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Microscopic accident prediction models for signalized intersections Quintero Toscano, Mario Alberto 2000

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MICROSCOPIC ACCIDENT PREDICTION MODELS FOR SIGNALIZED INTERSECTIONS By MARIO ALBERTO QUINTERO TOSCANO B.A.Sc. (Civil Engineering), Instituto Tecnologico y de Estudios Superiores de Monterrey, Monterrey, Mexico, 1997 A THESIS SUBMITTED IN PARTIAL FULFILMENT 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 APRIL, 2000 © Mario Alberto Quintero Toscano, 2000 In presenting this thesis in partial fulfilment of the requirements for an advance 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 the permission for the extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. 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 2324 Main Mall Vancouver, B. C. Canada, V6T 1Z4 Date: Mk-roseopk A c c k k i i t Prediction Mode ls i 'w S i g n a l l e d Ink i ' sec t i ons ABSTRACT The main objective of this thesis is to develop microscopic accident prediction models for estimating the safety potential of 4-leg signalised intersections in the City of Vancouver, B.C. and describes the applications of these models in traffic safety analysis. The aim, therefore, is to examine the traffic variables that appear to underlie the occurrence of accidents of these intersections and explain, in a statistical sense, the generation of accidents as a function of these variables. Generalised linear regression was employed to develop the models because of its superiority over conventional linear regression in modelling accident occurrence. The statistical software package GLIM4 was used to accomplish this task. The study made use of a sample of 8466 accidents that occurred at 170 4-leg signalised intersections during the years of 1994-1996. The data on accident frequencies and traffic volumes were obtained from the City of Vancouver. Several models that have different applications in the field of traffic safety were developed in this study for the 4-leg signalised intersections of the City of Vancouver. Different error structures that can be utilised to model the relationship between accidents and traffic flows are reviewed. Microscopic models for different accident types were developed in conjunction with macroscopic models for Total, Severe and Property Damage Only accidents. The microscopic models are presented in conjunction with the three macroscopic models which all resulted in statistical significance. Several model applications are discussed. Examples of how to obtain location-specific safety estimates, how to identify accident prone locations, how to rank the accident prone locations, and how to conduct a before and after safety evaluation are presented. Microscopic and macroscopic models are used simultaneously to determine which intersections should be regarded as accident ii M i c r o s c o p i c A c c i d e n t Prediction M o d e l s 1'ttr Signalised In tersect ions prone locations according to specific accident patterns that can be effectively treated by engineering countermeasures. M i f r o s c o p k Aerifleur Prediction M o d e l s f o r Signal ise si iotersecrkors TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iv LIST OF TABLES vii LIST OF FIGURES ix ACKNOWLEDGEMENTS xi 1. INTRODUCTION 1 1.1 Background 1 1.2 Thesis Objectives 2 2. PREVIOUS WORK 3 2.1 Introduction 3 2.2 Conventional Linear Regression Models (CLRM) 3 2.3 Generalised Linear Regression Modelling (GLIM) 4 2.4 Microscopic Accident Prediction Models (APMs) 8 2.4.1 Model Evolution 9 2.4.2 Model Form 9 2.5 Outliers 10 2.6 Testing the Models Significance 12 M i c r o s c o p i c Aedficut Prediction M o d e l s for Signal ise*! Intei 'set ' t imis 2.7 The Empirical Bayes Refinement (EB) 13 2.8 Accident Prone Locations (APLs) 14 2.9 APMs for 4-leg signalised intersections 17 3. DATA DESCRIPTION 21 3.1 Introduction 21 3.2 Data Analysis 21 4. Models Development 34 4.1 The Models 34 4.2 Outlier Analysis 37 4.3 Testing the Models' Statistical Significance 37 5. Selection of APM type 40 5.1 Introduction 40 5.2 Poisson Model 40 5.3 Negative Binomial Model 42 5.4 Accident Database Evidence 45 6. APPLICATIONS 47 6.1 Introduction 47 6.2 Location specific prediction: EB safety estimates 47 iVi i iToscofise A e c k k i s t Predict io is M o d e l s for Sisjmiissed In lot 'seci ions 6.3 Identification of APLs 52 6.4 APLs Ranking 55 6.5 Critical Accident Frequency Curves 58 6.6 Before and After studies 71 6.7 Recommended Research 72 7. CONCLUSIONS 73 Bibliography 75 APPENDIX I INTERSECTION LIST 79 APPENDIX II DATA ANALYSIS 81 APPENDIX III OUTLIERS ANALYSIS SAMPLES 88 APPENDIX IV GLIM OUTPUT SAMPLE 91 APPENDIX V STATISTICAL SIGNIFICANCE TESTS 95 vi Microscopic Accident Prediction Models for StgnaJiscsl l o f e r s w t K m : LIST OF TABLES FIGURE 2.1 APL IDENTIFICATION PROCESS 16 FIGURE 3.1 TIME OF ACCIDENTS 24 FIGURE 3.2 1994 ACCIDENTS 25 FIGURE 3.3 1995 ACCIDENTS 25 FIGURE 3.4 1996 ACCIDENTS 26 FIGURE 3.5 TOTAL ACCIDENTS 26 FIGURE 3.6 3 YEARS ACCIDENTS PER MONTH 27 FIGURE 3.7 CONTRIBUTING CIRCUMSTANCES 28 FIGURE 3.8 T Y P E OF ACCIDENT 29 FIGURE 3.9 COMBINED T Y P E S 30 FIGURE 3.10 COMBINED T Y P E S PERCENTAGES 30 FIGURE 3.11 ACCIDENT SEVERITY 31 FIGURE 3.12 ACCIDENT SEVERITY PERCENTAGES 31 FIGURE 3.13 LIGHT CONDITION 32 FIGURE 3.14 ROAD CONDITION 32 FIGURE 3.15 W E A T H E R CONDITION 33 FIGURE 6.1 PREDICTED vs. E B REFINED NUMBER OF ACCIDENTS 51 FIGURE 6.2 APL IDENTIFICATION PROCEDURE 54 FIGURE 6.3 CRITICAL CURVE FOR AT1 59 FIGURE 6.4 CRITICAL CURVE FOR AT5 60 FIGURE 6.5 CRITICAL C U R V E FOR AT7 61 FIGURE 6.6 CRITICAL CURVE FOR AT13 62 FIGURE 6.7 CRITICAL CURVE FOR LEFT TURN ACCIDENTS 63 FIGURE 6.8 CRITICAL CURVE FOR RIGHT TURN ACCIDENTS 64 FIGURE 6.9 CRITICAL C U R V E FOR TOTAL ACCIDENTS 65 FIGURE 6.10 CRITICAL CURVE FOR SEVERE ACCIDENTS 66 FIGURE 6.11 CRITICAL CURVE FOR PDO ACCIDENTS 67 FIGURE 6.12 CRITICAL C U R V E S FOR VARIOUS K VALUES 70 FIGURE AIII.1 COOK'S DISTANCES BEFORE REMOVALS 89 FIGURE AIII.2 COOK'S DISTANCES AFTER 2 WARRANTED AND 2 UNWARRANTED REMOVALS 89 FIGURE All.3 COOK'S DISTANCES BEFORE REMOVALS 90 FIGURE All 1.4 COOK'S DISTANCES AFTER 4 UNWARRANTED REMOVALS 90 FIGURE AV.1 PREDICTED ACCIDENTS VS. PEARSON RESIDUALS FOR AT 1 96 FIGURE AV.2 PREDICTED ACCIDENTS VS. A V E R A G E SQUARED RESIDUALS AND ESTIMATED VARIANCE FOR AT 1 96 M i m w e o p i e Aeefrleitt P r e d i c t i o n M m M • i » • • • FIGURE A V . 3 PREDICTED ACCIDENTS VS. PEARSON RESIDUALS FOR A T 5 97 FIGURE A V . 4 PREDICTED ACCIDENTS VS. A V E R A G E SQUARED RESIDUALS AND ESTIMATED VARIANCE FOR A T 5 97 FIGURE A V . 5 PREDICTED ACCIDENTS VS. PEARSON RESIDUALS FOR A T 7 98 FIGURE A V . 6 PREDICTED ACCIDENTS vs. A V E R A G E SQUARED RESIDUALS AND ESTIMATED VARIANCE FOR A T 7 98 FIGURE A V . 7 PREDICTED ACCIDENTS vs. PEARSON RESIDUALS FOR A T 13 99 FIGURE A V . 8 PREDICTED ACCIDENTS VS. A V E R A G E SQUARED RESIDUALS AND ESTIMATED VARIANCE FOR A T 13 99 FIGURE A V . 9 PREDICTED ACCIDENTS vs. PEARSON RESIDUALS FOR TOTAL ACCIDENTS 100 FIGURE A V . 1 0 PREDICTED ACCIDENTS VS. A V E R A G E SQUARED RESIDUALS AND ESTIMATED VARIANCE FOR TOTAL ACCIDENTS 100 FIGURE A V . 1 1 PREDICTED ACCIDENTS vs. PEARSON RESIDUALS FOR SEVERE ACCIDENTS 101 FIGURE A V . 1 2 PREDICTED ACCIDENTS VS. A V E R A G E SQUARED RESIDUALS AND ESTIMATED VARIANCE FOR S E V E R E ACCIDENTS 101 FIGURE A V . 1 3 PREDICTED ACCIDENTS VS. PEARSON RESIDUALS FOR P D O ACCIDENTS 102 FIGURE A V . 1 4 PREDICTED ACCIDENTS VS. A V E R A G E SQUARED RESIDUALS AND ESTIMATED VARIANCE FOR P D O ACCIDENTS 102 FIGURE A V . 15 PREDICTED ACCIDENTS VS. PEARSON RESIDUALS FOR LEFT TURN ACCIDENTS 103 FIGURE A V . 1 6 PREDICTED ACCIDENTS VS. A V E R A G E SQUARED RESIDUALS AND ESTIMATED VARIANCE FOR LEFT TURN ACCIDENTS 103 FIGURE A V . 17 PREDICTED ACCIDENTS VS. PEARSON RESIDUALS FOR RIGHT TURN ACCIDENTS 104 FIGURE A V . 1 8 PREDICTED ACCIDENTS VS. A V E R A G E SQUARED RESIDUALS AND ESTIMATED VARIANCE FOR RIGHT TURN ACCIDENTS 104 viii M i c r o s c o p i c A c c i d e n t Prediction Models for Signalise;! Intersection* LIST OF FIGURES FIGURE 2.1 APL IDENTIFICATION PROCESS 16 FIGURE 3.1 TIME OF ACCIDENTS 24 FIGURE 3.2 1994 ACCIDENTS 25 FIGURE 3.3 1995 ACCIDENTS 25 FIGURE 3.4 1996 ACCIDENTS 26 FIGURE 3.5 TOTAL ACCIDENTS 26 FIGURE 3.6 3 YEARS ACCIDENTS PER MONTH 27 FIGURE 3.7 CONTRIBUTING CIRCUMSTANCES 28 FIGURE 3.8 T Y P E OF ACCIDENT 29 FIGURE 3.9 COMBINED T Y P E S 30 FIGURE 3.10 COMBINED TYPES PERCENTAGES 30 FIGURE 3.11 ACCIDENT SEVERITY 31 FIGURE 3.12 ACCIDENT SEVERITY PERCENTAGES 31 FIGURE 3.13 LIGHT CONDITION 32 FIGURE 3.14 ROAD CONDITION 32 FIGURE 3.15 W E A T H E R CONDITION 33 FIGURE 6.1 PREDICTED vs. EB REFINED NUMBER OF ACCIDENTS 51 FIGURE 6.2 APL IDENTIFICATION PROCEDURE 54 FIGURE 6.3 CRITICAL CURVE FOR AT1 59 FIGURE 6.4 CRITICAL C U R V E FOR AT5 60 FIGURE 6.5 CRITICAL CURVE FOR AT7 61 FIGURE 6.6 CRITICAL CURVE FOR AT13 62 FIGURE 6.7 CRITICAL C U R V E FOR LEFT TURN ACCIDENTS 63 FIGURE 6.8 CRITICAL CURVE FOR RIGHT TURN ACCIDENTS 64 FIGURE 6.9 CRITICAL CURVE FOR TOTAL ACCIDENTS 65 FIGURE 6.10 CRITICAL C U R V E FOR SEVERE ACCIDENTS 66 FIGURE 6.11 CRITICAL CURVE FOR PDO ACCIDENTS 67 FIGURE 6.12 CRITICAL C U R V E S FOR VARIOUS K VALUES 70 FIGURE AIII.1 COOK'S DISTANCES BEFORE REMOVALS 89 FIGURE AMI.2 COOK'S DISTANCES AFTER 2 WARRANTED AND 2 UNWARRANTED REMOVALS 89 FIGURE All.3 COOK'S DISTANCES BEFORE REMOVALS 90 FIGURE AIII.4 COOK'S DISTANCES AFTER 4 UNWARRANTED REMOVALS 90 FIGURE AV.1 PREDICTED ACCIDENTS vs. PEARSON RESIDUALS FOR A T 1 96 (VHmwopte Accident P r e d i c t i o n M o d e l s f o r Signalised i n t e r s c M k o FIGURE AV.2 PREDICTED ACCIDENTS VS. AVERAGE SQUARED RESIDUALS AND ESTIMATED VARIANCE F O R A T 1 96 FIGURE AV.3 PREDICTED ACCIDENTS vs. PEARSON RESIDUALS FOR A T 5 97 FIGURE AV.4 PREDICTED ACCIDENTS VS. AVERAGE SQUARED RESIDUALS AND ESTIMATED VARIANCE FOR A T 5 97 FIGURE AV.5 PREDICTED ACCIDENTS VS. PEARSON RESIDUALS FOR A T 7 98 FIGURE AV.6 PREDICTED ACCIDENTS VS. AVERAGE SQUARED RESIDUALS AND ESTIMATED VARIANCE FOR A T 7 98 FIGURE AV.7 PREDICTED ACCIDENTS vs. PEARSON RESIDUALS F O R A T 13 99 FIGURE AV.8 PREDICTED ACCIDENTS VS. AVERAGE SQUARED RESIDUALS AND ESTIMATED VARIANCE F O R A T 13 99 FIGURE AV.9 PREDICTED ACCIDENTS vs. PEARSON RESIDUALS FOR TOTAL ACCIDENTS 100 FIGURE AV.10 PREDICTED ACCIDENTS vs. AVERAGE SQUARED RESIDUALS AND ESTIMATED VARIANCE FOR TOTAL ACCIDENTS 100 FIGURE AV.11 PREDICTED ACCIDENTS VS. PEARSON RESIDUALS FOR SEVERE ACCIDENTS 101 FIGURE AV.12 PREDICTED ACCIDENTS VS. AVERAGE SQUARED RESIDUALS AND ESTIMATED VARIANCE FOR SEVERE ACCIDENTS 101 FIGURE AV.13 PREDICTED ACCIDENTS VS. PEARSON RESIDUALS FOR P D O ACCIDENTS 102 FIGURE AV.14 PREDICTED ACCIDENTS VS. AVERAGE SQUARED RESIDUALS AND ESTIMATED VARIANCE FOR P D O ACCIDENTS 102 FIGURE AV. 15 PREDICTED ACCIDENTS VS. PEARSON RESIDUALS FOR LEFT TURN ACCIDENTS 103 FIGURE AV.16 PREDICTED ACCIDENTS VS. AVERAGE SQUARED RESIDUALS AND ESTIMATED VARIANCE FOR LEFT TURN ACCIDENTS 103 FIGURE AV. 17 PREDICTED ACCIDENTS VS. PEARSON RESIDUALS FOR RIGHT TURN ACCIDENTS 104 FIGURE AV.18 PREDICTED ACCIDENTS VS. AVERAGE SQUARED RESIDUALS AND ESTIMATED VARIANCE FOR RIGHT TURN ACCIDENTS 104 X i V i k r i w c o p k AU-M&PA Pred ic t ioH M o d e l * 1'or Signal ised In ierseer ions ACKNOWLEDGEMENTS My sincere thanks to Dr. Tarek Sayed, Assistant Professor, Department of Civil Engineering, University of British Columbia for the time he spent teaching me the theory behind traffic safety whose guidance and assistance were fundamental in the completion of this research. And to Dr. Frank Navin, Professor, Department of Civil Engineering, University of British Columbia for all the valuable insights I learnt from him about traffic flow and transportation planning. I would also like to thank The Insurance Corporation of British Columbia (ICBC) for providing the financial support for this thesis. I would also like to thank my parents, Dr. Mario Quintero and Guadalupe Toscano, and the rest of my family for their unwavering encouragement and support through the duration of this program. "De ustedes he aprendido lo que realmente importa" xi M i c r o s c o p i c Accident P r e d i c t i o n M o d e l s fo r Signal ised in tersec t ions 1. INTRODUCTION 1.1 Background The literature reviewed showed that several researchers have dealt with the development of accident prediction models for isolated intersections (signalised and unsignalised) in a macroscopic fashion. The accident frequency has been related to the product of the average daily traffic on the major and minor roads. In this study, microscopic models, which relate the accident frequency to the traffic flows to which the colliding vehicles belong, will be developed. This should enhance the use of the models for safety planning purposes. There are two approaches for dealing with traffic safety problems: the reactive approach and the proactive approach. The reactive approach, or retrofit approach, consists of making the necessary improvements to existing hazardous sites in order to reduce accident frequency and severity at these sites. The proactive approach, on the other hand, is an accident prevention approach that tries to prevent unsafe conditions from occurring in the first place. However, one obstacle associated with the delivery of proactive road safety measures is the lack of the necessary tools to evaluate road safety in a proactive manner. For road safety decisions to be made early in the planning or design stage, it is important to understand the impact of an action on safety performance. the development of microscopic safety prediction models, which help explain the relationship between accident occurrence and the geometric and traffic parameters, is therefore invaluable for the success of the proactive safety approach. These models can be used by transportation planners to assess the impact of various transportation policies on traffic safety. i Microscopic Accident Prediction Models for Signalised Intersection!! 1.2 Thesis Objectives The objective of this thesis is to develop microscopic safety prediction models for urban signalised intersections in British Columbia. The models will be capable of estimating the frequency of the total accidents and specific accident types as a function of the detailed traffic flows at the intersection, including turning movements. The models are developed using the Generalised Linear Regression Modelling (GLIM) approach. The GLIM approach addresses and overcomes the shortcomings of the conventional linear regression approach in modelling the occurrence of traffic accidents (as will be explained later). The data on accident frequencies and traffic volumes were obtained from the city of Vancouver. Several models that have different applications in the field of traffic safety are developed in this study for the 4-leg signalised intersections of the City of Vancouver. Microscopic models for different accident types are developed in conjunction with macroscopic models for Total, Severe and Property Damage Only accidents. Several model applications are discussed. 2 M i c r o s c o p i c A c c i d e n t P red i c t i on M o d e l s Cor Signal ised In tersect ions 2. PREVIOUS WORK 2.1 Introduction A literature review of the following areas was conducted: common practices used to assess safety at signalised intersections, the identification of Accident Prone Locations (APLs), and Accident Prediction Models (APMs) and their applications. It was found that the relationship between traffic accidents and traffic volume has been the subject of numerous studies. Most of the early studies used the conventional linear regression approach to develop models relating accidents to traffic volumes. However, several researchers have demonstrated the inappropriateness of conventional linear regression for modelling discrete and non-negative events such as traffic accidents. In this section, a brief outline of the major developments in accident prediction models will be presented. 2.2 Conventional Linear Regression Models (CLRM) Conventional Linear Regression Models (CLRM) were used by (Zegeer et al, 1986; Miaou and Lum, 1993) to described the empirical relationship between accidents, traffic, and road geometric design. The CLRM can be defined as follows: m Y = aQ+YJ aj xiJ +£. /=1,2,...,n (2.1) Where, Y = estimated or dependent variable 3 ao, B\ = estimated coefficients, obtained by the least squares or the maximum likelihood methods xy = independent variables si = estimated error, assumed to be normally distributed The shortcomings associated with this approach have been documented in the literature by several researchers. Jovanis and Chang (1986) pointed out three shortcomings. First, the relationship between the mean and the variance of accident frequencies does not remain constant as a normal distribution assumes it should, the variance actually increases as the volume of traffic does. Second, conventional linear regression models can result in estimating negative accident frequencies for a data set that contains low accident frequencies. Third, the error distribution non-normality. Similar conclusions were reached by Miaou and Lum (1993). In addition, Miaou and Lum (1993) found that using a Poisson distribution error structure produced models with a better fit when compared with CLRM. 2.3 Generalised Linear Regression Modelling (GLIM) Generalized Linear Regression Modelling (GLIM) has the advantage of overcoming the limitations associated with conventional linear regression in modelling traffic accident occurrence. GLIM computer packages can be used for modelling data that follow a wide range of probability distributions such as the Normal, Poisson, binomial, negative binomial, gamma, and many others. These computer packages also allow the flexibility of using several non-linear model forms that can be converted into linear forms through the use of several built-in link functions. 4 M i c r o s c o p i c A c c i d e n t P r e d i c t i o n M o d e l s far Sinttaiiscnf Inlet-sections The GLIM approach used herein is based on the work by work by Kulmala (1995) and Hauer et al (1988). It is assumed that Y is a random variable that describes the number of accidents at an intersection in a specific period of time, and y is the observation of this variable during that period of time. The mean of V is A which can also be regarded as a random variable. Then for A=A, Y is Poisson distributed with parameter A: P(Y = y\K = A) = i i f _ ; £ ( y | A = A);Var(Y\A = A) = A (2.2) Since each site has its own regional characteristics with a unique mean accident frequency A, Hauer et al, (1988) suggested that, for an imaginary group of sites with similar characteristics, A follows a gamma distribution described by the parameters *rand K/JU, where K is the shape parameter of the distributions, that is: / . W - ^ l ^ l (2.3, With a mean and variance of: E(h) = ju;Var(A) = — (2.4) K Kulmala (1995) showed that the point probability function of V based on equations 2.3 and 2.4 is given by the negative binomial distribution: P(Y = y) = n K + y ) ( K ^ K [ " ) KK+Uj With an expected value and variance of: (2.5) 5 Mienwcopie .Accident Prediction M o d e l s 1'or Signalised intersections E(Y) = rt;Var(Y) = ti + ?- (2.6) K It can be seen from equation 2.6 that the variance of the observed number of accidents is generally larger than its expected value, and it is composed of a first term (ju) from the variation of the number of accidents and by a second term ( ///K) from the variance of the predicted number of accidents. The only exception is when K goes to infinity, in which the distribution of A is concentrated at a point and the negative binomial distribution becomes identical to the Poisson distribution (Kulmala, 1995). When modelling traffic accident occurrence in the GLIM approach, as in the procedure described above, a Poisson or negative binomial error structure is assumed. The Poisson error structure offers the advantage of simplicity in the calculations because the mean and the variance are equal. Nevertheless, this advantage is also a limitation as it has been shown by shown by Kulmala and Roine (1988) and Kulmala (1995) when they demonstrated that most accident data are likely to be overdispersed. This means that the variance is greater than the mean (equation 2.6) which indicates that the negative binomial distribution is usually the more realistic assumption for the error structure. Three possible sources of overdispersion inherent to accident data were identified by Miaou and Lum (1993). The first is related to not including all explanatory variables when developing an APM. No model can include all of them because traffic accidents are the product of a complicated interaction between series of variables that sometimes are not available on the data set used to develop the APM. The second relates to the uncertainty involved in the vehicle exposure data and traffic variables that derive from errors during the data collection stage. And the third is due to using accident data that comes from a non-homogeneous roadway environment due to weather, light, road, traffic conditions, etc. 6 M i c r o s c o p i c Accident' P r e d i c t i o n M o d e l s for S i gna l i sed In tersect ions Bonneson and McCoy (1993) proposed a two step procedure to determine which error structure was the most appropriate for the data. First, a Poisson distribution error structure is used to determine the model coefficients (e.g. cc0, Pi, P2)- After this, a dispersion parameter ad defined as in equation 2.7 is calculated. Pearsonx2 ^ 7) n-p Where, n = number of observations p = number of model parameters Pearson x2 = Pearson test value defined as follow: Pearsonx2 = V ^ " ^ j f (2.8) ,=i Var(y:) Where, y, = observed number of accidents on the intersection / E ( A/) = predicted number of accidents on the intersection /' Var (y,) = variance of the observed number of accidents (as defined in equations 2.2 and 2.6) for the Poisson and negative binomial respectively If ad is greater than 1.0, then the data has greater dispersion than is explained by the Poisson distribution. Thus, further analysis with the negative binomial error structure is required. If ad is close to 1.0, then it can be said that the data approximately fits the Poisson error structure distribution. By using this method the model can be tested with an assumed Poisson distribution first, which requires less effort than the negative binomial distribution. 7 M i c r o s c o p i c A c c i d e n t Prediction M o d e l s i'or Signal ised In tersect ions The estimation of the shape parameter K when using the negative binomial error structure was studied by Kulmala (1995) who proposed an iterative approach employing the method of moments. Hauer et al (1988) used an iterative process in which an assumed value for *:was used to estimate the vector of coefficients with GLIM (e.g. a 0 , P i , P2), and after that the residuals were calculated. These residuals served as input into the program to obtain the maximum likelihood estimate of K. The new K was then fed back into the program to obtain a new set of coefficients until closure was reached. The GLIM software package used for this study (version 4.0) includes a macro library in which the parameter A:can be calculated with three methods: the maximum likelihood, the mean deviance estimate, and the mean %2 estimate (NAG, 1996). The method of maximum likelihood is used in this study. 2.4 Microscopic Accident Prediction Models (APMs) Detailed models that include several accident classifications, different impact types, or time periods have been studied previously. Lau et al (1989) proposed a method to estimate intersection safety by accident severity (Fatal, Injury, and PDO). Poch and Mannering (1996), and Al-Turk and Moussavi (1996) modelled the annual accident frequency for all accidents and for specific types (rear-end, angle, etc.). Hauer et al (1988) categorised accidents by the movement of the vehicle(s) before the collision and used the traffic flows to which the colliding vehicles belong to model specific accident types. For each recognised pattern, an accident prediction model was developed to estimate the expected number of accidents and the variance using the relevant traffic flows (through, left-turning or right-turning volume(s)). 8 2.4.1 Model Evolution Thorpe (1963), Smith (1970), and Worsey (1985) suggested that the total number of accidents at an intersection is proportional to the sum of all flows entering the intersection. The advantage of this approach is its simplicity; however, it fails in linking cause and effect of the accidents. Breuning and Bone (1959), Surti (1965), and Hakkert and Mahalel (1978) recommended relating accidents to the product of the entering flows. Webb (1955) and McDonald (1966) found in empirical research that accidents are related to the product flows with each flow raised to a power of less than one. Hauer et al, (1988), Nguyen and (1997) developed microscopic models for accidents at signalised intersections using the model of the product of flows raised to a power. 2.4.2 Model Form Hauer et al (1988) studied the general model intended for this project. This model relates accidents to the product of the traffic flows entering the intersection and it can be written as follow: E{m} = a0F/hFPle (2.9) Where E{m} Fi F 2 ao, P i , p2, bj = predicted accident frequency = major road traffic volume = minor road traffic volume = model coefficients X i = any of the additional explanatory variables 9 Detailed APMs were also developed by (Bonneson and McCoy, 1993; Belanger, 1994) and had the following forms: E{m} = a0Ffi> (2.10) E{m} = a0FPiFp2 (2.11) Where, E {m} = estimated parameter ao , P i , P2 = estimated coefficients, obtained by Generalised Linear Interactive Modelling Software F, Fi, F 2 = specific traffic flows which define the specific traffic pattern In these equations the coefficients P i and P2 represent the changes of traffic flows and a 0 represents the changes of any other factors. As it can be seen, equation 2.10 is a special case of 2.11 and applies for all accident patterns that occur within the same traffic flow. As reported by (Persaud and Dzbik, 1993), these equations agree with the logic "no traffic flows, no accidents" and allow using non-linear relationships between accidents and traffic flows. 2.5 Outliers Outliers are observations from the data that diverge or are very distinct from the rest of the data. They can occur due to irregularities or inaccuracies incurred during the data gathering process or in the case when the data genuinely diverge from the rest of the observations. The leverage statistics, as proposed by Kulmala (1995), can be used to decide whether the observation should be removed or not. The leverage can be 10 Mierwscon te A c c i d e n t P r e d i c t i o n M o d e l s to r S igna l i sed ( intersections defined as the measure of how far the x-value of the point is away from the average of the rest of the x-values (NAG, 1994). NAG (1994) showed that the leverage by itself is not a good measure to warrant the removal of an observation. NAG (1994) also showed that a measure that can help to determine whether the parameter estimates are being affected by the specific observation is the Cook's distance. This distance is the measure of how the observations influence the model. Therefore, an observation which exhibit a high Cook's distance compared to the rest of the observations would have a stronger influence on the model than an observation that is clustered with the rest. The Cook's distance can be calculated as follow: h.(rps')2 c,= A ' ' (2.12) />(!-*,) Where, c t = Cook's distance h t = Leverage rvPS' = Standardised residuals p = Number of parameters No clear evidence of a procedure to determine which observations should be considered as having a high Cook's distance was found in the literature. NAG (1994) suggested sorting the data according to the Cook's distance and in an iterative process remove the points that exhibit the highest values. This procedure should be accompanied by the assessment of the change in the scaled deviance. Maycock and Hall (1984) found that change in the scaled deviance from a model that includes the complete data set (with dofi) to a model in which an observation has been removed (dof2), is x2 distributed with parameters dof rdof 2. By doing this the removal of an observation would be u ' . ' I n l e r s M ' r k s n s warranted if it exhibits a higher Cook's distance than the %2 value for the desired level of confidence. 2.6 Testing the Models Significance The Scaled Deviance (SD) and the Pearson x2 are usually utilised to assess the significance of the GLIM models. The SD is defined as the likelihood test ratios measuring the difference between the log likelihood of the studied model, and the saturated model (Kulmala, 1995): SD = 2 log f(y, y) - 2 log f(E(A), y) (2.13) Where, 21og /(£(A),>>) = The natural logarithm for the probability density function For a Poisson model, (McCullagh and Nelder, 1993) showed that the SD is defined by: (2.14) And for a Negative Binomial distribution is defined by: i=\ V £ ( A / ) / (yi+K)\n (2.15) The SD is asymptotically x2 distributed with n-p-1 degrees of freedom. 12 Another method to assess the statistical significance of the GLIM models is to use the Pearson %2 as in equation 2.8. In addition, several graphical methods can be used. They will be reviewed in Section 4.3. 2.7 The Empirical Bayes Refinement (EB) There are two types of clues to the safety of a location: its traffic and road geometric design characteristics, and its historical accident data (Hauer, 1992, Briide and Larsson, 1988). The Empirical Bayes (EB) approach makes use of both kinds of clues. The EB approach is 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 GLIM model to yield more accurate, location-specific safety estimate. The EB estimated number of accidents for any intersection can be calculated using (Hauer et al. ,1992): EBsafely = ccE(A) + (\-a)count (2.16) Where a - Var(E(A)) E(A) (2.17) 1 + and, count = observed number of accidents 13 M i e t w o n i c A c c i d e n t P red i c t i on M o d e l s for S ignal ised interseeiions E ( A) = predicted number of accidents as estimated by the GLIM model Var (E ( A)) = variance of the GLIM estimate Since Var(E(A)) = E ^ , then equation 2.16 can be rearranged to yield: K safely estimate r E(A) ^ (K +count) (2.18) K + E(A) The variance of the EB estimate can also be calculated by: Var(EBsafelv ) = safety estimate r E(A) V KK + E(A); {K +count) (2.19) As shown, the refinement estimates for any entity can be known by using the EB method if the predicted accident frequency and its variance from a reference population that shares similar physical characteristics with the entity and the accident history of the entity are known. These refinement estimates, that combine the information from both sources, should be used to decide whether the entity is unsafe. In addition, they can be used to assess the safety effects of countermeasures applied to the entity. 2.8 Accident Prone Locations (APLs) Accident-prone locations (APLs) are defined as locations that 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 used when identifying APLs. Higle and Witkowski 14 (1988) and Belanger (1994) proposed the use of the EB refinement method to account for this randomness as follow: Using the appropriate GLIM model estimate, determine the predicted number of accidents and its variance for the intersection. This would follow a gamma distribution (the prior distribution that represents the distribution of the expected accident frequency for a population of intersections with similar characteristics to the intersection analysed), that is defined by the parameters aj and Pi, where: P^V^ = 1^ B N D OCX=B,-E(K) = K (2.20) Determine the appropriate point of comparison based on mean and variance values obtained previously. Using the 50 t h percentile (P 5 0) is a practice commonly accepted, another possibility is to use the mean E ( A). P50 is calculated such that: KKlE(k)) -A -e d X = Q 5 Calculate the EB safety estimate and its variance using equations 2.18 and 2.19. This is also a gamma distribution, called posterior distribution, and defined by the parameters a2 and p2, where: EB = _ ^ _ + 1 a n d a - p EB = K + count (2.22) 2 Var(EB) E(A) Then, the probability density function of the posterior distribution is: 15 M i c r o s c o p i c A c c i d e n t P red i c t i on M o d e l s for S ignal ised i n te rsec t ions (KIE(A) + \ y K + c o m " ) . ^(K+c°un'-v . g - ( t f / £ ( A ) + i ) A r(/c + count) (2.23) Identify the intersection as APLs if there is significant probability that the intersection's safety estimate exceeds the P 5 o value. Thus, the location would be identified as APLs if: ' 50 1-f . T(K + count) dX > S (2.24) Where 8 represents the confidence level desired (usually 0.95) Prior and Posterior Distributions • Prior ' Posterior Probability of safety estimate less than Pso Figure 2.1 APL identification process 16 M i c r o s c o p i c Accident P r e d i c t i o n M o d e l s for Signalised intersections 2.9 APMs for 4-leg signalised intersections Microscopic APMs were developed by Hauer et al (1988) using data from 145 4-leg, fixed time, signalized intersections from Metropolitan Toronto. These intersections carried two-way traffic and had no turn restrictions. The majority of the intersections were on straight, level sites with a speed limit of 60 km/hr. Detailed flows for the AM peak, PM peak, and off-peak were collected manually (for weekday conditions). Only collisions involving two vehicles were analysed for the period between 1982 and 1984. APMs for rear end, intersection 90°, and opposing left turn were reported as significant models, and the rest of the accident patterns were provided but regarded as unreliable. The APMs that correspond to patterns studied in this project are summarised in Table 2.1. Similar APMs were developed by Nguyen (1997). A total of 254 signalised intersections maintained by the Central and Southwest regions of Ontario were used for this purpose. From these intersections, 197 were rural, 33 were semi-urban, and 24 were located in urban areas. Directional traffic flows for AM peak, PM peak, AADT and accident counts from 1988 to 1993 were used to develop the models. 25 Patterns were analysed, however, just 11 patterns had emerged as statistically significant models. Models for each of the different areas were developed for PDO and Severe accidents. The significant PDO and Severe APMs that correspond to patterns analysed in this project are presented in Tables 2.2 and 2.3. 17 M k r o s e o p i e A c d d e s i i P r e d i c t i o n M o d e l s for Signal ise*! i o fe rsee i i ons Pattern J U E{m}= 0.2052X10"6*F and E{m}= 0.1014X10"6*F 4.59 and 1.97 E{m}= 8.6129X10"9*F21 0 6 8 2 1,2 f i j " E{m}= 8.1296X10"6*F2 0-3 6 6 2 5.51 F1 (rn E{m}= 1.7741X10-9*F 1 1 U 2 1*F 2 0- 5 4 6 7 1,2 F l ? " E{m}= 0.4846X10-6*F 1 a 2 7 6 9*F 2 0- 4 4 7 9 1,2 E{m}= 2.6792X10"6*F2 0 '2 4 7 6 1 ,2 H < f = F ^ E{m}= 0.2113X10- 6*F 1°^*F 2 0-' ' 0 5 1 1,2 E{m}= 0.0418X10"6*F2 0-4 6 3 4 2.10 * Table 2.1 APMs for 4-leg signalised intersections (Hauer et al, 1988) Where, * = Models studied in detail, the rest of the models were regarded as "unreliable" because of the limited data F = Sum of flows Fi = Major flow (for the third and fourth models), straight flow (for models that have a turning flow) F 2 = Minor flow or turning flow (refer to Fi explanation for) All flows in vehicles per hour Note: As observed in Table 2.1, some of the models suggest the use of a K range instead of giving a specific value. These models were regarded as unreliable by the authors. 18 M i c r f t s c o n i c A c c i d e n t P r e d i c t i o n M o d e l s f o r S i gna l i sed In tersect ions Pattern >1AW5 K E { m } = 2 9 . 9 X 1 0 " 6 * F 11 0 7 7 and E { m } = 4 1 . 8 X 1 0 ' 6 * F 10 9 2 5 3.15 and 1.52 F l ^ ' i E { m } = 16929X10" 6 *F 2 0 2 5 5 0.99 E { m } = 47 .3X10" 6 *F 1 ° -6 7 6 *F 20 - 2 0 6 0.43 E { m } = 269.3XIO- 6*F 1°- 3 4 0*F 2 0- 3 4 4 1.17 E { m } = 2 3 7 . 4 X 1 0 " 6 * F 1a 5 6 7 * F 20 0 8 3 1.21 E { m } = 2 3 3 1 . 4 X 1 0 " 6 * F 1 ° -3 6 9 * F 20 0 3 2 1.55 li <^  F2 E { m } = 38.7X10- 6 *F 1 ° - 5 7 *F 2 0 - 6 0 4 2.40 Table 2.2 PDO APMs for4-leg signalised intersections (Nguyen, 1997) Pattern APMs SEVERE K c=>"c=> E { m } = 3 . 3 8 X 1 0 ' 6 * F 2 1 2 4 9 and E { m } = 5 7 . 4 X 1 0 " 6 * F 2 a 7 8 8 4.05 and 0.77 111 B E { m } = 3 5 4 9 . 8 X 1 0 " 6 * F 2 0 4 2 3 0.089 iTVn E { m } = 2 .6X10- 6 *F 1 ° - 5 7 6 *F 2 0 - 2 0 6 0.43 E { m } = S l . l X l O - 6 ^ ! 0 - 3 4 0 ^ , 0 3 4 4 1.17 E { m } = 70.3X10" 6 *F 1 ° -5 6 7 *F 20 0 8 3 1.21 E { m } = l S l Q . e X l O - ^ F ^ ^ ^ F z 0 ' 0 3 2 1.55 E { m } = H . e X l O - 6 ^ ! 0 - 6 9 9 ^ , 0 5 4 2 2.05 Table 2.3 SEVERE APMs for 4-leg signalised intersections (Nguyen, 1997) Where, F, Fi, and F 2 are as described for the first set of models Traffic volumes in AADT 19 Microscopic A c c i d e n t P red i c t i on Models for S ignal ised In tersect ions Both studies differentiated rear end accidents that occur before the intersections and in the intersection. These two models are shown in the same cell (first row cell), their K'S are reported in the next cell respectively. Similar models for the Greater Vancouver Regional District were developed by Feng (1997). 72 urban intersections form the City of Vancouver and 67 from the City of Richmond were used for this purpose. The APMs developed in this study intended to relate accidents to their traffic and road variables. Models for specific accident types and aggregate models were developed. They included severity types, different time during the day, and peak periods. The models that are relevant for this project are summarised in Table 2.4. Pattern APMs K E{m}= 0.0627*F 1 ' 2 3 6 0 26.43 TOTAL E{m}= 2.1813*F 1 0 3 2 8 6 *F 2 0 - 4 4 1 8 8.4 SEVERE E{m}= 0.9049*F 1 a 2 8 3 6 *F 2 0 - 4 4 9 0 18.45 PDO E{m}= 1.3012*F 1 a 3 4 0 9 *F 2 a 4 4 1 2 11.27 LEFT TURN E{m}= 0 .5572*F 1 a 2 9 9 9 *F 2 a 4 9 5 0 3.42 Table 2.4 APMs for signalised intersections (Feng, 1997) Where, F = Sum of all entering flows Fi = Major road volume F 2 = Minor road volume All volumes are AADT in thousands 20 M k r o s f o p i c A c t - i f l c M t P red i c t i on M o d e l s for Signal ised In tersect ions 3. DATA DESCRIPTION 3.1 Introduction The accident data used for this project consists of 8466 accidents reported in a set of 170 four-legged signalised intersections in the City of Vancouver from 1994 to 1996. The MV104 accident reporting form, British Columbia's accident police report, provided the information needed to group the accidents into the 17 accident types according to vehicle's movements before the crash (16 accident types plus the type "other" explained by police comments). The MV104 form categorises accidents into the 17 types shown in Table 3.1. The volume data was obtained from the City of Vancouver. 3.2 Data Analysis After summarising the accident information provided by the MV 104, several tables were developed. They are included in the Data Analysis Appendix. The time, date and the accident contributing factors are summarised in Tables AIM, All.2 and All.3. As shown, more accidents occurred during the daytime period, reaching a maximum between 16:00 and 16:59 and a minimum between 4:00 and 4:59. Accidents seem to occur less during the summer and to be fairly constant over the rest of the year. The contributing circumstances were not reported for all accidents. More than two thirds (67.77%) of the accidents did not have a contributing circumstance reported. The most frequent circumstance was failing to yield. 21 M i c r o s c o p i c A c c i d e n t P red i c t i on M o d e l s f o r S ignal ised intersections Accident Type Description Pattern AT 1 Rear end AT 2 Head on AT 3 Side swipe AT 4 Backing AT 5 Intersection 90° AT 6 Overtaking AT 7 Right Turn AT 8 Right Turn AT 9 Right Turn AT 10 Right Turn AT 11 Left Turn AT 12 Left Turn AT 13 Left Turn • -AT 14 Off road right Off road left One way street t t Other "explained by police comments " N. A. Table 3.1 The MV 104 Form: Accident types, traffic patterns 22 M i c r o s c o p i c A c c i d e n t P red i c t i on iVfotiels for Signalised interseci losrs Accident types are summarised in Table AII.4 where it can be observed that Rear end (Accident Type 1), Left turn (Accident Type 13), and Intersection 90° (Accident Type 5) were the most common types; these three types alone represent 77.4% of the all accidents. This finding agrees with previous research (Hauer et al, 1988) where it was pointed out that these three accident types accounted for 83% of the accidents in their study. Light, road, and weather conditions are shown in Tables All.5, All.6 and All.7. The most frequent light, road, and weather conditions were daytime, clear, and dry respectively. Tables All.8, All.9 and All. 10 contain the volume information. The volume information was summarised according to movement type (e.g. Northbound Through, Northbound Right turn, Northbound Left Turn, etc.), approach, and by intersection. Some useful statistics were calculated when possible. The maximum and minimum count, average, standard deviation, median, and variance were calculated for all tables. Count percentages were calculated for Tables 6.3 through 6.7. Unreported information was a problem and is noted in Table 6.3 and 6.6. Table 6.3, Contributing Circumstances, indicates that just 32.23% of the accidents analysed for this report had a contributing circumstance assigned by the police officer (more than two thirds of the accidents did not have a reported contributing circumstance). Table 6.6, Road Conditions, indicates 25 unreported accidents (0.295% of the accidents) which did not have a road associated contributing circumstance. In addition, several graphs were used to summarise the data. Figure 3.1 shows the accidents that occurred at each hour during the day. As mentioned before, the period from 16:00 to 16:59 presented the highest count and the period from 4:00 to 4:59 the lowest. Daytime hours have higher accident counts, as would be expected. 23 V i i f r o s c o j i i c A c c i d e n t Prediction M o d e l s fo r S igna l i sed in te rscc t tons Figure 3.1 Time of Accidents 800 0 £ 600 £ 9 400 § o 1 ^ 200 0 HOUR OF ACCIDENTS iin 0 1 2 3 4 5 6 7 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 HOUR Figures 3.2 to 3.4 show the accident count for each month from 1994 to 1996. The average accident counts per month for the 3 years studied was 235 accidents/month. 24 YiieroM'isjtic Accident Prediction Models for Signalised Intersections Figure 3.2 1994 Accidents 1994 ACCIDENTS 400 £ £ 300 on LU LU m 9 2 0 0 o o < 100 0 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC MONTH 400 U- (J) O hi 300 g Q 200 | ^ 100 0 Figure 3.3 1995 Accidents 1995 ACCIDENTS JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC MONTH 25 M i r r w c o i i k A c c i d e n t P red ic t i on M o d e l s f o r Signal ised In tersect ions 300 u_ (/) 250 ° Z 200 S = 150 18 1 0 0 z < 50 0 Figure 3.4 1996 Accidents 1996 ACCIDENTS JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC MONTH Figure 3.5 shows the accident count per year for the 1994 to 1996 period. Figure 3.6 compares the 3 years accident count simultaneously. Figure 3.5 Total Accidents 3200 £ CO 3000 g m 2800 | § 2600 I < 2400 2200 3 YEARS ACCIDENTS 1995 Y E A R 26 Microscopic A c c i d e n t P red ic t i on M o d e l s f o r Signalised Intersections Figure 3.6 3 years Accidents per Month 350 3 YEARS ACCIDENTS JAN FEB M A R APR M A Y JUN JUL A U G SEP OCT NOV DEC MONTH The contributing circumstances are shown on Figure 3.7. As stated before, just 2729 out of the 8466 accidents had a reported contributing circumstance. Failing to yield was the most frequently reported contributing circumstance. Figure 3.8 shows the accidents by type according to the MV 104. Rear end (type 1), Head on (type 2), Intersection 90° (type 5) are the single types with the highest count. 27 c/) UJ o z < I -o o CQ O O s c aa is O N O M M H3H1V3AA 2 El a a i v d r a s i A ascno ooi n v d a u u . DNI033dS siuaaa avou aiOOV A3Hd 3ZISd3AO HSRLO NOUVOICBW an aynossNi N y n i d o u d w i SSVd dOddl/MI SS3N11I sonua 11 NCO d l UN9I UOIddO UN9I Nns-auvio d i i d v - g a v n o d33nSVTBd a n o i i v d a i i u i v d s i s nivd dxaNi Aiya iiVNI Alda i H o n a H daa H o n x a a d sa 3 » v a a d i a Niino j/io>iovia Q a d v s N n x o v a » | a/A / da ioAv " lOHCOIV 5 C o o o o o o o o o o o o o o o o o o o O C O S C D l O ^ - C O C v l ^ -CO z LU Q o LL o LU Q. >-CN O CM CM LO LO CO CO O OO M" O CD CM Gi CO CO 0 0 n LO • CD [ o g co — 1 a3HU0 91 ±V Sl IV IV Cl IV 31 IV 11 IV 01 ±v 61V 81V i l V 9 IV S1V LO t l V I CIV 31V I IV o o o O o o o o o o O o o o LO o LO o LO CO CN CM Microscopic Accident Prediction Models for Signalised Intersections Figures 3.9 and 3.10 include accident types that showed higher counts: Accident types 1, 2, 5, left turn (types 11, 12 and 13) and right turn (types 7, 8, 9, and 10) by count and by percentage respectively. 3000 2500 2000 1500 1000 500 0 Figure 3.9 Combined Types TYPES OF ACCIDENT 2692 2385 1655 497 REAR HEAD RIGHT LTs END ON ANGLE 372 RT's Figure 3.10 Combined Types Percentages TYPES OF ACCIDENTS 30 Microscopic Accident Prediction Models for Signalised Intersections Figures 3.11 and 3.12 summarise accidents by severity, count, and percentage respectively. 8000 6000 4000 2000 0 Figure 3.11 Accident Severity ACCIDENTS BY SEVERITY 5963 2493 10 Fatal Injury P D O Figure 3.12 Accident Severity Percentages ACCIDENTS BY SEVERITY 0% • Fatal • Injury • PDO 31 Microscopic Accident Prediction Models for Signalised Intersections Finally Figures 3.13, 3.14, and 3.15 show the different light, road, and weather conditions at which accidents occurred. Day (light), Dry (road surface), and Clear (weather) were the most frequently reported accident conditions. 7 0 0 0 6 0 0 0 5 0 0 0 4 0 0 0 3 0 0 0 2 0 0 0 1 0 0 0 0 2 7 < Q Figure 3.13 Light Condition LIGHT CONDITION 5132 9 5 < Q < Q 6 2 9 X Z L Q 1 7 0 7 2 5 6 2 0 co Z) Q Figure 3.14 Road Condition 3 ROADWAY SURFACE CONDITION 6 0 0 0 5000 4 0 0 0 3000 2 0 0 0 1000 0 5311 >-LY. Q 16 LU o 12 Q D 12 x CO D _] CO 55 O z CO 3 0 3 5 h-iu 32 Microscopic Accident Prediction Models for Signalised Intersections Figure 3.15 Weather Condition WEATHER CONDITION 4 6 8 5 33 Microscopic Accident Prediction Models for Signalised Intersections 4. MODELS DEVELOPMENT 4.1 The Models Accident Prediction Models (APMs) were developed for all accident types as specified in the MV104 form excluding accident type 3,14,15,16, and 17 (See Table 4.2). In addition to the 12 accident types, 2 models that combined the turning movements and 3 macroscopic models were developed. The combined models are left turn accidents (types 11, 12, and 13 combined), and right turn accidents (types 7, 8, 9, and 10 combined). The macroscopic models developed correspond to the total number, severe (fatal plus injury), and PDO. The criterion of major and minor volumes was used to find the flows for the macroscopic models (i.e. the larger volume was regarded as the major volume and the smaller as the minor). The models and traffic patterns used are summarised in Tables 4.1 and 4.2. Equation 2.10 is used for accident types related to a single flow in which F denotes the sum of all volumes entering the intersection. Equation 2.11 is used for models that are described by two traffic volumes, where Fi is the larger volume. Head on accident (AT 2) traffic volumes are defined as a special case; Fi is the larger volume between the combination of southbound through/westbound through and northbound through/eastbound through and F2 the smaller of the two volumes. By doing this all the possible head on colliding trajectories are covered. Also, for Intersections 90° accidents (AT 5), the treatment of traffic volumes is a special case. Southbound through (SBT) and Northbound through (NBT) volumes are added to form a combined volume, and Westbound through (WBT) and Eastbound through (EBT) volumes are also added to constitute the second volume. These combined volumes are then sorted as major and minor volumes. All models use major and minor road volumes in 1000s AADT. 34 Microscopic Accident Prediction Models for Signalised Intersections Accident Type Pattern F F. F2 Equation AT 1 ' Approaches E {m} = a 0 P 1 AT 2 Major Minor E {m} = a 0 Fi151 F/ 2 AT 3 Major Road Minor Road E {m> = a 0 F / 1 F/ 2 AT 4 L Approaches E {m} - ao P 1 AT 5 Major Minor E {m} = a 0 F^1 F/ 2 AT 6 . Approaches E {m} = a 0 P 1 AT 7 <=>tf> i Through Z Right Turn E {m} = a 0 F/ 1 F/ 2 AT 8 \ Through Z Right Turn E {m} = a 0 F/ 1 F/ 2 AT 9 1 Through Z Right Turn E {m} = ao F/ 1 F/ 2 AT 10 l Through Z Right Turn E {m} = a 0 F^1 F/ 2 Al 11 l Through £ Left Turn E {m} = a 0 F/2 AT 12 Through z Left Turn E {m} = a 0 F/ 1 F/ 2 >. Through Z Left Turn E {m} = a 0 Fx"1 F/ 2 AT'14 ', ; V I Approaches N. A. AT 15 Z Approaches N. A. AT it) N. A. N. A. N. A N. A. N. A. * Major and minor between the combination of SB-WB and NB-EB ** Major and minor between SBT+NBT and WBT-EBT (refer to previous page for further explanation about these cases) Table 4.1 Accident types, defining patterns and flows, and equations used to develop the models (for accident description refer to table 3.1) 35 Microscopic Accident Prediction Models for Signalised Intersections 9 8 7 6 5 4 10 •mJ&' 12 1 2 3 NB+SB 1+2+3+7+8+9 WB+EB 4+5+6+10+11+12 SB+WB 7+8+9+4+5+6 NB+EB 1+2+3+10+11+12 RT LT F i AT 1 X of all Flows AT 2 Highest of SB+WB and NB+EB AT 3 Highest of NB+SB and WB+EB AT 4 Z of all Flows AT 5 Highest of 2+8 and 5+11 AT 6 S of all Flows AT 7 2+5+8+11 AT 8 2+5+8+11 AT 9 2+5+8+11 AT 10 2+5+8+11 AT 11 2+5+8+11 AT 12 2+5+8+11 AT 13 2+5+8+11 2+5+8+11 2+5+8+11 TOTAL Highest of NB+SB and WB+EB I SEVERE Highest of NB+SB and WB+EB PDO Highest of NB+SB and WB+EB Not used Smallest of SB+WB and NB+EB Smallest of NB+SB and WB+EB Not used Smallest of_2+8 and 5+11 Not used | 3+6+9+12 3+6+9+12 3+6+9+12 3+6+9+12 1+4+7+10 1+4+7+10 1+4+7+10 3+6+9+12 1+4+7+10 Smallest of NB+SB and WB+EB Smallest of NB+SB and WB+EB Smallest of NB+SB and WB+EB Table 4.2 Traffic flows graphical explanation 36 Microscopic Accident Prediction Models for Signalised Intersections 4.2 Outlier Analysis The GLIM software used for this study has the capacity to calculate the values of leverage and Cook's distance. The procedure followed in this study to identify the outliers in a systematic manner involves the visual examination of the observed number of accidents, at any given intersection, versus the Cook's distance. The intersection that exhibits the highest the Cook's distance (ct) is identified and consequently removed. After the removal of the intersection, a new model is developed and the change in scaled deviance with respect to the previous model is assessed. This difference is compared to the %2 value for the desired level of confidence with a dof (degrees of freedom) that corresponds to the number of intersections removed. The previous procedure was applied to all models in this study. After the analysis was performed two of the models had intersections which warranted removal from the original database. Intersection (INT) 87 (refer to the intersection list appendix Table AI.1) was removed for the AT 7 (left turn) model and INT 169 and INT 170 for the PDO model. A Sample analysis is provided in the Outlier Analysis Samples Appendix. 4.3 Testing the Models' Statistical Significance Several ways to test the statistical significance of the models can be used. For this project it was decided to use the Scale Deviance, the Pearson %2 and the t-ratio test. The scale deviance (SD) should be approximately equal to the degrees of freedom (dof). The Pearson %2 obtained from the model should be less than the x2 distribution value that corresponds to the degrees of freedom for the model tested. The t-ratio test can be used to asses the significance of the model's coefficients, in this case the ratio of the estimated coefficient and its 37 Microscopic Accident Prediction Models for Signalised Intersections standard error should be greater than the t-test value (1.96 for a 95% confidence). In addition, graphical methods can be used to test the models. Two methods were used for this project. The first is the plot of the Average Squared Residuals (ASR) versus the Predicted Accident Frequency (PAF). In this case the data points of a model with a good fit should be clustered around the variance function line for the NB distribution. The second is the plot of the Pearson Residuals (PR) versus predicted accident frequency. A model with a good fit should have the PR values clustered around the zero axis over the range of E (A) (Bonneson and McCoy, 1997). The Pearson Residuals are defined as follow (NAG, 1994): £ C A ^ (4.1) JVar(yi) Where, E (A,-) = predicted accident frequency y, = observed accidents ^Var(yt) = standard deviation A total of 17 models were developed (14 microscopic and 3 macroscopic) After applying the statistical tests used to asses the model's significance, 9 models were considered to be statistically significant. All models developed are presented in Table 4.2. The models shown in shaded rows are not statistically significant according to at least one of the tests used for this research, thus they may not provide reliable estimates of accidents. 38 MkTi>sfojiit-Aixiikiit Prediction Models ft., . • - ,• Atad^nt Type Pattern (MV104) Model t-test SD ( dof ) K Pearson / ("/.Jest J,.-s .il 1 > 1.5210 0 0418 x V a 0 -8.23 Pi 15.36 179.7 ( 168 ) 5.560 167.7 ( 198 ) AI S _ J , 0.3177 0.1501 2 5971 x MA x MI a 0 3.23 Pi 3.42 P2 5.40 187.9 ( 167 ) 3.588 184 ( 197 ) 0. »-AT 7 =" (f> 0.6149 0.5462 0.0283 x T x RT a 0 -4.01 Pi 2.46 p 2 2.07 161.7 ( 166 ) 4.581 171 ( 196 ) t o cc o 0 9327 0.2441 0.3027 x T < LT a 0 -2.89 p, 7.89 P2 2.19 192.9 ( 167 ) 2.160 165 ( 197 ) CO I 1 COMBINED LEt T 7URU TYPe* 0.8480 0.2532 0.4423 x T x LT u 0 -2.06 .-. 7 * 7 ,. 2.33 189.8 ( 167 ) 2.242 166 ( 197 ) RI COMBINED RIGHT TURN TYPE:', 0.4740 0.5468 0.1703 x T x RT -3.16 Pi 2.92 .. 2.94 192.1 ( 167 ) 2.136 159 ( 197 ) MICROSCOPIC AP. NOT SIGNIFICANT APMs \ AT 2 — — 0.3036 0.3959 0 '38P - MA HI p ! 0.95 1.36 193.4 ( 1 6 7 ) 2.8/4 184.8 ( 197 ) MICROSCOPIC AP. NOT SIGNIFICANT APMs \ AT * 0.8822 0 X 8 S v .j.- 2.07 ".. 2.26 : LGS ) 190 ( 298 ) MICROSCOPIC AP. NOT SIGNIFICANT APMs \ AT 6 1.4790 0.0003 x V a 0 -2.73 h .1.99 ( if-s) 182 . ( 198 ) V MICROSCOPIC AP. NOT SIGNIFICANT APMs \ ATS ft " 0.7334 0.3831 C" J I S ' J T RT « 0 -4.06 [i, 2.53 0.28 192.4 ( 167 ) 236 ' ( 197 ) , MICROSCOPIC AP. NOT SIGNIFICANT APMs \ AT 9 0.5642 fi 'Sb'l T RT . . . -3.15 1.8i :»j 2.20 168.9 ( l b / ) 1 522 159 MICROSCOPIC AP. NOT SIGNIFICANT APMs \ AT 10 <— -0 :038 (.' JP21 , - -2 07 -0 01 1.2: 128.1 ( ) 235 ( 197 ) MICROSCOPIC AP. NOT SIGNIFICANT APMs \ AT11 ib . i , 2 2 / .. 1.7t .. ••• ( 167 ) 267 ' •• ' MICROSCOPIC AP. NOT SIGNIFICANT APMs \ AI ^) 0.1275 U 2087 0.2163 x T . ,< LT a Q -2.11 Pi 0.62 .. 0.2: 180.2 ( 167 1 176 i Wl \ 1 MACROSCOPIC APMs SIGNIFICANT APMs • OlAL fTERNS 0.6790 0.4387 1.4592 x MA x MI a 0 1.37 Pi 7.52 P2 6.84 179.5 ( 167 ) 5.064 136 ( 197 ) 1 MACROSCOPIC APMs SIGNIFICANT APMs scvrRE ALL PATTERNS 0.7330 0.4914 0.7628 x MA x MI a 0 -0.91 B, 7.63 p 2 7.24 181.2 ( 167 ) 4.728 141 ( 197 ) 1 MACROSCOPIC APMs SIGNIFICANT APMs rrj ALL PATTERNS 0.7152 0.3402 0.4958 x MA x MI a 0 -2.05 Pi 6.66 p 2 4.66 ( 165 ) 4.857 150 ( 195 ) Table 4.3 Accident Prediction Models for specific accident types and for aggregate types Shaded models are not statistically significant 39 M i c r o s c o p i c A c c k t e m P red ic t i on M o d e l s for S ignal ised in tersec t ions 5.SELECTION OF APM TYPE 5.1 Introduction Previous research by Miaou (1996) mentioned four types of regression models which are widely used. These types include the normal linear, the lognormal, the Poisson, and the Negative Binomial regression models. Linear regression models are discarded for this type thesis because the linear relationships are inappropriate to describe accidents and traffic flows (Hauer et al, 1988; Persaud and Dzbik, 1993; and Miaou et al, 1996). Also, it has been shown that the lognormal regression model is inadequate to model accident-flow-roadway design relationships (Miaou, 1996). Therefore, the Poisson and NB regression models are going to be considered for further analysis. 5.2 Poisson Model The probability function of a Poisson model can be written as: p\x\m, n (5.1) Where, x the number of accident counts at an intersection per n years p\x\m,n the probability that x accidents are expected to occur at an intersection per n years mn = the Poisson parameter, accident frequency expected to occur at an intersection per n years 40 Microscopic Arciotir t Pmlksiors M o d e l s f o r Si«swit«Hl isslerscci ions The Poisson model offers the advantage of capturing the two main properties of traffic accident counts, discrete and nonnegative, when modelling accident distribution at an intersection, m. However, for regression models the estimation of E{m} is of interest and not the accident frequency. As explained earlier, the estimation of E{m} is based on a group of intersections (reference population). It can be observed that because of the complexity of traffic accidents, and given that a high level of homogeneity has been achieved forming a reference population, the accident frequency still vary from intersection to intersection even when they are subject to the same traffic flows. According to Hauer and Persaud (1988) when a Poisson distribution is used to represent the accident frequency at each intersection and the distribution of the accident frequency around the E{m} is described by the Gamma probability distribution then: E{x}=E{m} (5.2) Var{x} = E{m}+ Var{m) (5.3) Where, E{x} = mean accident counts of the reference population Var{x} = variance of the accident counts in a reference population E{m} = mean of the reference population Var{m} = variance of the reference population It can be observed that in a reference population the E{x} and Var{x} are not equal, and that the Var{x} exceeds the E{x}. These equations have been accepted and used in previous research (Hauer et al, 1988; Persaud and Dzbik, 1993; and Belanger, 1994). Thus, to properly model accident counts in a reference population the previous equations have to be satisfied by the statistical model chosen. 41 The distribution of each accident frequency around the E{m} of the reference population has been described before as one of a family of "compound Poisson distribution" (Hauer et al, 1988). It can be implied form this description that the variance of accident frequencies around the E{m} is composed by the randomness of the accidents at each intersection and the differences among the accident frequencies in the reference population. The Poisson model is able to account for the randomness of the accidents but fails to account for the differences among the accident frequencies. In conclusion, it can be said that the use of Poisson models is inappropriate for regression models intended to predict the safety property of a reference population because it can not convey the fact that Var{x} > E{x}. However, it can be used as a platform to find a model that could be used for such a purpose. 5.3 Negative Binomial Model As shown by Hauer and Persaud (1988), the distribution of the accident frequencies in a reference population can be described by the Gamma probability distribution. Hauer et al, (1988), proved that if the accident counts at an intersection is Poisson distributed and if in a group of intersections the accident frequencies are Gamma distributed, then the distribution of accident counts in the group of intersections follows the NB distribution. The Gamma probability density function can be expressed as: f Vctr{m} E{m) V - M ^ f{m) = V (5.4) For m>0 and 0 otherwise 42 Microscopic Accident Prediction Models for Signalised intersections The Gamma probability density function is obviously appropriate to describe the distribution of accident frequencies around E{m} because it can take any positive values of accident frequencies. Also, as expressed in Equation 5.4, this distribution can be manipulated to fit a large number of shapes by stretching or skewing it as seen in Equation 5.6. As it was described before, p(x\m) can be calculated as: The distribution of m's in the reference population is indicated by Equation 5.4; therefore: Forx=0,1,2,... and where, from mathematical manipulation from Equation 5.4 m (5.5) p[x\m p(x)= jp(x\m)df(m) (5.6) f E{m} V ' - M Var^n\J df(m) = V \dm (5.7) r Thus, 43 Mieroseon fc A c c i d e n t P red ic t i on M o d e l s f o r Signal ised In tersect ions w -m x e m E{m] Var{m] Kor{n; Var{m) ' r 2 \ E{m] Var{m) \dm (5.8) E{III ^ Var{m} 2 \ r £{m} Var{m] JC! r g{»} I, f o r t +1 | x+ m E{n. Var \m) dm (5.9) f E{,„f f T?L.A \ Var{m} E{m} Var{m) 2 \ ' E{m) Var{m) V x + 1 x! Var{m] A V E{m} Var{m) + 1 Var {m } (5.10) Or, ix) = dw-{m} Var{m) E{m\ + 1 E{m}2 ( E{m} Var{m}\ Var{m} 2 \ r + 1 j E{m} Var{m] 2 ^ + 1 •x\ (5.11) Equation 5.11 is the Negative Binomial distribution. It can be shown by several methods that the mean and the variance of the NB models follow the forms sought in Equations 5.2 and 5.3. A comparison of the results obtained by models developed using the Poisson and NB distributions will be presented in the next section. 44 Miefwseo i t ie A c c i d e n t P r e d i c t i o n Mtidets for S i g n a l i s e d Intersections 5.4 Accident Database Evidence As reviewed in the previous sections, the NB model should be used to develop safety equations for traffic accidents. Nevertheless, it is worth while to verify the theory using the real data for this research. To start the database inspection dispersion parameters, ad , can be used to decide which error structure fits the data the best. If the dispersion parameter in the Poisson distribution model is greater than one, then the NB distribution might fit the data better. The Poisson distribution was then used as a first step to develop the models. Table 5.1 presents a comparative analysis between the two approaches and their results. Note that the dispersion parameter is greater than one and relatively high for the majority of the models, ranging from 1.03 for the Accident Type 7model, to 8.06 for the Total Accidents model. These high values can be explained by the lack of significance of the Pearson x 2 test which indicates that the data has greater dispersion than can be explained by the Poisson error structure and suggests the use of NB error structure. Calculations for the NB distribution show that Od range from 0.73 for the Total Accidents model to 0.98 for the Accident Type 5 model indicating that the data dispersion is satisfactorily explained by the NB distribution. In addition, other parameters can be observed in Table 5.1. The scaled deviance is considerably greater and it exceeds the degrees of freedom by far for all the Poisson models, while the NB models have close values for the scaled deviance and the degrees of freedom indicating a better fit. Also, the percentage of closer estimates by both models is presented, these percentages suggest that for the majority of the models the NB distribution fits the data better. 45 M i c r o s c o p i c .Acckienl ' P red i c t i on Models i'or Signal ised in tersec t ions 1 i i APM ATI ATS AT7 PARAMETERS Poisson NB Poisson NB Poisson NB <xp 0.05 0.04 2.30 2.60 0.03 0.03 Pi 1.49 1.52 0.34 0.32 0.60 0.61 Ih 0.17 0.15 0.53 0.55 3.20 0.89 3.43 0.98 1.03 0.91 K 5.56 3.59 4.58 SD 618.54 179.70 606.55 187.85 182.96 161.68 dof 168 168 167 167 166 166 Pearson •/ 602.00 167.70 641.40 183.70 193.20 170.90 (95%) 198 198 197 197 196 196 Closer Est. 45% 55% 53% 47% 38% 62% APM AT 13 LT RT PARAMETERS Poisson NB Poisson NB Poisson •:-}4NB:;M "o 0.41 0.30 0.56 0.44 0.21 0.17 /•'/ 0.90 0.93 0.83 0.85 0.46 0.47 r 0.12 0.24 0.14 0.25 0.45 0.55 op 6.24 0.88 1.04 0.89 1.72 0.85 2.16 2.24 2.14 SD 1091.50 192.92 1129.10 189.80 343.32 192.13 dof 167 167 167 167 167 167 Pearson •/ 1166.00 165.10 195.00 165.60 322.00 159.20 • (95%) 197 197 197 197 197 197 •'"loser Est. 51% 49% 52% 48%> 46% 54% APM TOTAL SEVERE PD O PARAMETERS Poisson NB Poisson NB Poisson NB <(o 1.75 1.46 1.05 0.76 0.61 0.50 • • » 0.66 0.68 0.68 0.73 0.67 0.72 /•*_. 0.39 0.44 0.43 0.49 0.33 0.34 • 8.06 0.73 6.28 0.75 3.21 0.80 • 5.06 4.73 4.86 SD ••• 1640.30 179.53 1273.10 181.22 644.17 185.69 dof 167 167 167 167 165 165 "•'"' Pearson 1508.00 136.30 1174.00 140.60 599.40 150.10 / (95%) 197 197 197 197 195 195 ; Closer Est. 49% 51% 47% 53% 44% 56%o I I \ es i Table 5.1Comparison between the Poisson and the NB distributions 16 M i c r o s c o p i c Acdikai P red i c t i on Model* for Signalised Intersections 6. APPLICATIONS 6.1 Introduction Five applications of APMs are discussed in this chapter. The first application relates to using the Empirical Bayes approach to obtain refined safety estimates. The second application refers to the identification of APLs. The third application refers to the ranking the identified APLs, and the fourth relates to before and after safety evaluations. A comparison between the use of microscopic and macroscopic models is provided. 6.2 Location specific prediction: EB safety estimates As reviewed in the Previous Work section, the main objective of using the EB refinement method is to yield more accurate, location specific safety estimates by combining the observed number of accidents at the location with the predicted number of accidents from the APMs. Consider the following volume and accident data: 47 IVlfrrosconk Arxuieirt Prediction Models for Signalised intersections MOVEMENT AADTK EBL 1247.5 EBT 9910.3 EBR 1024.5 WBL 738.7 WBT 9506.1 WBR 1832.9 NBL 725.2 NBT 11343.5 i\!3R 663.6 S3L 1183.6 SBT 9571.5 SBR 1874.6 Table 6.1 Example volume data Accident Accident Type count ATI 27 At 4 17 AT 7 3 Alls 38 40 RT 5 toiAL 103 SEVERE 78 PUU 25 Table 6.2 Accidents / 3 years Step 1. The flows needed for each APMs are calculated from the detailed volumes (see to Table 3.1): VOLUMES ( A A D T / 1 0 0 0 ) ! F. MICROSCOPIC APMs bv AT 1, 4, 6 49 62 21.25 19.08 25.36 24.26 20.91 19.42 7, 8, ? , 10 40.33 5.40 1 1 , 12, 13 40.33 3.90 MACROSCOPIC APMs TOTAL, S E V E R E , and PDO 25.36 24.26 "' LEFT TURN 40.33 3.90 RIGHT TURN 40.33 5.40 Table 6.3 APMs Flows 48 Microscopic- Aeeiaeat Prediction Models f«r Signalised intersections Step 2. Calculate the predicted number of accidents using the APM and its variance (equation 2.4). Sample calculations in equation form are shown for the total model only, see Table 6.4 for the rest of the results. 6790 „ 0 / 1 -i £ .4387 E(A) = 1.4592 x 25.23 x 24.26 = 53.\0acc/3years 53 102 Var(E(A)) = d±l2L. = 556.84(acc I'3yearsf 5.064 Step 3. Obtain the empirical safety estimate its variance for the intersection using equations 2.18 and 2.19 f 53.10 A FB safety estimate \ 5.064 + 53.10 (5.064 + 103) = 98.66acc 13years Var(EBsafclv ) = f 53.10 ^ safety estimate 5.064 + 53.10 (5.064 + 103) = 90.07 (acc/3years)1 Figure 6.1 shows the Observed number of accidents plotted versus the Predicted number of accidents and the EB refined number of accidents. It can be observed how the EB refined estimate is closer to the Observed number of accidents than the Predicted number of accidents. The EB refined estimates are closer to the 45° line than the Predicted number of accidents. 49 Microscopic Accident Prediction Models lor Signalised intersections Accident 1 Type APMs Accidents / 3 years APrfs .it, / iyens EBse : acc/3 years Var(EBse) MICROSCOPIC APMs ATI 1.5210 0.0418 x V 15.86 24.11 17.85 MICROSCOPIC APMs ATS 0.3177 0.1501 2.5971 x MA x MI 10.65 15.40 11.52 MICROSCOPIC APMs 0.6149 0.5462 AT ;< 10.0283 x T x RT 0.69 0.99 0.13 MICROSCOPIC APMs AT 13 0.9327 0.2441 0.3027 x T x LT 13.27 34.54 29.70 MICROSCOPIC APMs LT 0.8480 0.2532 0.4423 x T x LT 14.35 36.53 31.60 MICROSCOPIC APMs RT 0.4740 0.5468 0.1703 x T x RT 2.47 3.83 2.05 MACROSCOPIC APMs TOTAL 0.6790 0.4387 1.4592 x MA x MI 53.10 98.66 90.07 MACROSCOPIC APMs SEVERE 0.7330 0.4914 1 0.7628 x MA x MI 39.10 73.80 65.84 MACROSCOPIC APMs 0.7152 0.3402 PDO 10.4958 x MA x MI 14.82 22.49 16.93 Table 6.4 Predicted number of accidents and EB safety estimates 50 CO < 4< < o vO o o fNJ o o o CO o vO o o fNJ o o o o o o o o o o C O v O ^ r v J O C O v D ^ f N (sjeaA £ / 3 D B ) siuappDe jo jaquinu pauijsj 93 ' p a p i p a J d (Vlieroscopie A c c i d e n t P red i c t i on Models f o r Signalised In tersec t ions 6.3 Identification of APLs As mentioned earlier, APLs are locations that exhibit a significant number of accidents compared to a specific "norm". This "norm" is usually accepted as the 50 t h percentile of the prior distribution. For the example given in the previous section, the following steps are followed to identify APLs • Calculate P 5 0 using Equation 2.21 j(5.064/53.10)5064 • l ( 5 0 6 4 - ' ) • ^ ( 5 Q 6 4 / 5 3 1 0 ) " ^ l - Q 5 o r(5.064) Solving the integral for 0.5, P5o = 49.65 accidents / 3 years. Table 6.5 shows the P 5 0 values for the other models. 52 M i c r o s c o p i c Aa' .UIent P r e s l k t i o n M o d e l s f o r Signal ised in tersect ions Accident Type APMs Accidents / 3 years APMs . wicc/3 ye.irs Var (APMs e) (Ml/3yCMS}m Pso .ict.-3 ymiis MICROSCOPIC APMs ATI 1.5210 0.0-118 x V 15.86 61.08 14.92 MICROSCOPIC APMs ATS 0.3177 0.1501 2.5971 x MA x MI 10.65 42.26 9.68 MICROSCOPIC APMs AT 7 0.6149 0.5462 0.0283 x T x RT 0.69 0.79 0.64 MICROSCOPIC APMs AT 13 0.9327 0.2441 0.30;->7 x T x LT 13.27 94.74 11.28 MICROSCOPIC APMs Ll 0.8480 0.2532 0.4423 x T x LT 14.35 106.20 12.28 MICROSCOPIC APMs RT 0.4740 0.5468 0.1703 x T x RT 2.47 5.33 2.10 MACROSCOPIC APMs TOTAL 0.6790 0.4387 1.4592 x MA x MI 53.10 609.94 49.65 MACROSCOPIC APMs SEVERE 0.7330 0.4914 0.7628 x MA x MI 39.10 362.48 36.38 MACROSCOPIC APMs PDO 0.7152 0.3402 0.4958 x MA x MI 14.82 60.02 13.81 Table 6.5 P5o calculations Calculate the probability of exceeding P5o using equation 2.24 49.65 J (5 064/53 10 + 1 ) ( 5 ' ° 6 4 + 1 0 3 ) . ^( 5 0 6 4 + 1 0 3 - 1 ) .g-(5.064/53.10+l)A r (5 .064 + 103) dZ = 0.9999 This result indicates that this intersection can be regarded as an APL according to the total accident model because there is significant probability, 99.9999%, of exceeding the P 5 0 value. Figure 6.2 illustrates the APL identification procedure for this example. Table 6.6 shows the probabilities of exceeding P 5 0 according to the rest of the models. The last column indicates 53 M i c r o s c o p i c A c c i d e n t P red i c t i on Models f o r Signalised In tersect ions whether the intersection is APL or not according to each accident type (at the 95% confidence interval). Prior and Posterior Distributions 15 30 60 75 90 Accidents / 3 years Posterior / \ Probability of safety estimate less than I \ Pso=0.999999999907166 / \ Prior/^ Pso=49.65 'Prior " Posterior 105 120 135 150 Figure 6.2 APL identification procedure 54 Accident Type APMi ,,. .iter iyt. r Var (APMs fJ P. .icr J yturs Probability APL MICROSCOPIC APMs Al 1 1.5210 0 0418 x \l 15.86 45.23 14.92 99.33492696 APL MICROSCOPIC APMs Al 5 0.3177 0.1501 2 5971 x MA x MI 10.65 31.61 9.68 96.99417974 APL MICROSCOPIC APMs AT7 0.6149 0.5462 0 0283 x T x RT 0.69 0.10 0.64 84.09279633 NO MICROSCOPIC APMs •a 13 0.9327 0.2441 0.3027 x T x LT 13.27 81.48 11.28 99.99999984 APL MICROSCOPIC APMs LT 0.8480 0.2532 0.4423 x T x LT 14.35 91.85 12.28 99.99999986 APL MICROSCOPIC APMs KJ 0.4740 0.5468 0.1703 x T x RT 2.47 2.86 2.10 90.82578431 NO MACROSCOPIC APMs TOTAL 0.6790 0.4387 1.4592 x MA x MI 53.10 556.84 49.65 99.99999999 APL MACROSCOPIC APMs SEVERF 0.7330 0.4914 0.7628 x MA x MI 39.10 323.38 36.38 99.99999946 APL MACROSCOPIC APMs PDO 0.7152 0.3402 0.4958 x MA x MI 14.82 45.20 13.81 99.17899953 APL Table 6.6 APL according to the first method P5o It can be seen in Table 6.6 that, at the 95% confidence level, the intersection is an APL according to the microscopic models for AT 1 (rear end), AT 5 (intersection 90°), AT 13 (one of the three left turn types), and macroscopic models for Total, Severe, PDO, and Left Turn accidents. However, it is not an APL according to AT 7 (one of the four right turn types), and combined Right Turn. 6.4 APLs Ranking Two ranking criteria can be used to rank APLs (Rodriguez and Sayed, 1999). The first consists of calculating the difference between the EB estimates and the predicted frequency for the previously identified APLs. This is a good indication of the expected safety benefits and is helpful in carrying out the estimation of the pre-implementation safety benefits that are going to be obtained from implementing the countermeasure. For the second criterion the 55 r v i knweo jde Accident P r e d i e f i o o Models IVir Signalised Intersections ratio between the EB estimate and the predicted frequency has to be calculated. This ratio represents the deviation of the intersection from the "norm". In other words, the higher the ratio, the more accident prone the intersection is. The first criterion is useful to indicate potential economical benefits and the second guarantees that the safety level at each intersection is comparable to the other intersection with similar characteristics. A comparison of the two ranking criteria is shown in Table 6.7. Using the first method 48 intersections were identified as APLs at the 95% confidence interval. The table shows the calculations needed (EB - Predicted and EB / Predicted) and the ranking according each criteria, the difference in ranking is also included. The difference in rank between the two criteria ranges from 0 to 25, having an average of 6.29. This difference can be explained by the different goals sought by each criterion. While the first criterion tends to favour intersections that exhibit high accident occurrence which are usually more cost effective to treat, and the second considers the deviation from the expected values and its variance regardless of the number of accidents observed. Road authorities can consider the second criterion to ensure that the safety of different locations is within acceptable levels. 56 Mfcrwseoitio ' • *«{•• . '.-.--•iefioM Models, for Siana.iis.eil in te rsec t ions INT Observed Piedictud EB F.B - Prod EB / Prcd Rank Rank Difference; No Accidents Accidents (•stiruitcs EB - Prod EB / Prod in acc ; 3 voars acc / 3 years acc ;' 3 years acc / 3 years acc / 3 years Rank 1 128 57.46 122.29 64.82 2.13 1 2 1 2 160 71.43 154.14 82.71 2.16 2 1 1 3 117 61.97 112.84 50.87 1.82 3 6 3 1 103 53.10 98.66 45.55 1.86 4 9 5 5 97 49.40 92.57 43.17 1.87 5 5 0 6 58 24.57 52.29 27.72 2.13 6 4 2 82 43.03 77.90 34.87 1.81 7 8 1 C 138 79.94 134.54 54.60 1.68 8 3 5 9 99 54.62 95.23 40.61 1.74 9 7 2 10 122 84.49 119.88 35.39 1.42 10 16 6 11 74 45.02 71.07 26.05 1.58 11 10 1 12 101 72.02 99.10 27.07 1.38 12 20 8 13 85 58.20 82.85 24.66 1.42 13 15 2 14 79 54.81 76.95 22.14 1.40 14 18 4 15 81 57.64 79.11 21.47 1.37 15 21 6 16 56 36.09 53.55 17.46 1.48 16 14 2 17 55 35.25 52.52 17.27 1.49 17 13 4 18 85 61.22 83.18 21.96 1.36 18 23 5 19 67 46.15 64.94 18.79 1.41 19 17 2 20 98 73.91 96.46 22.54 1.31 20 28 8 21 41 24.64 38.21 13.57 1.55 21 11 10 22 100 76.31 98.53 22.21 1.29 22 29 7 23 110 85.58 108.64 23.06 1.27 23 33 10 24 113 89.61 111.75 22.14 1.25 24 36 12 25 39 23.87 36.35 12.48 1.52 25 12 13 26 106 83.35 104.70 21.35 1.26 26 34 8 27 72 52.91 70.33 17.43 1.33 27 25 2 . 28 69 50.72 67.34 16.62 1.33 28 26 2 29 109 87.73 107.84 20.11 1.23 29 41 12 30 96 75.72 94.73 19.01 1.25 30 35 5 31 84 66.74 82.78 16.05 1.24 31 37 6 32 58 44.08 56.57 12.49 1.28 32 30 2 ->3'~- 82 66.32 80.89 14.56 1.22 33 42 9 34 71 56.78 69.84 13.05 1.23 34 40 6 :s 36 25.36 34.23 8.87 1.35 35 24 11 36 31 21.00 29.06 8.06 1.38 36 19 17 37 49 37.19 47.59 10.39 1.28 37 32 5 = 38 67 54.15 65.90 11.75 1.22 38 44 6 439; 75 61.80 74.00 12.20 1.20 39 45 6 '-;4o; 63 50.86 61.90 11.04 1.22 40 43 3 41 81 68.22 80.12 11.90 1.17 41 47 6 42 88 74.93 87.17 12.24 1.16 42 48 6 4> 36 26.25 34.42 8.17 1.31 43 27 16 44 53 41.94 51.81 9.87 1.24 44 39 5 45 46 36.25 44.80 8.56 1.24 45 38 7 46 35 26.24 33.58 7.34 1.28 46 31 15 47 23 15.56 21.17 5.61 1.36 47 22 25 48 54 44.34 53.01 8.67 1.20 48 46 2 Table 6.7 Ranking according to the first method used to select APLs 57 M k ' n s s e opto A c c i d e n t P r e d i c t i o n M o d e l s l o r Signalised in te rsec t ions Alternatively accident priorities can be assigned according to the severity of each accident type, the average estimated accident cost, the specific accidents that are occurring at the intersection, or the effectiveness of the viable countermeasures that can be implemented at the intersection. 6.5 Critical Accident Frequency Curves The process of identifying accident prone locations requires considerable computational effort. To facilitate this process, critical accident frequency curves can be developed for any APM. A critical curve can be defined as the threshold that must be exceeded by the observed number of accidents in order to classify the location as APL for a given APM at a certain confidence level. The procedure to obtain these curves is iterative and follows the steps employed to identify APLs. The predicted number of accidents for any given APM and Kparameter is calculated and used as initial guess. Then, the P 5 0 is calculated as in equation 2.21. After this is done, the P5o value is used to identify the location as APL if the confidence level is exceeded as in equation 2.24. This procedure is repeated until the observed number of accidents yields to the confidence level desired. This means that the procedure is repeated until a count variable fits the level of confidence sought. The critical curve is then obtained by joining all the critical points in a Predicted versus Observed accidents graph. As an example, the following figures show the critical curves for all models developed for this thesis. Three curves are shown in each figure representing the 90%, 95%, and 99% confidence levels. In addition to the curves, the predicted versus observed number of accidents is shown. A triangle mark denotes an APL at the 99% confidence level and circle marks an intersection that is not accident prone at the same confidence level. 58 Microscopic Accident Pi rdk i ion Model* for Signalised Intersections Figure 6.3 Critical Curve for AT1 59 M i c r o s c o p i c etHretmii Models for Signalised Intersections Figure 6.4 Critical Curve for AT5 Critical Curve for AT5 60 Microscopic \ c e « l r a i P r e d i c t i o n M o d e l * fo r S igna l i sed Intersect ions Figure 6.5 Critical Curve for AT7 Critical Curve AT7 61 Microscopic A c c i d e n t P red i c t i on M o d e l s t o r S i g n a l l e d In tersect ions Figure 6.6 Critical Curve for AT13 Critical Curve for AT13 V l i r r o K C D i i i c A m d e i i t Prediction Model* for Signalised Intersections Figure 6.7 Critical Curve for Left Turn Accidents Critical Curve for Left Turn Accidents 63 M i c r o s c o p i c Accident Prediction Models for Signalised Itii.i-ectktns Figure 6.8 Critical Curve for Right Turn Accidents 64 MBawwcaplc A c c i d e n t P r e d i c t i o n M o d e l s fo r S i g n a l i s e d In tersect ions F i g u r e 6.9 C r i t i c a l C u r v e for T o t a l A c c i d e n t s Critical Curve for Total Accidents 175 150 99% 95% 90% 20 40 60 80 Predicted Accidents 100 90% — 95% — 99% * APLs 65 Microscopic- A c c i d e n t P r e d i c t i o n M o d e l s fo r Signalised i n te rsec t ions Figure 6.10 Critical Curve for Severe Accidents Critical Curve for Severe Accidents 66 V l k r o s c o n k AeekleiM P red i c t i on M o d e l s IVir S ignal ised In tersect ions Figure 6.11 Critical Curve for PDO Accidents Critical Curve for PDO Accidents 95% 3 6 9 12 15 18 21 24 27 30 Predicted Accidents 90% — 95% — 99% * APLs 67 To illustrate the use these curves, consider the example described before. Using the APM developed for this thesis, the predicted number of accidents for the given traffic volumes can be obtained as done for Table 6.4. For this number of accidents and for, lets say, the 95% confidence level the critical threshold should be obtained as described before. Calculations for the critical threshold are summarised in Table 6.8. The observed number of accidents, actual count, is then used to identify the location as accident prone if this value surpasses the threshold for the desired confidence level. THRESHOLD COJNT ACTiJAL APL APMs 95% COUNT A T I 22.7331 27 APL u ATS 16.0249 17 APL © u o §£ u t=a AT7 4.6917 3 NO AT13 17.6772 38 APL LT 18.9152 40 APL RT 5.7640 5 NO & r TOT 62.2219 103 APL O ,. w O SEV 47.3256 78 APL U PDO 21.2895 25 APL Table 6.8 Number of Accident Prone Locations 68 Table 6.9 shows the number of APLs identified by the APM developed for this thesis for different significance levels. COM FIDE6\SCE L EVE I APMs 90% 95% 99% A T I 33 21 13 AT 5 37 28 13 u 2 0 0 w 5 49 34 27 48 41 27 RT 21 12 5 u Q. , TOT 58 48 33 © u 0 SEV 55 41 21 PDO 39 32 17 Table 6.9 Number of Accident Prone Locations Furthermore, additional curves can be developed in a more general way for various K values. The main advantage of these curves is that they can be used for any negative binomial model. However, the results obtained from the use of these general curves are not as accurate as the results obtained by using the previous curves. 69 Microscopic Aecklesit Prediction Models 1 Inlersecrioos Figure 6.12 Critical Curves for various K values Critical Curves for different k 250 0 40 80 120 160 200 Predicted Accidents 0.5 1 2 5 10 25 50 100 70 M i c r o s c o p i c Aee ide is f P red ic t i on M o d e l s for Sco'iai is-eil Ir t icrsecfiofts 6.6 Before and After studies The benefits obtained from safety improvement programs are usually represented by the reduction in the number of collisions after implementing the program. To estimate this reduction, a comparison between the observed collision frequency after treatment and the expected collision frequency had no treatment taken place is undertaken. The easiest way to conduct this comparison is the simple before and after comparison method. In this method, a simple comparison between the number of collisions before and after treatment is undertaken using statistics such as the two sample T-test. The main assumption of this method is that the number of collisions would not change had no treatment taken place. However, because of the randomness inherent to accident occurrence (e.g. regression to the mean effect, accident migration, etc.), the observed number of accidents may not be a good estimate of what could have happened had no countermeasure been implemented. The EB refinement process can be used to overcome this limitation. To illustrate this, the example above is continued. To keep calculations simple, assume that traffic flows at the intersection remained unchanged. Suppose further that a countermeasure to reduce the number of rear end accidents (e.g. all red interval) was implemented and the observed number of accidents in a three year period after the implementation is 88. Thus, the safety would have been had the treatment not been implemented on the intersection can be calculated as: YdC MoE = \ - ^ ~ (6.1) pac Where, 71 Microscopic Accident Prediction Models for Signalised intersections MoE = Measure of Effectiveness rac = accidents count after implementation pac - previously observed accident count (before implementation) oo MoE = \ - ~ = 0.14 103 Hence, in regard to the number of total accidents, it can be said that the countermeasure yielded a positive effect; it decreased the number of accidents at the intersection by 14%. 6.7 Recommended Research A lack of detailed data to develop the APMs was found in the literature review and during the course of this thesis. It was observed that current microscopic models just take into account the traffic flows of the colliding vehicles but do not include other important factors that may contribute to traffic accidents. It is recommended that the models'developed in this thesis be further refined by adding Intersection Layout variables (e.g. number of lanes of each road, number of left and right turn lanes, pedestrian crosswalks, speed limit, etc). The addition of these variables should enhance the predictive ability of the models and contribute to improve our understanding of the relationships between accident occurrence and geometric design. 72 M i c r o s c o p i c A c c i d e n t P red i c t i on M o d e l s f o r S ignal ised in tersect ions 7.CONCLUSIONS This thesis summarises the results of a study undertaken to develop Microscopic Accident Prediction Models, APMs, for 4-leg urban signalised intersections based on data from the City of Vancouver, B. C. The Generalised Linear Interactive Modelling approach (GLIM) was used to overcome the shortcomings associated to the Conventional Linear Regression approach. The GLIM approach allows using non-normal (Poisson or Negative Binomial) error structure and the use of nonlinear relationships in the models. A comparison between the results obtained by models developed using the Poisson versus the Negative Binomial (NB) error structures was undertaken. It was observed that for the majority of the models the NB error structure provided closer safety estimates. Microscopic models for the accident types described in the MV 104 form, British Columbia's accident police report, were developed. Also Macroscopic models for total, severe and injury accidents were developed. Several statistical tests were applied to all developed models to determine their reliability. After applying the tests, six microscopic models were statistically significant. All macroscopic models were significant. Several applications were covered and a discussion regarding their use was provided. The Empirical Bayes, EB, method was used to overcome the problems associated with the regression to the mean phenomenon and to provide a more accurate site-specific safety estimates. The process to identify APLs, locations that exhibit a significant number of accidents compared to a specific "norm", was explained in detail. Two ranking criteria were used to rank the identified APLs. The first criterion is useful to indicate potential economical benefits and the second attempts to ensure that the safety level at each intersection is comparable to other intersection with similar characteristics. To 73 facilitate the APLs identification process, critical accident frequency curves were developed for each APM. A critical curve can be defined as the threshold that must be exceeded by the observed number of accidents in order to classify the location as APL for a given APM at a certain confidence level. The use of the models to perform before and after safety evaluations was described. A detailed example that includes the specific traffic flow calculations and all the applications previously discussed was provided for all statistically significant models. It was observed that the use of detailed microscopic accident prediction models could be beneficial when identifying accident prone locations that can be successfully treated. Microscopic models allow to identify locations that are accident prone to specific accident type(s) that can be effectively treated by some specific countermeasure that permits reducing that specific accident type as opposed to macroscopic models that don not provide such detail. By using microscopic models the safety of all road network users and the benefits of road improvement programs can be maximised. 74 M i c r o s c o p i c Accislcisf Predfctioss Models i'or SigmilistMl in tc rsac f ions BIBLIOGRAPHY Al-Turk, M. and Moussavi, M., (1996). 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"Safety effects of cross section design for two lane roads", Volume I, Final Report, FHWA-RD-87/008, Federal Highway Administration, Transportation Research Board 78 ikm IVedietinn Models for SfensSised Intel-sections APPENDIX I INTERSECTION LIST 79 Microscopic A c c i d e n t Prediction M o d e l s for Signalised i n te rsec t ions l|f||W^lllli INTERSECTION LIST 10th and Ash 49th and Boundary Cambie and King Edward Ave. 10th and Blanca 49th and Cambie Cambie and Marine 10th and Macdonald 49th and Elliott Commercial and Hastings 12th and Arbutus 49th and Fraser Commercial and Grandview 12th and Burrard 49th and Granville Commercial and Venables 12th and Cambie 49th and Kerr Cordova and Main 12th and Clark 49th and Knight Dunbar St and 16th 12th and Commercial 49th and Main Dunbar St and King Edward Ave. 12th and Fir 49th and Oak Dunbar St and S.W. Marine 12th and Fraser 49th and Tyne Earles and 29th Ave 12th and Granville 49th and Victoria Earles and 41st Ave 12th and Heather 49th and W. Boulevard Earles and Kingsway 12th and Hemlock 4th and Alma Fraser and King Edward Ave. 12th and Kingsway 4th and Arbutus Fraser and Marine 12th and Main 4th and Blanca Grandview and Victoria 12th and Oak 4th and Burrard Granville and King Edward 16th and Artubus 4th and Fir Granville and Park 16th and Blenheim 4th and Macdonald Hastings and Clark 16th and Burrard 4th and Main Hastings and Main 16th and Cambie 54th and Elliott Hastings and Nanaimo 16th and Fir 54th and Kerr Hastings and Renfrew 16th and Fraser 54th and Victoria Joyce and Kingsway 16th and Granville 57th and Fraser Kerr and S.E. Marine 16th and Macdonald 57th and Knight King Edward Ave. and Knight 16th and Main 57th and Main King Edward Ave. and Macdonald 16th and Oak 6th and Clark King Edward Ave. and Main 1st and Clark 70th and Granville King Edward Ave. and Oak 1st and Commercial 70th and Oak Kingsway and Broadway 1st and Main 70th and SW Marine Dr. Kingsway and Fraser 1st and Nanaimo Alma and 10th Ave Kingsway and Knight 1st and Renfrew Alma and Broadway Kingsway and Nanaimo 1st and Rupert Arbutus and Broadway Kingsway and Rupert 22nd and Boundary Arbutus and Cornwall Ave Kingsway and Slocan 22nd and Renfrew Arbutus and King Edward Ave. Kingsway and Victoria 22nd and Rupert Blenheim and 10th Macdonald and Kitsilano 33rd and Arbutus Blenheim and Broadway Main and 2nd 33rd and Cambie Blenheim and King Edward Ave. Main and Marine 33rd and Fraser Blenheim and S.W. Marine Main and Powell 33rd and Granville Boundary and 1st Main and Prior 33rd and Knight Boundary and 29th Main and Terminal 33rd and Main Boundary and Grandview Marine Dr and Cornish 33rd and Nanaimo Boundary and Hastings Marine Dr and Oak 33rd and Oak Boundary and Kingsway Marine Dr and Victoria 33rd and Victoria Boundary and Lougheed McGill and Renfrew 41st and Blenheim Boundary and Marine Nanaimo and Grandview 41st and Cambie Boundary and Moscrop Nanaimo and Dundas 41st and Clarendon Boundary and Rumble Oak and 19th 41st and Dunbar Broadway and Burrard Oak and Park 41st and Fraser Broadway and Cambie Point Grey and Alma 41st and Granville Broadway and Clark Renfrew and Grandview 41st and Knight Broadway and Commercial Rupert and 29th 41st and Main Broadway and Fir Rupert and Grandview 41st and Oak Broadway and Fraser Slocan and 29th 41st and Rupert Broadway and Granville Venables and Clark 41st and Victoria Broadway and Macdonald 41st and W. Boulevard Broadway Broadway Broadway Broadway Broadway and and and and and Main Nanaimo Oak Renfrew Rupert Table AI.1 Intersection List 80 APPENDIX II DATA ANALYSIS M i c r o s c o p i c A c x k i e i i t P red i c t i on Mosleis l o r Signal ised in tersec t ions HOUR FREQUENCY % 0:00 TO 0:59 195 2.30 1:00 TO.1:59 143 1.69 2:00 TO 2:59 140 1.65 3:00 TO 3:59 55 0.65 4:00 TO 4:59 40 0.47 5:00 TO 5:59 41 0.48 6:00 TO 6:59 174 2.06 7:00 TO 7:59 290 3.43 8:00 TO 8:59 460 5.43 9:00 TO 9:59 384 4.54 10:00 TO 10:59 428 5.06 11:00 TO 11:59 462 5.46 12:00 TO 12:59 483 5.71 13:00 TO 13:59 453 5.35 14:00 TO 14:59 539 6.37 15:00 TO 15:59 580 6.85 16:00 TO 16:59 616 7.28 17:00 TO 17:59 580 6.85 18:00 TO 18:59 601 7.10 19:00 TO 19:59 549 6.48 20:00 TO 20:59 357 4.22 21:00 TO 21:59 365 4.31 22:00 TO 22:59 287 3.39 23:00 TO 23:59 244 2.88 TOTALS 8466 100 MAX 616 MIN 40 AVERAGE 352.75 STDEV 188.19 MEDIAN 374.50 VAR 35413.67 Table Al 1.1 Time of Accidents 82 1994 1995 1996 COUNT % COUNT 0/ /o COUNT % J A N 257 9.14 270 8.82 210 8.10 FEB 260 9.25 229 7.48 256 9.87 MAR 254 9.04 283 9.24 253 9.76 A P R 210 7.47 228 7.45 239 9.22 MAY 218 7.76 238 7.77 207 7.98 J U N 225 8.00 218 7.12 178 6.86 J U L 228 8.11 248 8.10 213 8.21 A U G 124 4.41 255 8.33 192 7.40 SEP 188 6.69 273 8.92 199 7.67 OCT 289 10.28 259 8.46 256 9.87 NOV 283 10.07 287 9.37 210 8.10 DEC 275 9.78 274 8.95 180 6.94 TOTALS 2811 100.00 3062 100.00 2593 100.00 % 33.20 36.17 30.63 TOTAL 8466 MAX 289 287 256 MIN 124 218 178 AVERAGE 234.25 255.17 216.08 STDEV 46.59 23.05 28.39 MEDIAN 241.00 257.00 210.00 VAR 2170.57 531.42 805.90 Table All.2 Date of Accidents M i c r t w e o n k A c c i d e n t P red i c t i on M o d e l s f o r S ignal ised In tersect ions CONTRIBUTING CIRCUMSTANCES TOTAL % ALCOHOL 214 2.53 AVOID P/V/B 6 0.07 BACK UNSAFE 8 0.09 BLACKOUT 4 0.05 CUT IN 17 0.20 DEF BRAKE 11 0.13 DEF BRKLIGH 1 0.01 DEF HDLIGHT 3 0.04 DRIV INATT 282 3.33 DRIV INEXP 72 0.85 FAIL SIG 4 0.05 FAIL YIELD 789 9.32 FATIGUE 4 0.05 FELL ASLEEP 1 0.01 GLARE-ARTIF 1 0.01 GLARE-SUN 13 0.15 IGNR OFFICR 3 0.04 IGNRTR CON 499 5.89 ILL DRUGS 1 0.01 ILLNESS 7 0.08 IMPROP PASS 11 0.13 IMPROPTURN 44 0.52 INSECURE LD 1 0.01 MEDICATION 1 0.01 OTHER 490 5.79 OVERSIZE 3 0.04 PED ERROR 2 0.02 PREVACCID 1 0.01 ROAD DEBRIS 3 0.04 SPEEDING 60 0.71 TIRE FAIL 2 0.02 TOO CLOSE 96 1.13 VIS IMPAIRD 18 0.21 WEATHER 55 0.65 WRONG SIDE 2 0.02 REPORTED 2729 32.23 UNREPORTED 5737 67.77 TOTAL 8466 100.00 Table All.3 Contributing Circumstances diction Models for Signalised Intersections A CCIDENT IN FORM A TION TOTALS % MAX MIN AVERAGE STDEV MEDIAN VAR AT 1 2692 31.80 53 0 15.84 11.65 13 135.7 AT2 497 5.87 17 0 2.92 2.66 2.5 7.1 AT3 0 0.00 0 0 0.00 0.00 0 0.0 AT 4 45 0.53 2 0 0.26 0.54 0 0.3 AT5 1655 19.55 37 0 9.74 6.88 8.5 47.3 AT6 16 0.19 2 0 0.09 0.31 0 0.1 ATI 107 1.26 4 0 0.63 0.89 0 0.8 AT8 79 0.93 4 0 0.46 0.84 0 0.7 AT9 153 1.81 5 0 0.90 1.20 0 1.4 AT10 33 0.39 3 0 0.19 0.51 0 0.3 AT11 92 1.09 6 0 0.54 0.94 0 0.9 AT12 91 1.07 3 0 0.54 0.75 0 0.6 AT 13 2202 26.01 62 0 12.95 11.29 10 127.6 AT 14 0 0.00 0 0 0.00 0.00 0 0.0 AT 15 1 0.01 1 0 0.01 0.08 0 0.0 AT16 0 0.00 0 0 0.00 0.00 0 0.0 OTHER 803 9.48 23 0 4.72 3.40 4 11.5 Total 8466 100.00 LT's 2385 86.51 64 0 14.03 11.79 11 139.1 RT's 372 13.49 11 0 2.19 2.19 2 4.8 Total 2757 100.00 Fatal 10 0.12 1 0 0.06 0.24 0 0.1 Injury 5963 70.43 120 1 35.08 22.27 33 496.0 PDO 2493 29.45 53 0 14.66 9.31 14 86.6 Total 8466 100.00 Table All.4 Accident Information 85 M i c r o s c o p i c AceStteitt P red i c t i on M o d e l s f o r Si&»alb*<$ l o l c r s e c i i o n s LIGHT CONDITION DARK DAWN DAY DK-I DUSK ILLU UNKN TOTAL 27.00 95.00 5732.00 629.00 256.00 1707.00 20.00 % 0.32 1.12 67.71 7.43 3.02 20.16 0.24 MAX 2.00 5.00 107.00 14.00 7.00 32.00 2.00 MIN 0.00 0.00 2.00 0.00 0.00 0.00 0.00 AVERAGE 0.16 0.56 33.72 3.70 1.51 10.04 0.12 STDEV 2.09 7.28 436.26 47.89 19.53 129.97 1.56 MEDIAN 0.00 0.00 30.00 3.00 1.00 9.00 0.00 VAR 4.37 53.04 190326.40 2293.72 381.28 16892.81 2.44 I UNREPORTED I 0 Table All.5 Light Conditions ROAD CONDITION DRY ICE MUD SLUSH SNOW WET TOTAL 5311.00 16.00 12.00 12.00 55.00 3035.00 %* 62.92 0.19 0.14 0.14 0.65 35.96 MAX 97.00 1.00 1.00 2.00 4.00 73.00 MIN 0.00 0.00 0.00 0.00 0.00 0.00 AVERAGE 31.24 0.09 0.07 0.07 0.32 17.85 STDEV 404.22 1.25 0.95 0.96 4.23 231.02 MEDIAN 27.00 0.00 0.00 0.00 0.00 17.00 VAR 163397.52 1.56 0.90 0.92 17.87 53371.56 I UNREPORTED I 25 * Calculations according to reported accidents only Table All.6 Road Conditions WE A THER CONDITION CLOUDY CLEAR FOG RAIN SNOW UNKN WIND TOTAL 1375.00 4685.00 21.00 2259.00 96.00 29.00 % 16.24 55.34 0.25 26.68 1.13 0.34 MAX 25.00 87.00 2.00 57.00 4.00 . 3.00 MIN 0.00 0.00 0.00 0.00 0.00 0.00 AVERAGE 8.09 27.56 0.12 13.29 0.56 0.17 STDEV 104.68 356.58 1.63 171.97 7.34 2.25 MEDIAN 7.00 26.00 0.00 12.00 0.00 0.00 VAR 10957.15 127148.70 2.67 29575.10 53.92 5.06 I UNREPORTED 1.00 0.01 1.00 0.00 0.01 0.11 0.00 0.01 Table All.7 Weather Conditions 86 M k r o s c o p k A c c i d e n t P red i c t i on Models l o r Signalised in te rsec t ions MOVEMENT VOLUME STATISTICS MAX MIN AVERAGE STDEV MEDIAN VAR E L 15143.4 6.8 1371.3 1636.4 936.0 2677953.7 E T 37132.9 160.0 11563.1 7600.6 10315.0 57768374.9 E R 9892.6 53.7 1496.7 1593.5 1091.8 2539230.2 W L 8506.2 6.2 1337.4 1250.9 1015.3 1564818.1 WT 28739.6 395.0 9667.2 5882.3 9506.1 34600983.4 WR 5046.8 13.9 1116.4 912.0 964.0 831781.2 NL 5502.2 12.5 1169.9 1058.5 972.6 1120514.7 NT 23299.1 9.8 8161.1 5659.1 7523.8 32025608.2 NR 5442.6 27.2 1174.9 987.1 902.8 974339.3 S L 6208.9 5.0 1438.8 1306.4 1178.3 1706784.3 S T 30530.1 4.9 9671.3 7246.4 8716.6 52510878.3 S R 9205.0 15.0 1226.3 1284.6 865.0 1650089.8 Table All.8 Volume Data APPROACH VOLUME STATISTICS MAX MIN AVERAGE STDEV MEDIAN VAR E B 38823.6 135.0 14086.9 7918.1 12845.0 62696581.9 38847.0 113.7 11694.9 6644.9 10924.6 44154356.7 27260.4 152.0 10327.3 6179.6 9618.7 38187024.6 31046.6 100.0 12133.3 7533.1 10632.6 56748243.7 W B NB S B Table All.9 Approach Data INTERSECTION VOLUME STATISTICS MAX MIN AVERAGE | STDEV MEDIAN VAR 82213.0 10504.0 47787.4 18447.8 49050.5 340322842.1 Table All.10 Intersection Data 87 Microscopic A c c i d e n t Prediction Mode ls tor Xi|i»sdiseci intersections APPENDIX IIIOUTLIERS ANALYSIS SAMPLES 88 Microscopic Accident Prediction Models for Signalised Intersections Outliers Analysis (before removals) o) 0 . 2 5 o | 0 .2 £ 0 . 1 5 Q a 0.1 o 0 . 0 5 <S 0 —* 2 0 4 0 6 0 Observed Accidents Figure AIII.1 Cook's distances before removals Outliers Analysis (after removals) 6.00E-02 5.00E-02 4.00E-02 3.00E-02 2.00E-02 1.00E-02 0.00E+00 f • I* . .............. w . . . 4~ • 1 2 0 4 0 Observed Accidents 60 Figure AIII.2 Cook's distances after 2 warranted and 2 unwarranted removals Intersection Sample Scaled SD Cumulative SD x 2 STATUS Number Size Deviance Drop Drop 169 169 187.6 2.40 2.40 2.4 REMOVAL WARRANTED 170 168 185.69 1.91 4.31 4.2 REMOVAL WARRANTED 123 167 184.33 1.36 5.67 5.8 REMOVAL NOT WARRANTED 126 166 183.65 0.68 6.35 7.3 REMOVAL NOT WARRANTED Table Al 11.1 Outlier Analysis showing warranted and unwarranted removals 89 Mk ' t - o s d J i ) ! O ut I i er s A nal ysi s (bef or e r emoval s) u 8.00E-02 u « 6.00E-02 S 4.00E-02 M * 2.00E-02 o O O.OOE+00 50 100 150 Observed Accidents 200 Figure All .3 Cook's distances before removals O u t l i e r s Ana lys is (a f te r r e m o v a l s ) I D 0.05 r I 0 0 4 .£ 0.03 a 0.02 y o 0.01 <S o ' ^Sr t f r r f rF _* • • 50 100 150 Observed Accidents 200 Figure All 1.4 Cook's distances after 4 unwarranted removals i n t e r s e c t i o n S amp le Sca led SD Cumu la t i ve SD I 2 STATUS N u m b e r S ize Dev iance Drop Drop 123 169 178.31 1.22 1.22 3.8 REMOVAL NOT WARRANTED 97 168 175.44 2.87 4.09 6 REMOVAL NOT WARRANTED 126 167 174.43 1.01 5.1 7.8 REMOVAL NOT WARRANTED 128 166 173.39 1.04 6.14 9.5 REMOVAL NOT WARRANTED Table All 1.2 Outlier Analysis with unwarranted removals 90 Microscopic Accident Prediction Models for Signalised Intersections APPENDIX IV GLIM OUTPUT SAMPLE 91 M i c r o s c o p i c Accident P r e d i c t i o n Models for S i g n a l i s e d In tersect ions [o] GLIM 4, update B for IBM etc. 80386 PC / DOS on 19-Aug-1999 at 21:38:05 [o] (copyright) 1992 Royal Statistical Society, London l°l [i] ? $C Accident Type 1 (Rear End)$ (ij?$Units 170$ [ij ? $Data V ACS [i]?$DinpufAT1.txt'S [ij ? SCalc LV=%log(V)$ [ij ? $Yvar AC SError P SLink 1$ [i] ? $Fit LV $D E$ [o] scaled deviance = 618.54 at cycle 4 [oj residual df - 168 [o] estimate s.e. parameter [0] 1 -3.047 0.2335 1 [oj 2 1.489 0.05809 LV joj scale parameter 1.000 [o] [i] ? $Look %X2S [0] 602.0 [i] ? $lnput %plc 80 NEGBINS [ij ? SNumber thela=0$ [i] ? $Use NEGBIN theta SD ES [w] - model changed jwj - model changed jo) scaled deviance = 179.70 (change = -438.8) at cycle 2 [oj residual df = 168 (change = 0 ) [o] [0] ML Estimate of THETA = 5.560 [oj Std Error = ( 0.8789) [o] [o] NOTE: standard errors of fixed effects do not jo] take account of the estimation of THETA [o] [o] 2 x Log-likelihood = 10549. on 168 df [oj 2 x Full Log-likelihood = -1111.. [o] [o] estimate s.e. parameter [o] 1 -3.175 0.3856 1 [oj 2 1.521 0.09905 LV [oj scale parameter 1.000 [o] [ij ? SLook %X2S [o] 167.7 [i] ? SExtract %Lv : %cd$ [i] ? SCalc LEVERAGE=%lv : COOK=%cd$ [i] ? SLook LEVERAGE COOKS [o] LEVERAGE COOK [oj 1 0.008848 0.0039731790 [oj 2 0.011321 0.0022354159 [oj 3 0.007134 0.0011139343 [0] 4 0.010685 0.0002563359 [o] 5 0.016471 0.0064739888 [oj 6 0.010355 0.0006460811 [oj 7 0.007666 0.0008000573 [Oj 8 0.006871 0,0020213965 [o] 9 0.006289 0.0029712832 [oj 10 0.010039 0.0028329017 [oj 11 0.006487 0.0025161246 [oj 12 0.009867 0.0018433135 [oj 13 0.018130 0.0036192173 [oj 14 0.008718 0.0001919384 [oj 15 0.006734 0.0024754717 [o] 16 0.006279 0.0047664107 [oj 17 0.006491 0.0005430502 [oj 18 0.019282 0.0003298848 [oj 19 0.011780 0.0000565374 [oj 20 0.006356 0.0000291500 [oj 21 0.010890 0.0010440752 [o] 22 0.006427 0.0250271503 [0] 23 0.010504 0.0003394376 [oj 24 0.006596 0.0004500134 [0] 25 0.007444 0.0005880118 [oj 26 0.019063 0,0058863992 [0] 27 0.012653 0.0004235143 [oj 28 0.019915 0.0356851742 [oj 29 0.016406 0,0021061262 [oj 30 0.014895 0.0000047396 [oj 31 0.011488 0.0106990291 [0] 32 0.007922 0.0084114727 [o] 33 0.013474 0.0027754651 [oj 34 0.015372 0.0177618526 (oj 35 0.008561 0.0049488186 [oj 36 0.006391 0.0012198063 [o] 37 0.007973 0 0006500610 [oj 38 0.008006 0 0010331073 [o] 39 0.007628 0,0003803656 [oj 40 0.006568 0.0000318116 [oj 41 0.016317 0.0017054159 [oj 42 0.007885 0.0005298993 [o] 43 0.006263 0.0004679549 joj 44 0.022733 0.0004601932 [0] 45 0.010225 0.0028195812 92 [o] [o] [o] [o] [0] [0] [0] [o] [0] [0] [0] [0] [0] [o] [0] [0] to] [o] to] [o] W [0] [o] [o] [o] to] [0] [0] to] [0] to] [0] [o] to] to] [0] [0] [0] [o] to] to] 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 [o] 88 [o] 89 90 91 92 93 94 95 96 97 [oj 98 [0] 99 [0] 100 [o] 101 [o] 102 [o] 103 [o] 104 [0] 105 [o] 106 [o] 107 [o] 108 [o] 109 [o] 110 to] 111 [o] 112 [o] 113 [o] 114 [o] 115 [o] 116 [o] 117 [o] 118 [o] 119 [o] 120 [o] 121 [o] 122 [o] 123 [0] 124 [o] 125 jo] 126 W 127 [o] 128 [o] 129 [0] 130 [o] 131 [0] 132 [o] 133 [0] 134 0,006899 0.011359 0.009277 0.015769 0.016150 0.009150 0.019804 0.006222 0,011405 0,008038 0.011899 0.008605 0.009826 0.006310 0.006977 0.006280 0.013658 0.006280 0.012536 0.006733 0.006385 0.020555 0.009992 0.009904 0.026522 0.016598 0.006352 0.012911 0,017114 0.021484 0.015098 0.010147 0.014795 0.007297 0.013871 0.007411 0.013918 0.006461 0.011009 0.015050 0.009385 0.013517 0.006398 0.022472 0.022520 0.027128 0.016122 0.020877 0.006312 0.017879 0.008260 0.010962 0.007350 0.006225 0.006955 0.019647 0.018701 0.012231 0.006248 0.006226 0.009969 0.010351 0.008847 0.012234 0.006390 0.008905 0.008238 0.010764 0.011273 0.006353 0.017153 0.011259 0.019783 0.019750 0.017809 0.018621 0.023622 0.013313 0.008937 0.006359 0.007427 0011918 0.011273 0.007058 0.009374 0.007654 0.007208 0.008610 0.015863 [0] 135 0.006451 [o] 136 0.015745 [0] 137 0.007747 [0] 138 0.006232 0.0006760735 0.0018086425 0.0008052467 0.0017271583 0.0047979774 0.0080105886 0.0004602058 0.0000089103 0.0025070307 0.0000007956 0.0000483738 0.0001368830 0.0015173719 0.0001910912 0.0008590907 0.0004124309 0.0035824336 0.0004409757 0.0001394290 0.0044545904 0.0065230867 0.0068027927 0.0000711405 0.0000619128 0.0083447937 0.0016411453 0.0019203438 0.0014810637 0.0269276667 0.0000258306 0.0023696346 0,0032982950 0.0052510831 0.0002989127 0.0010458955 0.0000785961 0.0001293476 0.0014437840 0.0002028120 0.0054818625 0,0002578690 0,0001609665 0.0033163687 0.0038846249 0.0133653264 0.0025183051 0.0018106659 0,0077889333 0.0006634509 0.0149916941 0.0038020860 0.0138879567 00085223261 0.0064958008 0.0015455843 0.0006889208 0.0003617572 00001092878 0.0014125473 0,0007755669 0,0005964171 0.0010607555 0.0036933527 0.0031444493 0.0000004635 0.0002154496 0.0011997753 0.0379213169 0.0004830722 0.0003967240 0,0069388337 0.0058664731 0.0074054413 0.0001884097 0.0089123892 0.0082041239 0.0123640178 0.1928226203 0.0000692309 0.0002635980 0.0189390127 0.0015554284 0.0200481769 0.0023538608 0.0471574441 0.0006722647 0.0001392432 0.0023372041 0,0002407763 0.0000708950 0.0199013259 0.0000806124 0,0002460191 M i c r o s c o p i c A c c i d e n t P red i c t i on M o d e l s t o r Signal ised In tersec t ions l°] 139 0.007026 [0] 140 0.006866 [0] 141 0.011916 [0] 142 0.011296 [0] 143 0.009648 to] 144 0.007201 [0] 145 0.015438 [0] 146 0.016296 [ol 147 0.017114 to] 148 0.009064 [0] 149 0.018942 [0] 150 0.006410 [0] 151 0.019915 to] 152 0.015177 [0] 153 0.006924 [0] 154 0.015013 [0] 155 0.006988 [0] 156 0.006921 [0] 157 0.011837 [0] 158 0.014165 [0] 159 0.010434 [o] 160 0.014094 lo] 161 0.024030 [0] 162 0.015776 [0] 163 0.007718 [o] 164 0.023207 [0] 165 0.007037 [o] 166 0.007522 [o] 167 0.006951 [0] 168 0.009816 [0] 169 0.028560 [0] 170 0.028580 [i] ? $Stop$ 0.0039493907 0.0014397698 0.0085815061 0.0070819156 0.0007975730 0.0004957537 0.0145896049 0.0155632831 0.0223749205 0.0019125429 0.0132519715 0.0111316536 0.0050704558 0.0154527463 0.0005463689 0,0005292293 0.0042265719 0.0127476593 0.0194408149 0.0009196686 0.0056105438 0.0018650822 0.0371853895 0.0006301449 0.0000743885 0.0256439727 0.0104868412 0.0002143530 0.0090389494 0.0000098959 0.0181985386 0.0019639770 94 M i c r o s c o p i c A c c i d e n t P red i c t i on Models for S ignal ised In tersec t ions APPENDIX V STATISTICAL SIGNIFICANCE TESTS 95 VHcroseonie Accident Prediction Models I'or Signalised Hhiwemttfts Figure AV.1 Predicted Accidents vs. Pearson Residuals for AT 1 Predicted Accidents Figure AV.2 Predicted Accidents vs. Average Squared Residuals and Estimated Variance for AT 1 250 T J !> 200 4 3 rz 150 100 50 0 — 0 10 20 30 Predicted Accidents 96 Miersseonie Accident Prediction Models for Signalised Intersections Figure AV.3 Predicted Accidents vs. Pearson Residuals for AT 5 Predicted Accidents Figure AV.4 Predicted Accidents vs. Average Squared Residuals and Estimated Variance for AT 5 97 Figure AV.5 Predicted Accidents vs. Pearson Residuals for AT 7 Predicted Accidents Figure AV.6 Predicted Accidents vs. Average Squared Residuals and Estimated Variance for AT 7 0.5 1 Predicted Accidents i t . • • --'.-t'rions Figure AV.7 Predicted Accidents vs. Pearson Residuals for AT 13 Predicted Accidents Figure AV.8 Predicted Accidents vs. Average Squared Residuals and Estimated Variance for AT 13 10 15 20 Predicted Accidents 99 M i c r o s * l i j i i c Accident Prediction Models Figure AV.9 Predicted Accidents vs. Pearson Residuals for Total Accidents 3 <n 2 | 1 S o ce G -1 O i -2 ra o £ -3 -4 * • * • • * 9 Predicted Accidents Figure AV. 10 Predicted Accidents vs. Average Squared Residuals and Estimated Variance for Total Accidents 1800 1600 1400 1200 1000 800 600 400 200 0 10 20 30 40 50 60 70 80 90 100 Predicted Accidents 100 ,': :: .: . . i Figure AV.11 Predicted Accidents vs. Pearson Residuals for Severe Accidents _4 J Predicted Accidents Figure AV. 12 Predicted Accidents vs. Average Squared Residuals and Estimated Variance for Severe Accidents 101 Figure AV. 13 Predicted Accidents vs. Pearson Residuals for PDO Accidents Predicted Accidents Figure AV.14 Predicted Accidents vs. Average Squared Residuals and Estimated Variance for PDO Accidents 102 Y1icrisseo|dc .Accident Prediction Models for Signalised intersections Figure AV. 15 Predicted Accidents vs. Pearson Residuals for Left Turn Accidents Predicted Accidents Figure AV.16 Predicted Accidents vs. Average Squared Residuals and Estimated Variance for Left Turn Accidents 103 M i c r o s c o p i c A c c i d e n t P red ic t i on M Figure AV.17 Accidents Predicted Accidents vs. Pearson Residuals for Right Turn 2 • 1 i o e: -1 c j? -2 ra a: o- -3 ** «• * • » Predicted Accidents Figure AV.18 Predicted Accidents vs. Average Squared Residuals and Estimated Variance for Right Turn Accidents 1 2 3 Predicted Accidents 104 

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