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Accident prediction models for signalized intersections Feng, Shirley 1998

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ACCIDENT PREDICTION MODELS FOR SIGNALIZED INTERSECTIONS BY SHIRLEY FENG B.Sc, Beijing University of Aeronautics and Astronautics, 1988  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Civil Engineering) We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA December 21,1998 © Shirley Xiaoli Feng, 1998  In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her reporesentatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of Civil Engineering The university of British Columbia Vancouver, Canada  ABSTRACT  This thesis describes the development of accident prediction models for signalized intersections in the Greater Vancouver Regional District (GVRD). The traffic and roadrelated factors which appeared to underlie the occurrence of accidents are examined and models which explain, in a statistical sense, the generation of accidents as a function of these factors are developed. Recognizing the statistical and practical shortcomings associated with the use of the Conventional Linear Regression approach to develop accident prediction models, it was decided to utilize the Generalized Linear Regression Models (GLIM) approach. This approach addresses and overcomes the error structure problems that are associated with the conventional linear regression theory and allows for the use of nonlinear relationships in the model. In addition, the safety predictions obtained from the G L I M models can be refined using the Empirical Bayes' approach to provide, more accurate, site-specific safety estimates. The use of the complementary Empirical Bayes approach can significantly reduce the regression to the mean bias that are inherent in observed accident counts.  The study made use of sample accident, traffic and intersection design data corresponding to signalized intersections located in the Greater Vancouver Region. The accident data set contained 67 urban intersections from the City of Richmond and 72 urban intersections from the City of Vancouver giving a total of 139 intersections. Three different types of models were developed: (1) models relating the total number of accidents to traffic  volume; (2) models relating accidents of a specific type to traffic volume; and (3) models incorporating other geometric design variables such as the existence of left turn lanes, right turn lanes, pedestrian crossings, etc. The goodness of fit of the models was evaluated using two statistics: the Scaled Deviance (SD) and the Pearson % statistics. 2  The overall fit of the models was adequate. Three applications of the G L I M models and the Empirical Bayes refinement process were described. The first related to the identification of accident prone locations. The second related to the before and after safety analysis and the third to safety planning. The usefulness of the G L I M model estimates in accounting for the randomness inherent in the accident occurrence process and the regression to the mean bias was documented and discussed.  iv TABLE O F CONTENTS ABSTRACT  ii  TABLE OF CONTENTS  iv  LIST O F T A B L E S  vi  LIST O F FIGURES  vii  ACKNOWLEDGMENT  viii  1.0 I N T R O D U C T I O N .  1  1.1 BACKGROUND  1  1.2 AIMS AND OBJECTIVES  1  1.3 APPLICATION OF ACCIDENT PREDICTION MODELS  3  1.4 THESIS STRUCTURE  5  2.0 L I T E R A T U R E R E V I E W  5  2.1 CONVENTIONAL LINEAR REGRESSION MODELS  6  2.2 SHORTCOMINGS OF CONVENTIONAL LINEAR REGRESSION  9  2.3 GENERALIZED LINEAR REGRESSION MODELS WITH EMPIRICAL BAYES REFINEMENT . 10  2.3.1 Generalized Linear Regression Models  11  2.3.2 The Empirical Bayes Refinement  15  2.3.3 Applications  18  2.3.3.1  Accident Prediction Models for Intersections  18  2.3.3.2  Accident Classification at Intersections  22  2.3.3.3  Accident Prediction Models for Road Sections  23  2.4 SUMMARY 3.0 M O D E L D E V E L O P M E N T . 3.1 BACKGROUND  3.1.1 Accident Data  24 26 26  27  V 3.1.2 Traffic Exposure Measures  28  3.1.3 Intersection Characteristics  .  3.2 T H E MODELING TECHNIQUE  31  3.2.1 Measures of Significance 3.2.2 Model Development Models for the Total Number of Accidents  3.2.2.1. 3.2.2.2  Models for Specific Accident Types  3.2.2.3  Evaluation of the Effect of Intersection Layout Variables  4.0 APPLICATIONS  30  31 :  33 33 37 52  55  4.1 EMPIRICAL BAYES REFINEMENTS TO THE MODELS  55  4.2 IDENTIFICATION OF ACCIDENT PRONE LOCATION  59  4.3 BEFORE AND AFTER SAFETY EVALUATION  64  4.4 SAFETY PLANNING  65  5.0 CONCLUSION  67  REFERENCES  69  APPENDIX A  ACCIDENT DATA  APPENDIX B  MODELS FOR VANCOUVER AND RICHMOND  INTERSECTIONS  75  82  LIST O F T A B L E S  Table 3.1 Accident Variables from the MV104  28  Table 3.2 Statistical Summary for the Accident Traffic Volume Data for the Study Intersections.  29  Table 3.3 Intersection characteristics variables used in the study  30  Table 3.4 Models for the Estimation of the Total Number of Accidents  34  Table 3.5 Models for Specific Accident Types  38  Table 3.6 Parameter Estimates for the Model Incorporating Layout Variables  52  Table 3.7 Parameter Estimates for the Model Incorporating Layout Variables  53  vii  LIST OF FIGURES Figure 3.1 Model (1): Observed Versus Predicted Number of Accidents/yr  35  Figure 3.2 Model (2): Observed Versus Predicted Number of Accidents/yr  36  Figure 3.3. Model of PDO accidents  40  Figure 3.4. Model of injury accidents  41  Figure 3.5 Model of day time accidents  42  Figure 3.6 Model of night time accidents  43  Figure 3.7 Model of Rear-end accidents  44  Figure 3.8 Model of A M rush hour accidents  45  Figure 3.9 Model of P M rush hour accidents  46  Figure 3.10 Model of rush hour accidents  47  Figure 3.11 Model of Non rush hour accidents  48  Figure 3.12 Model of left turn accidents  49  Figure 3.13 Models of different types of Accidents  50  Figure 3.14 Models of time specific accidents  51  Figure 3.15 Observed number of accidents/yr vs. predicted number of accidents/yr  54  Figure 4.1 Predicted vs. E B Refined number of accidents  58  Figure 4.2 Identification of Accident Prone Locations  61  Figure 4.3. Accident Prone Locations for Total Model  63  Figure 4.4 Sensitivity Chart  66  Acknowledgment  It gives me great pleasure to thank my supervisor Dr. Tarek Sayed. Throughout my research, he carefully and patiently directed every step. He spent countless hours discussing the problem with me, and made numerous constructive suggestions to keep me in progress.  I must also thank the Insurance Corporation of British Columbia (ICBC) for the financial support. I also thank Hamilton Associates and the Engineering Division of City of Vancouver, City of Richmond for their data supply.  Finally, I thank my husband Jing Qu for his support and encouragement. This work is respectfully dedicated to Jing Qu.  1  1.0 I N T R O D U C T I O N  1.1 Background  Traffic safety in British Columbia is a serious concern of traffic analysts and road authorities. In 1995, there were 534 fatalities, about 50,000 injuries and 180,000 property damage only accidents. The direct annual cost to the province exceeds 2.0 billion dollars (ICBC 1995 Annual Report, Vancouver, B.C.).  Recognizing these safety problems and the need to reduce the social and economic costs associated with them, road safety authorities have established Road Safety Improvement Programs (RSIPs).  The objective of these programs is to monitor traffic conditions,  collect and analyze accident data, locate trouble spots with abnormally high accident occurrences and implement appropriate and effective countermeasures in order to improve the safety potential of these sites. The success of these Road Safety Improvement Programs can be enhanced by developing reliable accident prediction models, which provide accurate estimates for the long-term safety potential at the locations under study. The development of such models is the focus of this thesis.  1.2 Aims and Objectives  The main objective of this thesis is to develop accident prediction models for estimating the long-term safety potential for signalized intersections in the Greater Vancouver  2  Regional District (GVRD). This research stemmed from the apparent need to estimate the safety potential of these intersections on the basis of their traffic and road geometric design characteristics. The aim, therefore, is to examine the traffic and road-related factors that appeared to underlie the occurrence of accidents and explain, in statistical sense, the generation of accidents as a function of these factors. The specific objectives of the study evolved as follows:  •  Analyze the road, traffic and accident data available for signalized intersections under study in order to establish statistical models describing the empirical relationships between accident measures and the road and traffic variables. The accuracy and significance of each model and its coefficient estimates are investigated and discussed.  •  Investigate the importance of various road geometric design variables in affecting the safety of signalized intersections. Some of these variables include: number of lanes, existence of left/right turning lanes and pedestrian crosswalk.  •  Develop time-specific accident prediction models. These models may serve to address safety issues such as, comparing the intersection safety during peak hour to off peak hour, night to day, or A M peak hour vs. P M peak hour. Such time-specific models may be useful for evaluating strategies that affect traffic volumes or safety during certain time periods of the day and identifying potentially hazardous operating conditions.  3  •  Develop type-specific accident prediction models. These models may be useful in estimating the occurrence of specific types of accidents such as left-turn and rear-end accidents. Often these specific types of accidents can be targeted by specific measures. For instance, the safety of an intersection that exhibits an abnormally high occurrence of left-turning accidents may be improved by implementing a left-turn lane or phase.  •  Assess the reliability and interpret the significance of various models and detect any patterns in accident occurrence, identify and relate high-risk traffic and road characteristics to unsafety, estimate the safety potential of the implementation of certain road features and provide a global as well as local assessment of traffic safety of the G V R D signalized intersections.  There are several potential applications for the proposed accident prediction models. These applications are outlined in the next section.  1.3 Application of Accident Prediction Models  The development of reliable accident prediction models for signalized intersections offers a useful tool in a number of respects. Some of the potential applications of such models include:  •  Identifying accident prone signalized intersections: the occurrence of a relatively high number of accidents at a particular location, though undesirable, does not always  4  mean that the location concerned is a blackspot or would benefit from remedial treatment. Also, due to the regression to the mean phenomenon (as will be explained), an abnormally high number of accidents may not reflect the long-term accident occurrence at the location, and could in fact be followed by a relatively low number of accidents over a similar succeeding period of time even i f no changes were introduced to the location. Therefore, in identifying accident prone locations, the true accident potential for each site should be estimated. This can be achieved by using accident prediction models.  •  Before and after studies: to estimate the effectiveness of a safety measure, the expected number of accidents had the measure not implemented need to be estimated. This can be achieved by using accident prediction models and the Empirical Bayes refinement as will be discussed later.  •  In safety planning through identifying the traffic and geometric variables which have the most impact on the safety performance of signalized intersections so that the road authorities can focus their attention and investment on targeting these variables. Furthermore, the change of safety associated with the change in any traffic or geometric design variables can be estimated (e.g. the change of safety associated with increased traffic volume, etc.)  1.4 Thesis Structure  Chapter One provides an overview of the thesis and its structure. A literature review of the various techniques for developing accident prediction models and the main concepts behind the Generalized Linear Regression Modeling technique is outlined in Chapter Two. Chapter Three describes the accident data and the accident prediction models developed. Chapter Four discuses three applications of accident prediction models. Chapter Five provides suggestions for follow up work and the summary and conclusion of the thesis.  6  2.0 L I T E R A T U R E  REVIEW  The past decade has seen significant developments and advances in accident data analysis and modeling. Accident prediction models are no longer limited to the conventional linear regression assumptions, as more suitable and less restrictive nonlinear models can now be considered. The use of the Bayes' theory for developing accident prediction models has also been an important development in the data analysis literature. In this section, a brief outline of the major developments in accident prediction models will be presented.  2.1 Conventional Linear Regression Models  Conventional Linear Regression (CLR) models were initially used to describe the empirical relationships between accidents and traffic and road geometric design variables, such as traffic exposure (volume), horizontal curvature, vertical grade, lane width, and others (Zegeer et a l . , 1987, Miaou et al., 1991). The underlying assumption of these C L R models is that the number of accidents, 7/ at a site /, in a reference population of size n, varies linearly with a set of m traffic and road geometric variables, { x x^,...,^,,,} as follows: il;  m  where a , a], 0  a  (2.1)  m  are the model parameters to be estimated by the least squares or the  maximum likelihood method. These estimates can be obtained by using any of the  7  numerous standard linear regression computer software available. The term e/ measures the error, i.e. the difference between the model estimate and the observed accident number at the site. It is assumed that the error terms ej, i=l,2,...n, at all sites of the reference population are independently and normally distributed random variables with zero mean and variance cr >0 . As well be explained later, this assumption is restrictive and may not 2  be suitable for the typically discrete and non-negative accident data.  Lau and May (1988) investigated the use of conventional linear regression model to describe a simple relationship between the number of injury accidents per year and the average yearly traffic volume for a population of 2,488 intersections in California. It was demonstrated for this data set that traffic exposure was the most important single factor in predicting injury accidents, and the following model was found to provide the best fit:  Injury accidents/year = a + a\. (millions of entering vehicles/year) (2.2) 0  where a and a are the model parameters . The model was also used to predict the number 0  l  of property damage only (PDO) and fatal accidents.  Conventional linear regression models were also used in discriminate analysis to investigate whether a site is predicted to exhibit a potentially high number of certain types of accident. Al-Senan et al (1987) used the discriminate analysis technique to identify the significant predictors of head-on accident on highway sections. Although head-on crashes are relatively rare, this class of vehicular accidents accounted for 14.6% of highway fatalities in the  8  United States for the period 1982 to 1984 (Accident Facts, 1985). The authors proposed prediction models for head-on accidents sites to identify sites that are prone to head-on accidents based on traffic and road design variables.  The following traffic and road features were found to be significant predictors of head-on accident proneness of a highway section: pavement width, shoulder width, pavementshoulder combination, horizontal alignment, vertical alignment, combined horizontalvertical alignment, roadside elements, traffic control features.  The discriminate analysis  was used to distinguish between two groups of sites, head-on crash sites and control sites (which have less potential for head-on crashes), based on the above geometric and traffic control features. The discriminate analysis uses the following conventional linear regression model to describe the empirical relationship between the discriminate function D and a set of traffic and road geometric design variables {V ,V ,...,V } as follows: 1  D = a + a,V, + a V 0  2  2  +...+ a V p  p  2  p  (2.3)  The section is then assigned to the head-on group if the resulting discriminate score, D is less than zero; otherwise it is assigned to the control group. Discriminate analysis was shown to be a logical and convenient way to differentiate between sites with different accident contributing factors.  9  2.2 Shortcomings of Conventional Linear Regression  Conventional linear regression (CLR) models were initially widely used to develop accident prediction models, mainly because of their simplicity. However, their success has been generally limited. It has been demonstrated that C L R models lack the distributional property to adequately describe random, discrete, non-negative, and typically sporadic vehicle accident events on the road (Jovanis and Chang, 1986, Saccomanno and Buyco 1988, Miaou and Lam 1993).  Many difficulties were associated with the use of conventional linear regression to build accident prediction models for accident data (Persaud 1992). First, it is assumed, a priori, that accidents are proportional to traffic volume and the accident rate is used as the dependent variable. However, there has been much research to suggest that this assumption is not only inaccurate but it also leads to contradicting results (Mahalel, 1985). Secondly, most conventional regression modeling software assumes that the dependent variable has a normally distributed error structure. For accident counts, which are discrete and nonnegative, this is clearly not the case; in fact a negative binomial or Poisson error structure has been shown to be more appropriate (Persaud, 1989).  Another difficulty with the use of the conventional regression models lies in the unreliability of the estimates. Regression estimation suggests that two sites of the same reference population and which have similar independent variables (i.e. traffic and road geometric factors) values would have the same accident potential. However, in general, this  10  is not the case, since it is not possible to account for all the factors that cause differences in accident potential among similar sites.  The need to overcome these difficulties was  fundamental to developing a more reliable and accurate regression model describing the relationship between the unsafety and the traffic and geometric traits.  The use of Generalized Linear Regression Modeling (GLIM) theory and software to' develop accident prediction models has been investigated (Hauer et al., 1988; Persaud, 1989). The G L I M approach addresses and overcomes the error structure problems that are associated with the conventional linear regression theory and allows for the use of nonlinear relationships in the model. The use of the G L I M models for accident prediction is the topic of the following section. To distinguish between the safety estimates of similar sites, an Empirical Bayesian method that combines the regression model prediction with the observed short-term accident count at each location will also be discussed.  2.3 Generalized Linear Regression Models with Empirical Bayes Refinement  The need to overcome the statistical and practical shortcomings associated with the use of conventional regression models for accident prediction was fundamental to the development of realistic and more general empirical relationships between the accident potential, traffic and road design variables. To this end, use was made of a statistical generalized linear regression modeling package (GLIM) that allows for the flexibility of nonlinear accidenttraffic relationship and user specific error structure for the dependent variable. Use was also made of a complementary Empirical Bayesian procedure for improving the accuracy of the  11  regression model accident predictions. The theory of these developments will be briefly outlined next.  2.3.1 Generalized Linear Regression Models  The Generalized Linear Regression Method (GLIM) allows for non-linear empirical relationships between the dependent and the independent variables that can be linearized by taking its logarithmic function. G L I M is a recently developed statistical software package (Baker and Nelder, 1978) that is being widely used in accident data analysis. It also allows the specification of a negative binomial or a Poisson error structure for the dependent variable, which as noted earlier, is more appropriate for accident counts than the traditional normal distribution (Persaud, 1989).  The G L I M approach utilized in this study is based on the work of Kulmala (1995) and Hauer et al. (1988).  Assume that 7 is a random variable describing the number of  accidents at an intersection in a specific time period, and y is the observation of this variable during a period of time. The mean of Y is A which is itself can be regarded as a random variable. Then for A - A , Y is Poisson distributed with parameter A :  p(Y-y\A  = JL) =  -;E(Y\A  = A) = A; Var(Y\A = A) = A  (2.4)  12  Hauer et al. (1988) have shown that A follows a gamma distribution (with parameters K and K/U), where K is the shape parameter and u. is the mean of the distribution:  fW  =  h  ; £ (A) = ; Var(A) = ^ -  )  M  T(K)  (2.5)  K  Kulmala (1995) has also shown that the point probability function of Ybased on (2.4) and (2.5) is given by the negative binomial distribution:  (2  T(ic)y\  .  6)  K + JU K + JU  with an expected value and variance of:  M Var(Y) = //+—  2  E(Y)= ; M  (2.7)  As shown in equation (2.6), the variance of the expected number of accidents is generally larger than its expected value. The only exception is when K - * oo, where the distribution of  is concentrated at a point and the negative binomial distribution is identical to the  Poisson distribution (Kulmala, 1995).  As described earlier, for the generalized linear regression modeling approach, the error structure is usually assumed to be Poisson or negative binomial. The main advantage of  13  the Poisson error structure is the simplicity of the calculations (the mean and the variance are equal). However, this advantage is also a limitation. It has been shown (Kulmala, 1995) that most accident data will likely to be over dispersed (the variance is greater than the mean) which indicates that the negative binomial distribution is the more realistic assumption. However, the difference between the model parameters estimated using the Poisson and the negative binomial models was found to be very small (Kulmala, 1995). Therefore, the simpler Poisson error structure assumption will be used in this study.  The first step in developing a G L I M model is to subgroup similar sites that share a set of traffic or geometric characteristics in a reference population. The empirical relationship between the unsafety potential, measured in terms of the number of accidents at the location, and the traffic and road geometric design variables {x, x^...,^} is then described in an equation form as follows:  Accidents / year = function (traffic and road geometric design variables; x i , x , . . . x ) 2  m  where the form of the function is appropriately chosen so that it can be linearized by taking the logarithm. The above regression estimate would not be useful i f no measure of its variability is known. It has been shown that, using the generalized linear regression, the variance of the regression estimate is directly proportional to the square of the model safety estimate in the following fashion (Persaud, 1989):  14  (regression estimate) variance(regression estimate) = k  (2.8)  where the value of k depends of the structural error assumptions of G L I M . The estimation of the variance from the G L I M regression estimates is difficult since both k and the variance in equation (2.4) are unknown. Hauer et al. (1988) suggested an iterative process to calculate k using the maximum likelihood method. Kulmala (1995) showed that the method of moments (assuming a Poisson model) can also be used to calculate k as follows:  ZE\m,)  L  k  =  1 " r *" -U(x,-E(m,)) n  N  1=1  L  I  1 -E{m,)  ( 2  -  9 )  J  where E(m) is the regression estimate of the number of accidents at the location, and Xi is the location observed number of accidents.  Kulmala (1995) found that the method of moments produced accurate enough estimates that deviated less than 5% from those produced by the maximum likelihood method. He also indicated that the iterative process suggested by Hauer et al. (1988) was time consuming and changed the value of k only to a minor extent.  One of the main conceptual difficulties associated with the G L I M model involves the choice of the form of the function that describes the dependence of the unsafely potential of  15  the site on the traffic and road variables. Unlike conventional regression models, which restrict the function to be linear, The G L I M model allows nonlinear forms. Often, a scatter plot of the number of accidents as a function of the traffic and road geometric variables may provide some hints for choosing the function form. However, the choice of the form of the function is generally intuitive and lacks theoretical justifications. Since the choice of the function form is a matter of judgment, different analysts may make different choices, and therefore, arriving at discrepant estimates for the unsafely of the same entity.  Another difficulty with the use of regression models to estimate the unsafety of a site is that the above equation suggests that two sites in the same reference population that have similar values for the independent variables, x\ X2 •••,x are expected to have similar unsafety t  t  m  estimates. However, in general, this is not the case, since it is not possible to account for all the factors that cause differences in accident potentials among similar sites. The use of the Empirical Bayesian method, which combines the regression model safety estimate with the sites-specific accident count, to refine the G L I M estimates could alleviate these shortcomings.  2.3.2 The Empirical Bayes Refinement  There are two types of clues to the unsafety of an entity: its traffic and road geometric design characteristics, and its historical accident data (Hauer, 1992). The G L I M method provides an unsafety estimate of a site based on the first type of clues. The Empirical Bayes (EB) approach to unsafety estimation makes use of both kinds of clues. The E B method  16  combines the G L I M unsafety estimate with the site-specific accident history and yields better estimates of unsafety.  To illustrate, suppose that an entity is located in a reference population characterized by a set of traits. Let pred, and var (pred), be the G L I M model estimates for the location's safety and its variance respectively, as described in the previous section. Suppose that the observed number of accidents at this site during the specified period of time is given by count. The Empirical Bayesian method combines the G L I M model estimates with the observed accident count to obtain a more refined, site-specific unsafety estimate, in the following fashion (Hauer, 1992):  EB safety estimate = a.pred + (1 -a).count, where a =  var(pred) (2.10) pred  where a = +  — var(prect) pred  (2-11)  The form of the above equation is consistent with the following reasoning: if the reference population is homogeneous (i.e. its sites are very similar), then one would expect small variations among the G L I M model safety estimates (i.e. var (pred)  0), therefore  1, and  the above equation yields an EB safety estimate that is close to the G L I M model estimate, pred, as it should be since the effects of the accident counts should not influence the estimate as differences in accident counts among sites may be attributed to chance  17  variations. On the other hand, if the reference population is heterogeneous (i.e. the sites are diverse), then one would expect a relatively high G L I M estimates variations, var(pred), and a»0.  The above equation yields an EB estimate that is close to the observed accident count and what is known about the reference population exerts little influence on the estimation. This again is as x should be, since the locations are very diverse, then differences in accident counts should be attributed to the differences among sites and not to chance variations. In view of the absence of a theoretical justification for the above equation, it is important to know that the E B method combines the regression estimates with the site-specific accident counts in a practical manner (Hauer, 1992).  In addition to combining the two types of safety clues and providing site-specific safety estimates, it has also been shown that the EB method significantly reduces the regression to the mean effects that are inherent in observed accidents count (Brude and Larsson, 1988).  The regression to the mean is a statistical phenomenon by which a randomly large number of accidents for a certain entity during a before period is normally followed by a reduced number of accidents during a similar after period, even i f no measures have been implemented (while the opposite applies in the case of a randomly small number of accidents). As mentioned earlier, before and after studies are one of the main applications of the accident prediction models. For instance, in a before and after study of the effect of a particular action, one should not compare the after period accident counts to the before  18  period accident counts, this generally leads to misleading assessment of the safety improvement effect afforded by the undertaken measures. Instead, one should compare the after period accident counts to the prediction model safety estimates, had no measures been implemented.  2.3.3 Applications  The above two-step procedure, where the G L I M estimates are refined by using the Empirical Bayesian method has been used in the literature to estimate the unsafety of road sections and intersections. Next a brief review of some of these applications for developing the models for intersections will be presented.  2.3.3.1 Accident Prediction Models for Intersections  Although many studies have addressed the relationship between accidents and the traffic and geometric factors at road segments (Jovanis and Chang, 1986, Saccomanno and Buyco 1988, Miaou and Lam 1993), only a few studies have addressed this relationship for signalized intersections (Poch and Mannering, 1996). This is surprising given that accidents at signalized intersections represent a significant proportion (more than half) of the total accident population, especially in urban areas.  Thus, the development of accident  prediction models for intersections is of great importance. These accidents usually involve multiple vehicles. Numerous models to estimate the safety potential of an intersection on the basis of its traffic flow, geometric design features and accident history have been  19  suggested over the years. Initially, it was suggested that the number of all accidents at an intersection is proportional to the sum of flows that enter the intersection. The merit of this approach is its simplicity. However, it has several shortcomings. Such models assume a uniform traffic flow through the intersection. However, this is usually not the case, since most intersections consist of a major road, with a higher average annual daily traffic volume (AADT) crossing a minor one with a lower (AADT).  A commonly used G L I M accident prediction model for intersections that makes the distinction between the major and minor road traffic flow, relates the unsafety potential of the intersection as a function of the A A D T for the minor and the major roads as follows:  Accidents / year  — #0 X (AADTmajor road) X  (AADT  minor road)  (2-12)  Webb (1955) used data for rural signalized intersections in California developed the following model:  Accident I Year = 0M7(AAM ^)°'\AADT f» m  llM  (2.13)  Lau et. al. (1989) developed separate models for fatal, injury, and property damage only (PDO) accidents as follows:  PDO Accidents / year = 4.63 + 0,514 (the sum of entering vehicles in million)  20  Injury accidents / year = 0.62+0.169 (the sum of entering vehicles in million) Fatal Accidents /year = 0.018  (2.14)  Bonneson et al. (1993) developed a similar model to that of Webb (1955) using data for non-urban signalized intersections:  Accident I Year = QMnXAADT^^f^AADT^^?**  (2.15)  Bonneson and McCoy (1993) have found that this model provided a best fit for accident data from 125 rural unsignalized intersection in Minnesota for the three years 1985-1987. The same model was found to fit accident data from signalized urban intersections in Philadelphia (Persaud et al 1995), accident data from 149 unsignalized rural intersections in Quebec (Belanger, 1995), and accident data from signalized rural intersections in Virginia ( Hanna, 1976).  Brude and Larsson (1988) investigated a different form of combining the minor road and major road traffic flows. They suggested the following G L I M prediction model for fourlegged signalized junctions:  /  r.rr*  ACCidents/year = a ^(AADTmajor 0  i  road  +  j  r\Ti  \ai  AADTminor road)  AADT minor road  /  Xf AADL  major road  4AT)T + AAL)1  1 zr\  V - ,> z  -  mimrroaa  10  21  This model was found to provide a better fit for the accident data set under consideration than the previous model which relates the number of accidents to the traffic flow on the minor and the major road in a similar manner.  Kulmala (1992) used the following G L I M model to estimate the expected number of accidents for highway junctions in a reference population as a function of the total number of vehicles entering the junction, the minor road's portion of entering traffic and a set of variables describing the geometry and the environment of the junction, {x Xj,...^}, as l5  follows:  Accidents/year/km  = a x (AADJ\ tai) '*('AAD1\ wad)"' xe a  0  0  !  linor  '  (2-17)  In a similar fashion to the Empirical Bayes refinement discussed earlier, the above regression estimates are then combined with the accident history for each site to yield a sitespecific refined estimates. Kulmala (1992) has used this GLIM/EB accident prediction model to estimate the changes in the number of accidents due to road measures in a before and after study. Most notably, it was observed that the number of accidents at three-leg intersections was reduced by 44% and 48% by implementing stop signs and lighting respectively.  Similarly, for four-leg intersections, the above countermeasures reduced the accidents by 5% and 15% respectively. It is also worth noting that the implementation of a right turning  22  lane has resulted in a 17% increase in the number of accidents for three-leg junctions and a 17% decrease for four-leg junctions.  2.3.3.2 Accident Classification at Intersections  Hauer et al (1988) produced disaggregate models for 145 four-legged, fixed time, signalized intersections in Metropolitan Toronto. He estimated separate models for the most common accident types, such as rear-end, angle accidents, etc. He used the following model forms:  Rear-end:  Accidents = a x (approach traffic) 0  (2.18)  Other accident types:  Accidents - a x(V ) ' a  0  x  x(V ) 2  (2.19)  where Vj and V2 are the pattern specific traffic volumes.  Categorizing accident in this fashion has many advantages: better accident prediction models for specific types of accidents can be obtained; also from a remedy point of view, one could implement specific countermeasures to target specific types of accidents (such as  23  constructing a left-turn lane for intersections with high predicted frequency of left-turning and straight through accidents).  2.3.3.3 Accident Prediction Models for Road Sections  The traffic exposure for road sections is typically defined in terms of the average annual daily traffic in millions of vehicles per kilometer. The development of accident prediction models for road, especially freeway sections, is of great interest because of the severity and frequency of these, typically high speed, accidents. Persaud (1991) used the G L I M model with the two step E B refinement procedure described earlier to study the unsafety potential of three types of classes of Ontario road sections. The G L I M model suggested for the unsafety estimates as a function of the section length and the annual average daily traffic was given by:  Accidents / day = a x (section length) x (AADT)"' 0  (2.20)  where a, b are the model parameters estimated by G L I M . The G L I M estimates are then refined using the EB method.  It is worth mentioning that one would expect that the unsafety estimate of a road section would depend on many traffic and road geometric variables other than the section length and the average daily traffic. However, this model does indirectly account for many other  24  variables through using three highway classes. For instance, class 1 road sections consists of  freeway sections that are multi- lane, divided, have the same high geometric  standards, and are usually similar in other features such as speed limit.  Therefore, it seems unlikely that a model which incorporates some of these road geometric variables would be significantly better than the above traffic volume, road section length model. This is because, these variables are kept practically constant within the class 1 reference group. Indeed, the author has shown that many attempts to add more geometric variables to the model did not result in significant difference in the estimates. This is an important observation, since one can deduce that simple G L I M accident prediction models can be developed i f a reference population, in which some traffic and road geometric variables are kept constant, is appropriately chosen.  Several researchers have developed multi-variate models which consider other variables in addition to traffic flow. For example, Zeeger et al (1986) suggested the following multi-variate G L I M model for two-lane rural roads:  Accidents/(year - km) = ao AADT x  ai  average unpaved shoulder width  a.4  x  lane width x  ci2  x  average paved shoulder width  ci3  median recovery distance edge of shoulder  as  Example for other multi-variate models can be found in Forkenbrock et al. (1994).  (2.21)  25  2.4 Summary  Although many studies have addressed the relationship between accidents and the traffic and geometric factors at road segments, only a few studies have addressed this relationship for signalized intersections. This is surprising given that accidents at signalized intersections represent a significant proportion of the total accident population, especially in urban areas. Most of the existing research on accident prediction models for signalized intersections only consider the relationship between accidents and traffic flow.  Several researchers have shown that conventional linear regression models lack the distributional property to adequately describe random, discrete, non-negative, and typically sporadic events which are all characteristics of traffic accidents. The generalized linear modeling (GLIM) approach addresses and overcomes the error structure problems that are associated with the conventional linear regression theory and allows for the use of nonlinear relationships in the model. In addition, the safety predictions obtained from G L I M models can be refined using the Empirical Bayes approach to provide, more accurate, site-specific safety estimates. The use of the complementary Empirical Bayes approach can significantly reduce the regression to the mean bias that is inherent in observed accident counts.  There several applications of accident prediction models and the complementary Empirical Bayes refinement process including: the identification of accident prone locations, before, and after safety analysis and safety planning.  26  3.0 M O D E L  DEVELOPMENT  3.1 Background r The research made use of sample accident, traffic and intersection design data corresponding to signalized intersections located in the Greater Vancouver Region. The data set contained 67 urban intersections from the City of Richmond and 72 urban intersections from the City of Vancouver giving a total of 139 intersections.  Before the analysis is considered, it is essential to provide a description of the data. The need to describe the data stems from various principles of accident data analysis and prediction model development which include:  • Accident prediction models are often data dependent, in the sense that an empirical relationship that provides a best-fit for accident data for locations in the Greater Vancouver, may not be the best-fit model for accident data for sites in Metro Toronto, for instance. In the absence of a unique and conventional accident prediction model that fits accident data everywhere, the mathematical forms of these empirical relationships are often intuitive and data dependent, and are only meant to describe the original accident data under study.  •  The accuracy and reliability of these models also depend on the accuracy, the availability and the collection procedure of the accident data. The success of these  27  accident prediction models in reliably estimating the long term safety potential of the signalized intersections under study is directly proportional to the quality of accident data which are available. Consequently, the reliability of the applications of these models such as identification of accident prone locations and before and after studies is closely related to the accuracy and availability of the accident data.  Therefore, in order to develop reliable accident prediction models for the signalized intersection under study, time and effort was devoted to checking and validating the accident data. In this section, a brief description of accident data, traffic exposure measure and intersections characteristics is provided.  3.1.1 Accident Data  Three years of accident data was available for analysis on each intersection (1993 - 1995). The source of the accident data is the MV104 accident reporting form, the British Columbia's accident police report. The MV104 police report is the principal tool used to collect information giving accidents in British Columbia. In this form, there are about one hundred pieces of information giving the accident circumstances, type and outcome as well as the characteristics of the driver(s), the vehicle involved and the locations of the accident. The data set included a total of 6255 accidents.  28  Table 1 provides a summary of the variables extracted from the MV104 forms for use in this study. These variables will be considered as the dependent variables in the multipleregression accident prediction models. The data is included in Appendix A .  Variable  Description  TOTAL  Total number of accidents  PDO  Number of Property Damage Only (PDO) accidents  INJ  Number of injury accidents  PEAK(AM)  Number of morning peak hour accidents  PEAK(PM)  Number of afternoon peak hour accidents  NIGHT  Number of night-time accidents  DAY  Number of day-time accidents  LEFT  Number of left-turn accidents  REAR-END  Number of rear-end accidents  Table 3.1. Accident Variables from the MV104  3.1.2 Traffic Exposure Measures  The most commonly used intersection accident exposure is the total number of vehicles entering the intersection (sum of traffic flows). However, recent studies (Hauer et al. (1988) and others) have indicated that the "product-of-traffic-flows-to-power" model is  29  more suitable to represent traffic exposure than the "sum of traffic flows". In these models accident frequency is a function of the product of traffic flows raised to a specific power (usually less than one).  Hauer et al. (1988) considered only traffic flows related to the accident pattern. Others, related the accident frequency to the product of average daily traffic of the major and minor roads. This later approach will be used in this study because of the difficulty in obtaining accident patterns-related traffic flows.  Table 3.2 provides a statistical summary of the ranges of traffic volume and accidents for the intersections used in this study.  Statistics  Variable Minimum  Maximum  Mean  Std. Dev.  Major Road A D T  5700  67840  23790  9670  Minor Road A D T  300  54060  9420  7250  Accident/Year  2  42  15  9  Table 3.2 Statistical Summary for the Accident Traffic Volume Data for the Study Intersections  30  3.1.3 Intersection Characteristics  The objective of this research is to develop accident prediction models that link accident measures to not only traffic exposure but also to the intersection infrastructure and geometric design characteristics. The aim of accounting for these intersection attributes was to examine the road-related factors which appeared to underlie the occurrence of accidents. The intersection characteristics variables included in the model are summarized in Table 3.3. Some of the data was collected by direct observation of the intersections, and others are recorded from intersection layouts provided by the City of Vancouver and the City of Richmond.  Variable Description MANL  Number of Major Road lanes  MINL  Number of Minor Road Lanes  LT  Number of Left Turn Lanes  PRO  0, Unprotected; 1, Protected Left Turn Lane  RT  Number of Right Turn Lanes  PC  Existence of Pedestrian Crosswalk  Table 3.3 Intersection characteristics variables used in the study  31  3.2 The Modeling Technique  As described in Section 2.0, the use of Generalized Linear Regression (GLIM) models overcomes the statistical and practical shortcomings associated with the use of conventional regression models for accident prediction. The G L I M method allows for the flexibility of nonlinear accident-traffic relationship and the specification of a negative binomial or a Poisson error structure for the dependent variable which is more appropriate for accident counts than the traditional normal distribution. In the present study the Poisson distribution has been assumed for the error structure.  3.2.1 Measures of Significance  In conventional regression models, where a normal error structure is assumed, the coefficient of determination (R ) 2  is usually used to indicate the model significance.  However, the use of (R ) is not appropriate when the error structure is other than normal 2  (Belanger, 1993).  Significance tests for G L I M models are based on-a scaled deviance SD - a likelihood measure of discrepancy between two models.  In G L I M , the SD is defined as the  likelihood test ratios measuring the difference between the log likelihood of the studied model and the saturated model. Differences between scaled deviance for the two models, one of which is a submodel of the other, is attributed to the extra parameters included in the larger model. This provides a method of assessing the significance of each factor,  32  which is based on the decrease of the scaled deviance SD after the inclusion of a considered factor into the model. Thus the deviance fills the role of the residual sum of squares in the normal model, in providing a significance test for the importance of parameters omitted or added to the model. The difference in scaled deviance between two models is assumed to follow a Chi-Square ( % ) distribution, and the significance of such 2  difference is assessed based on the degrees of freedom (n-p) and the value read from the X table. 2  Another measure of significance is the Pearson % statistic defined as: 2  (3.2)  Miao et al. (1992), Bonneson and McCoy (1993), and Persaud and Dzbik (1993) evaluated their models using the Pearson % method which follows a % distribution. 2  2  Both the scaled deviance SD and the Pearson % will be used for the indication of the 2  goodness of fit in this research.  In addition, a useful subjective, measure of the model goodness of fit is to plot the predicted accident frequency versus the observed accident frequency. A well-fitted model should have all points in the graph clustered symmetrically around the 45° line.  33  3.2.2 Model Development  The main task of this research is to develop multivariate models to estimate the number of accidents. In this project, modeling was undertaken in three stages: (1) developing models relating the total number of accidents to traffic volume; (2) developing models relating accidents of a specific type to traffic volume; and (3) developing models incorporating intersection layout variables.  Because of the relatively small sample size, results for the development of separate models for the city of Vancouver and the city of Richmond is not reported. However, they are included in Appendix B.  3.2.2.1. Models for the Total Number of Accidents  As described earlier, there are different functional forms which can be used to relate the accident occurring at an intersection with the traffic volume. Two models where selected for use in this study:  (1) The model used by Hauer et. al. (1988) and Bonneson and McCoy (1993):  Accidents / year —  QO  x (AADT  j rraad)  ma 0  x (AADT j  where A A D T is the Average Daily traffic in 1000 veh/day.  m  d)  norroa  2  (3-3)  34  The model used by Brude and Larsson (1988)  Accidents/year =  OQ  X  (A.ADTmajor road  +  A  A  D  T  ^  r  AAD1  ^  r " t + AAL) L  major road  )  ai  (3-4)  MMORRAATL  where A A D T is the Average Daily traffic in 1000 veh/day.  The results of fitting the two models to the accident data are illustrated in Table 3.4.  Model 1  k  Pearson  2.1813  355.2  9.0  x 128.40*  a,  0.3286  (136)  a  0.4418 8.8  128.57*  a  E(m) = a x V°> x V "  2  0  2  SD (df)  Coefficient Estimates  Model Form  2  K E(m) = a x(v + v y> x ( — — )° v + v 2  0  K  1  2 2 J  0  2  1.9066  360.7  ai  0.7432  (136)  02  0.3622  2  2  J  x  2  » Denotes significance at a 95-percent confidence level  (^ o.o5,i36 2  = 163.8).  Table 3.4 Models for the Estimation of the Total Number of Accidents  Table 3.4 shows that the two models present a relatively good-fit with the first model having a slightly higher goodness of fit (a smaller SD). Figures 3.1 and 3.2 show the relationships between the observed number of accidents/year versus the predicted number of accidents/year for the two models. The results are symmetrically clustered around the 45° line to a reasonable extent; which is desirable. The dispersion of the results from the 45° line is also within acceptable limits.  35  10  20  30  40  Observed Accident (acc./yr)  Figure 3.1 Model (1): Observed Versus Predicted Number of Accidents/yr  Figure 3.2 Model (2): Observed Versus Predicted Number of Accidents/yr.  37  3.2.2.2 Models for Specific Accident Types  The "product-of-flow-to-power" model suggested by Hauer et. al. (1988) and Bonneson and McCoy (1993) was used to model the relationships between specific accident types and traffic flow. Table 3.5 shows different accident type models and their goodness of fit. In the case of rear-end accidents the total traffic volume entering the intersection was used instead of the "product-of-flow-to-power" model. A l l models have a relatively goodfit and the x values are significant at a 95-percent confidence level. 2  38  Accident Type  Model Form  Coefficient  SD  k  Chi-square  253.7  10.8  128.25  169.2  14.4  120.54  259.6  11.2  126.97  204.4  7.8  137.30  159.3  19.5  120.74  131.6  88.85  125.63  154.3  26.64  154.30  269.6  10.36  128.45  278.2  3.33  121.73  Estimates PDO Accidents  E(jn) = a x V"' x Vao  1.3012  a,  0.3409  a  0.4412  2  0  Injury Accidents  2  E(m) = a x V" x V 1  0  Day time Accidents  2 2  Rush  hour  0.4490  2  0  1.5470  ai  0.3470  a  0.4206  2  0.6714  a,  0.2547  aj  0.5007  ao  0.0627  a,  1.2360  E(m) = a x V"' x V ao  0.3256  ai  0.4266  a  0.3944  2  E(m) = a x V  th  2  2  2  E(m) = a x V"> x V ao " 2  0.4439  ai  0.4466  a  0.4025  0  2  2  E(m) = a x V"' x V a°  1.7400  ai  0.2873  a  0.4599  2  0  2  Accidents  Left-Turn Accidents  a  2  Accidents  Non-Rush Hour  0.2836  E(m) = a x V"> x V a»  0  Rush Hour Accidents  a,  2  0  PM  0.9049  2  0  Rear-end Accidents  ao  E(m) = a x V"' x V a Q  Night time Accidents  2  E(m) = a x V* x V 0  2 2  0  2  ao  0.5572  a,  0.2999  a.  0.4950  Table 3.5 Models for Specific Accident Types  39  The relationship between the types of accidents and major traffic volume for various minor road volumes are shown in Figures 3.3-3.12. Figure 3.13 shows the relationship between different accident types and the major traffic flow. Figure 3.14 shows the relationship between time specific accidents and the major road traffic flow. The minor road flow is kept at its mean value in Figures 3.13 and 3.14.  40  Figure 3.3. Model of PDO accidents  41  Minor Road ADT (veh/day) 5000  0  10000  10  15000  20  20000  30  25000  40  50  Major Street ADT(1000 veh/day)  Figure 3.4. Model of injury accidents  60  70  8  42  Figure 3.5 Model of day time accidents  43  Figure 3.6 Model of night time accidents  44  Figure 3.7 Model of Rear-end accidents  45  Figure 3.8 Model of A M rush hour accidents  46  Figure 3.9 Model of P M rush hour accidents  47  Figure 3.10 Model of rush hour accidents  48  Figure 3.11 Model of Non rush hour accidents  49  Figure 3.12 Model of left turn accidents  50  Major Street ADT (1000 veh/day)  Figure 3.13 Models of different types of Accidents  51  Figure 3.14 Models of time specific accidents  52  3.2.2.3 Evaluation of the Effect of Intersection Layout Variables  Table 3.6 shows the parameter estimates for a model including intersection layout variables, as well as the significance of the layout variables as given by the t-ratio. The tratio is the ratio between the parameter value and its standard error. The critical t-ratio at the 5% significance level is about 1.96. The following layout variables were included in the model:  M A N L = Average number of lanes on the major road, both approaches M I N L = Average number of lanes on the minor road, both approaches L T L = Number of left turn lanes, all approaches R T L = Number of right turn lanes, all approaches PC = Number of pedestrian crosswalks  T A B L E 3.6 - Parameter Estimates for the Model Incorporating Layout Variables  t-ratio  Model Form  K  Pearson  x  2  Accidents/year = 1.3708 x[  ^  J  Xft x = 0.0386MANL + 0.2086 MINL s  •  x[  ^  J  x  •  0.0264LTL-0.0403RTL-0.0123PC  Denotes significance at a 95-percent confidence level  MANL  0.50  MINL  3.20  LTL  -1.43  RTL  -1.09  PC  -0.38  (% 0.05,130 2  = 157.6)  14.42  134.13*  53  The variables PC, MANL, RTL, and LTL were not significant (t-ratio is less than critical value). A stepwise elimination procedure was used to remove insignificant variables from the model. The procedure involves removing one insignificant variable at a time starting with the one having the least t-ratio. The resulting model is shown in Table 3.7.  T A B L E 3.7 Parameter Estimates for the Model Incorporating Layout Variables  t-ratio  Model Form  K  Pearson  x  2  MINL Cedents/year =  YuPi  = 0.2381  1.3470 x[  ^  J  ,[  ^  J  LTL  x^'"  4.11  14.16  125.21*  -2.04  MINL - 0.03525 LTL  * Denotes significance at a 95-percent confidence level  (x o.o5,i33 2  = 160.9)  Figure 3.15 shows the observed versus predicted number of accidents for the model. The dispersion around the 45° line is much less than total accidents model (Figure 3.1) which indicates the significance of including these layout variables.  55  4.0 APPLICATIONS  4.1 Empirical Bayes Refinements to the Models  As described in section 2.0, the Empirical Bayes (EB) approach can be used to refine the estimate of the expected number of accidents at a location by combining the observed number of accidents at the location with the predicted number of accidents obtained from the G L I M model. The E B estimated number of accidents for any intersection can be calculated using (Hauer et al. ,1988):  EB safety estimate = a.pred + (1 -a).count,  a— 1+  1 var(pred)  (4.1) (4.2)  pred  where count = observed number of accidents at the location pred = predicted number of accidents as estimated from the G L I M model var(pred) = the variance of the G L I M estimates.  (pred)  2  Since var(pred) = yield:  k  as described earlier, equation 4.1 and 4.2 can be rearranged to  56  EB safety estimate = (  k k + pred  ) pred  +(  pred k + pred  -) count  (4.3)  In addition, the variance of the EB estimate can be calculated using:  var (EB safety estimate) =  pred (& + pred)  2 +  (  k + pred  -) count 2  (4.4)  As can be noted from Equation (4.3), the E B safety estimate lies between the observed number of accidents and the predicted number of accidents, taking into account the individual accident history of the location and the G L I M model prediction which combine data for similar locations. As noted in Section 2, the E B estimate is important since it provides correction for the regression to the mean phenomenon. Figure 4.1 illustrates the EB refinement estimation versus the value predicted from the G L I M model. The following is an example illustrating the use of Equation (4.3):  Assume that an intersection has the following data:  •  Major road A D T = 40,000 veh/day  •  Minor road A D T = 10,000 veh/day  •  Observed accidents/year = 29 acc/yr  57  Using the first model from Table 3.3, the predicted number of accidents for this intersection is:  pred = 2.181 x (40)  0329  x (10)  0442  = 20.27 acclyr  Using equation 4.3 and 4.4, the empirical safety estimate and its variance can be calculated as:  9 20.27 EB safety estimate = (-———-) x 20.27 + f 9 + 20.27 9 + 20.27 J  7  v  9x20.27 var (EB safety estimate) = — — „ „ , ^ (9 + 20.27)  2  v  J  y  J  22  x 29 = 26.31 acc I yr  20.27 , + ( r — r ^ r r ) x 29 = 18.22 (acc I yr) 9 + 20.27  1  v  y  In this example the expected number of accidents is reduced from 29 to 26.31 which corresponds to about 9.2 percent regression to the mean correction.  In the following section, two applications for using the G L I M models and the Empirical Bayes' estimates will be discussed.  58  10  20  30  40  Observed Accidents (acc./yr)  Figure 4.1 Predicted vs. E B Refined number of accidents  50  59  4.2 Identification of Accident Prone Location  Accident Prone Locations (APLs) are usually defined as locations which exhibit a significant number of accidents compared to a specific norm. Because of the randomness inherent in accident occurrence, statistical techniques that account for this randomness should be employed when identifying APLs. The E B refinement process discussed in the previous section can be used to identify accident prone locations as follows (Belanger, 1994):  1.  Estimate the predicted number of accidents and its variance for the intersection using the appropriate G L I M model and plot the probability density function of the distribution (gamma distribution)  2.  Determine the appropriate point of comparison based on the mean and variance values obtained in step (1). (usually the  50 percentile, P th  50  is used as a point of  comparison)  3.  Calculate the E B safety estimate and its variance from equations 4.3 and 4.4 and plot the distribution.  4.  Identify the location as accident prone i f there is a significant probability that the intersection's safety estimate exceeds the P value. 50  60  To continue with the previous example, the predicted number of accidents and its variance using model (1) can be calculated as 20.27 acc/yr and 45.65 (acc/yr) respectively. The P  5Q  2  value can be estimated from plotting the probability density  function of the gamma distribution. From Figure 4.2, P  50  = 19.53 acc/yr. The E B safety  estimate and its variance were calculated as 26.31 acc/yr and 18.22 (acc/ yr)  2  respectively. The updated distribution is also shown in Figure 4.2. From the figure, it can be shown that the probability of having accidents less thanP  50  is only 4.5 percent (the  shaded area in the figure). This means that there is a significant probability (95.5%) of exceeding the P value and the intersection can be considered accident prone. 50  0.1  0  10  20  30  40  Accidents/yr  Figure 4.2 Identification of Accident Prone Locations  50  62  To facilitate the process of identifying accident prone locations, critical accident frequency curves for different significance levels can be developed. A n example of these curves is shown in Figure 4.3 using the /rvalue for the model relating the total number of accidents to traffic flows (K = 9.0). To illustrate, using the same example as before, for a predicted accident value of 30 accidents/ year, the observed number of accidents at the intersection must exceed 43 accidents/ year to be identified as accident prone at the 99 % level, 39 accidents/ year at the 95% level, and 36 accidents/ year at the 90% level of confidence.  Figure 4.3. Accident Prone Locations for Total Model  64  4.3 Before and After Safety Evaluation  The effect of a safety measure is often studied by comparing the number of accidents observed after the implementation of the measure to the expected number of accidents had the measure not been implemented. In simple before and after studies, the observed number of accidents in the period before the implementation is used to estimate the latter value. However, because of the random variations in accident occurrence (e.g. the regression to the mean effect), the observed number of accidents before  the  implementation may not be a good estimate of what would have happened had no measure been implemented. A n alternative and more accurate approach is to use the E B refinement process discussed in Section 4.1.  Considering the same example as before, assume that a specific safety measure to reduce the number of accidents at the intersection was implemented. The observed number of accident in one year after the implementation is 20 accidents. The effectiveness of the measure can then be calculated as:  20 Measure of Effectiveness = 1 -  = 0.24  which indicates a 24 percent reduction in total accidents because of the treatment. If the data on specific accidents types is available, then the measure of effectiveness in reducing specific accident types can be estimated by applying the appropriate G L I M model from Table 3.4.  65  4.4  Safety Planning  Accident prediction models can be used in safety planning by identifying the traffic and geometric variables that have the most impact on the safety performance of signalized intersections. These variables should be the focus of road authorities attention and investment. As well, the models can be used to estimate the incremental safety benefit associated with the change in any traffic or geometric design variable.  For example, a sensitivity analysis was carried out for the variables included in the model shown in Table 3.6. Figure 4.4 shows the analysis in a non-dimensional form. For the variables examined, accident occurrence is found to be the most sensitive to the number of lanes of minor road followed by the total and minor road traffic volumes and the number of left turn lanes, respectively. A n increase in the first three variables will increase the expected number of accidents while an increase in the number of left turn lanes would cause a decrease in the expected number of accidents.  66  Figure 4.4 Sensitivity Chart  67  5.0 CONCLUSION  This thesis has documented the results of a study to develop accident prediction models for signalized intersections in the Greater Vancouver Regional District (GVRD). To avoid the shortcomings associated with the conventional linear regression approach, the models were developed using the Generalized Linear Regression Models (GLIM) approach. The G L I M approach addresses the error structure problems that are associated with the conventional linear regression theory and allows for the use of nonlinear relationships in the model.  Three types of models were developed: models for the total number of accidents, models for specific accident types, and models which account for variables other than traffic volumes. The models provided adequate goodness of fit for the accident data used. A n Empirical Bayes procedure was used to refine the estimates of the G L I M models to provide more accurate site-specific safety estimates. Application of the Empirical Bayes procedure included the identification of accident prone locations and performing before and after safety analysis.  There are several improvements which can enhance the models developed in this thesis. First, the sample size used in this study is relatively small (139 intersections). Therefore, expanding the sample size to include more intersections and re-calibrating the models is recommended.  Secondly, in developing the models for predicting specific accident types, the average daily traffic volumes on the major and minor roads were used. Hauer (1989) and others recommended the use of traffic flows related to the specific accident type. However, detailed traffic volume data were not available for this research. Therefore, collecting detailed traffic volume data and developing models relating accident types to the traffic flows generating these accidents may be of interest.  Thirdly, all signalized intersections included in this research are in the urban environment. The collection of data for other environments (e.g. rural, suburban) and the development of corresponding models are also suggested.  69  REFERENCES  1) Al-Bakri, A., Conaway, M . R. and Stoner, J., (1991). "Conditional analysis of accidents at four-approach traffic circles", Transportation Research Record, 1327, Transportation Research Board, National Research Council, Washington, D.C., pp. 42-46 2) Al-Senan, S. H . And Wright, P. H . , (1987). "Prediction of head-on accident sites", Transportation Research Record, 1122, Transportation Research Board, National Research Council, Washington, D.C., pp. 79-85 3) Belanger, C , (1994).  "Estimation of safety of four-legged unsignalised  intersections", Transportation Research Record, 1467, Transportation Research Board, National Research Council, Washington, D.C., pp. 23-29 4) Bonneson J. A . and McCoy P. T., (1993). "Estimation of safety at two-way stop-controlled intersection on rural highways", Transportation Research Record, 1401, Transportation Research Board, National Research Council, Washington, D.C., pp. 83-89 5) Bowman, B . L . , Vecellio, R. L . and Miao, J., (1995).  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"Safety and  highway investment", Midwest Transportation Centre, Iowa State University and University of Iowa, 10) Garder, P.,  Leden, L . and Thedeen, T., (1994).  "Safety implications of  bicycle paths at signalised intersections", Accident Analysis and Prevention. Vol. 26. No. 4, pp. 429-439 11) Golias, J. C , (1992). "Establishing relationships between accident and flows at urban priority road junctions", Accident Analysis and Prevention, Vol. 24, No. 6, pp. 689-694 12) Hanna, J. T., Flynn, T. E. and Tyler, W. K . , (1976).  "Characteristics of  intersection accidents in rural municipalities", Transportation Research Record 601, Transportation Research Board, pp. 79-82  13) Hauer, E., (1994). "The effect of resurfacing on the safety of two-lane rural roads in New York State", Paper 940541. Transportation Research Board, 73rd Annual Meeting. January 9-13, 1994, Washington D.C. 14) Hauer, E. and Papaioannou, p. (1991). "Prediction in road safety studies: an empirical inquiry", Accident Analysis and Prevention, Vol. 23, No. 6, pp. 595607 15) Hauer, E., Ng J. C. N . and Lovell, J., (1989).  "Estimation of safety at  signalised intersection", Transportation Research Record, 1185, Transportation Research Board, National Research Council, Washington, D . C , pp. 48-61 16) Hauer, E . , (1992). "Empirical Bayes approach to the estimation of 'unsafety': The multivariate regression method", Accident Analysis and Prevention, Vol. 24, No. 5, pp. 457-477 17) Kulmala, R., (1992). "Prediction models for accident at highway junction", ITE 1992 Compendium of Technical Papers 18) Kulmala, R., (1994).  "Measuring the safety effect of road measures at  junctions", Accident Analysis and Prevention, Vol. 26. No. 6, pp. 781-794 19) Lau, M . , May, A . D., and Smith, R. N . , (1989). "Applications of accident prediction models". Transportation Research Record, 1238, Transportation Research Board, National Research Council, Washington, D . C , pp. 20-30  72  20) Lau, M . and May A . D., (1988).  "Injury accident prediction models for  signalised intersections", Transportation Research Record, 1172, Transportation Research Board, National Research Council, Washington, D . C , pp. 58-67 21) Lin, T. , Jovanis, P. P. and Yang, C , (1994). "Time of day models of motor carrier accident risk", Transportation Research Record, 1467, Transportation Research Board, National Research Council, Washington, D . C , pp. 1-7 22) Maher, M . J., (1991). " A new bivariate negative binomial model for accident frequencies", Traffic Engineering and Control, Vol. 32, pp. 422-423 23) Margiotta, R. and Chatterjee, A., (1995). "Accident on suburban highwaysTennessee's experience", Journal of Transportation Engineering, /May/June, pp. 255-261 24) McCullagh, P., and J. A . Nelder. (1983). "Generalized linear models", The University Press, New York. 25) Miaou, S. and Lum, H . , (1993). "Modeling vehicle accident and highway geometric design relationships", Accident Analysis and Prevention, Vol. 25, No. 6, pp. 689-709 26) Miaou, S., Hu, P. S. and Wright, T., (1992). "Relationship between truck accidents and highway geometric design: A Poisson regression approach", Transportation Research Record, 1376 Transportation Research Board, National Research Council, Washington, D . C , pp. 10-18  27) Nembhard, D. A . and Young, M . R., (1995). "Parametric empirical estimates of truck accident rates", Journal of Transportation Engineering, /July/August, pp. 359-363 28) Persaud, B . , Hauer, E., Vallurupalli, R. and Mucsi, K . , (1995) "Study of the safety effect of removing traffic signals in Philadelphia", Draft Final Report, October 1995 29) Persaud, B . and Dzbik, L . , (1993).  "Accident prediction models for  freeways", Transportation Research Record, 1401, Transportation Research Board, National Research Council, Washington, D.C., pp. 55-60 30) Persaud, B . N . and Mucsi, K . , (1994).  "Microscopic Accident potential  models for two-lane rural roads", Transportation Research Record, 1485, Transportation Research Board, National Research Council, Washington, D.C., pp. 134-139 31) Persaud, B . N . , (1991). "Estimating accident potential of Ontario road section", Transportation research record, 1327, Transportation Research Board, National Research Council, Washington, D.C., pp. 47-53 32) Pursula M . and Pastinen V., (1993). " A Bayesian approach to update traffic flows  from  traffic  counts",  Transportation  and  Traffic  Theory. /C.F.  Daganzo(Editor). pp. 507-522 33) Seneviratne, P. N . and Islam M . N . , (1992). "Speed estimates for roadway design and traffic control", Transportation Research Record, 1375, Transportation Research Board, National Research Council, Washington, D.C., pp. 37-43  74  34) Stark, D. C , (1994). "Relating speed to accident risk at priority junctions", Traffic Management and Road Safety: Proceedings of Seminar J., Held at the PTRC European Transport Forum, University of Warwick, England Planning and Transport Research and Computation Co. Meeting (22 : 1994: Warwick, nd  England) 35) Tarko, A . and Tracz, M . , (1995). "Accident prediction models for signalised crosswalks", Safety Science, pp. 109-118 36) Webb, G. M . , (1955). "The relation between accidents and traffic volumes at signalised intersections", ITE Proceedings,, pp. 149-167 37) Zegeer, C. V . , Hummer, J., Reinfurt, D., Herf, L . , and Hunter, W., (1986). "Safety effects of cross section design for two lane roads", Volume I, Final Report, FHWA-RD-87/008. Federal Highway Administration, Transportation Research Board, Dec. 1986.  APPENDIX A  Accident data  CD CD (0 3 3" 3* CD CO  —i , - i  3  o —ai I 3 <" CD  O  0) Q.  <l CO  >  ol  D)  co  CD CD * ,  II o O  0)  O  o  S  -i CD (D  73 CD  1  <  3  3  53' co O SI CO c  1 CD  CD 3 3 -i O  col  3  CO  CD  Q. CO  o o  f-+ —1 CD  o 1  5'  o> >< •<  W CO  CD  3  Si <D  X et ro co X et (A o> X CA co 5" p» (A | ( Q  et CD X ro co X CA CO p* (A p»  (A  CD CD  (0  (A  (0  CA CA  CT|  CO CO p* fp  5" -t  si  CD CD  3 ro  x  , 1  3  CO ICQ CA CA  3  CD  (Q ,  o z  £'  CD CD  CD et et X X to 3 0) co C CD (A p*  5"  CO  p*  CQ CO  5"  ICQ  CO  2.2. CO co  ro ro o ro  4k co  CO  o CO  cn  co  ro  s  co  CO 00 ro  4k 00 r o to 4k CO CO CO 00 CO CO  Ul  cn co  ro  1  W| 4k 4k 00 CO 00 ro  co co  CO  4k c o co Ul o Ul Ul r o 4k ro co Ol ^1 o  co o  ro  co  ro cn  CO CO CO  oo  o  4k  o  ^1 CO Oil Ol -J -J  4k CO  to|  CO Oil -4  CO  ro  Ol 4k 4k  co  4k  ro  ro  co  CO  ro  co  ro ro o oo  00 CO  CO  ro  CO  o  CD  ro o  co  ro  12  o co  co  4k c o Ul Ul ro  00 4k  co  r o CO c o 4k a>  CO  4k  -J  E £» 3  co  4k O  4k  co  4k  ro  cn  Ol Ol  ro  co  CJ  CD  3 i.  CO Ol  6  CD  ° 3  2J o  ro co  co  W | co Ul Co |  co  u  co  ro  ro  ro Ol  ro  co In co  to  ro Ol  ro  Ol  8, 3  8 = 3 o to | o i l  co|  to |  Ul  co  cr CD  ro  ui  Ul  rol m l  ro  oil  o Co |  to |  to I  co  ro  ro|  Ul  o 0) * 3 O CD (A  to |  to I  ro  -* tJ CD Q.  to |  O  2 r o Ul Ul r o oo  9L  CO  Oil r o CO  co  ro  CO  Ol  ro co  ro  4k CO  < 65 3  ©  © 85  0)  CQ CO  w  (Q (A  o o >  3  3 CO  S'  O u  co  C  5' | ( 5' 5' Q  (Q  CQ  CD CD  o  8'  co  -1  CD  ll  CT »<  S i CO CD P*  CD  (0  |CQ CO  CD  3 CD <  $81et  1  (0  O  CO  1  CO  O  »<  1  I  o o 3  3  £ > CO co CO CD i-t- ro et ® , ro , 1 X x CD CD ^ tu D) x et X CD CO (0 (A CO X x (0 (A 0) B> c r + (0 CO X CQ  , a I' CO w  CD CD  Ul  TOTAL  o  o  o  cn  o  o  ro  o  4k  O  co  _k  co  O  o  o  o  o  o  o  o  o  o  o  o  O  o  o  o  o  O  o  4k  o  4k  o  o  4k  4k  4k  4k  4k  4k  ro  4k  4k  4k  4k  _ i cn CO c n  4k 4k  cn cn  ro co  4k  minor number # of Itl #of rtl #of ped lanes of lanes  _^ ro  ro  ro  o  _^  co  _^  ro  ro  ro  ro  co  ro  ro  _^  4k  4k  4k  o  ro  o  o  co  O  O  o  o  o  4k  o  4k  co  co  CO c n CO _^  CO c n  co  CO CO c n  ro cn _^ c n  _k  oJ  cn  to ro  4k  cn  4k  o  _^  o  ro  o  4k  4k  4k  4k  4k  4k  4k  ro  4k  o  4k cn  -J. -«l r o CO _ i  ro _^  co CD c o  4k 4k  co cn  CO c n  _k ro  4k  th  o ro cn  LOCATIONS  st  st  th  major number major minor total volume volume of lanes volume 28229 12374 40603 15838 12258 28096 27594 21978 5616 24118 2216 26334 23779 43344 19565 21641 3315 24956 24017 21669 45686 12014 22217 10203 33848 17208 51056 9247 12213 21460 29487 12768 42255 30167 12858 43025 2844 24582 21738 5176 26604 21428 1156 29165 28009 12817 8736 21553 30572 4762 35334 4861 16269 11408 14013 49778 35765 12224 45164 32940 11500 44580 33080 16138 13638 29776 26057 18637 44694 13837 13658 27495 12604 41492 28888 3846 56707 52861 14571 1045 15616 6947 20058 27005 34770 23403 58173 59284 36028 23256 33684 9345 43029  o ro  ro  cn  rd  Commercial Drive/broadway Fir Street/broadway Fraser Street/broadway Glen Drive/broadway Granville Street/broadway Heather Street/broadway Kingsway/broadway MacDonald Street/broadway Nanaimo Street/broadway Oak Street/broadway Renfrew Street/broadway Rupert Street/broadway Willow Street/broadway Yukon Street/broadway Commercial Dr/Powell St Commercial DrA/enables St Commercial Dr/E Hasting St Commercial Dr./N Grandview Hwy Kingsway / Victoria Dr. E 12 Ave / Commercial Dr. E 1 Ave/ Commercial Dr. E 33 Ave / Victoria Dr E 41 Ave / Victoria Dr E 54 Ave / Victoria Dr E Broadway St/Commercial Dr. SE Marine Dr. / Victoria Dr. Cambie Dr. / Pacific Bl. Cambie Dr. / Smithe St. Cambie Dr. A/V King Edward Ave W 12™ Ave/Cambie St. W 16™ Ave/Cambie St  LL  CO —^ 4k  co  4k  co  CO c o  ro  CO  co ro  ro co  ro ro  —X  ro  co ro ro ro ro  CO CO CO CO CO c o CO CO r o CO CO CO c o ro ro co ro ro cn  CO r o CO c o  minor major number minor number # of Itl # of rtl # of ped major total lanes of lanes volume volume volume of lanes 7187 39765 32578 28274 3986 32260 20887 51884 30997 28678 28981 28206 1319 29525 29997 15418 45415 31457 2766 34223 38229 5486 43715 28810 23546 52356 ro  —&  ro ro  Ul ro IO CO _k Ul  p  ro  o  ro _^  o  o  o  —*  Ul co Ul  co  TOTAL  Ul ro -J ro Ul 4k co 4k o> oo Ui Ul 4k 4k to  LOCATIONS W 29TH Ave / Cambie St W 33RD Ave / Cambie St W41ST Ave/ Cambie St W 43RD Ave / Cambie St. W 45TH Ave / Cambie St. W 49TH Ave / Cambie St. W 57TH Ave / Cambie St. W 7TH Ave / Cambie St. W Broadway St. / Cambie St. CO  o  co  co co w ro Ul CO Ul CO co Ul Ul co CO  4* 4k 4k 4k 4k 4k 4k 4k 4k  O O > H O  cr  5 o  z  CO  p a.  O » 85  ro co ro ro  ro 00 ro  Ul CT>  oil  ui Ul  Ul 4k  ro co o  CD  a> co co  4k  ro  co co  a>  ro o  co  Ul  co CD  o>  CO  ro oo  •Ol  co  co co oo CD CD CD ro  to Ul CD 4k ro CO to I00  22.  S  i  3  o  o  to I ro oo co co  ro  Ul Ul CD  o c CO  4k CO  co  CD CO 4k Ul  CD 00  ro to I CD CD o o ro CD 00 co Ul 4k  oo o 00  ro  4k Ul  ro o  «*| ro  -4 Ul Ul  oo to co Ul  o> -4 o  co o  Ul  4k 4k co CO  Ul Ul  co oo  to ro ro ro  to  CO  co  Is  Ul CD  CD  co  CD  CD  3  cr CD  ro  to IOl IO to Ito Iro to Iro ro to Iro Ul to to rol ro to to Ito Ito Ito I  to Iro ro to I  2.1  cr CD  Ul  to Ito Ul ro to Irol ro Ul  to | to | to | to | ro  Ul  Ul' Ul  o O) to I  00 00  ro o>| ro  oo  o  to  rol  CO|  3 O (D -* CO 73 (D  a. O  2 Ul 4k  6L  o  Ul to CO CD  4k  co ro CO  CD  ro 4k  4k  to  4k  |O O  > H O Z  CO  4k 4k 4k O to O cn cn O 0 0  CO _^ _l _^ ro CJ CO to ro -4 CO 4k co CO CO 4k CO CO CO 4k cn O CO cn -4 co CO CO O CO co CO CO CO 4> 4k O CO CO CO to CO 0  _^ CO 4k 4k cn CO 0 CO ro CO 4k 0 -k  CO ro ^ CO ro CO -4 00 CO _^ co CO CO CO to to CO CO O CO CO 0 4k 0 ^1 _l CO co  ro 00 4k ^4 4k  4k to CJ co CO -4 CO CO cn to CO 0  to -4 _i 4k  ro to 4k ^1  -k cn 4k •vj cn  ro _k CO 4k CO CO cn CO co  to ro to co CJ ->i CJ 0 0 co  cn to CO CO CO _k CO 0  _^ _k CO 4k CO cn 0 CO CO to CO CO CO -4 cn ~4 0 ^1 0  _k co 4k co  -k  -^  ^ _^  CO 4k ^4 co cn CO to CO _^ CO co 4k CO CO co co CJ CO 0 CO cn CO co ~4 CO -4 4k co 4k CD  C fi). CO CO to CJ CO CO to cn 3 4k 0 _^ CJ cn CO _x to <D ^1 cn 00 cn cn CO CO O 00 0 cn CO CO CO -4 cn cn 0 CO co to to  ro to CO ro to cn CO 0 4k 4k to CO co co co 4k CO CO co co _^ co _^ ^4 cn -* 00 00  CO -4 CO _^ -4 0 cn CO _^ ~4 ro -4 CO cn CO cn co 0 CO CO 0  _t co 4k 00 cn CO cn _^ •>! CO  to CO cn c CO 3 O cn CO CO 3 CO co CO CD 0 CO 00  cn CO ^1 to  O  maj or  4k 00 -4 -4 cn 0 cn CO -k 0  CO ro CO CO cn cn 0 to CO to 4k CO ^1 O ~4  vol  _^ CJ ro cn ro -4 _^ CO O -4 0 to  CO CO CO CO CO CO CO CO CO CO to to CJ CO to to ro to to -k _^ O CO CO to 0. CO CO ^1 cn 0 cn 0 cn cn co cn cn cn 0 0 0 cn 0 0 0 4k CO  ma  ro CO ro 4k co CO _k _k •>! 0 0 to cn ~4 0 cn  CO CO 4k CJ ro CO CO cn  vol  CO 4k 0 CO co CO co 04  CO CO CJ CO CO CO CO CO -4 -4 CO CO cn CJ CO O cn 0 0 0 0 O 0 0  tot  CO cn CJ CO CJ CO 0 CO to co CO •vl cn to cn cn 0 ro CJ oj CO CO CO -4  4k 4k 4k 4k co ro O cn cn O 0 0  [OA  4k 4k cn 0  niu  4k 4k 4k 4k CD CO CO cn O cn 0 cn o 0 0 0  _» cn c 0 4k 3 O 00 <D cn 00 CO CO  ne  —h 5T  ro  ^ to to ro ro to to to to to to to to co cn  CD  lane  to  ro 0  cn 0  -k ro cn  mi nor 1  0 —h  nu  ro ro to to to to ro to ro to ro to to to to to  3 c 3 cr  3 cr  CD  to cn cn 0  0  O  to to  cn cn ro to cn cn ro cn to _^ In In 0  —1  O  -* CO 4k 4k CO 4k to cn 0  0  ro CO to 4k CJ 0  0  to CO 4k to CO 0  4k CO 00 -k ro CO to 00 4k  *fc O  —h 4k _x _^ _ i  0  ro to _* _x 0  to to _k 0  -k _k 0  O  0  ro 0  to 0  0  ro O ST % 3 O  <D ^1 CO •O CD Q.  4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k H O H  > CJ - J CO . A co CO CD to CO to to co co CO to co 4k to CO CO co CO to CO co co cn cn CO cn CO CO to 4k 0 0 to O CO 0 CD CD CO cn ro _k 0 -* CO -4 ro 0 cn 0 0 .0. CO co _k  08  LOCATIONS 4950 4953 5050 5200 5300 5400 5458 5800 5805 5900 5909 6300 6400  major number minor major total volume volume volume of lanes 6093 29992 23899 3396 22993 26389 11822 31737 43559 8632 14286 22918 12207 37731 49938 14054 32677 18623 6266 29709 35975 8791 31212 40003 10085 28077 38162 3251 5702 8953 3976 11828 15804 8050 16890 24940 3116 14872 17988 _^ ro ro ro ro cn ro ro ro ro ro ro ro ro  # of Itl ro co cn ro ro 4k co 4k 4k CO 4k CO 0  minor number of lanes  # of rtl # of ped lanes  TOTAL  CO ro ro 4k ro CO CO co ro co CO 4k 04 CO CO - 4 •01 CO 4k co cn oJ 4k cn  - A  ro ro ro co ro co O 4k O 0  _^ cn ro ro ro ro _i In  4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k 4k  82  Appendix B  MODELS FOR VANCOUVER AND RICHMOND INTERSECTIONS  Total of 67 intersections in Vancouver were obtained for analysis.  Model 1. Bonneson and McCoy  Model Form  E(m) = a x V* x V 0  Coefficient Estimates 0.9 ao 0.6894 a, a 0.2738  2 2  SD  k  101.8  14.65  Chisquare 59.74  0.9 0.9630 0.2738  101.8  14.56  59.74  0.7 0.5617 0.1779 0.1532 0.07 0.04 0.1638 0.08 0.18 0.14 -0.06 -0.08 -0.08  62  11298  53.98  0.7 0.6117 0.4117 0.6 0.4958 0.408 0.7 0.6681 0.377 0.7 0.3302 0.5007 0.1 1.296 1.3 0.04 0.3697 0.1 0.9178 0.3373  59.7  16.43  2  2.Brude and Larsson  a ai  0  a 3. Kulmala  E(m) = a xV  ai  Q  xV "  xe^' '  2  x  2  2  ao a,  a  2  P, P  2  P  3  P4 P  5  P6 PT  Ps P 9  P. 4. PDO Accidents  E(m) = a x V"> x V"  a  5. Injury Accidents  E(m) = a x V* x V  2  3o  6. Day time Accidents  E(m) = a x V* x V  2  7. Night time Accidents  E{m) = a x V* x V  2  1  0  0  2  0  a.  ai  a 0  ao a.  2  a 0  2  2  ao a,  2  a. 8. Rear-end Accidents 9. Angle Accidents  E{m) = a x V'  10 A M Rush hour accidents  E(m) = a x V* x V "  ao ai  0  E(m) = a x V* 0  ao  x V2"  2  a>  a 2  0  2  2  ao a,  a.  72.1  69.7  54.41  12.34  73  66.43  67.3  16.66  44.6  65.52  24.1  69.3  123.17  83  Model  Model Form  11 PM Rush hour accidents  E(m) = a x V"' x V  12 Rush Accidents  E(m) = a x V" x V "  0  ai  Coefficient Estimates 0.1 a 0.8566 a 0.3318 0.33 0.4995 a 0.4461 1.2 ao 0.4605 a 0.4399 0.1 1.048 a. a 0.496 0  2  SD  k  Chisquare  41.3  2  Hour  1  0  2  2  41.3  2  13 Non Rush Hour Accidents  E(m) = a x V"' x V  14 Left accidents  E(m) = a x V"' x V"  0  2 2  82.1  23.6  57.34  71.9  4.08  46.07  2  turn  1  0  2  *  means the indicated value is a minus figure which shows a not proper form for the  corresponding model.  84  Total of 72 intersections in Richmond city were obtained for analysis.  Model  Model Form  1.Bonneson and McCoy  E(m) = a x V* x V 0  Coefficient Estimates 2.0 0.1537 a, a 0.6826  2 2  SD  k  127.8  14.65  Chisquare 64.06  127.6  14.56  64.06  73.6  11298  70.94  100.7  16.43  65.15  12.34  62.37  2  2. Brude and Larsson  a a,  0  Eim)-a x(V +V Tx( ^ r 0  l  2  v  v  3. Kulmala  1.7 0.8412 0.6336  0.3 a a, e^"'*'0.4033 a 0.2915 0.1536 P. 0.018 P2 0.093 P 0.1696 P4 0.062 P -0.156 P6 0.1233 P -0.017 Ps -0.1 P -0.026 PlO 0  E(m) = a x V > x V " x a  2  0  2  2  3  5  7  9  4. PDO Accidents  E(m) = a x V" x V  2  5. Injury Accidents  E(m) = a x Vf x V  2  6. Day time Accidents  E(m) = a x V* x V  1  0  32  1.0 0.3869 0.5141 0.7 0.3341 0.5148 1.2 0.3898 0.4771  a a, a  0.5 0.2985 0.5743  50.24  0.03 1.358  74.9  16.66  52.05  2.0 -0.235 0.5018  65.87  65.52  123.81  0.1 0.4843 0.4664  43.42  ao a,  2  32 0  3o  2  3, 3  7. Night time Accidents  Q  2  a  2 2  0  3i  E(m) = a x V"' x V 0  ai 2  0  2  8. Rear-end Accidents  E(m) = a x V  9. Angle Accidents  E(m) = a x V* x V  1  3o  0  3i  0  ai 2  3o 3l  a 10. A M Rush hour accidents  E(m) = a x V? x V "  2  0  2  2  3o  a, a  2  54.48  115.6  85  Model  Model Form  11 PM Rush hour accidents  E(m) = a x V"' x V  12 Rush Hour Accidents  E(m) = a x V > x V  13 Non Rush Hour Accidents  E(m) = a x V* x V  2  14 Left accidents  E(m) = a x V* x V  2  0  2  Coefficient Estimates 0.2 a 0.5095 ai a 0.4272 0.3 a 0.4995 a. 0.4461 1.3 ao 0.3368 a, a 0.5256 0.3 ao 0.4518 a, 0.577 32 0  2  k  SD  Chisquare  43.42  2  a  0  {  2  0  2  2  0  60  91.55  23.6  61.02  112.2  4.08  4.08  2  turn  0  2  

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