 95 percent probability level as shown in Figure 6.3.
Thus, we can test the hazardness of each site on the basis of various desired probability

levels using these results.

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6.3.2 Validation of the prediction model method

The conceptual foundation for identifying hazardous sites using an accident prediction
model are straightforward as discussed in the previous section. There are two main
advantages of this method over the rate quality control method. First, it diminishes the
scope of individual judgement through a logical procedure. Second, a large number of

reference sites for any particular site are not required.

Despite its advantages, the prediction model method can cause unreasonable
results since there may be significant errors in choosing the model structure and
calibrating the model parameters. For these reasons, it is important to illustrate
empirically that the prediction model method and reference method produce similar
results. However, we can not expect that the results of both approaches will be
coincident, because in the strict sense, the imaginary reference sites for the prediction

model method is a subset of the reference sites for the rate quality control method.

To demonstrate the results of both approaches, the two data sets that were
analyzed in chapter 4 were used. In Table 6.1, the 5th column presents the probability
that observed crashes exceed the expected crashes at a given site under the prediction
model method. The 6th column represents the probability that the observed accident rate
exceeds the reference accident rate under the rate quality control method. There is some
disagreement between the methods as expected. When these sites are identified at a high
probability level (i.e., 0.95), 3 sites out of 30 are identified by the rate quality control

method (marked by a "*" in the table), whereas there are no sites identified when using

114

 

the prediction model method. At a lower probability level (i.e., 0.90), 6 and 4 sites out of

30 are identified using the rate quality control method and prediction model method

respectively (noted by a " O " in the table).

In the prediction model method, the model parameters are calibrated through a
minimization of the sum of squared residuals, and hence there may be underestimates of
the variances for the special sites which have a larger values than the average sites as
shown in Table 6.1. Moreover, not all geometric elements (i.e., interchange size, ramp
length, et a1) and traffic elements (mainline traffic, on and off ramp traffic, truck traffic,
et al) were used in classifying the reference sites to design the upper control limit,
whereas the imaginary reference sites for the prediction model method match exactly the

characteristics of a special site.

Accordingly, it can be expected that the results of both approaches will be similar,
but not coincide in every cell in Table 6.1. To test similarity of the results by the rate
quality control method and prediction model method, the percentiles of sites were
calculated and were plotted in Figure 6.4. The results of both approaches are highly

correlated (correlation coefficient = 0.96).

All the sites were ranked by the probability and the top 10 sites were chosen from
the two data sets (5 sites at Diamond interchanges, and 5 sites at Par-Clo A or B 4 Q
interchanges). As shown in Table 6.1(noted by a "v" in the table), the prediction model

method identifies the same sites as the rate quality control method for the Diamond

115

interchanges. It also identifies 4 sites out of the 5 identified by the rate quality control
method for the Par-Clo A or B 4 Q interchanges. It is surprising that there is so little
difference between the rate quality control method and the prediction model method in

terms of determining the hazard ranking of several sites.

A practical application of the above results is that if the goal is to prioritize
several sites for a highway safety program, the prediction model method can be used as a
tool to produce very similar ranks as the rate quality control method. If the goal is to
evaluate a specific site for a purpose, we can approximately evaluate the hazardness of
the site under the desired probability level through the prediction model method. These
advantages imply that we can overcome the conceptual and practical problem associated
with the identification of sites where the crashes exceed the expected number of crashes
as discussed in the previous sections, through the use of the prediction model method.
The accuracy of this method depends on having the crash prediction model calibrated

under the appropriate error structure.

116

Table 6.1 A comparison of results

 

 

 

 

 

 

 

 

 

 

Site(i) Interchange type The number of crashes Probability
(3 yearS)
Observed Estimated By upper control limit By prediction model
1 Diamond 213 141.6 0.91 0 v 0.91 9 v
2 Diamond 322 204.2 099 * e V 0.93 o v
3 Diamond 137 113.8 072 Q75
4 Diamond 196 139.9 0.95 at Q V 0.37 V
5 Diamond 193 237.7 0.36 0.34
6 Diamond 194 251.5 0.42 0.29
7 Diamond 247 163.7 0.92 0 v 0.92 0 v
8 Diamond 164 138.7 0.79 0.73
9 Diamond 207 169.3 0.85 0.76
10 Diamond 160 166.5 0.45 0.51
11 Diamond 242 157.8 0.90 0 v 0.92 0 v
12 Diamond 102 177.7 0.05 0.10
13 Diamond 111 182.7 0.09 0.13
14 Diamond 158 179.5 0.31 0.42
15 Diamond 161 198.5 0.28 0.34
16 Diamond 121 188.4 0.11 0.16
1 Par Clo A 4 Q 39 44.8 0.43 0.43
2 Par Clo A 4 Q 45 69.1 0.12 0.20
3 Par Clo B 4 Q 53 87.1 0.07 0.15
4 Par Clo A 4 Q 62 93.3 0.09 0.21
5 Par Clo A 4 Q 127 85.4 0.77 v 0.89 v
6 Par Clo B 4 Q 120 135.1 0.59 0.44
7 Par Clo B 4 Q 157 127.8 0.83 v 0.76 v
8 Par Clo A 4 Q 111 102.5 0.55
9 Par Clo A 4 Q 103 125.7 0.33 0.64 v
10 Par Clo B 4 Q 117 134.8 0.33 0.36
0.42
11 Par Clo A 4 Q 131 166.1 0.39 0.33
12 Par Clo A 4 Q 226 221.5 0.75 v 0.58
13 Par Clo A 4 Q 286 275.9 0.81 v 0.59 v
14 Par Clo B 4 Q 403 285.3 095 * 9 v 0.86 v

 

 

 

 

 

 

 

* : Identified sites under 95 percent probability level
0: Identified sites under 90 percent probability level
v : Top 10 rankings (5 for Diamond, and 5 for Par C10 4 Q)

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6. 4 Evaluation of Michigan freeway interchanges on the basis of the prediction
model

As noted in the preceding section, the prediction model method can be used to identify
hazardous sites without the use of reference sites. Using this approach, the 199
interchanges which were utilized in the crash prediction model development were
assessed using the coefficients and k parameters estimated according to the interchange

type in chapter 3.

The sites which exceed the thick dotted line in Figure 6.2 are summarized in
Table 6.2. Under the 95 % upper control limit, there is one such site out of the 10
interchanges on L69, 4 sites out of 65 on I-75, 6 sites out of 90 on 1-94, and 1 site out of
34 on [-96, respectively. Therefore, a total 12 sites are identified out of 199. These results
are approximately consistent with the statistical concept that there may be 10 abnormal
sites out of 200 random sites using the 95 % upper control limit. Under the 90 % upper
control limit, 22 sites are chosen, which also supports the preceding conclusion. The

results of evaluating all interchanges are presented in detail in the Appendix.

The identified sites are candidates for improvement under highway safety
improvement program for freeway interchanges. These results could not be obtained
through the existing rate quality method because there are not enough reference sites as

discussed at length in section 6.1.

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Chapter 7

SUMMARY AND CONCLUSIONS

7.1 Summary

The Poisson distribution is a commonly accepted assumption in analyzing traffic crashes.
When freeway interchange crash data were examined, it was found that there is
substantially larger variability than would be expected if the distribution followed
Poisson's law, and that the Negative Binomial distribution provides a better fit. This
research focused on several linked issues which occur with the assumption that traffic

crashes follow the Negative Binomial distribution rather than the Poisson distribution.

7.1.1 Traffic crash distribution

To test the distribution on freeway interchange crashes, the year to year variances were
calculated for crashes that occurred during the 5 year-period 1994-1998. Throughout this
study, it was found that there is greater variability than would be expected under the
assumption of the Poisson distribution, and the Negative Binomial distribution fits the

data much better than the Poisson distribution. That is,

0 The correlation coefficients between observed and expected variances increased from
0.91 to 0.97 and from 0.84 to 0.90 in the analysis of 24 crash types and 84 Diamond
interchanges, respectively, when the data were fitted to the Negative Binomial

instead of the Poisson distribution.

121

0 The squared residuals between observed and expected variances were reduced by

more than 80 % when the Negative Binomial distribution is assumed.

7.1.2 Traffic crash prediction model
One objective of the research was to develop crash prediction models for freeway

interchanges using the Negative Binomial error structure.

0 Based on the results of ANOVA and correlation, mainline ADT, ramp ADT, and
truck percent were selected as traffic variables that effect freeway interchange
crashes. The number of on and off ramps, the number of lanes, shoulder width,
interchange length, and average spread-ramp length were determined to be geometric

variables that affect accidents.

0 A non-linear regression model was selected as the model structure for the crash

prediction model developed in this study, and the model is:

E(B) = A x ViBi ijCj x epo(Ck x Gk)

where,
E(6) : Expected number of crashes
Vi :Traffic variables

G j : Geometric variables(type 1)

Gk : Geometric variables(type 2)
A,Bi ,Cj,Ck : Parameters

122

0 To calibrate this model, the Generalized Linear Model (GLIM) approach that

prevents the Negative Binomial error assumption from being violated was used.

0 Using several measures of assessing the goodness of fit of models, such as Pearson
Chi-square(x2), Dispersion parameter (Dp), Coefficients of determination (R2),

Pearson Residuals(PR) and so on, 10 crash prediction models were developed, one

for each of the most common interchange types in Michigan.

0 Large reductions in the coefficient of variation of parameter estimates were found
when the traffic crash prediction models were calibrated based on the Negative
Binomial error assumption. For example, the coefficient of variation of parameter
estimates in the models for interchange type 11 and type 12 were reduced by an
average of 36 percent when the models were calibrated under the Negative Binomial

error assumption rather than the Normal one.

7.1.3 The rate quality control method based on the Negative Binomial distribution
Since the accidents follow the Negative Binomial distribution rather than the Poisson
distribution, the rate quality control method needed to be reexamined because it is based

on the Poisson distribution. The findings can be summarized as follows:

0 The rate quality control method under the Poisson assumption identifies more sites as
hazardous than should theoretically be expected because the variance of the Poisson

distribution is equal to the mean, whereas the variance of the Negative Binomial

distribution (and the observed data) equals the mean + mean 2 /k.

123

O The Negative Binomial distribution parameters that were necessary for the
identification of hazardous sites were calculated using the maximum likelihood

method of estimation.

0 On the basis of the Negative Binomial error structure, the framework of a rate quality
control method was proposed for the identification of hazardous sites. This
framework produced more reasonable results than the existing approaches, such as
the rate quality control method assuming the Poisson error structure, or Bayes

approach.

7.1.4 The Normal approximation method

Even though the rate quality control method based on the Negative Binomial distribution
produced conceptually more reasonable results than the existing approaches, the
application of this method may not be efficient because the parameters can not be easily

estimated.

0 In order to overcome the difficulties associated with the estimation of parameters of
the Negative Binomial distribution based rate quality control method, a Normal
approximation method was proposed, and is shown to produce good results when

identifying hazardous locations.

0 The Normal approximation method identified hazardous sites with no loss of
accuracy, even though it is a relatively simple method based on the Negative

Binomial distribution.

124

0 The validity of the Normal approximation method was shown to be contingent on two

assumptions. These assumptions are: 1) the standardized random variables X, follow

the Normal distribution and 2) the expected mean pi is large enough.

0 The testing of the two assumptions showed that Xi does follow the Normal

distribution, and pi is large enough to allow for the accuracy of the Normal

approximation method.

7.1.5 The prediction model method

In applying the rate quality control method to the identification of hazardous sites, two
limitations were identified in this study. The conceptual problem is that the selection of
reference sites is a matter of judgement. The practical problem is that a site can not be
efficiently evaluated unless there is a sufficient number of reference sites to assure the

accuracy of the results.

0 To overcome the limitations of the rate quality control method, the prediction model
method was tested, and it was found that there is little difference between the rate
quality control method and the prediction model method in identifying hazardous
sites. This implies that we can evaluate the safety of the sites in a statistical sense

without reference sites.

0 Recognizing the accuracy and the availability of the prediction model method, about
200 freeway interchanges in Michigan were evaluated, 12 sites were identified at the

95 % probability level.

125

7.2 Conclusions
This research focused on the issues occurring when an assumption is made that traffic
crashes follow the Negative Binomial distribution rather than the Poisson distribution.

The following is conclusions that were reached in this study.

0 Crash prediction models for freeway interchanges can be efficiently calibrated under

the assumption of the Negative Binomial error structure.

0 The rate quality control method using the Negative Binomial distribution identified a
more reasonable set of abnormal sites than the existing methods such as the Poisson

based rate quality control method, or Bayes approach.

0 The Normal approximation method proposed for user convenience identified
hazardous sites without loss of accuracy, even though it is relatively simple compared

to the Negative Binomial based rate quality control method.

0 The prediction model method developed accurately identified the safety of sites in the

statistical sense.

126

 

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10.

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132

APPENDICES

Appendix A - The results of accident prediction model calibration

Appendix B - The results of evaluating of freeway interchanges

133

APPENDIX A

THE RESULTS OF ACCIDENT PREDICTION MODEL CALIBRATION

134

THE RESULTS OF ACCIDENT PREDICTION MODEL CALIBRATION

Table A.l The results of accident prediction model calibration (Interchange type 11)

 

 

 

 

 

 

 

 

 

 

 

Coefficient Variable definition Unit Estimate Std error T -
statistic
A Constant - 3.448
Log(A) (1.238) (0.67) (1.85)
B] V]: Mainline traffic volume per lane (ADT/1000) 1-401 0-30 4-66
82 V2 : Ramp traffic volume (ADT/1000) 0.186 0.12 1.55
°/ . 2 . .

B3 V3 :Truck percent ( O) O 6 O O 19 3 26
C] G] : Interchange length (Mile) 0-733 0-15 4-92
C2 G2 : Average spread ramp length (Mile) '1'267 0'97 ’1'31
C3 G3 : The number of lanes -

C4 G4 : The number of total ramps -

(Feet) 0156 0.12 -l .30
C 5 G5 : Shoulder width
Model statistic

DP Dispersion parameter 1.0

X2 Pearson chi -square 23.34 (12 0.95, 27 = 40_11)

R2 Coefficient of determination 0.60

K Negative Binomial parameter 3_05

 

 

 

 

 

135

Table A.2 The results of accident prediction model calibration (Interchange type 12)

 

 

 

 

 

 

 

 

 

 

 

 

Coefficient Variable definition Unit Estimate Std error t -
statistic

A Constant - 31.343

Log(A) (3.445) (0.73) (4.72)
B] V1: Mainline traffic volume per lane (ADT/1000) 0.946 0-24 3-94
82 V2 : Ramp traffic volume (ADT/1000)

°/
B3 V3 :Truck percent ( 0)
Cl G1 ; Interchange length (Mile) 0.933 0.36 2.59
C2 G2 : Average spread ramp length (Mile) '3 '842 1'31 ‘2'93
C3 G3 : The number of lanes -
C4 G4 : The number of total ramps -
(Feet)
C5 G5 : Shoulder width
Model statistic

Dp Dispersion parameter 1.0
X2 Pearson chi -square 14.66 (x2 0'95, 14 = 23.68)
R2 Coefficient of determination 0.33
K Negative Binomial parameter 10.74

 

 

 

 

136

 

Table A.3 The results of accident prediction model calibration (Interchange type 13)

 

 

 

 

 

 

 

 

 

 

 

 

Coefficient Variable definition Unit Estimate Std error t -
statistic
A Constant - 3.614
Log(A) (1.285) (1.07) (1.20)
81 V1: Mainline traffic volume per lane (ADT/ 1000) 0947 0.47 2-01
32 V2 . Ramp traffic volume (ADT/1000) 0.187 0.16 1.17
0

B3 V3 :Truck percent (4)
C1 G] ; [rum-change length (Mile) 0.816 0.22 3.71
C2 G2 : Average spread ramp length (Mile)
C3 G3: The number of lanes - 0136 0'10 1‘36
C4 G4 : The number of total ramps -

(Feet)
C5 G5 : Shoulder width

Model statistic

DP Dispersion parameter 1.0
X2 Pearson chi -square 1932 (X2 095‘ 19 = 30.14)
R2 Coefficient of determination 047
K Negative Binomial parameter 543

 

 

 

 

137

Table A.4 The results of accident prediction model calibration (Interchange type 14)

 

 

 

 

 

 

 

 

 

 

 

 

Coefficient Variable definition Unit Estimate Std error t -
statistic
A Constant - 17.531
Log(A) (2.864) (1.25) (2.29)

B] V]: Mainline traffic volume per lane (ADT/1000) 0-91 1 0-43 2-12
B2 V2 : Ramp traffic volume (ADT/1000) 0.142 0.14 1.00

0
B3 V3 :Truck percent U0)
C1 G1 : Interchange length (Mile) 1-315 0-33 3-93
C2 G2 : Average spread ramp length (Mile) ’2‘278 L984 '1‘15
C3 G3 : The number of lanes -
C4 G4 : The number of total ramps -

(Feet)
C5 G5 : Shoulder width
Model statistic

[)p Dispersion parameter 1.0
X2 Pearson chi -square 9.07 (12 095’ 9: 16.92)
R2 Coefficient of detemtination 0.65
K Negative Binomial parameter 6.38

 

 

 

 

138

Table A5 The results of accident prediction model calibration (Interchange type 21)

 

 

 

 

 

 

 

 

 

 

 

 

Coefficient Variable definition Unit Estimate Std error I-
statistic
A Constant - 5.479
Log(A) (1.701) (1.02) (1.67)
B1 V1: Mainline traffic volume per lane (ADT/1000) 0-467 043 1-09
32 V2 : Ramp traffic volume (ADT/1000) 0.470 0.18 2.61
0

B3 V3 :Truck percent U0)
C) G) : Interchange length (Mile)
C2 G2 : Average spread ramp length (Mile)
C3 G3 : The number of lanes -
C4 G4 : The number of total ramps -

(Feet)
C5 G5 : Shoulder width

Model statistic

DP Dispersion parameter 1.0
X2 Pearson chi -square 6.35 (x2 0 95, 6: 12.19)
R2 Coefficient of determination 0.68
K Negative Binomial parameter 6.73

 

 

 

 

139

Table A.6 The results of accident prediction model calibration (Interchange type 31)

 

 

 

 

 

 

 

 

 

 

 

 

Coefficient Variable definition Unit Estimate Std error t -
statistic
A Constant - 3.494
Log(A) (1.251 ) (0.83) (1.52)
B] V]: Mainline traffic volume per lane (ADT/1000) 1-144 0-24 4-77
32 V2 : Ramp traffic volume (ADT/1000) 0.128 0.11 1.16
0

B3 V3 :Truck percent (/o) 0.138 0.12 1.15
C) G] : Interchange length (Mile) 0-319 0-19 1-68
C 2 G2 : Average spread ramp length (Mile)
C3 G3 : The number of lanes -
C4 G4 : The number of total ramps -

(Feet)
C5 G5 2 Shoulder width

Model statistic

[)p Dispersion parameter 1.0
x2 Pearson chi -square 37.68 (x2 0,95, 35 = 51.00)
R2 Coefficient of determination 0.72
K Negative Binomial parameter 7.02

 

 

 

 

140

Table A.7 The results of accident prediction model calibration (Interchange type 33)

 

 

 

 

 

 

 

 

 

 

 

 

Coefficient Variable definition Unit Estimate Std error t -
statistic
44.124
A Constant -

Log(A) (3.787) (0.87) (1.20)
B] V1: Mainline traffic volume per lane (ADT/1000) 0-515 0-24 2-15
32 V2 : Ramp traffic volume (ADT/1000) 0.244 0.12 2.03

°/
B3 V3 :Truck percent ( 0)
Cl Gl ; Interchange length (Mlle) 0.956 0.24 3.98
C2 G2 : Average spread ramp length (Mile) '2'500 0'98 '2'55
C3 G3 : The number of lanes -
C4 G4 : The number of total ramps -

(Feet)
C5 G5 : Shoulder width
Model statistic

DP Dispersion parameter 1.0
X2 Pearson chi -square 16.23 (12 0'95, 16: 26.30)
R2 Coefficient of determination 0.82
K Negative Binomial parameter 13.3 5

 

 

 

141

 

Table A.8 The results of accident prediction model calibration (Interchange type 35)

 

 

 

 

 

 

 

 

 

 

 

 

Coefficient Variable definition Unit Estimate Std error t —
statistic
A Constant - 8.619
Log(A) (2.154) (1.19) (1.81)
B] V1: Mainline traffic volume per lane (ADT/1000) 0-736 032 0-90
32 V2 :Ramp traffic volume (ADT/1000) 0.270 0.41 0.66
°/

B3 V3 :Truck percent ( o)
C 1 G1 : Interchange length (Mile)
C 2 G2 : Average spread ramp length (Mile)
C3 G3 : The number of lanes -
C4 G4 : The number of total ramps -

(Feet)
C5 G5 : Shoulder width

Model statistic

Dp Dispersion parameter 1.0
X2 Pearson chi -square 5.36 (x2 0‘95, 5 = 11.07)
R2 Coefficient of determination 0.37
K Negative Binomial parameter 4. 3 5

 

 

 

142

 

Table A.9 The results of accident prediction model calibration (Interchange type 41)

 

 

 

 

 

 

 

 

 

 

 

1 Coefficient Variable definition Unit Estimate Std error t -
statistic
A Constant - 28.247
Log(A) (3.341) (2.344) (1.43)
B] V): Mainline traffic volume per lane (ADT/ 1000) 0-839 0-29 2.89
B2 V2 : Ramp traffic volume (ADT/1000) 0.215 0.15 1.43
°/
B3 V3 :Truck percent ( 0)
C1 G1 : Interchange length (Mile)
C 2 G2 : Average spread ramp length (Mile)
C3 G3 : The number of lanes -
- 0.182 0.06 3.03

C4 G4 : The number of total ramps

(Feet) -0.238 0.18 -1.32
C5 G5: Shoulder width

Model statistic

DP Dispersion parameter 1.0
x2 Pearson chi -square 17.99 (12 095 17 = 27.59)
R2 Coefficient of determination 0.64
K Negative Binomial parameter 6.37

 

 

 

 

 

143

 

Table A.10 The results of accident prediction model calibration (Interchange type
51)

 

 

 

 

 

 

 

 

 

 

 

 

 

Coefficient Variable definition Unit Estimate Std error t -
statistic
A Constant _ - 3.658
Log(A) (1.297) (1.23) (1.05)

B( V1: Mainline traffic volume per lane (ADT/1000) 0478 0.65 0.73
132 V2 : Ramp traffic volume (ADT/1000) 0.506 0.33 1.53
B3 V3 :Truck percent (0/0)
C] G1 : Interchange length (Mile)
C2 G2 : Average spread ramp length (Mile)
C3 G3 : The number of lanes -
C4 G4 : The number of total ramps -

(Feet)
C5 G5 : Shoulder width

Model statistic

Dp Dispersion parameter 1.0
X2 Pearson chi -square 5.19 (12 0.95, 5: 11.07)
R2 Coefficient of determination 0_47
K Negative Binomial parameter 4.86

 

 

 

 

144

APPENDIX B

THE RESULTS OF EVALUATING OF FREEWAY IN TERCHAN GES

145

 

 

 

 

 

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