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USE OF ADVANCED TECHNIQUES TO ESTIMATE ZONAL LEVEL SAFETY PLANNING MODELS AND EXAMINE THEIR TEMPORAL TRANSFERABILITY

by

Alireza Hadayeghi

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Graduate Department of Civil Engineering University of Toronto

©Copyright by Alireza Hadayeghi 2009

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USE OF ADVANCED TECHNIQUES TO ESTIMATE ZONAL LEVEL SAFETY PLANNING MODELS AND EXAMINE THEIR TEMPORAL

TRANSFERABILITY

Doctor of Philosophy (2009) Alireza Hadayeghi

Department of Civil Engineering, University of Toronto

ABSTRACT

Historically, the traditional planning process has not given much attention to the road safety evaluation of development plans. To make an informed, defensible, and proactive choice between alternative plans and their safety implications, it is necessary to have a procedure for estimating and evaluating safety performance. A procedure is required for examining the influence of the urban network development on road safety, and in particular, determining the effects of the many variables that affect safety in urban planning.

Safety planning models can provide a decision-support tool that facilitates the assessment of the safety implications of alternative network plans. The first objective of this research study is to develop safety planning models that are consistent with the regional models commonly used for urban transportation planning. Geographically weighted Poisson regression (GWPR), full-Bayesian semiparametric additive (FBSA), and traditional generalized linear modelling (GLM) techniques are used to develop the models. The study evaluates how well each model is able to handle spatial variations in the relationship between collision explanatory variables and the number of collisions in a zone. The evaluation uses measures of goodness of fit (GOF) and finds that the GWPR and FBSA models perform much better than the conventional GLM approach. There is little difference between the GOF values for the FBSA and GWPR models.

The second objective of this research study is to examine the temporal transferability of the safety planning models and alternative updating methods. The updating procedures examine the Bayesian approach and application of calibration factors. The results show that the models are not

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temporally transferable in a strict statistical sense. However, relative measures of transferability indicate that the transferred models yield useful information in the application context. The results also show that the updated safety planning models using the Bayesian approach predict the number of collisions better than the calibration factor procedure.

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ACKNOWLEDGMENTS

This research required the assistance of a number of individuals. I am sincerely grateful to all of you for giving your time and efforts so generously in support of this task.

My sincere thanks are extended to my thesis supervisors, Professors Amer Shalaby and Bhagwant Persaud for their guidance, stimulating discussions, support and encouragement during the conduct of this research. It has been a great pleasure working with them for the past decade, not only for this research, but also for my undergraduate and Master’s degrees.

To my employer, I cannot express enough my heartfelt thanks to John McGill and Brian Malone at Synectics Transportation Consultants Inc for their endless support, encouragement, understanding, and providing me with a flexible and cooperative work environment. Without their support, I would not have been able to complete this lengthy assignment. I would also like to express my deepest appreciation to Doug Allingham, President of TSH, for providing me with financial support to complete this work.

To my dissertation committee members, Professors Eric Miller, Matthew Roorda and Tarek Sayed, I would like to say thank you for the insightful advice from different perspectives throughout the development of this research.

I must also thank Hossein Zarei for assistance and valuable comments on my thesis.

I would like to thank my fellow colleagues at Synectics Transportation Consultants for providing me with a wonderful working environment: Jeff, Greg, Oleg, Mathew, Brian, Ben, Renata, Altaf, Imtiaz, and Tariq.

I would like to thank my friends and the support staff at the University of Toronto, and Susanna in the Urban Transportation Research and Advancement Centre at the University of Toronto, for their time, ideas, and assistance with this study.

The contribution of Ann Khan and Steve Kodama of the City of Toronto for providing data and support is greatly appreciated.

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My deepest thanks go to my parents and my brothers in Iran for their tremendous support, patience, and encouragement.

Finally and most importantly, I need to thank my family. My beautiful daughters, Leili and Shali, have had to deal with a dad that was absent from time to time pursuing this task. I hope they have learned the value of education and the fact that nothing in life that is worth having ever comes easily. I will never forget the pride in their eyes and their genuine smiles of happiness when I told them that daddy’s “book” is done.

In all of this, the one who has stood beside me, encouraged me, given me the freedom to do what I needed to do, and loved me throughout the whole time, is my wife, Maggie. She has been nothing short of extraordinary in her support and encouragement as I followed my dream and pursued this degree. I was asked a few months ago by a work colleague about the most important thing that I learned during my PhD studies. Expecting me to respond with some educational-stock answer, I surprised him by saying that throughout all of this, I learned that I had married the most special person in the world.

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TABLE OF CONTENTS

ABSTRACT ...........................................................................................................................................II

ACKNOWLEDGMENTS......................................................................................................................... IV

TABLE OF CONTENTS ......................................................................................................................... VI

LIST OF TABLES ................................................................................................................................. IX

LIST OF FIGURES.............................................................................................................................. XIII

ABBREVIATIONS...............................................................................................................................XVI

1.0 INTRODUCTION....................................................................................................................... - 1 -

1.1 BACKGROUND.........................................................................................................................- 1 - 1.2 PROBLEM STATEMENT...............................................................................................................- 2 - 1.3 OBJECTIVE AND SCOPE .............................................................................................................- 4 - 1.4 ORGANIZATION OF DISSERTATION.................................................................................................- 8 -

2.0 LITERATURE REVIEW ............................................................................................................ - 10 -

2.1 ROAD SAFETY MANAGEMENT PROGRAMS .....................................................................................- 11 - 2.2 COLLISION PREDICTION MODELS................................................................................................- 14 - 2.3 PROACTIVE SAFETY PROGRAMS – A CONTEXT FOR SAFETY PLANNING MODELS.......................................- 21 - 2.4 SAFETY PLANNING MODELS ......................................................................................................- 29 - 2.5 TEMPORAL AND SPATIAL TRANSFERABILITY ...................................................................................- 34 - 2.6 SUMMARY AND CONCLUSIONS ...................................................................................................- 37 -

3.0 OVERVIEW OF THE METHODOLOGY ....................................................................................... - 39 -

3.1 METHODOLOGY FOR DEVELOPMENT OF SAFETY PLANNING MODELS.....................................................- 39 - 3.2 METHODOLOGY FOR ASSESSING TEMPORAL TRANSFERABILITY AND UPDATING OF SAFETY PLANNING MODELS.- 44 -

4.0 DATA ................................................................................................................................... - 46 -

4.1 DATA SOURCE ......................................................................................................................- 46 - 4.2 DESCRIPTIVE ANALYSIS OF THE 2001 DATA...................................................................................- 58 - 4.3 COMPARISON OF 1996 TO 2001 DATA .........................................................................................- 66 -

5.0 DEVELOPMENT OF SAFETY PLANNING MODELS USING THE GLM APPROACH .......................... - 70 -

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5.1 STATISTICAL PROPERTIES OF COLLISION DATA...............................................................................- 71 - 5.2 CALIBRATION OF GLM COLLISION PREDICTION MODELS....................................................................- 72 - 5.3 RESULTS OF SAFETY PLANNING MODELS......................................................................................- 80 - 5.4 SENSITIVITY ANALYSIS OF EXPLANATORY VARIABLES .......................................................................- 95 - 5.5 DISCUSSION OF RESULTS FOR SAFETY PLANNING MODELS ................................................................- 99 -

6.0 DEVELOPMENT OF LOCAL SAFETY PLANNING MODELS........................................................ - 100 -

6.1 BACKGROUND.....................................................................................................................- 100 - 6.2 CONCEPTUAL FRAMEWORK FOR GWPR MODELS..........................................................................- 101 - 6.3 SPECIFICATION OF GWPR MODELS ..........................................................................................- 102 - 6.4 CALIBRATION OF LOCAL SAFETY PLANNING MODELS ......................................................................- 105 - 6.5 RESULTS OF LOCAL SAFETY PLANNING MODELS...........................................................................- 108 - 6.6 DISCUSSION OF RESULTS FOR LOCAL SAFETY PLANNING MODELS .....................................................- 130 -

7.0 DEVELOPMENT OF SPATIAL SAFETY PLANNING MODELS ..................................................... - 132 -

7.1 BACKGROUND.....................................................................................................................- 132 - 7.2 CONCEPTUAL FRAMEWORK OF FULL BAYESIAN SEMIPARAMETRIC ADDITIVE (FBSA) MODELS ....................- 133 - 7.3 CALIBRATION OF SPATIAL SAFETY PLANNING MODELS....................................................................- 142 - 7.4 RESULTS OF SPATIAL SAFETY PLANNING MODELS.........................................................................- 144 - 7.5 DISCUSSION OF RESULTS OF THE SPATIAL SAFETY PLANNING MODELS ...............................................- 164 -

8.0 COMPARISON OF PERFORMANCE OF SAFETY PLANNING MODELS........................................ - 166 -

8.1 GOODNESS-OF-FIT MEASURES FOR COMPARATIVE ANALYSIS ...........................................................- 166 - 8.2 COMPARATIVE ANALYSIS OF GLM, GWPR AND FBSA MODELS ........................................................- 168 - 8.3 DISCUSSION OF THE RESULTS .................................................................................................- 172 -

9.0 TEMPORAL TRANSFERABILITY AND UPDATING OF SAFETY PLANNING MODELS .................... - 174 -

9.1 BACKGROUND.....................................................................................................................- 174 - 9.2 1996 MODEL SPECIFICATION ..................................................................................................- 176 - 9.3 MEASURES OF MODEL TRANSFERABILITY....................................................................................- 177 - 9.4 UPDATING SAFETY PLANNING MODELS ......................................................................................- 179 - 9.5 RESULTS AND DISCUSSIONS ...................................................................................................- 181 -

10.0 SUMMARY AND CONCLUSIONS ....................................................................................... - 199 -

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10.1 SUMMARY .....................................................................................................................- 199 - 10.2 CONCLUSIONS................................................................................................................- 201 - 10.3 RECOMMENDATIONS FOR FUTURE WORK ...............................................................................- 204 -

REFERENCES .............................................................................................................................. - 206 -

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LIST OF TABLES

Table 4-1 Sum of Collision Data for 1996 and 2001 within the City of Toronto........................... - 49 -

Table 4-2 Descriptive Statistics of TAZ Collision Data for 1996 and 2001.................................. - 49 -

Table 4-3 Descriptive Statistics of TAZ Street Network and Land Use Data for 2001 ................ - 51 -

Table 4-4 Survey Sample Statistic in the 2001 TTS.................................................................... - 52 -

Table 4-5 Descriptive Statistics of TAZ Socioeconomic and Demographic Data for 2001 ......... - 54 -

Table 4-6 Descriptive Statistics of TAZ Traffic Demand for 2001 Data....................................... - 54 -

Table 4-7 Descriptive Statistics of Data for TAZ Dwelling Unit and Employment Types............. - 58 -

Table 4-8 Changes in Urban Structure, Travel Behaviour and Socioeconomic and Demographic between 1996 and 2001 Data in the City of Toronto................................................................... - 66 -

Table 4-9 Changes in Number of Collisions between 1996 and 2001 Data in the City of Toronto- 68 -

Table 5-1 Regression Results Using GLM Approach for Traffic Intensity Based Models ........... - 81 -

Table 5-2 Correlation Matrix for Collision Data and Traffic Density Variables ............................ - 83 -

Table 5-3 Regression Results Using GLM Approach for Network Characteristics Based Models, Models T2-T4 .............................................................................................................................. - 84 -

Table 5-4 Regression Results Using GLM Approach for Network Characteristics Based Models, Models T5-T8 .............................................................................................................................. - 85 -

Table 5-5 Regression Results Using GLM Approach for Network Characteristics Based Models, Models S2-S4.............................................................................................................................. - 86 -

Table 5-6 Regression Results Using GLM Approach for Network Characteristics Based Models, Models S5-S8.............................................................................................................................. - 87 -

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Table 5-7 Regression Results Using GLM Approach for Land Use, Dwelling Unit and Employment Characteristics Based Models, T9-T11 ....................................................................................... - 90 -

Table 5-8 Regression Results Using GLM Approach for Land Use, Dwelling Unit and Employment Characteristics Based Models, S9-S11....................................................................................... - 91 -

Table 5-9 Correlation Matrix for Collision Data and Variables for T11 Model ............................. - 92 -

Table 5-10 Regression Results Using GLM Approach for Comprehensive Models.................... - 94 -

Table 5-11 Safety Index for Total Collisions Based on Sensitivity Analysis of Variables............ - 97 -

Table 5-12 Safety Index for Severe Collisions Based on Sensitivity Analysis of Variables ........ - 98 -

Table 6-1 Summary of Optimal Bandwidth Based on Number of TAZs for Total and Severe Collisions ................................................................................................................................... - 107 -

Table 6-2 Regression Results Using GWPR Approach for Traffic Intensity Based Models...... - 108 -

Table 6-3 Regression Results Using GWPR Approach for Network Characteristics Based Models, Models T2-T4 ............................................................................................................................ - 115 -

Table 6-4 Regression Results Using GWPR Approach for Network Characteristics Based Models, Models T5-T8 ............................................................................................................................ - 116 -

Table 6-5 Regression Results Using GWPR Approach for Network Characteristics Based Models, Models S2-S4............................................................................................................................ - 117 -

Table 6-6 Regression Results Using GWPR Approach for Network Characteristics Based Models, Models S5-S8............................................................................................................................ - 118 -

Table 6-7 Summary of Network Characteristics Based Models with Regard to Portions of TAZs with Unexpected Coefficient Signs for Total Models ........................................................................ - 120 -

Table 6-8 Summary of Network Characteristics Based Models with Regard to Portions of TAZs with Unexpected Coefficient Signs for Severe Models ..................................................................... - 121 -

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Table 6-9 Regression Results Using GWPR Approach for with Land Use, Dwelling Unit and Employment Characteristics Based Models, T9-T11 ................................................................ - 124 -

Table 6-10 Regression Results Using GWPR Approach for with Land Use, Dwelling Unit and Employment Characteristics Based Models, S9-S11................................................................ - 125 -

Table 6-11 Summary of Land use, Dwelling Unit and Employment Characteristics Based Models with Regard to Portions of TAZs with Unexpected Coefficient Signs for Total and Severe Models .. - 126 -

Table 6-12 Regression Results Using GWPR Approach for Comprehensive Models .............. - 128 -

Table 6-13 Summary of Comprehensive Models with Regard to Portions of TAZs with Unexpected Coefficient Signs for Total and Severe Models ......................................................................... - 129 -

Table 7-1 Regression Results Using FBSA Approach for Traffic Intensity Based Models ....... - 146 -

Table 7-2 Regression Results Using FBSA Approach, Models T2-T4...................................... - 151 -

Table 7-3 Regression Results Using FBSA Approach, Models T5-T8...................................... - 152 -

Table 7-4 Regression Results Using FBSA Approach, Models S2-S4 ..................................... - 153 -

Table 7-5 Regression Results Using FBSA Approach, Models S5-S8 ..................................... - 154 -

Table 7-6 Regression Results Using FBSA Approach, Models T9-T11.................................... - 157 -

Table 7-7 Regression Results Using FBSA Approach, Models S9-S10 ................................... - 158 -

Table 7-8 Regression Results Using FBSA Approach for Comprehensive Models .................. - 160 -

Table 8-1 Goodness of Fit Measures for Total Collisions ......................................................... - 170 -

Table 8-2 Goodness of Fit Measures for Severe Collisions ...................................................... - 171 -

Table 9-1 Results of Negative Binomial Regression for Yearly Total Collision Model (Model I)- 182 -

Table 9-2 Results from Transferring 1996 Models to 2001 Data Set for Model I ...................... - 183 -

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Table 9-3 Results of Negative Binomial Regression for Yearly Severe Collision Model (Model II) ... - 186 -

Table 9-4 Results from Transferring 1996 Models to 2001 Data Set for Model II ..................... - 187 -

Table 9-5 Results of Negative Binomial Regression for Models III & IV ................................... - 189 -

Table 9-6 Results from Transferring 1996 Models to 2001 Data Set for Models III and IV....... - 191 -

Table 9-7 Goodness of Fit Measures for Predicting 2001 Collisions from 1996, 2001 and Updated Models for Model I ..................................................................................................................... - 194 -

Table 9-8 Goodness of Fit Measures for Predicting 2001 Collisions from 1996, 2001 and Updated Models for Model II .................................................................................................................... - 195 -

Table 9-9 Goodness of Fit Measures for Predicting 2001 Collisions from 1996, 2001 and Updated Models for Models III and IV...................................................................................................... - 197 -

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LIST OF FIGURES

Figure 1-1 Traffic Deaths per Billion Vehicle Kilometres Travelled (OECD, 2001) ....................... - 2 -

Figure 2-1 Framework for Proactive Road Safety Planning (de Leur & Sayed, 2003)................ - 23 -

Figure 2-2 Application of Road Safety Audits in Road System Lifecycle (McGill et al., 2005) .... - 24 -

Figure 2-3 Overview of Transportation Planning Process in Context of Safety (AECOM Consulting Transportation Group et al., 2000) .............................................................................................. - 28 -

Figure 2-4 Process Followed to Develop Planning Level Collision Prediction Models (Washington et al., 2006) ................................................................................................................................. - 33 -

Figure 3-1 Process for Development of Safety Planning Models................................................ - 42 -

Figure 4-1 Histograms of Toronto TAZ level Total and Severe Collision Frequencies for Year 2001 - 50 -

Figure 4-2 Spatial Distributions of Collision Density for Total Collisions in the City of Toronto for 2001............................................................................................................................................. - 59 -

Figure 4-3 Spatial Distributions of Collision Density for Severe Collisions in the City of Toronto for 2001............................................................................................................................................. - 59 -

Figure 4-4 Ratio of Total Collisions to VKT in the City of Toronto for 2001................................. - 61 -

Figure 4-5 Ratio of Severe Collisions to VKT in the City of Toronto for 2001 ............................. - 61 -

Figure 4-6 Population Density for the City of Toronto in 2001 .................................................... - 62 -

Figure 4-7 VKT in the City of Toronto in 2001............................................................................. - 63 -

Figure 4-8 VKT Density in the City of Toronto in 2001................................................................ - 64 -

Figure 4-9 Signalized Intersection Density in the City of Toronto in 2001................................... - 65 -

Figure 4-10 Road Density in the City of Toronto in 2001 ............................................................ - 65 -

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Figure 4-11 Percentage Change in Average Traffic Volume from 1996 to 2001 in the City of Toronto Based on 1996 TAZ Definitions ..................................................................................... - 67 -

Figure 4-12 Percentage Change in Total Number of Collisions from 1996 to 2001 in the City of Toronto Based on 1996 TAZ Definitions ..................................................................................... - 69 -

Figure 4-13 Percentage Change in Severe Number of Collisions from 1996 to 2001 in the City of Toronto Based on 1996 TAZ Definitions ..................................................................................... - 69 -

Figure 5-1 CURE Plot for Traffic Intensity Models Based on GLM Approach............................. - 82 -

Figure 5-2 CURE Plot for Comprehensive Models Based on GLM Approach ............................ - 95 -

Figure 6-1 Coefficients of Zonal VKT for T1 Model ................................................................... - 109 -

Figure 6-2 Coefficients of Zonal VKT for S1 Model................................................................... - 109 -

Figure 6-3 t-Statistics of Zonal VKT for T1 Model ..................................................................... - 112 -

Figure 6-4 t-Statistics of Zonal VKT for S1 Model ..................................................................... - 112 -

Figure 6-5 CURE Plot for Traffic Intensity Models Based on GWPR Approach........................ - 114 -

Figure 6-6 CURE Plot for Comprehensive Models Based on GWPR Approach....................... - 130 -

Figure 7-1 The “map object” Information for the Study Area Using BayesX ............................. - 144 -

Figure 7-2 Posterior Mean for Spatial Covariates of Intensity Based Model, T1....................... - 148 -

Figure 7-3 Posterior Mean for Spatial Covariates of Intensity Based Model, S1 ...................... - 149 -

Figure 7-4 CURE Plot for Traffic Intensity Models Based on FBSA Approach ......................... - 150 -

Figure 7-5 Posterior Means for Spatial Covariates of Comprehensive Model, T12 .................. - 162 -

Figure 7-6 Posterior Means for Spatial Covariates of Comprehensive Model, S11.................. - 163 -

Figure 7-7 CURE Plot for Comprehensive Models Based on FBSA Approach......................... - 164 -

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Figure 9-1 Comparison of CURE Plots for Applying 1996 and 2001 Safety Planning Models to 2001 Data for Total Collisions ............................................................................................................ - 185 -

Figure 9-2 Comparison of CURE Plots for Applying 1996 and 2001 Safety Planning Models to 2001 Data for Severe Collisions......................................................................................................... - 188 -

Figure 9-3 Comparison of CURE Plots for Applying 1996 and 2001 Safety Planning Models to 2001 Data for Total Collisions for Morning Peak Period .................................................................... - 192 -

Figure 9-4 Comparison of CURE Plots for Applying 1996 and 2001 Safety Planning Models to 2001 Data for Severe Collisions for Morning Peak Period ................................................................. - 192 -

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ABBREVIATIONS

AADT AASTO AIC AICc CAR CBD CMF CURE DRS DUE EB FBSA FHWA FSA GEE GIS GLM GOF GWR GWPR GTA IHSDM LHRS LR MAE MCMC MAD MSE MSPE MTO

Annual Average Daily Traffic American Association of State Highways & Transportation Officials Akaike Information Criterion Corrected Akaike Information Criterion Conditional Auto Regressive Central Business District Collision Modification Factor Cumulative Residual Data Retrieval System Deterministic User Equilibrium Empirical Bayes Full-Bayesian Semiparametric Additive Federal Highway Administration Forward Sortation Areas Generalized Estimating Equations Geographic Information Systems Generalized Linear Modelling Goodness of Fit Geographically Weighted Regression Geographically Weighted Poisson Regression Greater Toronto Area Interactive Highway Safety Design Model Linear Highway Referencing Systems Nested Likelihood Ratio Test Multiple Account Evaluation Markov Chain Monte Carlo Mean Absolute Deviation Mean Squared Error Mean Squared Prediction Error Ontario Ministry of Transportation

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O-D PDO PLANSAFE RSA SAFETEALU SCP SD TAZ TI TSU UTRAC V/C VDF VKT VMT

Origin-Destination Property Damage Only Planning Level Forecasting Models Road Safety Audit Safe, Accountable, Flexible, Efficient, Transportation Equity Act A Legacy for Users Safety Conscious Planning Scaled Deviance Traffic Analysis Zones Transfer Index Traffic Safety Unit in City of Toronto Urban Transportation Research and Advancement Centre Volume over Capacity Ratio Volume Delay Function Vehicle Kilometres Travelled Vehicle Miles Travelled

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CCHHAAPPTTEERR OONNEE

1.0 INTRODUCTION

1.1 Background

Road safety is regarded as one of the most important transportation concerns in urban areas. Many transportation agencies responsible for road safety have recently increased their formal efforts to mitigate the adverse impacts of traffic collisions on urban mobility, economy, and health. The ultimate goal of these practices and standards is to promote public health and safety by reducing transportation-related deaths, injuries, and property damage.

Road collisions are a major cause of concern to road users and organizations responsible for road construction and maintenance. Collisions may involve the loss of human life, injuries, and damage to vehicles. They also increase travel costs by delaying traffic. According to Canadian Motor Vehicle Traffic Collision Statistics (Transport Canada, 2001), there were 2,778 deaths in Canada due to motor vehicle traffic collisions in 2001 (a rate of 8.9 deaths per 100,000 population). In 2000-2001, there were 24,403 hospital admissions for traffic-related injuries, (a rate of 79 hospitalizations per 100,000 population) (Transport Canada, 2004). The annual economic cost to Canadian society of injury-producing and property damage traffic collisions is estimated at between

Use of Advanced Techniques to Estimate Zonal Level Safety Planning Models and Examine Their Temporal Transferability Chapter 1: Introduction

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$11 and $27 billion, depending on the calculation method used (Transport Canada, 2005). Road transportation fatalities and injuries are unacceptably high. Fatality rates for Canada are reported to be 8.94 per billion Vehicle Kilometers Travelled (VKT) as shown in Figure 1-1.

Figure 1-1 Traffic Deaths per Billion Vehicle Kilometres Travelled (OECD, 2001)

While Canadian statistics on traffic fatality are comparable to those of other developed countries, there remains room for improvement. Many organizations have responded by initiating road safety management programs. These programs have identified numerous factors that contribute to collisions. Although these traditional, or reactive, safety improvement programs have been shown to be successful (within certain limitations), they are expensive and require significant collision data history before any action can be taken. There is a clear need to consider road safety from a wider and more proactive perspective.

1.2 Problem Statement

As highlighted by Sayed et al. (1995), a failure in one or more of the three road system components; the driver, vehicle and road, can create a collision. Roads or transportation networks are designed to be safe, but the planning and engineering processes can nevertheless result in a dangerous road environment. Decisions made by government officials, planners, and engineers


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can also affect the level of safety in transportation systems. The ultimate goal is to promote public health and safety by working to minimise the number of collisions and transportation-related deaths, injuries, and property damage. This goal can only be achieved through an active and conscious approach to monitoring the transportation system and by anticipating safety problems before they emerge.

In recent years, the inclusion and consideration of road safety has emerged as a central part of the regional transportation planning process and strategy for improving safety on the transportation network. This emphasis on safety has been accomplished through the introduction of the concept of safety conscious planning (SCP), which is based on the idea that safety should be considered as an explicit part of the transportation planning process. The approach requires safety-conscious practices to be comprehensively, routinely, and effectively incorporated into the overall transportation planning process, and expects safety to receive the same priority accorded to strategies aimed at improving, for example, mobility (Dumbaugh et al., 2004).

Urban transportation planning has traditionally focused on capacity and congestion, and also given some attention to operation and management. Alternative transportation network scenarios and policies are typically evaluated in terms of their expected effects on congestion, pollution, etc., and the most favourable network alternative and supporting policies are proposed and implemented. Road safety has traditionally received little attention during this planning process with the result that it is only assessed after the planning process, when the transportation facility has been used and safety problems have emerged. Even when safety is considered during the design stage, it is seldom treated as an explicit planning priority in the decision making process.

The explicit consideration of road safety in the planning process is a complex and a challenging task. Several studies (AECOM Consulting Transportation Group et al., 2000; Chatterjee et al., 2004 and Washington et al., 2006) have discussed the issues involved in incorporating safety considerations into the transportation planning process, and recommended methods for overcoming the problems. The researchers noted that major challenges include the lack of data and suitable models for evaluating road safety in the planning process using conventional transportation planning modelling platforms, such as Equilibrium Multimodal, Multimodal Equilibrium (EMME/2) (INRO, 1998) or similar ones.


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1.3 Objective and Scope

As mentioned in Section 1.2, the comprehensive incorporation of safety considerations into the transportation planning process has emerged as a central strategy for improving transportation safety. Lord (2000), for example, worked to develop a tool that would allow the estimation of traffic collisions on computerized or digital transportation networks during the planning process. To accomplish this aim, he developed micro-level collision prediction models and applied them to two sample digital networks created by EMME/2 (INRO, 1998). The results showed that it is possible to predict collisions on computerized transportation networks, but that accuracy is directly related to the precision of the traffic flow predictions obtained from the transportation planning software programs. The collision predictions are limited because micro-level collision prediction models essentially predict the level of safety at a single location where traffic volume can be accurately estimated using relatively simple, short-term projections. As highlighted by Lovegrove (2005), traffic forecasts for planning-level analyses are often conducted by using strategic-level transportation planning models calibrated only coarsely on regional transportation data and not on individual streets or intersection counts. The coarse level, longer-term time frame and broad focus of planning-level traffic forecasts mean that forecasts at any one specific location are usually inaccurate.

Several research efforts have recently been undertaken to forecast the level of road safety of a transportation network in the transportation planning process. The purpose of these studies is to provide a safety planning decision-support tool which facilitates a proactive approach to assessing the safety implications of alternative network planning initiatives and scenarios. Some of the studies (Hadayeghi et al., 2003; Washington et al., 2006; Lovegrove and Sayed, 2005 and de Guevara et al., 2004) have developed models that operate at an aggregate zonal level and are compatible with the travel demand models typically used in the transportation planning process.

Given the gaps in knowledge concerning safety planning tools, and the paucity of relevant research, the author of this thesis previously developed a series of collision prediction models that estimated the number of collisions in planning traffic analysis zones (TAZs) as a function of traffic intensity, socioeconomic and demographic variables, and network data variables. The models were tested using City of Toronto data (Hadayeghi, 2002). However, due to data limitations and


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availability, some important variables were not included in the calibrated models. As a result, the models developed have some limitations, and a number of improvements were suggested for future work.

The goal of this thesis is to develop a series of zonal-level collision prediction models (safety planning models) using advanced statistical techniques, and examine their temporal transferability. The models are to be consistent with the regional ones commonly used in urban transportation planning. The research has two objectives: the development of safety planning models (discussed in Section 1.3.1), and an examination of their temporal transferability (discussed in Section 1.3.2).

1.3.1 Objective #1: Development of Safety Planning Models

The first objective of this research study is to develop a series of zonal-level collision prediction models, known as safety planning models. These models are to be consistent with the regional models commonly used in urban transportation planning. The study uses data from Toronto, Canada, to examine the relationship between zonal collisions and factors (explanatory variables), such as traffic intensity, land use, employment type, socioeconomic and demographic factors, and traffic network characteristics.

Different statistical methods were used in the development of the models. A common technique used in research into the calibration of safety planning models is the Generalized Linear Modelling (GLM) procedure with the assumption of a Negative Binomial error distribution. In this technique, fixed coefficients are estimated to represent the average relationship between the dependent variable, typically number of collisions per TAZ, and each explanatory variable. As the stationary relationship assumed by the GLM technique may hide some important spatial factors that affect the number of collisions in a particular TAZ, the accuracy of the models for explaining the relationship between the dependent variable and explanatory variables may leave room for doubt. In reality, it is likely that collision frequency is influenced by many spatially defined factors, such as land use, demographic characteristics, and traffic volume patterns.

To investigate the spatial variations in the relationship between the number of zonal collisions and potential transportation planning predictors, two additional techniques are used to develop safety


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planning models in this study. The two techniques are geographically weighted Poisson regression (GWPR), and full-Bayesian semiparametric additive (FBSA) modelling.

The GWPR technique is used to allow for local analysis of the relationships between collision and transportation planning data at the TAZ level. The technique captures the local variations by calibrating a multiple regression model that allows different relationships between variables to exist at different points in space. The main advantage is that the independent variable coefficient estimates can vary locally. This is a very attractive feature of a model that attempts to capture the relationship between collision occurrence data and their spatial location. It allows for a locally varying parameter representation of a particular point in space. GWPR is considered to be a fully local model.

The FBSA modelling technique is used for the development of spatial safety planning models. It is adopted after attempting to use the generalized estimating equation (GEE) procedure to develop spatial safety planning models.

Whereas GWPR models produce a set of local parameters for each independent variable for each TAZ, FBSA models produce a set of global parameter estimates similar to traditional GLM models. Spatial relationships are incorporated into the FBSA modelling framework through the covariance of the error terms. FBSA models can be thought of as “semi-local” (or spatial) rather than as fully local like the GWPR models.

The predictions of the three models for the number of collisions in each TAZ are compared and the performance of each model is evaluated. The GWPR and FBSA models are compared with the GLM model, and with each other.

The safety planning models developed in this study involve a large number of variables requiring a large amount of data from various sources. The applicability and reliability of the models depend on the ability to integrate data from various sources. Thanks to spatial referencing, good data management, and the visualization power of Geographic Information Systems (GIS), the integration of the data can be easily managed.


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The first step in the development of the safety planning models is to identify the various potentially explanatory factors that are related to collision frequency. The second step is to collect the data required to capture the relationship between collision frequency (collision data) and the explanatory factors. The data required for the explanatory factors include traffic volume, traffic network variables, socioeconomic and demographic variables, traffic demand variables, and land-use variables (employment types, dwelling units, etc). The third step is to develop statistical models which best fit and explain the variation within the available data.

1.3.2 Objective #2: Examine the Temporal Transferability of Safety Planning Models

The second objective of this research study is to examine the temporal transferability of the safety planning models. Temporal transferability refers to how well the relationship between the dependent and independent variables holds across time. Data for transportation planning variables are usually obtained from periodic surveys. For example, the Transportation Tomorrow Survey (TTS) is conducted once every five years in the Greater Toronto Area (GTA). The surveys usually cover a comprehensive area and are therefore, expensive in terms of data collection and analysis. Once a model is well specified, it is clearly cost effective to be able to transfer the model to other applicable contexts, including where the data available may be inadequate for the specification of a model (Badoe and Miller, 1995a).

Three aspects of temporal transfer are considered in this study: the extent of temporal transferability of the safety planning models; alternative updating methods for temporal transfer; and the effect of sample size on transfer bias.

The examination of temporal transferability uses appropriate evaluation measures of predictive performance to assess whether the relationship between the dependent and independent variables holds reasonably well across time. Models estimated using 1996 data, obtained from the author’s previous study (Hadayeghi, 2002), are used with a 2001 dataset to predict 2001 collisions.

The comparison of alternative updating methods of temporal transfer includes estimating the effects of the size of the application context data on model transferability. Two updating procedures


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are examined: Bayesian and that based on recalibration factors. Models calibrated for the 2001 samples are also explored, but found to be inadequate. Hence, they are not tested in the 2001 application context.

Transfer bias caused by sample size has not been addressed in previous studies. Previous studies focused mainly on using only one method and sample size. Transfer bias caused by sample size is important since it is believed that a large portion of the differences in coefficients and predictions is caused by random variations that can be attributed to the selection of the data. In this study, the GOF measures obtained in the updated models are compared with those obtained in the models estimated using the entire data set in the application context. This part of the research makes a key contribution by employing a variety of validation methods and measures in the testing and comparing of the models.

The importance of obtaining good and accurate collision prediction models is much emphasized in the literature (see Chapter Two). The quantity and quality of the data used to calibrate collision prediction models is very important to their success or failure. As the predictive accuracy of models calibrated with limited data will be inadequate, a good data set is very important. The specification of an appropriate functional model form for the calibration of collision prediction models is a complex task which becomes even more complicated when the data are of limited quantity and/or quality.

1.4 Organization of Dissertation

The primary focus of this thesis is to encourage and make feasible, the proactive inclusion of road safety in the transportation planning process. The thesis attempts to develop a methodology that makes it possible during the initial stages of the planning process to estimate the future road safety of an urban area. The approach is part of the effort to develop urban transportation networks that are more safety-conscious.

This thesis is organized into ten chapters. This first chapter provides an introduction to the dissertation by presenting the background, research problems and goals, and objectives. Chapter Two provides a brief literature review of existing research into the research problems and proposed


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subject. The literature review includes an overview on road safety improvement programs, collision prediction models, proactive safety programs, safety planning programs and the transferability of collision prediction models. Chapter Three describes the general format of the methodology framework used in this study. Chapter Four describes the data used for the development of the safety planning models. It presents a descriptive analysis of the variables used in the study. Chapter Five discusses the statistical properties of collision data, and presents the safety planning models developed using the traditional GLM approach. Chapter Six discusses the GWPR approach to the development of local safety planning models, and presents the GWPR models developed. Chapter Seven discusses the FBSA regression approach to spatial safety planning models (“semi-local” models). Chapter Eight describes the GOF of the models, interprets their associations, and compares the accuracy of GWPR, FBSA and GLM models in predicting the number of collisions in each TAZ. Chapter Nine discusses the methodology used to examine temporal transferability and the updating of safety planning models. The chapter also discusses the effect of sample size on transfer bias. Finally, Chapter Ten presents the conclusions and recommendations for future research in developing safety planning models.

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2.0 LITERATURE REVIEW

The objective of this chapter is to review the literature in several subject areas related to urban transportation planning, road safety, collision prediction models and relevant issues, including transferability of planning and collision prediction models. The purpose of this literature review is to describe the research context and theoretical foundations which have built this dissertation. To that end, this chapter is divided into five main sections. The chapter starts with a broad literature review that is relevant to this research, but not directly relevant to safety planning models and their related issues. In the latter part of this chapter, a more focused review directly relevant to this thesis is presented.

The first section describes the traditional concepts and techniques used for road safety improvement programs and it also includes an introduction to “black spot” programs. This is followed in the second section by a review of existing micro-level collision prediction models used to predict the number of collisions at intersections and on arterial roads. These models, as explained later, are commonly used in a black spot improvement program. Also, this section is partly dedicated to the literature related to the development of collision prediction models. The third section mainly deals with the concept of proactive safety programs, including brief discussions of

Use of Advanced Techniques to Estimate Zonal Level Safety Planning Models and Examine Their Temporal Transferability Chapter 2: Literature Review

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road safety audit (RSA) programs, sustainable road safety programs and concepts of SCP. Thereafter, in the fourth section, recent macro-level collision predictions, or safety planning models are presented. The fifth section provides a summary and conclusions of previous research on safety planning models. Finally, the last section concludes this chapter by presenting a review on transferability of planning and collision models, with special emphasis on their temporal transferability, in order to provide the context for this aspect of the research.

2.1 Road Safety Management Programs

In general, road safety management programs are used to improve the understanding of the state of practice in road safety and consequently, improve the safety performance of the road component of a transportation network.

The road safety management program at a transportation agency, in general, is comprised of four interrelated processes:

Black Spot Identification: the main goal is to identify road locations that have poor safety performance and need safety investigation. The underlying assumption is that road design attributes often play a significant contributory role in collision occurrences. As explained later in this section, there are many methods to identify black spots in a transportation network.

Diagnosing Safety Problems: this process examines the causes of safety problems at an identified black spot location. To complete this task, a systematic methodology, known as “in-service road safety review”, should be conducted to ensure thoroughness and accuracy of analysis. To conduct them, two complementary tasks have to be performed. First, at least three years of historical collision data have to be collected and analyzed to identify over-represented clusters of particular collision types. Secondly, location-specific data, both physical and operational, are gathered and analyzed, accompanied by some consultation with local road agencies, enquiries on site traits, and observations of driver characteristics.


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Countermeasure Selection and Economic Appraisal: this task involves selection of treatments deemed necessary for potential safety improvement of a black spot following diagnosis. In the course of this selection process, more than one countermeasure with the potential to remedy the problem is often identified. A subsequent economic appraisal will evaluate all options for all black spots in optimizing an improvement program for a given budget.

Monitoring: this step involves monitoring implemented improvements to assess the safety effectiveness.

As evident from the above procedures, identifying sites that require investigation for safety treatments is the first step taken by a transportation agency as an essential part of its safety management program. However, it is very expensive and time consuming. In the absence of any systematic approach, identifying road locations with the greatest potential for safety improvements at the network level is often impossible. This is mainly due to the fact that results of safety improvements in one road group are not directly comparable to the others. Hence, there is a need to establish a quantitative traffic safety approach in order to identify problematic locations and eventually, rank the candidate projects.

As indicated above, to ensure that resources are primarily spent on the locations with the highest potential for safety improvements, it is vital that a sound procedure be in place to screen the road network. This procedure will properly identify and rank black spots for diagnosis and treatment purposes. A black spot or a location with high potential for safety improvements exhibits a collision potential that is significantly higher than typical potential values for a group of similar locations. The collision potential of a location is commonly described by several measures, including: collision frequency, rate, severity, or a combination of such. However, there are several problems and issues with these measures which can lead to incorrect identification of road elements.

Many road agencies describe the collision potential of a location by using collision frequency, which is defined as the number of collisions occurring at a location during a specific time period. Typically, time periods between one to three years are used to minimize the effects of random fluctuations and sensitivity to changes over time.


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A common measure used for identifying locations for safety improvement is the use of collision counts. It is the simplest of techniques and requires much less data than the more sophisticated techniques. However, it suffers from the regression-to-mean bias in which an unusually high count is likely to decrease subsequently even if no improvement is implemented. Also, this method does not account for traffic volumes or exposure when comparing the potential for safety improvement in different locations. This can lead to a bias in favour of high traffic volume locations which tend to have more collisions.

To address the drawbacks of collision frequency, McGuigan (1982) recommended a traffic rate measure for identifying black spots. Collision rate is used directly or indirectly to identify locations in need of safety investigation. Collision rate is defined as collision frequency divided by some unit of traffic exposure, usually million-vehicle-kilometres for road sections, and million-entering-vehicles for intersections. However, care should be exercised that if collision rates are calculated based on the observed counts, the regression-to-mean bias may still exist. Moreover, some other problems still remain. First, the measure may identify low-volume roads as hazardous even though their collision frequencies may be low. Secondly, its simple ratio of collisions per unit of traffic volume assumes that a linear relationship exists between collisions and traffic volumes, which has already been shown as incorrect (Hauer, 1995). Thirdly, previous research has shown biased results for short road segment lengths (Zegeer, 1982 and Nicholson, 1980).

To overcome the difficulties with the above conventional methods, an Empirical Bayes (EB) approach has been developed to compensate for the random fluctuations in collision occurrences by combining the collision count of an element and its expected safety performance. This method is becoming widely used in road safety evaluation and increasing the accuracy of the safety estimates (Hauer, 1992; Hauer, 1997; Hauer et al., 2002 and Persaud et al., 2002a). Moreover, EB techniques can be applied not only to the identification of black spots, but also the evaluation of candidate countermeasures or assessment of potential savings earned as a result of site safety improvements.

The EB estimate is based on a weighted combination of the observed and predicted collision frequencies of a group of locations acting as a reference whose safety performance is similar to


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that of the location. This reference group-based estimate can be obtained using multivariate regression methods to provide a site-specific estimate of the regional reference group mean collision frequency and its variance (Hauer, 1992 and Hauer, 1997).

As evident, the EB estimate relies heavily on the use of collision prediction models. The collision prediction models must be properly developed for programs which use the EB method. Therefore, it is imperative that these models be developed in accordance to methods recommended in the literature.

2.2 Collision Prediction Models

2.2.1 Background

Collision prediction models that are used in safety planning, predict the number of collisions on transportation networks. They are similar in principle and in their development to models for intersections, and highway and arterial road sections. Hence, this section reviews collision prediction models in general.

Collision prediction models are developed by finding the relationship between collision frequency as the dependent variable, and a number of independent or explanatory variables. These models are capable of incorporating a large number of variables and their interactions to explain differences in collision frequency among entities.

Variations in road collisions consist of two components: systematic, which can be explained statistically, and random, which cannot be done so. Collision prediction models account for systematic variation by fitting mathematical relationships to observed data. Collision prediction models have the following general form:

( ) { }X,fAE β= (2.1)

where

E(A) = expected number of collisions per unit of time;


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X = a series of independent variables; and

β = coefficients of individual covariates.

Equation 2.1 is used to predict the number of collisions per unit of time for a specific road element (e.g. intersection, road segment, or transportation network) as a function of a set of independent variables. These variables can be comprised of several independent covariates, such as traffic intensity (Annual Average Daily Traffic (AADT) or VKT), lane configurations, types of traffic control devices, speed limits, etc. However, the most common models use traffic intensity as the only independent variable. In general, these models can be developed to predict the number of collisions by severity or impact type.

The main goal in calibrating the above equation is to find the estimate of the coefficients, β, associated with the covariates or independent variables. There exist many methods and techniques used in the literature for estimating such coefficients. Studies have calibrated linear models (Myer, 1990 and Bowerman and O’Connell, 1990), GLM (McCullagh and Nelder, 1989) and non-linear models (Saber and Wild, 1989).The following subsections introduce previous research related to the development of statistical models in the transportation context and specifically in road safety. These sections describe the research context and provide theoretical foundations that serve as a basis for calibration of the safety planning models in this dissertation.

As explained in Chapter One of this dissertation, three different methods were used for estimation of coefficients in this study. First, the coefficients of the models were estimated by adopting the traditional GLM approach. The second approach is the GWPR and it was used for the local analysis of relationships between collision data and the explanatory variables. Finally, spatial safety planning models were calibrated using the FBSA approach based on Markov Chain Monte Carlo (MCMC) simulation for sampling the posterior probability distribution.


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2.2.2 Generalized Linear Models

Road collisions are rare and random events, and therefore, the number of occurrences on a road element in a short time period can fluctuate considerably. This property of collision data, along with the fact that counts are non-negative and discrete, makes conventional linear regression models, with a normally distributed error structure, inappropriate for modelling. Instead, Poisson and NB regression have been adopted, usually in the GLM framework.

The Poisson distribution has been shown to be reasonable for modelling collision data at a given single site, but in reality, collision data over a number of sites often exhibit a large variance and a small mean, displaying overdispersion with a variance-to-mean value greater than 1. For this reason, the NB distribution, also known as the Poisson-gamma distribution, has become the most commonly used probabilistic distribution for modelling collisions at a series of sites. The NB distribution is considered to be better at handling overdispersion effects than any other distribution patterns (Miaou, 1994).

An extensive number of research efforts have been made to develop statistical collision models for intersections and road segments using the GLM approach. For example, in a study of collision occurrences at signalized intersections, Persaud and Nguyen (1998) developed aggregate and disaggregate models to estimate the safety performance of three-legged and four-legged signalized intersections in the Province of Ontario using the NB regression. The models predicted the number of collisions for specific conflicting movements, such as collisions between through and left-turning vehicles as well as right-angle collisions. Such models are very useful in investigating the relationship between collision occurrences and the movement of vehicles. However, in order to develop such models, massive amounts of data, such as traffic counts for every single intersection under study, are required. Poch and Mannering (1996) used NB regression to develop a different model for total, rear-end, right-angle and turning collisions. The results of their study show the importance of road geometric design and traffic-related variables in increasing and decreasing collision frequency. They also compared the use of NB regression with Poisson regression in the modelling of collision data. They found that the NB regression is better suited for road collision counts than the Poisson distribution due to the “overdispersion” phenomenon.


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Miaou and Lum (1993) compared two conventional linear regression models and two Poisson regression models in terms of their ability to model the relationship between the number of vehicle collisions and highway geometric design variables. The authors found that the Poisson regression models perform better than the linear regression ones. They concluded that the conventional linear regression models lack the distributional properties required to thoroughly describe random, discrete, non-negative and typically sporadic natures of vehicle collisions on the road network and must be avoided as a technique for the development of collision prediction models .They also found that using the Poisson model may not be appropriate for overdispersed collision data. Due to the issues caused by the overdispersion phenomenon, the authors concluded by suggesting the use of a more general probability distribution, such as the NB.

In addition to the traditional Poisson and NB regression for development of collision prediction models, a GEE procedure introduced by Liang and Zeger (1986), and Zeger and Liang (1986), has been used lately in the literature for examining the trend in the expected number of collisions. The GEE procedure is commonly used for parameter estimation of correlated data, such as estimating trends in temporally correlated data. The GEE models are an extension of traditional generalized linear models. Their random component for each marginal response with a common link and variance function are similar to that in the GLM approach. However, unlike GLMs, GEEs account for the covariance structure of the repeated measures (Abdel-Aty and Abdalla, 2004).

Lord and Persaud (2000) used the GEE procedure to obtain the coefficients of time effect models for 4-legged signalized intersections in the City of Toronto. Three different models were calibrated using a simple model form that includes the major and minor road flows as the only explanatory variables. Model 1 was calibrated from aggregated data, i.e., using average AADT over the study period and total collisions and adopting the regular GLM approach. Model 2 was calibrated from the data disaggregated by year and with the regular GLM approach. Model 3 was calibrated from disaggregated data, but with the GEE procedure so that it incorporated both trend and temporal correlation. The results showed that temporal correlation increases the standard error, but the coefficients are still significant. The authors mentioned the possibility of some coefficients becoming insignificant due to their inflated variance when accounting for temporal correlation and trend.


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Hadayeghi et al. (2007a) used the GEE approach and assumed an NB error distribution for development of safety performance functions of signalized intersections, unsignalized intersections, and road segments in the Regional Municipality of Halton. GEE was used to capture the yearly temporal correlation changes. The results of the developed models indicated that the yearly coefficients are statistically significant. They also indicated that the GEE models with temporal trend perform better than traditional models.

2.2.3 Geographically Weighted Regression Models

The geographically weighted regression (GWR) technique has been recently developed to allow for local analysis of relationships in multivariate data sets. The GWR technique attempts to capture the local variations by calibrating a multiple regression model that allows different relationships between variables to exist at different points in space. This statistical method incorporates the influence of surrounding observations into coefficient estimates (Fotheringham et al., 2002). Data from observations close to the observation being regressed are weighted greater and assumed to exert more influence on the relationships between variables than those which are further away. It is a technique that extends the traditional regression framework so that the coefficients in the model are specific to a given location rather than being global estimates. This technique allows for a better understanding of variations in the strength of relationships in space.

Several research efforts have been conducted to apply the GWR technique to the development of statistical models. However, these were mostly in the areas of epidemiology and health science, and only very few models have been developed in transportation engineering applications, with none directed towards road safety, or specifically, road safety planning models. In an earlier research (Hadayeghi, 2002), the author of this dissertation investigated how much improvement can be achieved using the GWR model over conventional normal regression analysis. The author also used the GWR technique to explore the relationship between the number of zonal collisions and various explanatory planning variables. However, it was an exploratory study in nature since it assumed a normally distributed error structure for collision data, which is not a correct assumption due to the non-negative and discrete properties of collision counts.


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Of most relevance to the current effort is the work of Nakaya et al. (2005) who proposed the GWPR technique for relating the number of deaths in TAZ to socioeconomic covariates, such as the proportion of elderly people, rate of house-ownership, unemployment rate, and proportion of professional and technical workers in each TAZ. The results indicated that there are significant spatial variations in the relationships between the dependent variable and some of the independent variables. Consequently, the application of traditional global models would yield misleading results.

Zhao et al. (2005) is one of the few studies that applied the GWR technique in transportation engineering applications. The study investigated the spatial variations in the relationships between transit use and potential ridership variables, including socioeconomics and demographics, land use, accessibility and transit supply features. The GWR models developed were compared with a global model estimated with the ordinary least squares method. The results indicated that the GWR models have better predictive power and provide an improved understanding of the spatial variations.

Park and Zhao (2004) applied the GWR technique to the estimation of the AADT. The result indicated that the GWR models are able to better explain the variation in the data and predict AADT with smaller errors than the ordinary linear regression models. Additionally, the GWR technique provided capability for development of spatial models by considering the non-stationary effects of data.

Du and Mulley (2006) used the GWR techniques to examine the relationship between transit accessibility and land value. They concluded that transit accessibility may have a positive effect on land value in some areas, but a negative or no effect on others. The use of GWR allows capturing of such spatially varying relationships while a global model with an average or constant coefficient for land value is deemed inappropriate for this type of analysis.

In summary, previous research reviewed suggest that GWR models perform much better than the global regression models in terms of accuracy. However, all have used GWR techniques that assume a normally distributed error structure in the calibration of regression models. Such an assumption is not optimal for calibrating regression models of count data (such as collision data).


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2.2.4 Models Accounting for Spatial Correlation

Recently, a number of studies have been conducted to provide a regression framework which takes spatial dependency into account. These approaches are generally described as spatial regression models. These models recognize the fact that spatial data, such as collision counts, are not generally independent. Therefore, statistical inference based on the results of conventional normal regression analysis on spatial data may produce biased results.

While such models are generally not regarded as local models (i.e. GWR), they recognize the local nature of spatial data by relaxing the assumption that the error terms for each observation are independent. The output from these models consists of a set of global parameter estimates similar to conventional normal regression. However, spatial relationships are incorporated into the modelling framework through the covariance of the error terms (Fotheringham et al., 2002). In this case, these models can be thought of as “semi-local” as opposed to fully local models. In this dissertation, these semi-local models are called spatial models.

Previous research has dealt with the spatial component of road collision in different ways. Aguero-Valverde and Jovanis (2005) developed spatial models for road collisions for the State of Pennsylvania at the county level based on socioeconomic, transportation infrastructure, traffic exposure, and environmental factors. Using injury and fatal collision data for 1996–2000, full Bayes (FB) hierarchical models with spatial and temporal effects and space–time interactions were compared to traditional NB estimates of annual county-level crash frequency. The results of the study indicated that the model coefficients estimated by FB hierarchical models are generally consistent with the ones based on traditional GLM models. It also revealed that spatial correlation, time trend, and space–time interactions are significant in the FB injury collision models.

Miaou et al. (2003) developed a series of spatial models of collisions at the county level for data from the state of Texas. Poisson-based full hierarchical Bayes models of fatal (K) collisions along with incapacitating (A) and non-incapacitating (B) injury collisions were estimated using both frequency and rate values (using VMT as an offset term). The conditional auto regressive model (CAR) was used to model spatial correlation and MCMC simulation was used to sample the posterior probability distribution.


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Levine et al. (1995) examined the spatial relationship between trip generating activities and motor vehicle collisions in the City and County of Honolulu. A spatial aggregate lag model was developed to examine the zonal relationship of motor vehicle collisions to population, employment and road characteristics. The spatial lag model is defined as:

( ) ε+β+ρ= XYWY ji (2.2)

where

Yi = N * 1 vector of observations on the dependent variable for all locations,

N = number of zones,

W(Yj) = A weighted matrix of N * 1 vector of values for the dependent variable summed over all locations j, where i≠j (the “spatial lag”)

ρ = coefficient of the spatial lag (the spatial autoregressive term),

X = N * K matrix of observations on the explanatory variables,

β = K * 1 vector of regression coefficients, and

ε = N * 1 vector of normally distributed random error terms, with mean 0 and constant variance σ2.

The authors stated that the level of overall predictability of the model is quite reasonable (R2= 0.55). They also suggested that the effect of introducing the spatial lag increases the predictability of the model by 6.7 %. The foregoing model form assumes a linear relationship between collision frequency and the independent variables. However, in the following chapters, we will show that this is not a correct assumption.

2.3 Proactive Safety Programs – a Context for Safety Planning Models

Traditional road safety management programs have been very effective in identifying and treating hazardous locations. However, this reactive approach is expensive and requires a significant collision data history before any action can be taken. Such programs are, of course, too late for those involved in the collisions that generated the data. Also, as highlighted by Lovegrove (2005),


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road safety improvement programs in reaction to existing hazardous locations can only address unacceptably high collision frequencies. Several research studies have argued that the reactive safety management program is perhaps not the best method to address road safety effectively (Washington et al., 2006; Herbel, 2004; de Leur and Sayed, 2003; Roberts, 2001 and Van Schagen and Janssen, 2000). There is a need for a more proactive analytical approach in order to prevent such unsafe situations from arising in the first place.

To address this problem, the “safety planning” process is introduced to complement the traditional or reactive methods.

Safety planning is a proactive approach that incorporates road safety at the planning stage. The purpose is to prevent occurrence of unsafe situations, such as traffic collisions, in the first place. The safety planning strategy addresses a range of activities, including:

programming safety improvements to address roadway “hotspots” or collision-prone locations;

introducing multi-disciplinary programs (i.e. integrating engineering, enforcement and educational activities);

reflecting road safety considerations as a key decision-making parameter in evaluating projects and programming expenditures; and

establishing inherently safe transportation networks or SCP.

However, to establish such a process, a framework is required. de Leur and Sayed (2003) developed a framework to address road safety in the planning process as illustrated in Figure 2-1.


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Figure 2-1 Framework for Proactive Road Safety Planning (de Leur & Sayed, 2003)

They used three elements to explain the safety risk phenomenon (exposure, probability and consequences). These elements, which in turn include several guiding principles, were used to quantify road safety risk. They provided a systematic process for safety inputs into the planning stage. They argued that it would be much more efficient to evaluate safety, detect potential problems, and revise designs proactively as part of the planning process before any construction begins. At the post-planning stage, the planning options could be improved by deploying an optimization process known as Multiple Account Evaluation (MAE) and a RSA process.

This section reviews the state of development of current proactive empirical tools, including RSAs, sustainable road safety programs, and SCP.


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2.3.1 Road Safety Audit

As highlighted by McGill et al. (2005), a RSA is a formal examination of safety performance for existing or future locations of a transportation network by an independent audit team. It estimates and reports on potential road safety issues and identifies opportunities for improvements in safety for all road users. The results of an RSA are brief reports identifying the safety problems and potential solutions.

RSAs are widely recognized as a proactive and low-cost tool to improve safety at each stage in the lifecycle of a transportation facility as shown in Figure 2-2. RSA is a proven road safety engineering tool to provide an explicit, formalized evaluation of road projects of any size (McGill et al., 2005).

Figure 2-2 Application of Road Safety Audits in Road System Lifecycle (McGill et al., 2005)

Conducting RSAs earlier in a road project’s lifecycle (e.g. during planning) results in less implementation costs than being performed later in the process, for instance, during detailed design or construction (McGill et al., 2005). RSA can be used as a tool to advance and support proactive safety programs at the planning level. However, its application is currently undefined within the transportation planning process, as it has been mostly employed in the design process.


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Due to the desire to use RSA as a tool, there is potential for establishing synergy between transportation planning and RSA.

2.3.2 Dutch Sustainable Road Safety Programs

In a review of road safety in the early 1990s, the Dutch government set a series of quantitative targets for road safety to reduce severe collisions through better-integrated community and transportation planning (Wegmen, 1997 and Van Schagen and Janssen, 2000). It was concluded that the targeted numbers cannot be achieved with traditional approaches, even if traditional policies are greatly intensified. Therefore, a new approach was required. In order to do that, a new policy for systematic improvement in road safety, known as a “sustainably safe traffic system”, was introduced by the Institute for Road Safety Research (SWOV). Sustainability was linked with road safety in response to suggestions that the main cause of present road safety problems lies in inadequate community planning, which created a built form that nurtures auto dependence.

The objective of “sustainable safety” was to significantly reduce the probability of collisions in advance and reduce serious injury collisions (Koornstra, 1990). It was found that the key to achieving a sustainably safe road system lies in the systematic and consistent application of three safety principles:

rationalization and deployment of the functional use of the road network with the objective of preventing unintended use of a roadway;

homogenous use of the road network by preventing large differences in vehicle speed, operating characteristics and travel direction; and

development of a predictable road system to prevent uncertainties among road users, thereby improving their overall behaviour.

Basically, the concept of sustainably safe roads can be achieved by removing all functional combinations of roads and making them “mono-functional”, i.e. by creating categories, such as pure through, pure distributor and pure access roads. Based on understanding the functional


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requirements for each road category, a set of planning criteria has been suggested for achieving a sustainable safe traffic network (Van Minnen and Slop, 1994). These criteria are:

creating residential areas as large as possible;

assigning every trip as long as possible to the safest type of roads;

making the length of trip as short as possible;

preventing search behaviour for destinations;

making road types recognizable;

reducing and uniforming design characteristics;

preventing conflicts between oncoming traffic;

preventing conflicts between crossing traffic;

separating different transport modes;

reducing speed where conflicts occur; and

preventing obstacles alongside a road.

Several studies have been conducted to develop safety empirical tools that work with a traditional, four-stage, regional transportation planning model (Poppe, 1995, 1997a, 1997b). However, as explained later in this dissertation, developing a tool that predicts the number of collisions on any road requires some measures of exposure. Initial analysis toward the development of such models revealed a problem in predicting safety on inner roads, that is, local roads that are not modelled inside each TAZ. To address this, Dutch researchers refocused their efforts by looking at ways to infer the level of road safety in a TAZ directly from other proxy indicators, such as the number of inhabitants and jobs, the surface area, and/or overall street pattern. Van Minnen (1999) also studied proxy indicators when trying to quantify the optimal size of a neighbourhood core (i.e. the maximum inner neighbourhood area not bisected by major roads) from a road safety perspective. His recommended proxy variables include: neighbourhood core size, journey length, traffic volumes, choice of route, car speeds, and accessibility.


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2.3.3 The U.S. Safety Conscious Planning Program

In recent years, the consideration and inclusion of road safety in regional transportation planning has emerged as a strategy for improving the overall transportation network safety in the US. This has been accomplished through the introduction of SCP, which is based on the rationale that safety should be considered as an explicit part of the transportation planning process. The vision is that safety-conscious practices will be comprehensively, routinely and effectively incorporated and should receive the same priority often accorded to strategies aimed at increasing mobility of transportation networks (Dumbaugh et al., 2004). This was complemented by introducing a new legislation known as The Safe, Accountable, Flexible, Efficient, Transportation Equity Act A Legacy for Users (SAFETEALU) which institutes a requirement for active, explicit consideration of safety on all public roads as part of the traditional planning process and requires each state to prepare a Strategic Highway Safety Plan.

To address this, the Federal Highway Administration (FHWA) developed an institutional framework for considering safety in the transportation planning process (AECOM Consulting Transportation Group et al., 2000). It was argued that safety can be incorporated into the traditional transportation planning process in many areas. Figure 2-3 depicts the simplified cyclical transportation planning process in the context of safety.


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Figure 2-3 Overview of Transportation Planning Process in Context of Safety (AECOM Consulting Transportation Group et al., 2000)

In addition to the FHWA report, the National Cooperative Highway Research Program initiated a project (NCHRP Project 8-44: Guidebook for Integrating Safety into Transportation Planning and Decision Making). The objective was to develop a guidebook for practitioners that identifies and evaluates alternative ways to more effectively incorporate and integrate safety considerations into long range statewide and metropolitan transportation planning and decision-making processes (Washington et al., 2006). As explained in the following section, a tool named PLANSAFE was developed as part of the research to forecast safety effects at the TAZ level or higher. Further evidence that the relevance of incorporating safety into the planning process and need for tools to do so was recognized by a new project under the National Cooperative Highway Research Program (NCHRP Project 8-44(02): Transportation Safety Planning: Forecasting the Safety Impacts of Socio-Demographic Changes and Safety Investments). The objective, according to the project statement, is to develop analytical tools for effectively forecasting the safety impacts of socioeconomic and demographic changes and safety investments in the transportation planning process. However, development of such tools is still in preliminary stages.


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2.4 Safety Planning Models

Incorporating safety considerations into the transportation planning process in a comprehensive way has emerged as a strategy for improving transportation safety in recent years. Several research efforts have been undertaken recently to forecast the level of road safety. The general idea of such studies is to provide a safety planning decision-support tool, which facilitates a proactive approach to assessing safety implications of alternative network planning initiatives and scenarios. Collision prediction models are used for the development of such tools, increasingly known as “safety planning models”. Most models found in the literature are at a TAZ or macro-level. As highlighted in the first chapter of this dissertation, the reason for using an aggregate approach is to be able to estimate safety implications of various alternative transportation network scenarios in a manner consistent with the use of conventional aggregate travel demand models. This is to evaluate mobility and environmental impacts of such scenarios. Moreover, it is economically feasible and easily understandable to use the aggregated data. This section reviews the results of studies conducted in the past on the development and use of safety planning models, including methodological issues.

Lord (2000) developed a series of collision prediction models that would allow the estimation of traffic collisions on digital networks. The digital networks in this study are defined as a series of nodes (i.e. vertices or points) and links (i.e. arcs or edges) that connect these nodes. The nodes and links represent physical structures of a road network, such as streets, intersections and interchanges. It was argued that the digital representation of physical urban transportation infrastructures is very important in transportation analysis since these networks are used by transportation planning software programs, such as EMME/2 (INRO, 1998). In the second section of this research, the author has developed collision prediction models that predict the number of collisions at nodes and links. Then, developed models are applied to three sample digital networks. The results indicated that it is possible to predict the number of collisions on digital networks, but accuracy is contingent upon precision of transportation planning software programs. The main drawback is that the developed models cannot be used for safety planning at the TAZ level since they are developed for predicting number of collisions in link and node levels. As highlighted by


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Washington et al., (2006), the safety planning models should not be used for corridor or project-level-related assessments and analyses.

Hadayeghi et al. (2003) developed a series of macro-level collision models to estimate the number of total and severe (fatal and non-fatal injury) collisions in planning zones in the City of Toronto as a function of zonal characteristics. The GLM approach was employed in which NB regression models were developed separately for total and severe collisions as a function of socioeconomic/demographic, traffic demand and network data variables. The zonal variables that had significant effects were the number of households, major road kilometers, VKT, intersection density, posted speed limit and volume-capacity ratio. This research pioneered a tool for planners to assess road safety in the planning process. However, due to data limitations and availability issues, some of the important variables were not included.

Other researchers that have worked in the analysis of collisions with consideration of the transportation planning components are Noland and Quddus (2004). In their work, analysis was performed at a “ward” (census track) level. NB models were estimated with dependent variables selected as total fatalities, serious injuries, and slight injuries. The independent variables were classified into four different categories: land use indicator variables (employment and population density), road characteristics, demographic characteristics, and traffic flow proxies (total employment). The main limitation of their study is in the use of proxy variables for traffic flow estimation. In additional research with county-level data, Noland and Oh (2004) estimated the expected number of collisions by using infrastructure characteristics and demographic indicators as independent variables in an NB model.

de Guevara et al. (2004) presented planning-level zonal crash prediction models for Tucson, Arizona. The models were developed separately for fatal, injury and property-damage-only collisions. These models were developed as a non-linear, exponential function, with NB error distribution, using simultaneous equation and log-linear transformation techniques. Geo-statistical tools were used to aggregate road network and socioeconomic geo-coded data for each of the 859 traffic zones. Also for ease of use in the road planning and design process, zonal road data were aggregated according to each of the seven AASHTO criteria (American Association of State


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Highways & Transportation Officials, 2001). All road classes were included in the data, from local roads to state highways. However, roads on zonal boundaries were not assigned due to uncertainty over collision and zonal boundary geo-coding errors. This was estimated to introduce an error of less than 5%. The results of this study indicated that the variables with significant effects on fatal collisions are population density, percentage of the total population with 17 years of age or younger, and intersection density. The significant variables for the injury and property-damage-only collision models were population density, number of employees, intersection density, percentage of miles of principal arterials, minor arterials and urban collectors. However, the major issues with this study are the reliance on population density as a proxy for exposure, and not using a leading exposure variable (e.g. VKT, AADT) in the model form.

Lovegrove (2005) developed a series of macro-level collision prediction models using data from 577 neighbourhoods across the Greater Vancouver Regional District. Several models were presented for total and severe collisions in rural and urban zones, and the resulting models indicated that neighbourhood characteristics, such as traffic volume and demographic characteristics along with network shape and type of transportation demand management strategy that was in place, had a significant effect on collision frequency occurrences. Then, the developed models were used to provide guidelines for use by planners and engineers. Therefore, road safety can be explicitly considered and reliably estimated in all stages of the road planning process with provision of this tool. To test whether the developed models and guidelines are practical and relevant for practitioners, the author demonstrated the use of macro-level collision prediction models in several reactive and proactive case studies. The proactive use guidelines made recommendations on determining the zones of influence, data extraction, and interpretation of results. The author used the GLM procedure with the assumption of NB error distribution for development of the safety planning models. In this technique, fixed coefficients that represent the average relationship between the dependent variable and each explanatory variable are estimated. However, as shown in subsequent chapters of this dissertation, the stationary relationship assumed by the GLM models may ignore some important spatial factors of the number of collisions at a particular TAZ. Consequently, their accuracy for explaining the relationship between the


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dependent variable and explanatory variables may be doubtful since collision frequency is likely to be influenced by many spatially defined factors.

Washington et al. (2006) introduced a series of zonal planning level forecasting models (PLANSAFE) that included the following collision frequency models: total, property-damage-only, fatal, incapacitating injury and fatal, nighttime, pedestrian, injury and one involving bicycles. The models were developed based upon data from the cities of Tucson and Phoenix metropolitan areas, Arizona and some cities in the state of Michigan. The authors also described the procedure of detailed development of the planning level safety prediction models; this process is presented in Figure 2-4.

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Figure 2-4 Process Followed to Develop Planning Level Collision Prediction Models (Washington et al., 2006)


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Similar to the previous safety planning models, Washington et al. (2006) used the GLM procedure with the assumption of NB error distribution for development of models which ignore the spatial characteristics of collision data for a TAZ.

Cheung et al. (2008) developed safety planning models for arterial roads that incorporate characteristics applicable to urban transit planning. The GLM approach with an NB regression error structure was employed using a dataset from the City of Toronto. The safety planning models indicated that VKT, bus or streetcar kilometers traveled, arterial road kilometers, bus stop density, stop location, and average posted speed limit have significant associations with transit-involved collision occurrences. It was argued that the developed models can provide transit agencies with decision-support tools for considering safety implications in the strategic and service planning processes.

An important assertion that was found from the above research efforts is that all of those planning level prediction models were developed using the GLM technique with the assumption of an NB regression error structure. This is because the collision data had been found to display extra variation or overdispersion, and NB regression models were known as the best choice in order to accommodate such large overdispersion effect in the data.

2.5 Temporal and Spatial Transferability

As explained in Chapter One, one of the main objectives of this research study is to examine temporal transferability of safety planning models. A review of the literature on transferability of collision and transportation planning models, with special emphasis on temporal transferability of these models, is presented in this section.

There are two aspects to model transferability; temporal and spatial. The first results from application of a model estimated at one point of time for prediction at another point of time for the same spatial environment. This occurs whenever estimated models are applied in a forecasting framework. The second aspect; spatial transferability, involves the application of a model estimated in a specific geographic region for prediction in a different one for the same time period. Of course, there can be a combination of temporal and spatial transfers.


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Several research efforts have been conducted to address the temporal and spatial transferability of transportation planning models. Most is on trip generation, and disaggregate and aggregate mode choice models (Ben-Akiva, 1981; Koppelman and Wilmot, 1982; Atherton and Ben-Akiva, 1976; Gunn et al., 1985; Gun, 2001; Badoe and Miller, 1995b; Ben-Akiva and Bolduc, 1987 and Parody, 1977). Only a few studies (Persaud et al., 2002 and Turner at al., 2007) have examined the transferability of collision prediction models with none of those efforts focused specifically towards road safety planning models. This is quite understandable due to the fact that only few safety-planning models have been calibrated so far.

2.5.1 Transferability of Transportation Planning Models

Badoe and Miller (1995a) investigated the long-range temporal transferability of disaggregate logit models of work trip mode choices for the GTA between 1964 and 1986. The 1964 estimated models were used to predict the 1986 travel choices. The relationship between model specifications and transferability was tested on different models, ranging from the simplest possible market share models to the more complex market segmentation models. The authors concluded that: 1) the model parameters are not temporally stable, 2) improved model specification improves the extent of the model’s transferability, and 3) the models that best fit the 1964 data do not always transfer well to the 1986 conditions.

Badoe and Miller (1995b) presented methods for updating disaggregate travel choice models in order to examine their temporal transferability from the estimation context (the year and geographical area for which the models were developed) to an application context (differences from the original estimation context geographically, temporally, or both). Four updating procedures, namely Bayesian updating, transfer scaling, combined transfer estimation, and joint context estimation, were applied to a “small” data set of observed travel choice for the application context. The study concluded that updating a model estimated in another context based on a small sample from a new context significantly improves the model’s temporal transferability.

Parody (1977) examined the short temporal stability of logit mode choice models using data collected before and after implementation of changes in a transit system. The author suggested


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that the parameter estimates do not change over time and local models can reasonably predict the future mode choice behavior. A simple level-of-service model was compared in terms of the predictive performance to one which had socioeconomic variables specified in addition to the level of service varieties (fully specified model). The author concluded that the fully specified model performs better than the simple model.

2.5.2 Transferability of Collision Prediction Models

As indicated above, only a few studies have examined the transferability of collision prediction models. Of most relevance to the current effort is that conducted by Oh et al. (2003) who validated intersection collision prediction models intended for use in the IHSDM1 from two perspectives: internal and external validity. Internal model validity examines the strength of the original models in explaining the underlying phenomenon. External validity examines the ability of the models to predict the number of collisions over space and time. The authors recommended that the validation over time can suffer from omitted variables that influence safety, such as poorly measured and surrogate variables, and model functional forms. On the other hand, internal validity and validation over space can also suffer from omitted variables: site selection and countermeasure selection bias, poorly measured and surrogate variables, and misspecification of model functional form.

Also of relevance is work by Persaud et al. (2002a) who investigated the transferability of collision prediction model coefficients in a spatial context by applying models calibrated on Toronto data to the City of Vancouver and State of California using the IHSDM procedure (Harwood et al., 2000). A calibration factor was calculated by dividing the number of collisions in the “new” jurisdiction by the sum of the collisions predicted for the sample by the original model. The authors used cumulative residual (CURE) plots to show how well the recalibrated models fit the data for Vancouver and California. The results were mixed. The authors suggested that a single calibration factor as

1 Interactive Highway Safety Design Model (IHSDM), Federal Highway Administration, US Department of Transportation


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specified in the IHSDM for recalibration of the models over space may not be appropriate and that disaggregation by traffic volume might be preferable.

Turner et al. (2007) used the same methodology as Persaud et al. (2002) to assess the spatial transferability of collision prediction models from the United States to New Zealand where design standards and driver characteristics are different. Based on the results, it was concluded that it is possible to transfer models from one country to the next, but there are a number of differences in collision patterns and definitions. Many factors can bring about these differences, including differences in the definition for intersection-related collisions (i.e. the proximity of collisions to the intersection to be related to its operation of the intersection), intersection design standards and signal configurations/practices, speed limits and operating speeds in each country, and their climatic conditions. There may also be differences in urban form (e.g. new and old world cities) and driver behaviors in terms of driving skills and propensity to take risks.

2.6 Summary and Conclusions

Recent research has stressed the importance of developing sustainably safe road networks in a proactive manner since the traditional reactive road safety improvement programs have some limitations. The reactive safety programs require a significant collision data history, treat a limited set of black spots based on available funding, and involve costly retrofits in existing communities. Furthermore, such programs are certainly too late for those involved in the collisions that generated the data.

There is a need for a more proactive engineering approach to road safety in order to prevent such unsafe situations from arising in the first place. This requires the consideration and inclusion of road safety in the planning process of road networks. However, this is a complex and challenging task. Several studies discussed the general concerns of incorporating safety considerations into the transportation planning process and recommended methods for overcoming such challenges. The most common conclusion has been that the lack of data and suitable macro-level models are considered as major impediments in evaluating road safety during the planning process which


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commonly uses conventional transportation planning modelling systems that are calibrated at an aggregate spatial level, typically the traffic zone.

To address these problems, the development and use of macro-level collision prediction models can provide a safety planning decision-support tool, which facilitates a proactive approach to assessing safety implications of alternative network planning initiatives and scenarios. This research study developed safety planning models using several different statistical techniques, such as the application of GWPR technique for development of the local models, FBSA technique for spatial models in addition to the use of the GLM technique for traditional safety planning models. Consideration of the spatial variations in the relationship between the number of zonal collisions and potential transportation planning predictors in the safety planning models is one major contribution of this study. Also, as part of this research, temporal transferability of safety planning models has been closely investigated and can also be considered as another major task accomplished that was not addressed in any of the previous related studies. The models are at the TAZ levels and require spatial planning data that are obtained and analyzed with the aid of GIS tools.

It is worth mentioning here that some of the variables, such as vehicle characteristics, road conditions, weather information etc. which are believed to be important in the development of collision prediction models, are irrelevant as independent variables for the development of safety planning models. This is because it was hypothesized that if the models were to be useful for strategic urban transportation planning, which is the stated objective of this study, the inclusion of such variables would not be appropriate because of their irrelevance in estimating the safety implications of transportation planning strategies and decisions.

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CCHHAAPPTTEERR TTHHRREEEE

3.0 OVERVIEW OF THE METHODOLOGY

This chapter presents an overview of the methodology used in this research study. As explained in the first chapter, the primary objective of this study is to develop a series of safety planning models for the City of Toronto. The second objective is to examine the temporal transferability of the safety planning models by using appropriate evaluation measures of predictive performance of the transferred models. The relationship between the dependent and independent variables will be assessed to determine if it holds reasonably well across time. The first section of this chapter presents an overview of methodologies and techniques for the development of the safety planning models. The second section is dedicated to an overview of methodologies which are used for examining the temporal transferability and updating of the concerned models.

3.1 Methodology for Development of Safety Planning Models

Over the last few decades, the development of collision prediction models have enabled traffic engineers and road safety researchers to identify important factors related to the occurrence of collisions at intersections, on arterial road sections, highways or transportation networks. For transportation network applications, these models are usually developed at a macro-level,

Use of Advanced Techniques to Estimate Zonal Level Safety Planning Models and Examine Their Temporal Transferability Chapter 3: Overview of the Methodology

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specifically at the TAZ level. The objective of these models is to establish a relationship between collision frequency and a number of explanatory variables, such as traffic volume, road network characteristics, socioeconomic-demographic features, etc.

Several researchers have recently developed macro-level collision prediction models at the planning level to provide empirical tools for planners and engineers to conduct proactive road safety planning (Hadayeghi, 2002; Washington et al., 2006; Lovegrove and Sayed, 2005; de Guevara et al., 2004 and Hadayeghi et al., 2003). The primary objective of such studies is to provide a decision-support tool, which facilitates a proactive approach to assessing road safety implications of alternative network planning initiatives and scenarios.

A common technique used in the previous research, including that by the author for calibration of safety planning models, is the GLM procedure with the assumption of an NB error distribution pattern. In this technique, a constant coefficient for each explanatory variable is calibrated. The model parameters are estimated globally for the entire study area with the implicit assumption that the relationship between the dependent variable and each independent variable does not vary across the geographic area.

However, as the number of collisions vary across a geographic area, it is also likely that the strength of the relationships between collision frequency and independent variables also change across space. Some variables may have a strong explanatory power at certain locations and weak at other locations. To investigate the spatial variations in the relationship between the number of zonal collisions and potential transportation planning predictors, spatial and local modelling techniques need to be developed to investigate important issues related to unmeasured confounding variables, and spatial correlation.

The statistical modelling approaches used in this dissertation include GLM, GWPR, and FBSA models. The GWPR and FBSA techniques are used for examining the local and spatial variation of zonal collisions. The error structures for all the calibrated models are assumed to have Poisson and NB distributions. In the last decade or so, there has been a steady realization of the advantage of using Poisson and NB regression over conventional normal distribution-based regression


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models for modelling road collision frequency. The theory behind these two models will be discussed quite extensively in the following chapters.

The goal of these collision prediction models is to estimate the number of collisions and its association with independent variables, satisfying two conditions. First, they must yield logical results, meaning that they must not lead to a prediction of negative values for the number of collisions. The second condition is that to use generalized linear regression as the modelling procedure, a link function should be used that can linearize this form for the purpose of coefficient estimation. These conditions are satisfied by a model form that consists of the product of powers of the exposure measures multiplied by an exponential incorporating the transportation planning explanatory variables. Such a model can be linearized by a logarithm link function (Sawalha and Sayed, 2001).

However, prior to calibration of such models, a series of steps has to be followed. Figure 3-1 presents the procedure for calibration of the safety planning models. As evident from Figure 3-1, the process consists of four basic steps:

data collection,

data preparation,

model calibration, and

comparison analysis.


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Figure 3-1 Process for Development of Safety Planning Models


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In the first step, data were collected from different sources. For the purpose of this study, the data were obtained from five main sources. The first dataset included data related to traffic collisions within the borders of the City of Toronto in 2001, provided by the Traffic Safety Unit (TSU) of the Transportation Service Department of the City of Toronto in an electronic version. The second dataset consisted of road network data obtained from the University of Toronto Map Library. The third database included land use data obtained from the City of Toronto. The fourth and fifth databases, the 2001 TTS and EMME/2 (INRO, 1998) data bank, were obtained from the Urban Transportation Research and Advancement Centre (UTRAC) at the University of Toronto.

Once all the required data were collected, a database in a matrix form at the TAZ level was prepared. During the data preparation process, MapInfo, a GIS software, was utilized to develop the datasets. A total of 42 explanatory variables that describe the road network, land use, dwelling units, employment types, socioeconomic characteristics and travel demand for 481 TAZs within the City of Toronto were identified as possible explanatory variables for collision occurrence. Once the matrix was prepared, all variables were examined individually to determine whether the possible inclusion of each in the proposed model was sensible or not. As highlighted by Washington et al. (2006), checks must be performed to detect possible flaws in the data and examine their reasonableness. These include computing and assessing the means, medians, modes, and maximum and minimum values of all variables in the database.

With the completion of data extraction and preparation, the next step was to develop safety planning models. A systematic procedure was followed for the development of GLM, GWPR and FSBA safety planning models. As shown in Figure 3-1, this procedure includes four steps, namely explanatory analysis of the data, model form selection, error structure selection, and determining suitable measures of goodness of fit (GOF) for assessing the performance of each model. The procedure of calibration of the safety planning models will be discussed quite extensively in the following chapters.

Upon completion of the model calibration, measures for evaluating the performance of GLM, GWPR and FBSA models were defined. These measures were used to compare the accuracy of GWPR and FBSA models to GLM models, and each other, in predicting the number of collisions


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for each TAZ. The measure used for conducting the comparison analysis will be discussed extensively in the following chapters.

3.2 Methodology for Assessing Temporal Transferability and Updating of Safety Planning Models

As explained previously, collision prediction models are used to predict the number of collisions at intersections, on arterial road sections, highways and transportation networks. For planning applications, these models are aimed at establishing a relationship between collision frequency and a number of explanatory variables, such as traffic volume, road network characteristics and socioeconomic-demographic features. These models enable transportation planners to examine the relationship between the number of collisions and any of the planning variables. The purpose is to assess safety implications of planning options that involve changes in such variables over time. This requires the availability of accurate models that are transferable across time.

The importance of collision prediction models makes it crucial that they be properly calibrated (Persaud et al., 2002a). Hence, the quantity and quality of the data used is very important to the success or failure of these models. Models calibrated with limited data will not perform adequately in terms of predictive accuracy. Therefore, a reasonable dataset is essential in the development of collision prediction models. Moreover, the specification of an appropriate functional model form is another crucial task and this gets further complicated for planning models because of the paucity of these models in the literature.

This aspect of the research has two basic objectives. First, the temporal transferability of the safety planning models is investigated using appropriate evaluating measures of their predictive performance. This is to assess whether the relationship between the dependent and independent variables holds reasonably well across time.

The two temporal contexts are the years 1996 and 2001, with models estimated based on 1996 data being used to predict the number of zonal collisions in 2001. As explained in the previous section, a series of safety planning models based on a 1996 database were calibrated in the


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previous study (Hadayeghi, 2002) to estimate the number of collisions in planning TAZs in the City of Toronto as a function of planning variables. These models were used for examining the temporal transferability of safety planning models in this study.

The second objective of this aspect of the research is to compare alternative updating methods of temporal transfer and estimate effects of the size of the application context data on model transferability. The updating procedures examined are the Bayesian and those based on recalibration factors.

In addition, the transfer bias due to sample size is examined. This issue is not addressed in previous studies which focused mainly on using only one method and sample size. This is important since it is believed a large portion of differences in coefficients and predictions is caused by random variation due to selection of data. Also, the GOF measures resulting from updated models are compared to those estimated using the entire data set in the application context. The methodology and measure used for the examining of temporal transferability and updating the safety planning models will be discussed in more detail in later chapters.

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CCHHAAPPTTEERR FFOOUURR

4.0 DATA

The objective of this chapter is to describe the data used. Section one introduces the sources and provides detailed information for each data set. Section two presents a descriptive analysis of the data, discussing the spatial analysis. Section three examines the changes that occurred in travel characteristics and collision occurrences between 1996 and 2001, the two years used in assessing temporal transferability of the models.

4.1 Data Source

The data used in this study are comprised of collision, street network, land use, traffic demand, socioeconomic and demographic characteristics, dwelling units and employment types for 463 TAZs in 1996 and 481 TAZs in 2001, in the City of Toronto. The TAZ definitions for 1996 and 2001 are not identical.

For the purpose of this research study, the 2001 TAZ definition is adopted for both 1996 and 2001 since only the data in 2001 were used for development of safety planning models. The 1996 data and consequently, the collision prediction models developed from such in a previous study

Use of Advanced Techniques to Estimate Zonal Level Safety Planning Models and Examine Their Temporal Transferability Chapter 4: Data

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(Hadayeghi, 2002), were used only for examining the temporal transferability of safety planning models. Data from 1996 are not used for development of the safety planning models in the current study.

4.1.1 Traffic Collision Data

The collision data were obtained from two main sources. The first dataset, consisting of the 1996 and 2001 collision data for the City of Toronto, was provided electronically by the TSU. The database includes collisions at intersections, non-intersections and other locations, such as private lots. An initial evaluation of the collision data indicated that entries in the database are defined by the person involved in the collision. For instance, if a collision involves three drivers and a pedestrian, each person is coded as a separate entry. Therefore, four different rows would be recorded in the collision database. However, for the purpose of this research, only one row for each collision record containing the worst scenario in terms of severity is selected. The City’s collision data are geocoded, which allows grouping of collisions by TAZ using GIS tools.

A major issue with the City’s collision data is that the coordinate systems assigned to some of the collisions are not accurate. Therefore, these collision data points do not overlay properly on top of the street files. To address this problem, a macro was developed in the GIS to assign such collisions to a corresponding street by matching the street address location, which is obtained from the “street names” column in the collision database, with the “location description” column from the street network file. Moreover, in cases where the collisions occurred on a major road forming the boundary of two TAZs, the data points were assigned to the nearest direction of travel on the major road and consequently, to the corresponding TAZ. It must be noted here that as indicated by Jonsson et al. (2006), even when a road is on a boundary of a TAZ, the match of the zone boundary and road location is not absolutely perfect, since they came from different sources and were therefore, coded separately. For purposes of this study, it is assumed that the geocoding for all GIS files is reasonably accurate. Therefore, aggregation of data can be based on geocoded location in relation to TAZ boundaries. This is not an unreasonable assumption since previous research (de Guevara et al., 2004 and Lovegrove, 2005) used a very similar assumption for assigning collisions and other data near boundaries to TAZs. de Guevara et al. (2004) examined


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the “boundary effect” with respect to assignment of collision data to their respective TAZ boundaries. The authors concluded that the number of collisions involved on boundaries is less than 5%, and does not significantly impact the results of the analysis.

The second dataset, consisting of the 2001 collision data for all of the provincial highways within the City of Toronto, was obtained from the Ontario Ministry of Transportation (MTO). The MTO collision database is assigned to each section of road based on the Linear Highway Referencing System (LHRS). The LHRS is the official referencing system for the MTO and used as the indexing system in the majority of the MTO’s existing databases. The LHRS is used to locate road sections and points on the Ontario highway network. In this system, each highway is divided into consecutive sections that have unique reference numbers assigned in ascending order and an offset value is used to refer to a certain point (e.g., n collision location) along specific highway sections. Zero offsets; the starting point of any section, are at major geographical key points, such as intersections, structures, and railway grade crossings.

Since the collision data from the MTO were not geocoded, their collision database was manually assigned to each road segment by using the LHRS and offset information for each collision. Subsequently, the collision data from the City were merged with that of the Ministry to produce a master collision database.

It is worth mentioning that the major freeways and their related ramp collisions (i.e., Highway 401, Highway 400, Don Valley Parkway, QEW, etc.) are excluded from the 1996 collision data used for development in an earlier study of the safety planning models (Hadayeghi et al., 2002). The reasons are that MTO’s collision data were not available at the time of development and also assumed that these collisions do not really represent the characteristics of the related TAZ. Table 4-1 presents a summary of total and severe collisions for 1996 and 2001 within the City of Toronto.


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Table 4-1 Sum of Collision Data for 1996 and 2001 within the City of Toronto

2001 Collision Data Statistics 1996 Collision

Data* MTO Freeway Data

City Data without Freeways

Freeway Data for the City

Sum

Total Collisions 53,286 8,906 73,043 153 82,102

Severe Collisions 15,818 1,955 16,764 18 18,737 *MTO and City’s freeway collision data are excluded

As evident from Table 4-1, a total of 82,102 traffic collisions occurred in 2001 in the City of Toronto. The percentages of severe collisions (collisions with fatalities and injuries) and Property Damage Only (PDO) are 30% and 70%, respectively. Table 4-2 presents descriptive statistics for collision data for 1996 and 2001 per TAZ.

Table 4-2 Descriptive Statistics of TAZ Collision Data for 1996 and 2001

Category Statistics 1996 2001

Average 115.09 171.04

Minimum 6.00 14.00

Maximum 388.00 880.00

Tota

l Col

lisio

ns

Standard Deviation 63.30 123.44

Average 34.16 39.03

Minimum 0 2.00

Maximum 106.00 192.00

Seve

re C

ollis

ions

Standard Deviation 19.50 27.97

Figure 4-1 presents the histograms of total and severe collisions. The histograms are highly skewed to the left, as expected from count data, suggesting that the Poisson or NB distribution may be appropriate in both cases.


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Figure 4-1 Histograms of Toronto TAZ level Total and Severe Collision Frequencies for Year 2001

4.1.2 Street Network and Land Use Data

The GIS database for the street network and land use data were provided by the City of Toronto. Using GIS, manual aggregations were performed to extract data on intersection types, types of road classifications and road-lane-kilometres. Other street network data, such as the number of schools, total rail kilometers and land use, were obtained from CanMap Street files at the University of Toronto Map Library. Table 4-3 presents descriptive statistics of street network and land use data for 2001 per TAZ.


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Table 4-3 Descriptive Statistics of TAZ Street Network and Land Use Data for 2001

Category Variable Average Minimum Maximum Standard Deviation

Area of Each TAZ (1000 m2) 1,317.88 59.37 8,735.03 937.70

Commercial (1000 m2) 45.03 0.00 1,726.21 109.02

Government & Institutional (1000 m2) 97.10 0. 00 2,829.58 221.89

Residential (1000 m2) 669.53 0.00 2,816.70 535.05

Open Area (1000 m2) 132.17 0.00 6,510.81 435.76

Parks and Recreational (1000 m2) 142.01 0.00 2,278.35 257.98

Resource & Industrial (1000 m2) 220.62 0.00 2,445.53 449.97

Land

Use

Dat

a

Water body (1000 m2) 11.04 0.00 612.93 42.66

Number of Rail Stations 0.20 0.00 4.00 0.51

Total Rail Kilometers 0.43 0.00 4.31 0.72

Number of Schools 2.20 0.00 23.00 1.99

Total Arterial Road Kilometers 2.36 0.00 10.00 1.47

Total Expressway Kilometers 0.74 0.00 9.96 1.62

Total Collector Kilometers 1.42 0.00 7.99 1.38

Total Laneway Kilometers 0.53 0.00 9.47 1.16

Total Local Road Kilometers 7.19 0.00 25.44 4.99

Total Ramp Kilometers 0.51 0.00 10.01 1.16

Number of 4-Legged Signalized Intersections

2.52 0.00 10.00 1.91

Netw

ork D

ata


1.00 0.00 8.00 1.17


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4.1.3 Transportation Tomorrow Survey (TTS) Data

The TTS is a one-day travel survey conducted in the Greater Toronto and surrounding areas in five year intervals starting in 1986. The most recent survey was conducted in 2006. However, only the 1996 and 2001 TTS data are used for the purpose of this study. In this research study, the 1996 TTS data are employed in the transferability investigation. The TTS database is an extensive and detailed travel information base, collected at the household level in the GTA.

The survey is conducted to provide detailed demographic and travel information of GTA household members to local, regional, provincial and transit operating agencies. The data collected from this survey are maintained by the Data Management Group (DMG) within the UTRAC. The data can be used in projects involving transportation planning and travel behavioural modelling.

The 1996 and 2001 TTS data consist of demographic and travel information collected from a 5% and 5.5% random sample, respectively, of households in the survey area. The sample selection for the 2001 TTS was based on Forward Sortation Areas (FSA), which are the first 3 characters of the postal code (DMG, 2003). Since the data represent only a small portion of the actual population in the survey area, an expansion factor was applied to achieve a representative total population (DMG, 2003). For calculating the expansion factor, the Census Canada data were used. The survey statistics for the 2001 TTS are shown in Table 4-4.

Table 4-4 Survey Sample Statistic in the 2001 TTS

Category 2001 TTS

Number of Households in the survey area 2.51 Million

Household records 136,379

Person Records 374,182

Trip Records 817,744

The geographical area covered by the 1996 and 2001 surveys includes Metropolitan Toronto, Region of Durham, Region of York, Region of Peel, Region of Halton, Hamilton-Wentworth Region,


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Region of Niagara, Region of Waterloo, City of Guelph, Region of Wellington, Town of Orangeville, City of Barrie, Simcoe County, Victoria County, City of Peterborough and Peterborough County. The geographical areas are further divided into TAZ units, in which TAZ boundaries are defined by existing street networks, rail lanes and natural barriers used by the individual agencies. This is accomplished by converting the entire street network into a custom map projection using GIS.

While the TTS database covers an immense geographic area, for the purposes of this research study, only the City of Toronto area that consists of 463 TAZs for 1996 and 481 TAZs for 2001 is used. As explained previously, the 2001 TAZ definition is adopted for both 1996 and 2001 in this study.

The TTS database was used in order to extract traffic demand, dwelling unit, and employment type for each TAZ within the City of Toronto. In the following subsections, the data extraction process will be described.

4.1.3.1 Socioeconomic and Demographic Data

Socioeconomic and demographic data were obtained from the 2001 TTS. Table 4-5 presents the list of socioeconomic and demographic data that are considered for this research study, along with descriptive statistics of each variable.


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Table 4-5 Descriptive Statistics of TAZ Socioeconomic and Demographic Data for 2001

Variable Average Minimum Maximum Standard Deviation

Number of Females 2,541.54 0.00 11,194 1,922.15

Number of Males 2,386.20 0.00 105,210 1,804.79

Number of People Not Possessing a Driver’s License 1,945.03 0.00 110,97 1,731.61

Number of People Possessing a Driver’s License 2,986.19 0.00 12,499 2,125.87

Number of People 4,931.60 0.00 21,764 3,720.28

Number of People Younger than 17 Years Old 1,026.86 0.00 5,429 911.87

Number of People Older Than 65 Years Old 685.88 0.00 3,504 542.88

Number of People Not Holding Transit Pass 4,535.12 0.00 1,9812 3,383.22

Number of People Holding Transit Pass 394.47 0.00 3,422 412.67

4.1.3.2 Traffic Demand Data

The traffic demand data include the VKT, and the average of the following: posted speed, 85th percentile operating speed, and Volume over Capacity (V/C) per TAZ. Table 4-6 presents descriptive statistics of traffic demand data for 2001 per TAZ.

Table 4-6 Descriptive Statistics of TAZ Traffic Demand for 2001 Data

Variable Average Minimum Maximum Standard Deviation

Average 85% Operating Speed 37.13 26.65 53.35 5.02

Average Posted Speed 48.80 40.00 67.78 6.53

Vehicle Kilometer Traveled 7,365.64 100.51 6,1321.0 9,112.29

Average Volume over Capacity* 0.49 0.00 1.00 0.17 Note: The results of EMME/2 (INRO, 1998) could produce V/C greater than one. In such cases a value of 1 was assigned since the V/C greater than one is unreasonable.


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All data were obtained by conducting an auto trip assignment in the EMME/2 (INRO, 1998) software package. A brief description of the application of the EMME/2 (INRO, 1998) software is provided in the following section. The input requirements, functionalities and output results will also be explained.

EMME/2 Software

EMME/2 (INRO, 1998) is an interactive graphic, state of the art multimodal urban transportation planning system developed at the Centre of Research on Transportation (CRT) at the University of Montreal (INRO, 1998). It is designed to provide a flexible framework for demand modelling, multimodal network modelling and analysis, and as an evaluation procedure to determine the effectiveness of an existing transportation network.

The network data, network performance functions and trip travel patterns, also known as Origin-Destination (O-D) tables, are three key components of a typical EMME/2 (INRO, 1998) databank.

In other words, the databank is a representation of the transportation infrastructure, economic activities and socioeconomic characteristics of the population in the urban area studied (INRO, 1998). Any type of modelling and analysis will be based on the information contained in the databank.

The transportation network represents transportation infrastructure. It consists of the modes, base network, transit vehicles and lines, and turns. Nodes and links represent the base network corresponding to intersection/transit stops and roads/transit lanes. The node and link in a digital network corresponds to the intersection/transit stops and roads/transit lines in real life infrastructure. For this research study, the GTA digital network, created by UTRAC, is used as the input network.

Another type of input required by EMME/2 (INRO, 1998) is the O-D trip matrix. This basically consists of trips occurring between each pair of TAZs. The O-D matrix was obtained from the TTS database. The O-D matrix was retrieved from the TTS database using cross tabulation because it can produce 2 or 3 dimensional tables, which is especially useful for creating O-D matrices such that the row and column attributes can be the TAZs of origin and destination. The queries were


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performed by the Data Retrieval System (DRS) developed by the DMG, UTRAC. To start the query, the following instructions were entered: year (2001), start time of trip (6:00 – 8:59 a.m.), primary travel mode of trip (auto-drive trip), row variable (2001 GTA TAZ of origin), and column variable (2001 GTA TAZ of destination). By executing the query, the O-D matrix result is automatically generated in an output file that is ready for input to EMME/2 (INRO, 1998). Before proceeding to the trip assignment, the O-D matrix was multiplied by a peak hour proportion factor (0.405) to generate the number of trips occurring in one hour for each O-D pair. This can be done within the EMME/2 (INRO, 1998) platform using the matrix calculation function.

Link performance functions (also known as volume-delay functions) provide estimates of delay on each link given the level of traffic volume. For the network performance function, EMME/2 (INRO, 1998) has a volume delay function (VDF) to represent the link performance. When the link information was entered, the VDF would calculate the travel delay based on the level of volume and the capacity on that given link. In this research, the VDF is obtained from UTRAC.

Trip Assignment

The travel demand on each link can be estimated by running the trip assignment in EMME/2 (INRO, 1998). The road network information, VDFs for each link and O-D matrix created by the DRS query are initially read by EMME/2 (INRO, 1998), serving as the basis for the trip assignment model. By specifying the type of assignment and criteria when the assignment is done and executing the trip assignment function, EMME/2 (INRO, 1998) will choose the route or path for each individual trip. For this study, the deterministic user equilibrium (DUE) assignment is used. For this, EMME/2 (INRO, 1998) searches for routes for all vehicle trips that will minimize the travel time of each trip. The assignment continuously runs until a user-equilibrium condition is obtained, whereby all trips are using as the best possible route and there is no improvement on individual travel time through shifting routes by any user.

Traffic Demand Forecasting

After the trip assignment was completed, EMME/2 (INRO, 1998) generated a list of trip assignment results that include the following: the origin node, destination node, length of link in kilometers,


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VDF of the link, travel time in minutes, number of lanes of each link, volume of travel on the link, capacity of the link, modelled speed in km/hr on the link, and free flow speed in km/hr on the link. The VKT, which is used to represent exposure in this research, is calculated by multiplying the volume and length of the given link. By following the macro that was provided by the UTRAC, each link and its travel demand information were then assigned to a longitude and latitude coordination.

The output data were then imported to the MapInfo GIS software. By running the spatial join function, each item of link travel demand information was assigned to a TAZ based on the given longitude and latitude (x and y coordinates). Then, the data were entered into Microsoft Access. By executing a query function, the required variables of each TAZ were calculated for the City of Toronto. These were: the total VKT average posted speed, average actual speed, and consequently, the average 85% operational speed of each TAZ, and the average V/C.

4.1.3.3 Dwelling Unit and Types of Employments

The data on dwelling unit and employment types were extracted from the 2001 TTS database. By running the SQL query in MapInfo, the dwelling unit and employment types for each TAZ were calculated. Table 4-7 presents the descriptive statistics of data.


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Table 4-7 Descriptive Statistics of Data for TAZ Dwelling Unit and Employment Types

Category Variable Average Minimum Maximum Standard Deviation

Number of Single Houses 2,621.29 0.00 13,273 2,097.01

Number of Apartments 2,106.29 0.00 19,535 2,739.20

Dwell

ing

Units

Number of Townhouses 204.00 0.00 2,765 356.83

Employed in: General Office 304.77 0.00 1,539 246.32

Employed in: Management 1,150.07 0.00 5,867 888.34

Employed in: Sales/Service 513.33 0.00 2,934 426.12

Employed in: Manufacturing /Construction/Trades

509.81 0.00 3,242 518.70

Number of Full time Employees 2,062.65 0.00 8,971 1,523.69

Number of Part time Employees 420.88 0.00 2,351 335.34

Empl

oym

ent T

ypes

Number of Not employed 2445.98 0.00 11,548 1,961.92

4.2 Descriptive Analysis of the 2001 Data

This section presents a descriptive analysis of the 2001 data. The first section discusses the spatial analysis of the collision data, while section 4.2.2 presents the spatial analysis of some of the selected independent variables. The spatial analyses for the rest of independent variables are presented in Appendix A.

4.2.1 Spatial Analysis of Collision Data

Figures 4-2 and 4-3 show the spatial distribution of total and severe collision densities. It is evident from the figures that the densities are quite high for the zones in the Central Business District (CBD) area. A possible reason for such a trend is that the traffic intensity is higher in the downtown area compared to the suburban TAZs. Higher traffic intensity leads to more exposure and more exposure in turn, leads to higher probability of collision occurrence.


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Figure 4-2 Spatial Distributions of Collision Density for Total Collisions in the City of Toronto for 2001

Figure 4-3 Spatial Distributions of Collision Density for Severe Collisions in the City of Toronto for 2001


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Figures 4-4 and 4-5 provide the spatial distributions of the ratio of total and severe collisions to traffic intensity, respectively. It is evident that the risk of involvement in a collision is relatively high for most of the TAZs in the downtown area. Another interesting observation is that the collision risk for most of the TAZs containing the city’s freeway network (indicated by the green lines in Figures 4-4 and 4-5) is quite low. The reason can be explained by the curvilinear relationship between collision frequency and traffic flow (e.g. VKT in this case). It is well known from previous research (Hauer, 1997) that as traffic flow increases after a certain threshold value, the number of collisions starts declining.


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Figure 4-4 Ratio of Total Collisions to VKT in the City of Toronto for 2001

Figure 4-5 Ratio of Severe Collisions to VKT in the City of Toronto for 2001


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4.2.2 Spatial Analysis of Selected Independent Variables

4.2.2.1 Analysis of Socioeconomic and Demographic Data

The population density map for the City of Toronto reveals that downtown Toronto is characterized by a high-density urban form as shown in Figure 4-6. From Figure 4-6, it is evident that the downtown TAZs have a population density of over 15,000 persons per square kilometer. In contrast, some suburban areas in the east and west of Toronto have population densities of less than 1,400 persons per square kilometer. A general pattern observed from Figure 4-6 is that population density declines as the distance from the downtown core increases. The number of observations recorded for each range is reported within parentheses in the legend.

Figure 4-6 Population Density for the City of Toronto in 2001 The spatial analyses for the rest of the socioeconomic and demographic data are presented in Appendix A.


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4.2.2.2 Analysis of Traffic Demand Data

Figure 4-7 provides the spatial distribution of VKT production during the morning peak period, by TAZ, in the City of Toronto. The VKT is the total number of kilometers covered by auto drivers in vehicles for their trips. As explained in the previous chapter, the VKT is obtained by assigning 2001 TTS auto-drive trip data to the EMME/2 (INRO, 1998) GTA network. The VKT of each link is calculated by running a trip assignment for the morning peak period (6:00-8:59 A.M). Then, each link is assigned to a TAZ. The summation of VKT for all links in an assigned TAZ produces the total VKT in that specific TAZ.

From Figure 4-7, it is evident that there is a pattern between the VKT and distance from the CBD. As the distance from the CBD increases, the total VKT per TAZ also increases. Also, from Figure 4-7, it is evident that the VKT values for the TAZs containing the city’s freeway network (indicated by the green lines) are quite high. This is an expected result since freeways generally carry more traffic volume than the city streets.

Figure 4-7 VKT in the City of Toronto in 2001


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The VKT density map for the City of Toronto reveals that the TAZs in the CBD of downtown Toronto is characterized by a high-density VKT as shown in Figure 4-8. This confirms the hypothesis presented in the previous section in which higher traffic intensity in the TAZs within the CBD area causes higher observed collision density.

Figure 4-8 VKT Density in the City of Toronto in 2001

Additional spatial analyses for the traffic data are presented in Appendix A.

4.2.2.3 Analysis of Road Network Data

Figures 4-9 and 4-10 present the signalized intersection and road densities in the City of Toronto for 2001. The figures show that the densities of the road and signalized intersections are quite high in downtown Toronto. The reason is the higher compact nature of blocks in downtown Toronto and some outlying areas. In addition, the TAZs in the central area have less green space and more roads compared to outlying areas. It is evident from the figures that some TAZs have densities of over 50 signalized intersections and over 26 kilometers per square kilometer. The higher number collisions can be expected since the densities of these variables are higher in the downtown area.


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Figure 4-9 Signalized Intersection Density in the City of Toronto in 2001

Figure 4-10 Road Density in the City of Toronto in 2001


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4.3 Comparison of 1996 to 2001 Data

Table 4-8 presents the magnitude of changes in various characteristics in the City of Toronto between 1996 and 2001. It is worth mentioning here that some of the variables that were used for development of the 2001 safety planning models were not available when the 1996 safety planning models were developed in the previous study (Hadayeghi, 2002). For comparing 1996 to 2001 conditions, only variables available in the 1996 data were used.

Table 4-8 Changes in Urban Structure, Travel Behaviour and Socioeconomic and Demographic between 1996 and 2001 Data in the City of Toronto

Attribute for Independent Variables* 1996 2001 Difference Percent

Change

VKT 1,728,775 2,089,257 360,482 17.25%

Average V/C 0.22 0.31 0.09 30.18%

Average Posted Speed 42.08 42.48 0.40 0.95%

Total Population 2,305,600 2,368,700 63,100 2.66%

Number of Households 908,499 943,252 34,753 3.68%

Number of Part-time Employees 181,224 187,890 6,666 3.55%

Number of Full-time Employees 875,225 943,373 68,148 7.22%

Total Number of Employees 1,056,449 1,131,263 74,814 6.61%

Total Minor Road Kilometer 4,172.66 4,406.24 233.58 5.30%

Total Major Road Kilometer 859.83 900.73 4.90 0.54%

Total Number of Intersections 15,498 16,649 1,151 6.91% * The independent variables were used in the 1996 safety planning models in the previous study (Hadayeghi, 2002)

As evident from Table 4-8, the number of households has grown by 3.68% from 1996 to 2001. The population of the City of Toronto has increased by 2.7% in the 5-year period, while the total population of the GTA increased by 9.3% (DMG, 2001). The difference in growth percentage


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between the GTA and City of Toronto is indicative of the reality that more people live in the suburbs in 2001 as a result of urban sprawl. During this period, the total employment in the City of Toronto has increased by more than 7%.

Total VKT per person has increased by 15%, from 0.75 in 1996 to 0.88 in 2001. However, there is little change in total major road kilometers in the same period, suggesting that traffic conditions may have worsened. This observation is confirmed by the V/C ratios in Table 4-8, which have increased by almost 30% from 1996 to 2001. The average percentage changes in zonal traffic volume from 1996 to 2001 in the City of Toronto, based on the 1996 zonal definitions, are presented in Figure 4-11. As seen, most of the TAZs experience an increase in traffic volume.

Figure 4-11 Percentage Change in Average Traffic Volume from 1996 to 2001 in the City of Toronto Based on 1996 TAZ Definitions

Table 4-9 presents the magnitude of changes in occurrence of collisions in the City of Toronto between 1996 and 2001. As highlighted in the previous sections, the major freeways and MTO collision data were not used for development of the 1996 safety planning models (Hadayeghi,


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2002). Therefore, for comparative purposes, MTO and major freeway collision data are excluded and not shown in Table 4-9.

Table 4-9 Changes in Number of Collisions between 1996 and 2001 Data in the City of Toronto

Attribute for Dependent Variables* 1996 2001 Difference Percent Change

Total Collisions 53,286 73,043 19,757 24.60%

Severe (Fatal & Injury) Collisions 15,818 16,764 946 2.10%

Total Collisions, Morning Peak Period 5,317 8,691 3,374 33.39%

Severe Collisions, Morning Peak Period 1,664 1,971 307 13.29% * The dependent variables were used in the 1996 safety planning models in the previous study (Hadayeghi, 2002)

Table 4-9 also shows that the total number of collisions has increased by 27% from 1996 to 2001, an average annual increase of 5.41%. On the other hand, the number of severe (i.e., fatal plus injury) collisions has increased only by 5.64% in the entire 5-year period.

Figures 4-12 and 4-13 present the zonal percentage change in total and severe number of collisions for the City of Toronto, respectively. As evident from the figures, most of TAZs outside of downtown have experienced a higher percentage change in total number of collisions than the downtown TAZs.


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Figure 4-12 Percentage Change in Total Number of Collisions from 1996 to 2001 in the City of Toronto Based on 1996 TAZ Definitions

Figure 4-13 Percentage Change in Severe Number of Collisions from 1996 to 2001 in the City of Toronto Based on 1996 TAZ Definitions

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CCHHAAPPTTEERR FFIIVVEE

5.0 DEVELOPMENT OF SAFETY PLANNING MODELS USING THE

GLM APPROACH

The objective of this section is to document the development of a series of safety planning models using the traditional GLM approach. The first section discusses statistical properties of collision data. In the second section, the conceptual framework for the development of the GLMs is introduced. Then, the results of developed safety planning models are presented, followed by a sensitivity analysis for each explanatory variable in the GLMs. At the end, a detailed discussion of the results is provided.

It must be mentioned that in developing collision prediction models, the interest is in predicting the frequency of collisions over some period of time (e.g. one year for this particular research). The number of collisions per year serves as the dependent variable for all of the developed models in this study.

Use of Advanced Techniques to Estimate Zonal Level Safety Planning Models and Examine Their Temporal Transferability Chapter 5: Development of Safety Planning Models Using the GLM Approach

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5.1 Statistical Properties of Collision Data

Road collisions are rare and random events. By rare, it is meant that the frequency of their occurrence is very low in comparison to traffic volume; by random, it is meant that their occurrence is very uncertain over time and space. A given location may experience a large number of collisions during a specified period of time, but may not observe the same number of collisions during the same time interval in the future. This implies that any prediction made regarding the expected number of collisions at a given location has some associated uncertainty and can be expected to be true within some specified confidence limits.

In the absence of any other reliable method, statistical models have been extensively used as a useful tool for the prediction of collision frequency. Collision frequency can be modelled by various statistical methods, including conventional linear regression with normal error structure or GLM with Poisson or NB error structure. However, Jovanis and Chang (1986) found that the three properties of collision occurrence, when related to traffic volume, cannot be modelled using multiple linear regression. They argued that as the Vehicle Mile Traveled (VMT) increases, so does the variance of collision frequency. This result clearly violates the homoscedasticity assumption of linear regression, which assumes the error term has equal variance for the entire range of the predictor variables. The consequences are that any hypothesis test concerning the significance of parameters is invalid because confidence intervals are erroneous. They concluded that if the objective of a study is to determine the influence of particular predictor variables on collision occurrence, the failure to properly test for parameter significance is a serious flaw. The second problem deals with non-negativity of collision frequency. Linear regression is not restrained from predicting negative collision frequency. This property of conventional linear regression leads to meaningless models, especially when a substantial proportion of the locations in a calibration database has experienced no collisions at all during the study period. The third problem with using conventional linear regression is that the error terms of discrete dependent variables (collision frequency) are small. Small integer counts, often zero or close to zero, do not typically follow a normal distribution. In fact, the Poisson and NB distributions are often more appropriate for discrete counts of events that are likely to be zero or a small positive integer during a given time period.


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5.2 Calibration of GLM Collision Prediction Models

With completion of data extraction and preparation, the next step is to develop safety planning models. For the development of GLM models, the following procedures are followed:

selection of an appropriate model form,

selection of an appropriate error structure,

selection of candidate explanatory variables by conducting an exploratory analysis, and

selection of measures of the GOF.

5.2.1 Model Form

Generally, collision prediction models are used to explore the relationships between the number of collisions and traffic intensity, and road characteristics. For this research, safety planning models are calibrated to explain the relationship between number of collisions in a TAZ and traffic intensity, road network, land use, socioeconomic and demographic characteristics of the TAZ.

Following a review of various model forms used in the literature (Sawalha and Sayed, 2001; Hadayeghi et al., 2003 and de Geuvara et al., 2004) and application of a variety of model forms to the data in an exploratory analysis, the following model form is used for the development of GLM models:

( ) i

n

ii XVKTYE ∑

=

β β∗∗β=1

0 exp)( 1 (5.1)

E(Y) = predicted collision frequency per TAZ,

VKT = vehicle kilometers travelled,

Xj = jth explanatory variables (j=2,…..,p), and

βj=jth model parameters (j=0,…..,p).


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The GLM version of Equation 5.1 can be written as follows:

( )[ ] pp XXXVKTYELn β++β+β+β+β= ......)ln()ln( 332210 (5.2)

The model form in Equation 5.1 is consistent with the non-linear relationship between collisions and traffic intensity (measured in this case by VKT) which has been suggested by other researchers. Also, the model in Equation 5.1 assumes a linear relationship between collisions and other independent variables. Such an assumption prevents the model from predicting zero collision for a TAZ where some of these independent variables might be zero. The model uses the linear regression model with logarithmic transformation of the dependent variable as shown in Equation 5.2.

5.2.2 Error Structure for Collision Prediction Models

The GLM approach employed assumes that the error structure for developed collision models is either Poisson or NB. The following subsections provide detailed theoretical basis for each error structure.

5.2.2.1 Poisson Distribution

The Poisson distribution is an adequate model for collisions since they are random, discrete events that are sporadic in nature and the frequency is non-negative. In collision models with a Poisson distribution, it is assumed that the dependent variable Y, the number of collisions in each TAZ, has a Poisson distribution given the independent variables Xi1, Xi2,..., Xip,

!

),...,,( 21i

yi

ipiiii ye

XXXyYPii μμ−

== i=0, 1, 2, ......, (5.3)

where

P (Y=y| Xi1, Xi2,..., Xip)= the probability of Y collisions occurring in a TAZ,

yi = the number of collisions occurring in the ith TAZ during a time period, and


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μi = the expected number of collisions during a time period.

The log of the mean μ is assumed to be a linear function of the independent variables. That is,

( ) ippiii XXXLn ×++×+×+= ββββμ ......22110 (5.4)

which implies that μ is the exponential function of independent variables,

)......exp( 22110 ippiii XXX ×++×+×+= ββββμ (5.5)

The model coefficients, β0, β1, β2,……., βp are estimated by the maximum likelihood method

(Lawless, 1987). The likelihood function is:

∏=

−

=n

i i

ii y

eL

1 !)(

μμμ (5.6)

The parameters are estimated by maximizing the likelihood, or more commonly, the logarithm of the likelihood (denoted by the log likelihood). Equivalently, the estimation can be done by minimizing the negative of the log likelihood. The log likelihood is given by the equation:

( ) ( ) ( )[ ]∑ −μ−μ×=μ !iiiii yLnLnyLn (5.7)

The Poisson distribution has been shown to be reasonable for modelling collision data (Miaou, 1994). However, there is a limitation as the variance of the data is restrained to be equal to the

mean, E(Yi) = Var(Yi)= μi. In many applications, collision data were found to display extra variation

or overdispersion relative to a model with Poisson distribution. That is, the variance of the data is greater than indicated by the model with Poisson distribution. The extra variation can come from several possible sources, such as omitted variables, uncertainty in exposure data, and non-homogenous highway environment (Miaou and Lum, 1993). Due to the overdispersion in models with Poisson distribution, the variance of the estimated parameters tends to be underestimated, and the estimated coefficients themselves are also biased. This problem can be resolved by assuming that collision data are distributed by NB (Miaou, 1994).


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5.2.2.2 Negative Binomial Distribution

As mentioned above, a limitation of the Poisson distribution is that the mean equals the variance of the distribution. The NB, also known as the Poisson-Gamma distribution, is used for calibration of collision models in this study and has the following form:

)1()1(

)1()(

α

α

Γ+Γ

+Γ==

i

i

ii

y

yyYP iy

i

i

i

)1()

11(

1

αμαμ

αμα

++ yi =1, 2, 3,… (5.8)

where

)( ii YE=μ i=1, 2, 3,…, n (5.9)

and the variance of Yi is

2)](1)[()( iiiii YEYEYVar αμ+μ=×α+= (5.10)

where α ≥ 0 is usually referred as the overdispersion parameter. From Equation 5.10, one can see

that this model allows the variance to exceed the mean. In other words, the NB allows for extra

variation. If α goes to zero, the NB distribution is reduced to the Poisson distribution.

The coefficients, β0, β1, β2,……., βp and the dispersion parameter α, are estimated by the method

of maximum likelihood (Lawless, 1987). The likelihood function is:

1

)11()

1(!)()(

),(1

1

1−

++Γ+Γ

=∏=

−

−α

αμαμαμ

αα

αμi

y

i

in

i i

ii

i

yy

L (5.11)

The estimation of the model parameters can be done by minimizing the negative of the log likelihood. For the NB distribution, the log likelihood is given by the equation:

))11ln()

1ln()

!)()(

(ln(),ln( 1

11

1

ii

ii

n

i i

ii y

yy

αμ+α+

αμ+αμ

+αΓ

α+Γ=αμ −

=−

−

∑ (5.12)


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5.2.3 Exploratory Analysis

5.2.3.1 Parameter Estimation in the GLM Models

The PROCGENMOD procedure in SAS (SAS Institute Inc.) was used for estimating the independent parameters of the GLM models. The PROCGENMOD procedure fits a generalized linear model to the data by maximum likelihood estimation of regression parameters. The parameters are estimated numerically through an iterative fitting process. The value of dispersion parameter for the models with NB distribution is also estimated by using a maximum likelihood method. The theoretical background is as follows:

This method provides the maximum likelihood estimates of β and α. These are the values of β and

α that maximize the log-likelihood function l (β, α) given by:

( ) ( ) ( )∑ = ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ α+−⎟

⎠⎞

⎜⎝⎛

α+α−−αΓ−α+Γ=αβ

n

ii

ii

ii yLny

yLnyLnLnyLnl

111!),( (5.13)

The log-likelihood function depends on β through the terms ŷi = Ê (Yi ), which represent the model predictions or fitted values. The model form most commonly used in collision modelling given in Equation 5.1 specifies the fitted values as:

( )β′= ii Xy exp) (5.14)

where iX ′ is the vector of explanatory variables corresponding to the ith observation. The method

of maximum likelihood obtains β and α as the solution to the following two equations:

0=β∂∂l and 0=

α∂∂l (5.15)

Built-in link functions that accommodate a variety of probability distributions and associated variance functions are available in the PROCGENMOD procedure.


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It is worth mentioning here that the parameters in GLM models are estimated globally and do not change spatially with location within the calibration database. This approach ignores the fact that the data used for calibration of these models are collected from different points in space. The assumption of these models is that all TAZs for the study area can be accurately presented universally by a single variable even though the influence of some independent variables on the dependent variable may vary spatially. In Chapters Six and Seven, the spatial variations in the relationship between the dependent variable and the independent variables will be examined.

5.2.3.2 Selection of Explanatory Variables

The procedure used in this study for the selection of explanatory variables in GLM safety planning models is a forward procedure. In this procedure, a simple model with only an intercept term is used as a starting point and the explanatory variables added to the model one by one. Prior to incorporating variables into the basic model, a correlation matrix was set up to examine whether variables of interest were highly correlated with other variables or not. If two explanatory variables were substantially correlated, inserting them simultaneously into the same model was avoided. Also, the correlation matrix was used for selection of the most important variable in the model. For this study, the VKT is found to be the most important variable since it has the highest correlation with collision data.

The decision to keep a variable in the model was based primarily on two criteria. The first is whether the chi-square of the variable’s estimated coefficient is significant at the 95% confidence level. The second criterion is based on the improvement of the GOF of the model that includes the concerned variable. In cases where a variable satisfies these two criteria, but has a clearly counter-intuitive sign (e.g. flow variable having a negative sign), the concerned variable was omitted.

As will be explained in the subsequent section, two measures of GOFs, namely the mean Pearson chi-square and mean scaled deviance (SD), are used. These values should be within a range of 0.8 to 1.2, which indicates that the selected model form and variables included in the models are appropriate. Therefore, in this study, as long as the addition of a new variable in the model does


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not force the GOF measures outside the acceptable range, it is kept even if the goodness of measures is not significantly improved.

This variable selection procedure may not be entirely compatible with the principle of parsimony which calls for explaining as much of the variability of the data using the least number of explanatory variables based on improvement of GOF measures. However, it should be recognized that the safety planning models developed in this study are multi-purpose, intended for assessing the differential effects of transportation planning variables on the safety of a TAZ. As such, the focus is on short term policy analysis applications rather than long range forecasting for which the principle of parsimony is an essential component.

5.2.4 Measures of Goodness of Fit

GOF measures are used to evaluate accuracy of the selected model. However, the statistic of measures of GOF is not straightforward in terms of which GOF to use for the models with an NB error structure. Therefore, it should not be used as an absolute measure of model significance. Three statistical measures, the mean Pearson chi-square, mean SD, and the overdispersion

parameter (α) were used jointly to assess the fit of each model developed.

5.2.4.1 Mean Pearson Chi-Square

The Pearson chi-square is defined as:

∑=

−=χ

n

i i

ii

YVarYEY

Pearson1

22

)()]([ (5.16)

DFPearsonPearsonmean

22 χχ = (5.17)

where

Yi = observed number of collisions at each TAZ,

E(Yi) = predicted number of collisions at each TAZ,


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Var (Yi ) = variance of the collision frequency for each TAZ, and

DF = degree of freedom.

5.2.4.2 The Scaled Deviance

The SD is the likelihood ratio test statistic estimated as twice the difference between the maximized log-likelihoods of the studied models and the full or saturated model. The full model has as many parameters as observations, so that the model fits the data perfectly. It represents the maximum log-likelihood achievable for the given data. Therefore, the full model provides a baseline for assessing the GOF of an intermediate model with p parameters. For the NB error structure, the SD is:

∑= ⎥

⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛κ+

κ+κ+−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

n

i i

ii

i

ii YE

yy

YEy

ySD1 )(

ln)()(

ln2 (5.18)

DFSDSDmean = (5.19)

For these and all other descriptive statistics, 95% is the desired level of confidence used to assess GOF. The mean Pearson chi-square and mean SD values within an acceptable range of 0.8 to 1.2 would indicate that the selected model form is appropriate. The reason is that if the statistical

model is correct, then both SD and mean Pearson chi-square are asymptotically distributed as χ2

statistics with n-p degrees of freedom; where n is the number of subjects and p is the number of fitted parameters. Large ratio values may indicate model misspecification or an over-dispersed response variable; ratios less than one may also indicate model misspecification or an under-dispersed response variable. A consequence of such dispersion issues is that standard errors are incorrectly estimated, implying an invalid chi-square test statistic. Thus, if the regression model is adequate, the expected value of both the SD and mean Pearson chi-square is equal (or close) to

n-p (both the SD ≈ 1 and the mean Pearson chi-square ≈1), otherwise there can be doubt about

the validity of the model (Pedan).


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5.2.4.3 Cumulative Residuals (CURE) Plot

A descriptive method for evaluating the adequacy of a model in performing reasonable predictions across the range of the data is the method of CURE proposed by Hauer and Bamfo (1997). Once model parameters are estimated and initial assessment of the parameters (i.e. adequacy of the parameter sign and magnitude) is performed, it is desired to visually confirm such adequacy. This is to ensure a tolerable fit of the model for the full range of the calibration data. This method has a simple and comprehensive structure, yet is very powerful in performing such visual evaluation. The goal is to graphically observe how well the function fits the data. As highlighted by Lord and Persaud (2000), the CURE method has the advantage of not being dependent on the number of observations, unlike many other traditional procedures in statistical model evaluation. However, it must be noted here that traffic intensity is the only variable for which the cumulative residuals are examined in this study.

The CURE method is an examination of residuals after regression coefficients are estimated. The residual is defined as the difference between the observed and predicted number of collisions. Using cumulative residuals can provide potentially important information in the patterns when the usual plot of residuals does not show any systematic drift. It can be used for two purposes. The first examines whether the chosen functional form fits an explanatory variable along the entire range of its values represented in the data. The second ascertains whether an explanatory variable should be introduced in the model (Hauer and Bamfo, 1997).

5.3 Results of Safety Planning Models

Several collision prediction models were developed with the GLM technique to explore the relationship between collision frequency and zonal characteristics, such as traffic intensity, road network, land use and socioeconomic and demographic characteristics. For each model, the exposure variable (i.e., VKT) was initially considered due to its dominating prediction influence on collision data. Then, additional candidate variables were analytically, with the aid of results obtained from developed correlation matrix, selected from the lists of explanatory variables.


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Separate safety planning models were developed for each planning category (different categories of the planning variables were introduced in Chapter Four) and for combinations of categories in order to ensure applicability according to the availability of planning data.

In this section, the results of safety planning models and their performance are summarized and presented, along with a sensitivity analysis for each explanatory variable. These results have been published in Hadayeghi et al. (2007b).

5.3.1 GLM Model with Traffic Intensity as Independent Variable (Model T1 for Total Collisions and Model S1 for Severe Collisions)

The regression results of the traffic intensity-based models for total (T1) and severe (S1) collisions are presented in Table 5-1.

Table 5-1 Regression Results Using GLM Approach for Traffic Intensity Based Models

Model Number Parameter

T1 S1

ln (β0) 2.5148

(0.228, 121.3) 2.3092

(0.196, 138.6)

ln (VKT) 0.3068

(0.027, 129.6) 0.2431

(0.024, 101.2)

α 0.3401 0.232

Mean Scaled Deviance 1.0598 1.0526

Mean Pearson Chi-Square 0.9875 1.1704 (Standard Error, Chi-Square) ln (β0) = Natural Logarithm of Intercept ln(VKT ) = Natural Logarithm of VKT

As shown, the relationship between collisions and VKT is highly significant at the 95% confidence level. The positive coefficient of the VKT in the model indicates that as the VKT increases, so does


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the number of collisions. The explanation is that as the total kilometers travelled increase within a TAZ, the exposure to potential collisions as well as the number of conflicts also increases.

The dispersion parameters for both models (total and severe) are significantly different from 0 and in the range of 0.35. The Pearson chi-square and SD for both models are within the acceptable range of 0.8-1.2, another indication that the choice of the NB error structure appears to be appropriate.

Figure 5-1 illustrates the cumulative residual plots for the covariate VKT for the total and severe collision models. These figures show that the severe collision model fits the data with better accuracy than the model for total collisions. However, the cumulative residuals for both models

oscillate around the value of 0 and lie between the two standard deviation (±2σ*) boundaries.

Therefore, it can be concluded that the assumption of a non-linear relationship between collisions and traffic intensity is appropriate.

a) Total b) Severe

Figure 5-1 CURE Plot for Traffic Intensity Models Based on GLM Approach

Other traffic variables, such as average for zonal posted speed, 85% operational speed and V/C, were found to be insignificant contributors in occurrence of total and severe collisions. Also, the results of correlation analysis, which are presented in

Table 5-2, indicate pairs of variables that are so correlated that they should not be used in the same model. V/C and VKT, with a correlation coefficient of 0.69, constitute an example of such a variable pair.


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Table 5-2 Correlation Matrix for Collision Data and Traffic Density Variables

Total Collisions

Severe Collisions

Average 85% Operational

Speed Average

Posted Speed Average

VKT Average

V/C

Total Collisions 1 --- --- --- --- ---

Severe Collisions 0.937653 1 --- --- --- ---

Average 85% Operational Speed 0.257769 0.264646 1 --- --- ---

Average Posted Speed 0.365273 0.362334 0.870977 1 --- ---

Average VKT 0.481349 0.439664 0.51419 0.711924 1 ---

Average V/C 0.059365 0.040859 -0.39624 -0.02539 0.690089 1

5.3.2 GLM Models with Network Characteristics as Independent Variables (Models T2 to T8 for Total Collisions and Models S2 to S8 for Severe Collisions)

Several models were developed using the TAZ road network variables. These network-based models rely primarily on explanatory variables describing the TAZ road network. The regression results of the traffic network characteristic models for total collisions are presented in Tables 5-3 and 5-4. The results for severe collisions are presented in Tables 5-5 and 5-6.


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Table 5-3 Regression Results Using GLM Approach for Network Characteristics Based Models, Models T2-T4

Model Numbers Parameters

T2 T3 T4

ln (β0) 3.4844

(0.259, 181.7)

2.7441

(0.221, 154.5)

2.0913

(0.20, 109,9)

ln (VKT) 0.1111

(0.032, 12.5)

0.2357

(0.028, 73.2)

0.2964

(0.023, 162.7)

Total Arterial Road Kilometers 0.1951

(0.019, 110.4) --- ---

Total Expressway Kilometers 0.0868

(0.23, 14.4) --- ---

Total Collector Kilometers 0.0599

(0.019, 10.4) --- ---

Total Laneway Kilometers 0.0778

(0.022, 12.8) --- ---

Total Local Road Kilometers -0.009

(0.005, 3.1) --- ---

Total Ramp Kilometers 0.0759

(0.033, 5.3) --- ---

Total Road Kilometers --- 0.0276

(0.004, 48.4) ---

Number of 4-Legged Signalized Intersections --- --- 0.1185

(0.012, 91.4)


(0.0202, 64.9)

α 0.2459 0.3094 0.2491

Mean Scaled Deviance 1.059 1.577 1.0509

Mean Pearson Chi-Square 1.0364 0.9966 1.2833


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Table 5-4 Regression Results Using GLM Approach for Network Characteristics Based Models, Models T5-T8


T5 T6 T7 T8

ln (β0) 2.1244

(0.198, 114.8) 2.3052

(0.196, 138.1) 2.2378

(0.198, 128.4) 2.3092

(0.196, 138.6)

ln (VKT) 0.2923

(0.023, 160.4) 0.2425

(0.024, 100.7) 0.2479

(0.024, 104.9) 0.2431

(0.024, 101.2)


(0.004, 34.6 0.0176

(0.004, 22.7) 0.0204

(0.004, 34.3)

Number of 4-Legged Signalized Intersections ---

0.1049 (0.012, 71.7)

0.1033 (0.012, 70.2)

0.1059 (0.012, 72.4)


0.1653 (0.020, 71.8)

0.1658 (0.019, 73.3)

0.1665 (0.020, 72.5)

Total Number of Signalized Intersections

0.1316 (0.010, 168.4)

--- --- ---

Total Rail Kilometers --- --- --- -0.0261

(0.0316, 0.7)

Number of Schools --- --- 0.0279

(0.012, 5.3) ---

α 0.2507 0.2323 0.2297 0.232

Mean Scaled Deviance 1.049 1.0505 1.0522 1.0526

Mean Pearson Chi-Square 1.2873 1.1709 1.1747 1.1704 (Standard Error, Chi-Square) ln (β0) = Natural Logarithm of Intercept ln(VKT ) = Natural Logarithm of VKT


- 86 -

Table 5-5 Regression Results Using GLM Approach for Network Characteristics Based Models, Models S2-S4


S2 S3 S4

ln (β0) 2.1495

(0.266, 65.4)

1.4599

(0.231, 40.0)

0.7614

(0.210, 13.1)

ln (VKT) 0.0864

(0.032, 7.1)

0.2055

(0.029, 50.9)

0.2756

(0.024, 127.1)

Total Arterial Road Kilometers 0.2234

(0.019, 135.7) --- ---

Total Expressway Kilometers 0.115

(0.023, 24.1) --- ---

Total Collector Kilometers 0.0506

(0.019, 7.1) --- ---

Total Laneway Kilometers 0.0667

(0.022, 9.1) --- ---

Total Local Road Kilometers -0.0061

(0.005, 1.29) --- ---

Total Ramp Kilometers 0.0226

(0.033, 0.48) --- ---


(0.004, 60.4) ---


(0.013, 97.2)


(0.021, 60.9)

α 0.2404 0.3129 0.2516




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Table 5-6 Regression Results Using GLM Approach for Network Characteristics Based Models, Models S5-S8


S5 S6 S7 S8

ln (β0) 0.7841

(0.210, 14.0) 1.0134

(0.206, 24.2) 0.9168

(0.209, 19.3) 1.0157

(0.206, 24.3)

ln (VKT) 0.2728

(0.024, 125.7) 0.2147

(0.025, 71.7) 0.2232

(0.025, 76.9) 0.2161

(0.025, 72.6)


(0.004, 39.48) 0.020

(0.004, 24.3) 0.023

(0.004, 39.0)


0.1105 (0.013, 72.1)

0.1089 (0.013, 71.0)

0.1119 (0.013, 73.5)


0.1648 (0.020, 66.11)

0.1653 (0.020, 67.74)

0.1671 (0.020, 67.6)


0.1394 (0.011, 172.2)

--- --- ---

Total Rail Kilometers --- --- --- -0.041

(0.033, 1.6)

Number of Schools --- --- 0.0336

(0.013, 6.4) ---

α 0.2527 0.2299 0.2265 0.2291

Mean Scaled Deviance 1.0565 1.0594 1.0605 1.0614

Mean Pearson Chi-Square 1.2113 1.0925 1.0872 1.0891 (Standard Error, Chi-Square) ln (β0) = Natural Logarithm of Intercept ln(VKT ) = Natural Logarithm of VKT


- 88 -

The models reveal that collision frequency increases with the increase in the following explanatory variables: total arterial road, collector, laneway, ramp, and road kilometers; number of signalized, 4-legged, and 3-legged signalized intersections; and number of schools in each TAZ. However, the total rail kilometers and total local road kilometers are found to be inversely related to collision frequency. A similar inverse relationship between the total local road kilometers and collision frequency was found by Lovegrove and Sayed (2005). This may be explained by the fact that the operating speeds are usually lower for the local roads.

The dispersion parameters for all of the models are significantly different from 0, again confirming the appropriateness of NB regression. The GOF statistics show Pearson chi-square and SD values close to 1, again indicating that the models fit the data reasonably well. The CURE plots for the covariate VKT for the total and severe safety planning models are shown in Appendix B.

5.3.3 GLM Models with Land Use, Dwelling Unit and Employment Characteristics as Independent Variables (Models T9 to T11 for Total Collisions and Models S9 to S10 for Severe Collisions)

In this section, the relationship between collision frequency and land use, dwelling unit and employment is explored. This type of analysis is beneficial to urban planners and other analysts who deal with issues related to zoning and development of neighbourhoods.

The regression results of the land use, dwelling unit and employment characteristic models for total and severe collisions are presented in Tables 5-7 and 5-8, respectively. The models reveal that collision frequency increases with the increase in the following explanatory variables: commercial, residential and industrial areas; and number of houses, townhouses, apartments, and employees who work in management and manufacture sectors. All the variables, with exception of “townhouse”, show positive and significant relationships with collision frequency at the 95% confidence level for both total and severe collisions.

The GOF statistics show that the developed models fit the data reasonably well. The dispersion parameters for all the models are significantly different from zero. The values for dispersion


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parameters found to be in the range of 0.295 and 0.329 confirming the appropriateness of the NB. The mean SD and mean Pearson chi-square values are close to one, indicating that the models fit the data very well.


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Table 5-7 Regression Results Using GLM Approach for Land Use, Dwelling Unit and Employment Characteristics Based Models, T9-T11


T9 T10 T11

ln (β0) 2.6651

(0.224, 141.7)

2.369

(0.225, 111.2)

2.4426

(0.218, 125.8)

ln (VKT) 0.2608

(0.028, 88.4)

0.3017

(0.026, 134.0)

0.2735

(0.026, 107.9)

Commercial ( 1000 m2) 0.0013

(0.0003, 14.5) ---

0.0012

(0.0003, 15.0)

Residential (1000 m2) 0.0002

(0.0001, 18.9) --- ---

Resource & Industrial (1000 m2) 0.0002

(0.0001, 13.9) ---

0.0003

(0.0001, 18.1)

Other (1000 m2) -0.0001

(0.046, 4.6) ---

-0.0001

(0.000, 5.0)

of Houses × 10-3 --- 0.0314

(0.013, 6.0) ---

Number of Apartments × 10-3 --- 0.0385

(0.011, 13.3) ---

Number of Townhouses × 10-3 --- 0.0719

(0.081, 0.80) ---

Number of People Employed in Management × 10-3 --- ---

0.1072

(0.033, 10.6)

Number of People Employed in Manufactures-Constructions-Trades × 10-3 --- ---

0.2464

(0.053, 21.4)

α 0.3129 0.3213 0.2925




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Table 5-8 Regression Results Using GLM Approach for Land Use, Dwelling Unit and Employment Characteristics Based Models, S9-S11

Model Number Parameters

S9 S10

ln (β0) 1.3043

(0.2339, 31.1)

0.9971

(0.235, 18.0)

ln (VKT) 0.2405

(0.029, 69.6)

0.2835

(0.027, 108.7)

Commercial (1000 m2) 0.0009

(0.0003, 8.2) ---

Residential (1000 m2) 0.0004

(0.0001, 26.0) ---


(0.0001, 16.2) ---

Government & Institutional

(1000 m2)

0.0004

(0.0004, 4.9) ---

Other (1000 m2) -0.0001

(0.0001, 5.4) ---

Number of Houses × 10-3 --- 0.0442

(0.014, 10.7)

Number of Apartments × 10-3 --- 0.0399

(0.011, 13.6)

Number of Townhouses × 10-3 --- 0.1251

(0.083, 2.3)

α 0.323 0.3266


Mean Pearson Chi-Square 0.968 0.9759

(Standard Error, Chi-Square), ln (β0) = Natural Logarithm of Intercept, ln(VKT ) = Natural Logarithm of VKT


- 92 -

Models T10 and S10 explore the relationship between the number of collisions and dwelling units. For both total and severe collisions, these models show positive relationships between collisions and the type of dwelling units. It must be noted here that the number of townhouses for models T10 and S10 is found to be statistically insignificant.

For total collisions, model T11 is calibrated using land use data and the number of people living in each TAZ categorized by employment type. Before calibration of such a model, a correlation analysis is conduced, as shown Table 5-9, in order to examine multi-collinearity among the potential explanatory variables. The results of the correlation analysis indicate that the independent variables considered in this model are not highly correlated to each other.

Table 5-9 Correlation Matrix for Collision Data and Variables for T11 Model

Total Collisions

Severe Collisions Commercial Residential Industrial VKT Management Manufacture

Total Collisions

1.00 --- --- --- --- --- --- ---

Severe Collisions

0.94 1.00 --- --- --- --- --- ---

Commercial 0.19 0.14 1.00 --- --- --- --- ---

Residential 0.14 0.20 -0.08 1.00 --- --- --- ---

Industrial 0.21 0.20 0.02 -0.26 1.00 --- --- ---

VKT 0.48 0.44 0.09 0.02 0.17 1.00 --- ---

Management 0.12 0.12 -0.08 0.51 -0.31 -0.07 1.00 ---

Manufacture 0.23 0.30 0.01 0.45 -0.09 -0.01 0.44 1.00


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5.3.4 GLM Comprehensive Models (Model T12 for Total Collisions and Model S11 for Severe Collisions)

These models are calibrated using all of the variables introduced in Chapter Four, including land use, network, traffic intensity and socioeconomic and demographic variables. It is worth mentioning that applicability of these models for use in other regions may be limited due to the large number of explanatory variables and the reality that data for some of these may not be available. The regression results of the comprehensive models for total and severe collisions are presented in Table 5-10.


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Table 5-10 Regression Results Using GLM Approach for Comprehensive Models


T12 S11

ln (β0) 2.2817

(0.195, 136.5) 0.9456

(0.206, 21.1)

ln (VKT) 0.2392

(0.024, 98.1) 0.2165

(0.025, 72.8)

Total Road Kilometers 0.0183

(0.003, 22.0) 0.0200

(0.004, 22.6)


0.097 (0.012, 63.07)

0.1027 (0.013, 63.0)


0.1569 (0.019, 69.0)

0.1557 (0.020, 62.0)

Commercial (1000 m2) 0.0009

(0.0003, 10.9) 0.0005

(0.0002, 4.1)


(0.0001, 6.4) 0.0001

(0.0001, 6.0)

Government & Institutional (1000 m2)

--- 0.0002

(0.0001, 2.2)

Other (1000 m2) -0.0001

(0.0000, 15.9) -0.0002

(0.0000, 13.9)

Population × 10-3 0.0172

(0.007, 5.6) 0.0191

(0.008, 6.2)

α 0.3266 0.2168


Mean Pearson Chi-Square 0.9759 1.106


- 95 -

All of the variables, with exception of “other”, show positive and significant relationships with collision frequency at the 95% confidence level for both total and severe collisions. The “other” variable for both models is the total area of open space and parks. This variable is found to have a negative, but significant association to collision data.

The dispersion parameter values for both models are different from 0, which confirms that the data are overdispersed relative to the Poisson distribution and that the NB error structure is justified. The GOF statistics show mean Pearson chi-square and SD values close to 1, again indicating that the models fit the data reasonably well.

The cumulative residuals are produced for model T12 for total collisions and model S11 for severe collisions. The cumulative residuals are presented for the total and severe collisions in Figure 5-2 respectively. The figures show that the cumulative residuals stay between the two standard

deviation (±2σ*) boundaries for both models.

a) Total b) Severe

Figure 5-2 CURE Plot for Comprehensive Models Based on GLM Approach

5.4 Sensitivity Analysis of Explanatory Variables

A sensitivity analysis is conducted for the key variables within each developed safety planning model in this chapter. The objective is to determine whether these variables have a positive or negative impact on the safety of each TAZ and examine the sensitivity of safety of TAZ to changes in these variables. Specifically, the analysis provides the percentage change in the dependent variable, number of total and severe collisions in each TAZ for this study, subject to one unit


- 96 -

change in one explanatory variable with all else held constant. This analysis can be comparable to conducting elasticity analysis in cross-sectional travel demand models which are used for examining the impact on travel demand of changes in values of certain variables, such as travel cost or transit frequency.

This sensitivity analysis provides conceptually similar information to the conventional collision modification factors (CMFs). CMFs are used in road safety to estimate the safety implications of design and operational changes to roads in service or under design. Simply stated, a CMF is used to estimate the change in crash frequency as a result of the design or operational change.

It must be noted here that no impact analysis can be conducted for the exposure variable since relationships between collisions and VKTs are non-linear. Therefore, one single value or percentage change cannot be reported for the VKT.

The results of the sensitivity analysis are reported in the form of a safety index. The relationship between the safety index of each explanatory variable and percentage changes in collisions can be expressed as follows:

( ) 100IndexSafety1ChangeCollisionPercent ×−= (5.21)

An index in this study can have one of two “states”:

Safety index > 1: when the index is greater than 1, the variable is associated with increased collision frequency in a TAZ.

Safety index < 1: when the index is less than 1, the variable is associated with reduced collision frequency in a TAZ.

Tables 5-11 and 5-12 present the results of the safety index as a result of sensitivity analysis for all the variables in total and severe collision models, respectively.


- 97 -

Table 5-11 Safety Index for Total Collisions Based on Sensitivity Analysis of Variables

Model Number Variables

T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12

Total Arterial (km) 1.22 --- --- --- --- --- --- --- --- --- ---

Total Expressway (km) 1.09 --- --- --- --- --- --- --- --- --- ---

Total Collector (km) 1.06 --- --- --- --- --- --- --- --- --- ---

Total Laneway (km) 1.08 --- --- --- --- --- --- --- --- --- ---

Total Local Road (km) 0.99 --- --- --- --- --- --- --- --- --- ---

Total Ramp (km) 1.08 --- --- --- --- --- --- --- --- --- ---

Total Road Kilometers --- 1.03 --- --- 1.02 1.02 1.02 --- --- --- 1.02

Number of 4-Legged Sig. Intersections --- --- 1.13 --- 1.11 1.11 1.11 --- --- --- 1.10

Number of 3-Legged Sig. Intersections --- --- 1.18 --- 1.18 1.18 1.18 --- --- --- 1.17

Number of Signalized Intersections --- --- --- 1.14 --- --- --- --- --- --- ---

Total Rail (km) --- --- --- --- --- --- 0.97 --- --- --- ---

Number of Schools --- --- --- --- --- 1.03 --- --- --- --- ---

Commercial (1000 m2) --- --- --- --- --- --- --- 1.0013 --- 1.0012 1.0009

Residential (1000 m2) --- --- --- --- --- --- --- 1.0002 --- --- ---

Resource & Industrial (1000 m2) --- --- --- --- --- --- --- 1.0002 --- 1.0003 1.0001

Number of Houses × 10-3 --- --- --- --- --- --- --- --- 1.03 --- ---

# of Apartment × 10-3 --- --- --- --- --- --- --- --- 1.04 --- ---

# of Townhouse × 10-3 --- --- --- --- --- --- --- --- 1.07 --- ---

# of Population × 10-3 --- --- --- --- --- --- --- --- --- --- 1.02


- 98 -

Table 5-12 Safety Index for Severe Collisions Based on Sensitivity Analysis of Variables

Model Number Variables

S2 S3 S4 S5 S6 S7 S8 S9 S10 S11

Total Arterial Road (km) 1.25 --- --- --- --- --- --- --- --- ---

Total Expressway (km) 1.12 --- --- --- --- --- --- --- --- ---

Total Collector (km) 1.05 --- --- --- --- --- --- --- --- ---

Total Laneway (km) 1.07 --- --- --- --- --- --- --- --- ---

Total Local Road (km) 0.99 --- --- --- --- --- --- --- --- ---

Total Ramp (km) 1.02 --- --- --- --- --- --- --- --- ---

Total Road Kilometers --- 1.03 --- 1.02 1.02 1.02 --- --- 1.02

# of 4-Legged Signalized Intersections --- --- 1.14 --- 1.12 1.12 1.12 --- --- 1.11

# of 3-Legged Signalized Intersections --- --- 1.18 --- 1.18 1.18 1.18 --- --- 1.17

Number of Signalized Intersections --- --- --- 1.15 --- --- --- --- --- ---

Total Rail (km) --- --- --- --- --- 0.96 --- --- ---

Number of Schools --- --- --- --- --- 1.03 --- --- ---

Commercial (1000 m2) --- --- --- --- --- --- --- 1.0009 --- 1.0005

Residential (1000 m2) --- --- --- --- --- --- --- 1.0004 --- ---

Resource & Industrial (1000 m2) --- --- --- --- --- --- --- 1.0003 --- 1.0001

Government & Institutional (1000 m2) --- --- --- --- --- --- --- 1.0004 --- 1.0002

# of Houses * 10-3 --- --- --- --- --- --- --- --- 1.05 ---

# of Apartments * 10-3 --- --- --- --- --- --- --- --- 1.04 ---

# of Townhouses * 10-3 --- --- --- --- --- --- --- --- 1.13 ---

Population * 10-3 --- --- --- --- --- --- --- --- --- 1.02


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5.5 Discussion of Results for Safety Planning Models

Twenty-three safety planning models are developed in order to examine the relationships between several types of transportation planning variables and collision frequency. Models are calibrated for each planning category separately and in combination with other categories. It is believed a good forecasting model should be parsimonious (i.e. with few variables) and sensitive enough to examine the impact of different policies affecting the safety of a network. Therefore, separate models are developed for each category.

A comparison of the relative performance of the developed models indicates that the comprehensive models are statistically performing better than other models since they have lower dispersion parameters. Therefore, the comprehensive models are better tools for the prediction of the number of zonal collisions. However, as highlighted above, applicability of these models to transportation planning and policy analysis may be limited for jurisdictions that do not have abundant and diverse data, such as those used in this study.

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CCHHAAPPTTEERR SSIIXX

6.0 DEVELOPMENT OF LOCAL SAFETY PLANNING MODELS

6.1 Background

The GLM models from the previous chapter assumed that the relationship between the dependent variable and independent variables does not vary across space, i.e., there is a constant coefficient for each explanatory variable which reflects the average relationship between the dependent variable, typically the number of collisions per TAZ for safety planning models, and each explanatory variable. The model parameters are estimated globally for the entire study area based on the assumed stationary relationship. However, this relationship may hide some important spatial factors. Consequently, the accuracy of such models for explaining the relationship between the dependent variable and explanatory variables for specific TAZs may be doubtful because collision frequency is likely influenced by many spatially defined factors. Some of these factors may have strong predictive power for estimating the number of collisions at certain locations, but weak predictors at other locations. To address this problem, several studies conducted in the past have developed techniques to consider spatial autocorrelation in multivariate regression models. Specifically, the GWR was developed by Fotheringham et al. (2002) to allow calibrating

Use of Advanced Techniques to Estimate Zonal Level Safety Planning Models and Examine Their Temporal Transferability Chapter 6: Development of Local Safety Planning Models

- 101 -

multivariate regression models of spatially non-stationary processes. The GWPR is a special case for count data. Through the explicit treatment of spatial coordinates, GWR provides a set of local spatial parameters where the weights are linked to the distance between the observation and the location where independent variables in the models are measured. These local parameters are estimated using a geographically weighted likelihood principle described by Nakaya (2001). Such an approach is in line with the Tobler’s First Law of Geography (Tobler, 1970) which states that “everything is related to everything else, but near things are more related than distant things”.

The primary objective of the research documented in this chapter is to investigate the local variations in the relationship between the number of zonal collisions and potential transportation planning predictors, including traffic volume, road network characteristics, socioeconomic and demographic features, land use, dwelling unit, and employment type, through the development of GWPR models. As highlighted above, the main advantage of the GWPR or GWR models in general is that the independent variable coefficient estimates vary locally. This is a very essential feature as it adds distinctiveness to the spatial location of data, such as collision occurrence. It allows for a locally varying parameter representation of a particular point in space. The results presented in the following sections are also documented in Hadayeghi et al. (2008).

6.2 Conceptual Framework for GWPR Models

As explained above, the GWPR technique permits the parameter estimates to vary locally rather than globally. These parameter estimates have spatial coordinates and can therefore, be mapped to show how the relationship between the dependent variable and particular independent variables varies over space (Fotheringham, 2000). The basic GWR model has the following form (Fotheringham et al., 2002; Fotheringham et al., 1997a; Fotheringham et al., 1997b and Fotheringham et al., 1998):

( ) ( ) ( ) ik

iki XiYE εββ ++= ∑ iuu (6.1)

where ui(=(uxi, uyi)) denotes the coordinates of the ith data point in space and is a vector of two dimensional coordinates describing the location of i with x and y coordinates. GWR is a technique


- 102 -

developed to calibrate a multiple regression model that allows different relationships to exist at different points in space. The basic idea of GWR in general and GWPR in this study is that the observed data near point i have more of an influence on the estimation of the βj (ui)’s than data located farther from i. This influence around i is described by the weighting function.

The mechanism of GWR in general and GWPR in this study is illustrated in the following simplified example. Suppose data were collected from 6 TAZs. Each TAZ is a data point with one observation. A traditional GLM model would result in a single set of parameter estimates based upon the 6 observations obtained from the 6 TAZs. In contrast, a GWR model would produce 6 sets of parameters with one set for each TAZ. To estimate β for TAZ #1, 5 different weights would be given to the rest of the 5 data points depending on their distances to the first TAZ. To estimate β for TAZ #2, another 5 weights are employed, depending on their distances to the second TAZ. In the GWR approach, the end result is 6 sets of regression equations, one for each observation, and each set influenced by the nearness of the other observations.

6.3 Specification of GWPR Models

The discussion presented in this section is extracted from Nakaya et al. (2005) and Fotheringham et al. (2002).

Consider a traditional Poisson regression model written as:

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛≈ ∑

kikkii XEPoissonY ,exp β (6.2)

where Yi is observed number of collisions in place i.

The parameters of the Poisson regression are estimated by maximum likelihood function aiming at maximizing the logarithm of the likelihood. The log likelihood is given by the following equation:

( )∑=i

ii YlLmax β


- 103 -

( ) ( )( )∑ +−=i

iii YLnYY ββ (6.3)

where β= (β0, β1, …) is the vector of the regression parameters and li is the log-likelihood of

sample i.

As explained previously and shown in Equation 6.1, the GWPR allows parameter values for a model to vary with geographical location. The Poisson model in Equation 6.2 for GWPR models can be written as follows:

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛≈ ∑

kikikii XEPoissonY ,exp uβ (6.4)

To estimate GWPR parameters, a variant of the local likelihood principle (Loader, 1999) which is referred as a geographically weighted likelihood principle and consistent with geographically weighted least squares for conventional Gaussian GWR was employed. Following the local likelihood methodology, parameters at location i are estimated by solving the following maximization problem of the geographically weighted log-likelihood.

( ) ( )( ) ( )( )( ) ( )jiijijji

N

jji wYLnYYLmax uuuβuu −×+−= ∑ β (6.5)

where ( )( )ijY uβ is the predicted number of collisions at TAZ j with parameters at regression point

i.

( )( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= ∑ j,kikjij XˆexpEY uuβ β (6.6)

wij is the geographical weight of the jth observation at the ith regression point. The weighting function is used to place different emphasis on different observations in generating the estimator of the parameters (Fotheringham et al., 2002).


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As evident from the above equations, the GWPR method attempts to capture spatial variation by fitting a regression model at each subject point (TAZ in this study) in the data, weighting all neighboring observations by a function of distance from that subject. It is evident that the spatial kernel function and window size; the bandwidth, used in the model fitting process impact the GWPR coefficient estimates. Therefore, the selection of the spatial kernel function and consequently, bandwidth, is an important step in the implementation of GWPR (Guo et al., 2008).

The bandwidth is the number of observations around each subject point and controls the distance decay in the weighting function. For the special case of GWPR models, the Gaussian and bi-square functions are commonly used to produce the weighting scheme for each data point.

Gaussian: ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −×−=

G21expw ji

ij

uu

(6.7)

Bi-square:

( )[ ]

⎪⎪⎩

⎪⎪⎨

⎧ −−−

=otherwise

GifG

wijiiji

ij

0

/122

puuuu

The parameter Gi (called the bandwidth) manages the kernel size. The results of GWPR models are sensitive to the bandwidth of a given weighting function. The bandwidth might be constant (fixed kernel), as in the Gaussian function, or variable (adaptive kernel) as shown in the bi-square function.

The shape and magnitude of the weighting functions will be the same for every sample point over space when the fixed kernel weighting scheme is used. As a result, the fixed kernel weighting scheme can produce inaccurate results for the estimation of model parameters since bandwidths may be larger for regions with dense data points and smaller for others. On the other hand, adaptive kernel allows the kernel to vary spatially, i.e., larger for regions with scarce sample data and smaller for regions with more substantial data (Zhao et al., 2005).


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Computational details for calibration procedures of GWPR, specifically local scoring procedures and calculation of standard error of parameters in GWPR models, can be found in Nakaya et al. (2005).

6.4 Calibration of Local Safety Planning Models

As explained previously, the GLM approach using Poisson and NB regression provides a more appropriate basis for the collision dataset analysis than conventional linear regression, especially for entities where observed numbers of collisions are low. The use of Poisson regression in conjunction with GWR represents an advance in the calibration of safety planning models compared to previous studies which used the GLM approach. Although it would be beneficial to consider geographically weighted Negative Binomial regression, available GWR software, most notably “GWRx3.0”, does not support the calibration of GWR models with an NB error structure.

It is worth noting that the use of Poisson regression instead of NB regression does not produce inaccurate model estimates for safety models in general since the model coefficients are quite similar for the two error distributions (Miaou, 1994). The main reason for using NB regression is that it facilitates the estimation of an overdispersion parameter that is necessary for applying the models in an EB analysis (Hauer, 1997). Given all of these considerations, it was deemed reasonable to adopt the GWPR technique for the development of collision models using the “GWRx3.0” software package.

The model used for development of GWPR safety planning models in this study is similar to Equation 5.2 for the GLMs, and is of the form:

pip2i2i1i0 X)(......X)()VKTln()()((ln)Y(Ln uuuu ββββ ++++= (6.8)

Here, unlike Equation 5.2, βj (j=0,1,…,p) is a function of location (ui), denoting the coordinates of the ith point (TAZ centroid in this study) in space. This suggests that Equation 6.3 produces different predicted collisions for each TAZ depending on the location where VKT and the Xj are measured.


- 106 -

For calibration of collision frequency GWPR models, both fixed and adaptive kernels were examined. To do that, the Akaike Information Criterion (AIC) was used as an indicator, not only for selection between fixed and adaptive kernels, but also for bandwidth selection in adaptive kernels. Nakaya et al. (2005) showed that the AIC of GWPR models with bandwidth G can be defined as:

)G(K)G(D)G(AIC += (6.9)

where D and K are the deviance and effective number of parameters in the model with bandwidth G, respectively. The best GWPR model is the one with the lowest AIC. However, Nakaya et al. (2005) argued that since the degrees of freedom for GWPR models are likely to be small, a modest sample bias adjustment in the AIC definition should be done. They introduced the corrected AIC (AICc) to address this bias. AICc is defined as follows:

( ) ( ) ( ) ( ) ( )( )( ) 1GKN

1GKGK2GK2GDGAICc−−+

++= (6.10)

If the effective number of parameters, K, is small relative to the number of observations, N, then the difference between AIC and AICc is negligible.

As indicated in the above discussion, selection of the bandwidth and consequently, effects of this selection on the kernel function may change the coefficient estimates. As the bandwidth increases, the GWPR coefficient estimates approach those of a global model. On the other hand, if a smaller bandwidth is used, the GWPR coefficient estimates depend on the observations in close proximity to the subject point and they change rapidly over space (Fotheringham et al. 2002). This may over-fit the developed local models and produce a large coefficient variability (Farber and Pa’ez, 2007). Therefore, the selection of the kernel function and bandwidth is an important step in the implementation of GWR.

For this study, both fixed and adaptive kernels are applied for the development of each model. The AICc selection process, as highlighted above, is used to determine the optimal TAZ size required for GWPR modeling. For all cases, the models with the adaptive kernels provide lower AICc, which is an indicator of a better fit model. However, for the selection of bandwidth for the adaptive


- 107 -

kernels, the “GWRx3.0” software package does not provide an optimal model with the best selected bandwidth. The solution was to manually insert a series of bandwidths, examine results of each, and finally, select a model which performs the best. Table 6-1 contains a summary of the optimal bandwidth size in terms of the number of TAZs for all the GWPR models in this study.

Table 6-1 Summary of Optimal Bandwidth Based on Number of TAZs for Total and Severe Collisions

Model Number for Total Collisions

Bandwidth (Number of TAZs)

Model Number for Severe Collisions

Bandwidth (Number of TAZs)

T1 21 S1 10

T2 157 S2 134

T3 39 S3 26

T4 19 S4 15

T5 21 S5 13

T6 28 S6 23

T7 30 S7 23

T8 111 S8 110

T9 54 S9 44

T10 72 S10 94

T11 124 S11 33

T12 33 --- ---


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6.5 Results of Local Safety Planning Models

6.5.1 GWPR Model with Traffic Intensity as Independent Variable

As identified above, GWPR provides the ability to examine local relationships that can be hidden in a global GLM model. Therefore, the results of the GWPR models are a set of local parameters for each independent variable. In this case, the results of the traffic intensity-based GWPR models for total and severe collisions are different parameter estimates for the independent variable, VKT, for each TAZ. The local parameter estimates are described by 5-number summaries and presented for total and severe collisions in Table 6-2, respectively. The 5-number summary of a distribution presents the median, upper and lower quartiles, along with the minimum and maximum values of the data. As evident from Table 6-2, the signs of coefficients of VKT for each TAZ are not always the same. Traffic exposure (i.e., VKT) is expected to have a positive effect on the number of collisions in each TAZ; therefore, its coefficient should be positive.

Table 6-2 Regression Results Using GWPR Approach for Traffic Intensity Based Models

Minimum, Maximum (Lower Quartile, Median, Upper Quartile)

Figures 6-1 and 6-2 depict the local parameters for both total and severe collision models for each TAZ, respectively.


T1 S1

ln (β0) -6.4 , 8.10

(1.09, 2.4,8.1) -13.50 , 15.17 (-1.2,1.14,3.8)

ln (VKT) -0.36 , 1.37

(0.12,0.33,0.47) -1.585 , 2.037

(0.03,0.28,0.57)

AICc GWPR 16,304 3,195

AICc Global 27,410 6,742


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Figure 6-1 Coefficients of Zonal VKT for T1 Model

Figure 6-2 Coefficients of Zonal VKT for S1 Model


- 110 -

It is clear from the figures that the parameters demonstrate spatial variation. The figures also show that in some of the TAZs, the sign of VKT coefficient is negative. The number of TAZs with negative sign for the VKT coefficient are more for severe than total collision models.

As indicated by Zhao et al. (2005) and Wheeler (2007), the problem with the counterintuitive signs is not uncommon in GWR or GWPR models. There are two possible reasons for this:

1. There exists multi-collinearity among some variables in the data set for some TAZs, or locations, i.e., due to correlation in the estimated coefficients. These variables may be correlated locally and not globally. Wheeler and Tiefelsdorf (2005) indicated that “while GWR coefficients can be correlated when there is no explanatory variable collinearity, the coefficient correlation increases systematically with increasingly more collinearity. The collinearity in explanatory variables can apparently be increased by the GWR spatial kernel weights, and moderate collinearity of locally weighted explanatory variables can lead to potentially strong dependence in the local estimated coefficients, which makes interpreting individual coefficients problematic”.

2. Some variables may be less significant or insignificant at all or some locations. This might be the case due to the method used for estimating the standard errors in the GWR models. The standard errors of the GWR models may only be rough estimates due to the reuse of the data at multiple locations (Congdon, 2003) and also because of using the data to estimate both the bandwidth and regression coefficients (Wheeler and Calder, 2007).

Both of the above phenomena can result in unexpected coefficient signs for some TAZs. A good diagnostics tool is not available to examine multi-collinearity in GWRs. Therefore, this possibility cannot be examined. To examine the level of significance of each local variable, local t-statistics were computed in order to determine where relationships are significant and where they are not.

Figures 6-3 and 6-4 show the results of the t-statistics for both total and severe collision models for each TAZ, respectively. The results show that the t-values for most of the TAZs with negative coefficients are insignificant at the 90% confidence level (TAZs coloured in light grey in Figures 6-


- 111 -

3 and 6-4). Another observation from Figures 6-3 and 6-4 is that the TAZs with insignificant t-values are in close proximity to each other in small pockets distributed throughout the study area.


- 112 -

Figure 6-3 t-Statistics of Zonal VKT for T1 Model

Figure 6-4 t-Statistics of Zonal VKT for S1 Model


- 113 -

Unexpected coefficient signs for the rest of TAZs with significant t-values can be due to missing or mis-specified explanatory variables in the model. For instance, VKT data were not available by type of road. So the model could not explain the possibility that a TAZ with more freeway kilometers and higher VKT can have fewer collisions than a neighbouring TAZ with lower VKT that is mostly on arterial roads. These arterial roads are known to be less safe than freeways, a possibility that can produce a counterintuitive sign for the VKT exponent for both TAZs.

Also, the size of TAZs may play some role for unexpected coefficient signs with significant t-values since they are arbitrarily sized. Finally, aggregation of the data (VKT in this case) into the TAZ level can also be another reason behind this unexpected behaviour of a rather small number of TAZs. To address this, further research needs to be conducted to explore the effect of TAZ size on the GWPR model results.

The AICc associated with the total and severe GWPR models are 16,305 and 3,196 respectively. The AICc values for the total and severe global Poisson models are found to be 27,410.2 and 6,742.0 respectively. A comparison of the AICc values suggests that the GWPR models outperform the global models, an indication that there is a spatial non-stationary feature in the relationships being examined.

Figure 6-5 illustrates the cumulative residual plots for the covariate VKT for the total and severe collision models. These figures show that the severe collision model fits the data better than the model for total collisions. However, the cumulative residuals for both models oscillate around the

value of 0 and lie between the two standard deviation (±2σ*) boundaries. Therefore, it can be

concluded that the assumption of a non-linear relationship between collisions and traffic intensity is appropriate.


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a) Total b) Severe

Figure 6-5 CURE Plot for Traffic Intensity Models Based on GWPR Approach

6.5.2 GWPR Models with Network Characteristics as Independent Variables

Similar to the previous section, several models that rely primarily on explanatory variables describing the TAZ road network were developed. The 5-number summaries for these models are presented in Tables 6-3 and 6-4 for total collisions, and in Tables 6-5 and 6-6 for severe collisions, respectively.

The total rail and total local road kilometers were found to be inversely related to collision frequency in the previous section when the GLM approach was used. Interestingly, the lower quartiles for these two variables were found to be negative values, indicating that most of the TAZs have negative coefficients.

The AICc values associated with GWPR and GLM models are shown in Tables 6-3 to 6-6. A comparison of the AICc values suggests that the GWPR models outperform the global models in all the cases, again indicating that there is a spatial non-stationary feature in the relationships being examined.


- 115 -

Table 6-3 Regression Results Using GWPR Approach for Network Characteristics Based Models, Models T2-T4


T2 T3 T4

ln (β0) 1.06 , 4.60

(3.31,3.84,4.14)

-3.23 , 6.92

(1.2,2.3,3.3)

-6.42 , 7.41

(1.32,2.53,3.74)

ln (VKT) -0.009 , 0.38

(0.03,0.07,0.13)

-0.33 , 0.89

(0.14,0.27,0.38)

-0.32 , 1.298

(0.06,0.23,0.38)

Total Arterial Road Kilometers 0.07 , 0.33

(0.17,0.22,0.26) --- ---

Total Expressway Kilometers -0.15 , 0.23

(0.05,0.08,0.12) --- ---

Total Collector Kilometers -0.02 , 0.17

(0.05,0.09,0.12) --- ---

Total Laneway Kilometers -0.023 , 0.48

(0.04,0.06,0.10) --- ---

Total Local Road Kilometers -0.048 , 0.016

(-0.02,-0.009,0.002) --- ---

Total Ramp Kilometers -0.29 , 0.26

(0.01,0.07,0.13) --- ---

Total Road Kilometers --- -0.04 , 0.16

(0.03,0.04,0.06) ---

Number of 4-Legged Signalized Intersections --- ---

-0.165 , 0.487

(0.08,0.14,0.21)


-0.514 , 0.564

(0.04,0.15,0.23)

AICc GWPR 13,404 14,602 8,137

AICc Global 20,431 24,373 20,041



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Table 6-4 Regression Results Using GWPR Approach for Network Characteristics Based Models, Models T5-T8


T5 T6 T7 T8

ln (β0) -4.58 , 7.31

(1.55,2.67,3.74) -3.81 , 5.88

(1.50,2.56,3.47) -3.97 , 6.23

(1.41,2.42,3.47) -2.839 , 5.526

(1.49,2.50,3.40)

ln (VKT) -0.26 ,1.17

(0.07,0.22,0.35) -0.224 , 0.979

(0.08,0.21,0.34) -0.22 , 0.99

(0.09,0.21,0.34) -0.165 , 0.857

(0.10,0.21,0.34)


(0.01,0.03,0.05) -0.043 , 0.156

(0.01,0.03,0.05) -0.038 , 0.118

(0.01,0.03,0.05)


-0.083 , 0.372 (0.05,0.10,0.17)

-0.098 , 0.350 (0.05,0.10,0.16)

-0.069 , 0.347 (0.05,0.09,0.15)


-0.195 , 0.543 (0.06,0.13,0.19)

-0.23 , 0.474 (0.06,0.12,0.18)

-0.269 , 0.419 (0.06,0.13,0.19)


-0.067 , 0.348 (0.09,0.13,0.18)

--- --- ---

Total Rail Kilometers --- --- --- -0.56 , 1.12

(-0.11,0.01,0.15)

Number of Schools --- --- -0.26 , 0.24

(-0.06,0.00,0.05) ---

AICc GWPR 9,910 8,220 7,629 8,032

AICc Global 20,161 18,805 18,634 18,779



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Table 6-5 Regression Results Using GWPR Approach for Network Characteristics Based Models, Models S2-S4


S2 S3 S4

ln (β0) -0.488 , 3.761

(2.0,2.47,2.80)

-6.800 , 7.130

(-0.31,0.93,2.35)

-7.703 , 7.932

(-0.57,1.10,2.74)

ln (VKT) -0.089 , 0.376

(0.01,0.04,0.1)

-0.459 , 1.131

(0.08,0.24,0.38)

-0.846 , 1.237

(0.02,0.21,0.41)

Total Arterial Road Kilometers 0.087 , 0.338

(0.20,0.25,0.27) --- ---

Total Expressway Kilometers -0.167 , 0.282

(0.08,0.12,0.16) --- ---

Total Collector Kilometers -0.038 , 0.148

(0.03,0.07,0.09) --- ---

Total Laneway Kilometers -8.11 , 0.996

(0.02,0.06,0.09) --- ---

Total Local Road Kilometers -0.043 , 0.028

(-0.01,-0.00,0.00) --- ---

Total Ramp Kilometers -0.479 , 0.280

(-0.03,0.02,0.12) --- ---


(0.03,0.05,0.07) ---


-0.229 ,0.736

(0.07,0.15,0.23)


-0.700 , 0.676

(0.02,0.14,0.25)

AICc GWPR 3,228 3,251 2,492

AICc Global 4,861 5,848 4,885



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Table 6-6 Regression Results Using GWPR Approach for Network Characteristics Based Models, Models S5-S8


S5 S6 S7 S8

ln (β0) -10.139 , 11.700 (-0.52,1.38,2.81)

-4.886 , 7.160 (-0.19,1.20,2.27)

-6.89 , 8.275 (-0.38,1.17,2.37)

-3.561 , 7.120 (0.02,1.2,2.1)

ln (VKT) -1.075 , 1.568

(0.00,0.18,0.41) -0.438 , 0.896

(0.05,0.19,0.34) -0.562 , 1.165

(0.04,0.18,0.36) -0.490 , 0.779

(0.07,0.19,0.31)

Total Road Kilometers ---

-0.080 , 0.228 (0.00,0.03,0.05)

-0.090 , 0.213 (0.00,0.02,0.06)

-0.038 , 0.118 (0.00,0.03,0.05)


--- -0.145 , 0.478

(0.05,0.11,0.18) -0.215 , 0.473

(0.05,0.11,0.19) -0.097 , 0.392

(0.05,0.09,0.16)


--- -0.255 , 0.735

(0.03,0.11,0.20) -0.336 , 0.747

(0.03,0.12,0.19) -0.356 , 0.436

(0.05,0.12,0.19)


-0.110 , 0.599 (0.08,0.14,0.21)

--- --- ---

Total Rail Kilometers --- --- --- -0.696 , 0.991

(-0.11,0.01,0.14)

Number of Schools --- --- -0.308 , 0.288

(0.007,0.01,0.07) ---

AICc GWPR 2,404 2,295 2,275 2,358

AICc Global 4,905 4,492 4,448 4,485


The coefficients of all the variables, except total arterial road kilometers in Models T2 and S2, vary from negative to positive values. For each variable in each model, a similar analysis to the one for the coefficient of VKT in Models T1 and S1 in the previous section was conducted. The results of


- 119 -

the t-values indicated that the most of TAZs with unexpected coefficient signs are insignificant at the 90% confidence level.

Tables 6-7 and 6-8 present the summaries pertaining to network characteristics based models for portions of TAZs with unexpected negative coefficient signs for explanatory variables in total and severe collisions respectively. Each cell in the two tables contains 2 numbers. The upper number indicates the total number of TAZ with unexpected coefficient signs and the lower number (in parentheses) represents the percentage of the TAZ with unexpected coefficient signs for which the t-values are insignificant at the 90% confidence level. For instance, Table 6-7 illustrates that for Model T7, there are 35 TAZs with unexpected coefficient signs for the VKT coefficient, of which 17 (49% of the 35 TAZs) have insignificant t-values at the 90% confidence level.

As evident from the tables, the results show that the t-values for most of the TAZs with negative coefficients are insignificant at the 90% confidence level. As explained in the previous section, the size of TAZs may also play some role for unexpected coefficient signs. Unexpected coefficient signs for the rest of TAZs with significant t-values can also be possibly due to the effect of multi-collinearity.


- 120 -

Table 6-7 Summary of Network Characteristics Based Models with Regard to Portions of TAZs with Unexpected Coefficient Signs for Total Models

Parameters T2 T3 T4 T5 T6 T7 T8

ln (VKT) 3

(100%)

26

(35%)

73

(53%)

57

(46%)

36

(69%)

35

(49%)

30

(100%)

Total Arterial Road Kilometers 0

(0%) --- --- --- --- --- ---

Total Expressway Kilometers 59

(10%) --- --- --- --- --- ---

Total Collector Kilometers 17

(53%) --- --- --- --- --- ---

Total Laneway Kilometers 13

(92%) --- --- --- --- --- ---

Total Local Road Kilometers 344

(9%) --- --- --- --- --- ---

Total Ramp Kilometers 83

(70%) --- --- --- --- --- ---

Total Road Kilometers --- 12

(17%) --- ---

80

(34%)

79

(24%)

78

(88%)

Number of 4-Legged Signalized Intersections --- --- 23

(52%) ---

43

(44%)

55

(33%)

52

(87%)


(26%) ---

65

(42%)

63

(43%)

65

(32%)

Total Number of Signalized Intersections --- --- --- 9

(56%) --- --- ---

Total Rail Kilometers --- --- --- --- --- --- 220

(85%)

Number of Schools --- --- --- --- --- 228

(25%) ---

Number of TAZs with Unexpected Signs (Percentage of TAZs with Unexpected Signs and Insignificant at 90% Confidence Level)


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Table 6-8 Summary of Network Characteristics Based Models with Regard to Portions of TAZs with Unexpected Coefficient Signs for Severe Models

Parameters S2 S3 S4 S5 S6 S7 S8

ln (VKT) 100

(76%)

70

(57%)

109

(52%)

113

(60%)

75

(69%)

89

(70%)

56

(80%)

Total Arterial Road Kilometers 0

(0%) --- --- --- --- --- ---

Total Expressway Kilometers 59

(15%) --- --- --- --- --- ---

Total Collector Kilometers 40

(70%) --- --- --- --- --- ---

Total Laneway Kilometers 77

(57%) --- --- --- --- --- ---

Total Local Road Kilometers 287

(31%) --- --- --- --- --- ---

Total Ramp Kilometers 179

(56%) --- --- --- --- --- ---

Total Road Kilometers --- 28

(64%) --- ---

109

(63%)

137

(62%)

93

(81%)


(79%) ---

48

(63%)

59

(66%)

43

(65%)


(60%) ---

81

(75%)

87

(67%)

59

(78%)

Total Number of Signalized Intersections --- --- --- 27

(67%) --- --- ---

Total Rail Kilometers --- --- --- --- --- --- 251

(52%)

Number of Schools --- --- --- --- --- 218

(44%) ---



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6.5.3 GWPR Models with Land Use, Dwelling Unit and Employment Characteristics as Independent Variables

The relationship between collision frequency and land use, dwelling unit and employment was explored by developing GWPR models. This type of analysis is beneficial to urban planners and other analysts who deal with issues related to zoning and development of neighbourhoods.

The 5-number summaries for the developed models in this section are presented in Tables 6-9 and 6-10 for total and severe collisions, respectively. As evident from these tables, the signs of coefficients in the developed models for all TAZs are not always the same since the minimum values are negative for all of the coefficients. However, the lower quartile values for most coefficients are positive except for “other” in Model T9 and number of townhouses in both Models T10 and S10. The positive value of the lower quartile indicates that most TAZs have positive coefficients.

Models T10 and S10 explore the relationship between the number of collisions and dwelling units. For both total and severe collisions, these models show positive relationships between collisions and the type of dwelling units for most TAZs, as their lower, median and upper quartile coefficient values are positive. The exception is townhouses for which most TAZs have a negative coefficient sign, contrary to expectations. It is expected that the coefficient for the number of townhouses would be positive for most TAZs because as the number of dwelling units, including townhouses, increases, exposure and consequently, the number of collisions, also increases. As explained in the previous section, a possible reason for a coefficient with a counterintuitive sign is that the variable is insignificant in most TAZs. Interestingly, this is in accordance with the finding of the previous chapter in which the coefficient for the number of townhouses, although positive, was statistically insignificant in the GLM models. Table 6-11 presents the summaries pertaining to land use, dwelling unit and employment characteristics based models. These are with regard to portions of TAZs with unexpected negative coefficient signs for explanatory variables in both total and severe collisions.


- 123 -

The AICc values indicate that the developed GWPR models fit the data better than the global GLM models since their values are smaller. However, in the subsequent chapter, a detailed analysis will be conducted in order to examine and compare the GOF measures from the GWPR and FBSA models from Chapter Seven with the one from the GLM models.


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Table 6-9 Regression Results Using GWPR Approach for with Land Use, Dwelling Unit and Employment Characteristics Based Models, T9-T11


T9 T10 T11

ln (β0) 0.110 , 4.842

(1.41,2.37,3.05)

-2.177, 6.450

(0.69,1.93,3.08)

0.345 , 4.486

(1.32,2.11,3.04)

ln (VKT) -0.038 , 0.490

(0.20,0.26,0.35)

-0.202 , 0.725

(0.21,0.34,0.47)

0.018 , 0.477

(0.20,0.29,0.37)

Commercial ( 1000 m2) -0.0004 , 0.005

(0.001,0.002,0.003) ---

-0.0002 , 0.005

(0.001,0.002,0.003)

Residential (1000 m2) 0.000087 , 0.0007

(0.0003,0.0003,0.0004) --- ---

Resource & Industrial (1000 m2) -0.00009 , 0.0024

(0.0003,0.0004,0.0005) ---

-0.0002 , 0.002

(0.0003,0.0004,0.0005)

Other (1000 m2) -0.0012 , 0.0009

(-0.0002,0.0001,0.0002) ---

-0.0008 , 0.0008

(-0.0001,0.0001,0.0002)

Number of Houses × 10-3 --- -0.119 , 0.465

(0.03,0.07,0.11) ---

Number of Apartments × 10-3 --- -0.126 , 0.164

(0.01,0.04,0.07) ---

Number of Townhouses × 10-3 --- -2.108 , 2.311

(-0.19,0.08,0.34) ---

Number of People Employed in Managements × 10-3 --- ---

-0.092 , 0.294

(0.01,0.08,0.15)

Number of People Employed in Manufactures-Constructions-Trades × 10-3

--- --- -0.134 , 0.680

(0.06,0.27,0.44)

GWPR AICc 17,667 17,103 17,065

GLM AICc 25,071 25,794 23,536


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Table 6-10 Regression Results Using GWPR Approach for with Land Use, Dwelling Unit and Employment Characteristics Based Models, S9-S11


S9 S10

ln (β0) -1.191 , 3.641

(0.20,0.85,1.84) -4.007 , 5.332

(-0.80,0.56,1.88)

ln (VKT) -0.080 , 0.466

(0.18,0.26,0.33) -0.259 , 0.744

(0.17,0.33,0.46)

Commercial (1000 m2) -0.0001 , 0.004

(0.001,0.0014,0.0019) ---

Residential (1000 m2) 0.00000 , 0.0008

(0.0003,0.0004,0.0005) ---


(0.0003,0.0005,0.0006) ---


-0.0009 , 0.002 (0.00002,0.0004,0.0009)

---

Other (1000 m2) -0.001 , 0.001

(-0.000,0.0001,0.00003) ---

Number of Houses × 10-3 --- -0.125 , 0.667

(0.04,0.08,0.12)

Number of Apartments × 10-3 --- -0.240 , 0.192

(0.01,0.04,0.09)

Number of Townhouses × 10-3 --- -2.90 , 2.71

(-0.22,0.12,0.42)

GWPR AICc 4,496 4,048

GLM AICc 6,087 6,162



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Table 6-11 Summary of Land use, Dwelling Unit and Employment Characteristics Based Models with Regard to Portions of TAZs with Unexpected Coefficient Signs for Total and

Severe Models


T9 T10 T11 S9 S10

ln (VKT) 5

(60%) 21

(19%) 0

(0%) 11

(82%) 38

(45%)

Commercial ( 1000 m2) 10

(50%) ---

4 (75%)

3 (82%)

---

Residential (1000 m2) 0

(0%) --- ---

0 (0%)

---

Resource & Industrial (1000 m2) 3

(100%) ---

14 (93%)

3 (100%)

---


--- --- --- 103

(75%) ---

Other (1000 m2) 176

(16%) ---

171 (12%)

197 (22%)

---

Number of Houses × 10-3 --- 66

(29%) ---

---

46 (67%)

Number of Apartments × 10-3 --- 87

(24%) ---

--- 88 (52%)

Number of Townhouses × 10-3 --- 182

(21%) ---

--- 181 (41%)

Number of People Employed in Managements × 10-3

--- --- 100

(43%)

--- ---


43 (51%)

--- ---



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6.5.4 GWPR Comprehensive Models

These models were calibrated using all components of planning variables, i.e., land use, network, traffic intensity and socioeconomic and demographic variables. Although they include all available planning variables, their potential for use in some other regions may be limited due to the relatively large numbers of explanatory variables and the practical reality that data for some of these independent variables may not be available. The 5-number summaries for the developed models in this section are presented in Table 6-12 for total and severe collisions.

Similar to the models presented in the previous section, unexpected signs for the coefficients pertaining to some TAZs were encountered. Table 6-13 presents the summaries pertaining to comprehensive models for portions of TAZs with unexpected negative coefficient signs for explanatory variables in both total and severe collisions. The results show that the t-values for most of the TAZs with negative coefficients are insignificant at the 90% confidence level.


- 128 -

Table 6-12 Regression Results Using GWPR Approach for Comprehensive Models


T12 S11

ln (β0) -0.181 , 4.414

(1.57,2.44,3.18) -1.162 , 3.297 (0.33,0.93,1.7)

ln (VKT) -0.020 , 0.502

(0.13,0.21,0.30) -0.053 , 0.437

(0.13,0.21,0.28)

Total Road Kilometers -0.015 , 0.075

(0.02,0.03,0.04) -0.013 , 0.069

(0.02,0.03,0.04)


-0.089 ,0.209 (0.04,0.09,0.12)

-0.055 , 0.182 (0.06,0.09,0.12)


-0.109 , 0.251 (0.07,0.12,0.15)

0.003 , 0.313 (0.10,0.13,0.16)

Commercial (1000 m2) -0.0007 , 0.006

(0.001,0.002,0.003) -0.0017 , 0.004

(0.000,0.000,0.0000)


(0.0000,0.0003,0.0004) -0.00079 , 0.001

(0.000,0.000,0.0000)


--- -0.0024 , 0.0018

(0.000,0.000,0.0000)

Other (1000 m2) -0.001 , 0.0009

(-0.000,0.000,0.00001) -0.0008 , 0.001

(0.000,0.000,0.0000)

Population × 10-3 -0.091 , 0.123

(-0.003,0.013,0.03) -0.044 , 0.1067

(-0.0009,0.009,0.03)

GWPR AICc 9,288 2,839

GLM AICc 18,073 4,338



- 129 -

Table 6-13 Summary of Comprehensive Models with Regard to Portions of TAZs with Unexpected Coefficient Signs for Total and Severe Models


T12 S11

ln (VKT) 7

(100%) 11

(100%)

Total Road Kilometers 25

(56%) 37

(100%)


54 (24%)

35 (66%)


25 (56%)

0 (0%)

Commercial (1000 m2) 5

(20%) 42

(69%)

Resource & Industrial (1000 m2) 111

(27%) 127

(65%)


--- 157

(70%)

Other (1000 m2) 299

(17%) 314

(34%)

Population × 10-3 134

(27%) 130

(61%) Number of TAZs with Unexpected Signs (Percentage of TAZs with Unexpected Signs and Insignificant at 90% Confidence Level)


- 130 -

CURE plots were produced for Model T12 for total collisions and Model S11 for severe collisions. The cumulative residuals are presented for the total and severe collisions in Figure 6-6. The

figures show that the cumulative residuals stay between the two standard deviation (±2σ*)

boundaries for both models. However, both models tend to underestimate the collision counts for lower traffic flows, and overestimate for higher traffic flows, suggesting that there might be a need to introduce additional explanatory variables which must be weighed against the disadvantages of over-fitting.

a) Total b) Severe

Figure 6-6 CURE Plot for Comprehensive Models Based on GWPR Approach

6.6 Discussion of Results for Local Safety Planning Models

A common technique used for the calibration of safety planning models is the GLM procedure with the assumption of NB or Poisson error distribution. In this technique, a fixed coefficient, which represents the average relationship between the dependent variable, typically number of collisions per TAZ, and each independent variable, is calibrated under the assumption that this relationship does not vary across space. However, collision frequency is influenced by many spatially defined factors.

The primary objective of this chapter was to investigate the local variations in the relationship between the number of zonal collisions and potential transportation planning predictors through the development of GWPR models. The main advantage of the GWPR models is that the model coefficient estimates vary locally. This is a very essential trait as it captures the uniqueness to the


- 131 -

spatial location of an event, such as collision data. It allows for a locally varying parameter representation of a particular point in space.

Although the developed local safety planning models seem to perform well in explaining the relationship of collisions with considered variables, there is still a need for further improvement since the GWPR models produce unexpected signs for the coefficients pertaining to some TAZs. Nevertheless, the developed GWPR models can be used as tools for predicting the number of zonal collisions for a future planning year if they are conceptually proven to be temporally transferrable. However, these models are not spatially transferable, since they produce a local coefficient for each TAZ in a specific geographic region. This is a relatively minor difficulty in that most jurisdictions tend to develop their own models.

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CCHHAAPPTTEERR SSEEVVEENN

7.0 DEVELOPMENT OF SPATIAL SAFETY PLANNING MODELS

7.1 Background

Recently, several attempts have been made to provide a regression framework in which spatial dependency of data is taken into account. This includes development of spatial collision prediction models since many factors affecting collisions have spatial attributes. Therefore, it is reasonable to explore the use of spatial collision prediction models for measuring the safety of a location.

While such models are generally not considered as local models in the same sense as GWR models presented in the previous chapter, they do recognize the local nature of spatial data by relaxing an assumption that the error terms for each observation are independent. The output from these models consists of a set of global parameter estimates similar to conventional normal regression. However, spatial relationships are incorporated into the modelling framework through the covariance of the error terms (Fotheringham et al., 2002). In this case, these models can be considered as “semi-local” as opposed to fully local models.

Use of Advanced Techniques to Estimate Zonal Level Safety Planning Models and Examine Their Temporal Transferability Chapter 7: Development of Spatial Safety Planning Models

- 133 -

Two modelling techniques were initially attempted for the development of the spatial safety planning models. First, a GEE procedure was used in order to consider spatial dependency of data for estimating the number of collisions. For the development of these models, the PROCGENMOD function in SAS (SAS Institute Inc.) was used. The model form considered for calibration of the GEE models is as follows:

( )zonepp XXVKTYLn β+β++β+β+β= ln......)ln()ln()( 2210 (7.1)

The model form is similar to Equation 5.2 which was used for the development of the GLM models. However, in Equation 7.1, the spatial effects of the TAZs are considered by introducing a spatial coefficient, ln(βzone), in the model. Unfortunately, it was found that the algorithms for these models do not reach convergence in the GEE framework. Therefore, no further analysis was conducted.

The FBSA technique was then attempted, using the software package, “BayesX” (BayesX 1.51), which can estimate structured additive regression models. MCMC simulation techniques were used for estimation of regression models.

In the beginning of this chapter, the conceptual framework for the development of the FBSA models is introduced. The model description along with all of the necessary conditional distributions for the parameters of the prior distributions (known as hyper-parameters), used for calibration of the spatial safety planning models, are introduced in a later section. Then, the results of safety planning models based on the FBSA technique are reported. At the end of the chapter, a detailed discussion of the results is provided.

7.2 Conceptual Framework of Full Bayesian Semiparametric Additive (FBSA) Models

As highlighted previously, full Bayesian semiparametric structured count data regression has been used for development of spatial models in this study. Miaou et al. (2003) emphasized that the overall strength of this approach is its ability to structure complicated models, inferential goals, and analysis. Bayesian inference is the process of fitting a probability model to a set of data and


- 134 -

summarizing the results by a probability distribution on the parameters of the model and unobserved quantities, such as predictions for new observations.

The Bayesian framework is based on determination of a prior distribution that represents the best guess about the parameters before incorporating information from the data. The prior can be either objective, e.g., non-informative or elicited from historical data or expert opinions. With the information gathering from samples, distribution of unknown parameters can be updated. The updated prior is referred as the posterior distribution. This information infusion is a major advantage of the Bayesian method. In general, the procedures of a Bayesian analysis can be divided into three steps:

choosing an appropriate prior distribution of unknown parameters;

getting information from samples, i.e., conditioning distribution of unknown parameters given observed data; and

evaluation and interpretation of the result.

Prior to discussing the conceptual framework of Bayesian semiparametric and its use in spatial modelling, the general Bayesian inference approach is presented.

7.2.1 Bayesian Inference

The Bayesian inference is based on the posterior distribution of model parameters.

( ) ( ) ( )( )zp

ypyzpzyp || = (7.2)

where y is the vector of parameters, z is the set of observed data, p(y | z) is the posterior distribution, p(z | y) is the likelihood function, and p(y) is prior distribution of y, and p(z) is the marginal distribution of data. The p(z) is of less interest and not essential in most cases. The key to a Bayesian setup is selecting an appropriate likelihood function and prior distributions. The posterior distribution p(y | z) combines information from both data (through likelihood function) and prior distribution.


- 135 -

In general, methods for summarizing posterior distributions are divided into two categories: EB and FB methods. Examples of EB methods include moment’s estimators, likelihood approximations and Laplace asymptotic integral approximations (Lawson et al., 2003). On the other hand, FB methods include maximum a posteriori estimation, estimation of posterior functional and posterior sampling. The advantage of the FB approach over the EB approach is that it takes into account the uncertainty associated with the estimates of the parameters and can provide exact measures of uncertainty.

7.2.2 Bayesian Structured Count Data Models

7.2.2.1 Semiparametric Model Forms

The discussion presented in this section is extracted from Fahrmeir and Osuna (2003), and Lang and Brezger (2000). In some of the subsections, excerpts from the above two documents are used and slightly modified for applicability to this study.

Consider regression collision data (yi,zi), i= I,…,n where yi are collision observations on a variable y and zi are observed values of an explanatory vector z. Similar to the previous developed models, the NB regression models are used as the starting point. Given the predictor

ηI and a scale or an overdispersion parameter δ>0, the NB model assumes conditionally

independent observations.

( )δμ≈δη ,,| iii NBy (7.3)

with probability function given by:

( )δ

⎟⎟⎠

⎞⎜⎜⎝

⎛δ+μ

δ⎟⎟⎠

⎞⎜⎜⎝

⎛δ+μ

μδΓ+Γ

δ+Γ=δη

i

y

i

i

i

iii

i

yy

yP)()1(

)(,| (7.4)

where

[ ] iiiyE μ=δη ,| (7.5)


- 136 -

and the variance is

δμ

+μ=δη2

),|( iiiiyVar (7.6)

It is worth mentioning that Equation 7.4 is the same as Equation 5.8 in Chapter Five. The

dispersion parameter (α) in Equation 5.8 is equivalent to the inverse function of δ (α=1/δ) in

Equation 7.4. The reason for this difference is that SAS, which is used for the development of the GLM models in this study, calculates the overdispersion parameter. However, BayesX (BayesX 1.51), which is used for development of the spatial models, calculates the inverse value of the overdispersion parameter.

Also, in order to modify the parametric linear predictor to a semiparametric structured additive

predictor, the linear predictor ηi has to be much more flexible. The basic structured additive

predictor ηi has the form:

( )ispati sf+γω=η ' (7.7)

i =1,…n. The ω’γ term in Equation 7.7 is the linear part of the predictor, with fixed effects. The

function fspat(s) represents the effect of TAZ s ∈ {1,….,S}, with S=481 TAZs for this study. The

spatial effect further is split up into the sum:

( ) ( ) ( )sfsfsf unstrstrspat += (7.8)

where

fspat(s) = total spatial function;

fstr(s) = structured or spatially correlated function; and

funstr(s) = unstructured or uncorrelated function.

As highlighted by Fahrmeir and Osuna (2003), the reason for this decomposition is that a spatial effect is usually a proxy of many basic unobserved important factors. Some of them may be present only locally, while others are correlated with neighboring effects. By estimating structured


- 137 -

and unstructured spatial effects, both kinds of influential factors are taken into account. Also by comparing values of these two effects, the spatial dependency of the data can be assessed. When the structured covariates are larger than the unstructured covariates, this can be an indication that local effects are much smaller than the spatial effects and vice-versa.

7.2.2.2 Prior Dispersion Parameter

As mentioned previously, the Bayesian framework is based on determination of a prior distribution. In a Bayesian approach, priors have to be assigned to all unknowns. Priors for the dispersion

parameter δ>0 are assumed to be the Gamma form.

( )baG ,≈δ (7.9)

with density

( ) ( ) ( )δ−δΓ

=δ − babg aa

exp1 (7.10)

and with mean and variance

[ ]baE =δ [ ] 2b

aV =δ (7.11)

“The parameters a and b can be chosen such that the Gamma distribution has a flat prior. As a more data driven alternative, they are considered hyper-parameters and introduced as a hyper-prior in a further stage of the hierarchy” (Fahrmeir and Osuna, 2003). For this study, the precision of the overdispersion parameter is assigned a prior gamma distribution G (1, 0.001).

7.2.2.3 Prior for Fixed Effects

In the following, f denotes the vectors of all function evaluations, including structured and

unstructured spatial effects, ϕ is the vector of all variances.


- 138 -

Random variables with suitable priors are considered for Bayesian inference of all functions and

parameters in the predictor. To formulate priors in summarized notation, the predictor vector η=(ηi)

is expressed in matrix notation by:

unstrstr ffW ++γ=η (7.12)

where fstr, funstr are the vectors of corresponding function values and W=(ωi) is the design matrix for

fixed effects. Each function vector can always be expressed as the product of a design matrix X and a high-dimensional parameter vector β. Using f=Xβ as a generic notation for functions, Equation 7.7 becomes:

γβη W....X.... +++= (7.13)

A vague non-informative normal prior distribution is assigned to γ and a uniform prior distribution is

assigned to the intercept. The reason for assigning a vague non-informative distribution to γ is that

its exact distribution is not known due to lack of any empirical or theoretical information about this unknown parameter.

Unstructured random effects (funstr) are modelled through independently and normally distributed random effects as shown.

( )2unstr

2unstrunst ,0N)s(f ϕϕ ≈ (7.14)

where

2unstrϕ = the variance of unstructured random effects.

Structured spatial effects are modelled through Markov random field priors in which spatial neighbourhood relations are considered. Constructions of the design matrix X and priors depend on the type of the function and degree of smoothness. Priors for the vectors β for spatial covariates in Equation 7.13 have the general Gaussian form:


- 139 -

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ϕββ′

−∝ϕβ 22

2exp| Kp (7.15)

The penalty matrix K penalizes roughness of the function and its structure depends on the type of covariate and smoothness of the function. The variance φ2, which corresponds to the inverse of a smoothing parameter, controls the degree of smoothness. This can be accomplished by the trade–off between data fit and smoothness.

For a spatial covariate s, the values of s represent the locations or TAZs in connected geographical

regions and spatial neighbourhood relationships. It is assumed s∈ {1,…,S}, i.e., that the regions

are labelled by the numbers 1,…,S . Spatial covariates are calculated based on the assumption that neighbouring sites are more alike than two arbitrary sites. Therefore, for a valid prior definition a set of neighbours for each site, s must be defined. For geographical data, it is usually assumed that the two sites s and j are neighbours if they share a common boundary.

The simplest spatial smoothness prior for the function values fstr(s) = βs is:

⎟⎟⎠

⎞⎜⎜⎝

⎛ ϕβϕ≠ββ ∑

∂∈ sj sj

sjs NN

Nsj2

2 ,1~,| (7.16)

where Ns is the number of neighbouring sites and j∈∂s denotes that site j is a neighbour of site s.

Therefore, the mean of βs is an unweighted average of function evaluations of neighbouring sites which is known as a Gaussian intrinsic autoregression (Besag et al., 1991 and Besag and Kooperberg, 1995). The vector β = (β1,..,βs,..,βS)’ of spatial effects has a joint–distribution of the

form of Equation 7.15 with the elements of K defined by kss =ωs+, ksj=-ωsj for j∈δs, where ωs = 1

and + denotes summation over the missing subscript, and 0 else.

A more general prior including Equation 7.16 as a special case is given by:

⎟⎟⎠

⎞⎜⎜⎝

⎛≠ ∑

∂∈ ++sj s

2

js

sj2js ,N~,sj|

ωϕβ

ωω

ϕββ (7.17)


- 140 -

where ωsj are weights and + denotes summation over the missing subscript.

If the effect fstr of the spatial covariate s is expressed as the product of a design matrix X and the vector of unknown parameters β, then X is a 0/1 incidence matrix.

7.2.2.4 Markov Chain Monte Carlo Methods

As explained previously, an important feature of the Bayesian inference is that the posterior distribution is centered at a point that represents a compromise between the prior information and data, and the compromise increasingly is controlled by the data as the sample size increases.

It is almost impossible to usefully characterize posterior distributions analytically because they can be very complex in a high dimensional space. However, a sample of points drawn from such a distribution can provide a satisfactory picture. In particular, from such a sample, we can obtain Monte Carlo estimates for the expectations of various random functions of the variables.

MCMC methods allow the drawing of random numbers from the numerically intractable posterior distribution and in this way, the estimation of characteristics of the posterior, such as means, standard deviations or confidence intervals via their empirical analogue. The main idea is that instead of drawing directly from the posterior which is impossible in most cases, a Markov chain is created whose iterations of the transition kernel converge to the posterior (Lang and Brezger, 2000).

MCMC techniques (Gelfand and Smith, 1990 and Gilks et al., 1996) are used to generate samples from the posterior distributions. MCMC methods use computer simulation of Markov chains in the parameter space. The basic algorithms used in Markov chain simulation are the Metropolis and its extension, the Metropolis-Hastings (MH) and Gibbs sampler. Computation details about the MH and Gibbs sampler algorithms can be found in Green (2001).

7.2.2.5 Posterior Inference

Bayesian inference is the process of fitting a probability model to a set of data and summarizing the result by a probability distribution on the parameters of the model and unobserved quantities, such


- 141 -

as predictions for new observations. This is based on the posterior distribution of the usually very high–dimensional vector of all parameters. In the following, this vector is divided into the sub vector

ξ containing all parameters defining the predictor η, and remaining parameters specifying the

collision data distributions for given η. MCMC inference is repeatedly utilized to draw from full

conditionals of blocks of parameters given the remaining parameters and data. This results in the updating of each model separately.

For the models in this study, the posterior is defined by:

( ) ( ) ( ) ( ) ( )ξδδξ∝δξ pbgbgyPybp |,||,, (7.18)

where

P(y|ξ,δ) = likelihood of the NB model;

g(δ|b) = prior of the scale parameters δ given by Equation 7.10;

g(b) = hyper-prior (Equation 7.11) for b; and

p(ξ) = defined by the prior assumptions in the previous section.

Then the full conditional:

( ) ( ) ( )bgyPp |,|...| δδη∝δ (7.19)

( )( ) ( ) ( ) ( )δ−δ

Γ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛δ+μ

δ⎟⎟⎠

⎞⎜⎜⎝

⎛δ+μ

μδΓ+Γ

δ+Γ∝ −

=

δ

∏ bab

yy a

an

i i

y

i

i

i

ii

exp1

1

1

( )( )

( ) ( )δ−δδΓ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

δ+μ

δ+Γ∝ −+δ

=+δ∏ b

y annn

iy

i

ii

exp1

1


- 142 -

As highlighted by Fahrmeir and Osuna (2003), this expression has no analytical closed form, so

that an MH algorithm is implemented with a random walk proposal. Let δ* denote the proposed

value for δ in an iteration step. A Gamma function is selected as follows:

⎟⎟⎠

⎞⎜⎜⎝

⎛ δδδδ

δδ ppG ,~|*

2

(7.20)

with

[ ] δ=δδ |*E [ ] δ=δδ pV |* (7.21)

The parameter pδ is a tuning parameter that allows us to control the acceptance probability for the

MH algorithm. It is adapted in the burn in period to achieve acceptance probabilities for δ between

0.4 and 0.6.

For the hyper-parameter b, the full conditional is calculated as follows:

( ) ( ) ( )bgbgbp |...| δ∝ (7.22)

( ) ( ) ( ) ( )bbbab aa

21

1

21 expexp 11

θ−θΓθ

δ−δΓ

= −θθ

−

( )( )bexpb 21a1 θδθ +−∝ −+

( )21 , θ+δ+θ∝ aG

where the standard choice is a=1, θ1=1, θ2=0.001. Therefore, b can be updated in a Gibbs step.

7.3 Calibration of Spatial Safety Planning Models

The “bayesreg objects” function in the “BayesX” software (BayesX 1.51) was used to fit the FBSA safety planning models. The spatial safety planning models developed in this study are of the form:


- 143 -

( ) ( )TAZfTAZfX......X)VKTln(β)βln()Y(Ln unstrstrpp2210 ++++++= ββ

(7.23)

The model form is similar to Equation 5.2 which was used for the development of the GLM models. However, in Equation 7.22, the spatial effects of the TAZs are considered by using a spatially correlated part fstr(TAZ) and uncorrelated part funstr(TAZ) for TAZs. The rationale for this decomposition of the spatial effects is that a spatial effect is usually a surrogate for many underlying unobserved influential factors. Some of them may be present only locally, while others are correlated with neighboring effects. By estimating a structured and an unstructured spatial effect, both kinds of influential factors are taken into account (Fahrmeir and Osuna, 2003).

The “map objects” function in “BayesX” software (BayesX 1.51) was used to handle and store geographical maps. The main purpose of map objects in this context is to provide the neighborhood structure of the map, and compute associated weights. To create a map, the boundary information of a geographical map is read from an external file and stored in the map object. Based on the boundary information, the “map object” automatically computes the name of the object, number of regions, minimum and maximum number of neighbors and bandwidth of the corresponding adjacency or neighborhood matrix (Lang and Brezger, 2000). Figure 7-1 shows the “map object” information for the study area.


- 144 -

Figure 7-1 The “map object” Information for the Study Area Using BayesX

7.4 Results of Spatial Safety Planning Models

Several safety planning models were developed based on the FBSA regression technique using the “BayesX” software (BayesX 1.51). The spatial safety planning models included all the variables that were found significant in the GLM models as well as spatial effects.

Similar to the previous two chapters of this dissertation, models with traffic intensity as independent variable are first calibrated due to VKT’s dominating prediction influence on collision data. Then, a series of safety planning models are developed using the zonal road network variables. In this section, the relationship between collision frequency and land use, dwelling unit and employment is explored. In addition, comprehensive models are calibrated using all of the variables, including those related to land use, network, traffic intensity and socioeconomic and demographic characteristics.


- 145 -

7.4.1 FBSA Model with Traffic Intensity as Independent Variable

The posterior summary of the model parameters for the traffic intensity-based models for total (T1) and severe (S1) collisions is presented in Table 7-1. The results of the FBSA models include a set of global variables, which represent the average relationship between the dependent variable and each explanatory variable, and a spatial covariate. Typically, the effect of such a spatial covariate is incorporated into a model via an unstructured (funstr) and a structured (fstr) random effect. In the latter, a spatial smoothness prior for the spatial covariate is specified which penalizes overly quick changes of the estimated effect in neighbouring TAZs (Lang and Brezger, 2000).

The summary of the model parameters for global variables and variance of spatial covariates in Table 7-1 presents the posterior mean, posterior standard deviation, and 95% confidence level. The 95% confidence level is used to assess whether a parameter is significant at the 95% confidence level. If the 95% confidence level includes zero, the corresponding factor is not significant. The summary for variance of spatial covariates in Table 7-1 presents dispersion parameters for each model.


- 146 -

Table 7-1 Regression Results Using FBSA Approach for Traffic Intensity Based Models


T1 S1

ln (β0) 2.56, 0.252 (2.07, 3.07)

1.26, 0.274 (0.73, 1.79)

ln (VKT) 0.278, 0.029 (0.24, 0.36)

0.278, 0.032 (0.21, 0.34)

α 0.289 0.306

Variance of fstr 0.101, 0.049 (0.02, 0.24)

0.084, 0.046 (0.023, 0.204)

Variance of funstr 0.011, 0.01

(0.0009, 0.05) 0.017, 0.020

(0.0004, 0.071) Mean, Standard Deviation

(95% Confidence Level)

The posterior summary from Table 7-1 indicates that the VKT is significant at the 95% confidence level for both total and severe models. A comparison of VKT coefficients from the FBSA models with the VKT coefficients from the GLM models in Table 5-1 indicates that the coefficients are very similar.

The variances of structured and unstructured covariates for both total and severe models are significant since their 95% confidence level do not cover zero. Figures 7-2 and 7-3 show the posterior means for structured, unstructured, and total spatial covariates for both total and severe collision models for each TAZ, respectively. The posterior means for the structured and total spatial covariates display a clear regional pattern. This spatial structure indicates rather higher prevalence of collision occurrences in TAZs of CBD and those of northwest and mid-east areas as opposed to the TAZs in other parts of the city. Also, posterior means for the structured covariates have greater values than corresponding unstructured covariates. This observation indicates that number of collision occurrences in each TAZ is spatially correlated. In other words, underlying unobserved


- 147 -

influential factors on the prevalence of collision occurrences in each TAZ are to a greater extent, attributed to the effects of neighbouring TAZs than local effects.


- 148 -

a) Unstructured b) Structured

c) Total

Figure 7-2 Posterior Mean for Spatial Covariates of Intensity Based Model, T1


- 149 -


c) Total

Figure 7-3 Posterior Mean for Spatial Covariates of Intensity Based Model, S1


- 150 -

Figure 7-4 illustrates the cumulative residual plots for the variable VKT for the total and severe collision models. These figures show that the severe collision model fits the data with better accuracy than total collisions. However, the cumulative residuals for both models oscillate around

the value of 0 and lie between the two standard deviation (±2σ*) boundaries. Therefore, it can be

concluded that the assumption of a non-linear relationship between collisions and traffic intensity is appropriate.

a) Total b) Severe

Figure 7-4 CURE Plot for Traffic Intensity Models Based on FBSA Approach

7.4.2 FBSA Models with Network Characteristics as Independent Variables

Several models were developed using the TAZ road network variables. These network-based models rely primarily on explanatory variables describing the TAZ road network. Tables 7-2 and 7-3 present the posterior means, posterior standard deviation, and 95% confidence level for total collisions. Tables 7-4 and 7-5 present the posterior summary for severe collisions.


- 151 -

Table 7-2 Regression Results Using FBSA Approach, Models T2-T4


T2 T3 T4

ln (β0) 3.551, 0.26

(3.07, 4.05)

2.503, 0.25

(2.02, 3.00)

2.456, 0.23

(2.04, 2.95)

ln (VKT) 0.075, 0.03

(0.02, 0.13)

0.234, 0.03

(0.18, 0.29)

0.237, 0.03

(0.18, 0.29)

Total Arterial Road Kilometers 0.232, 0.02

(0.20, 0.27) --- ---

Total Expressway Kilometers 0.089, 0.02

(0.05, 0.13) --- ---

Total Collector Kilometers 0.085, 0.02

(0.05, 0.12) --- ---

Total Laneway Kilometers 0.080, 0.03

(0.03, 0.13) --- ---

Total Local Road Kilometers -0.004, 0.01

(-0.01, 0.006) --- ---

Total Ramp Kilometers 0.109, 0.03

(0.05, 0.17) --- ---

Total Road Kilometers --- 0.043,0.004

(0.03, 0.05) ---


0.144, 0.01

(0.12, 0.17)


0.167, 0.02

(0.13, 0.21)

α 0.131 0.199 0.086

Variance of fstr 0.205, 0.052

(0.126, 0.327)

0.235, 0.066

(0.123, 0.38)

0.131, 0.057

(0.040, 0.257)


(0.0006, 0.041)

0.007, 0.007

(0.0004, 0.029)

0.067, 0.039

(0.0008, 0.138)


- 152 -



T5 T6 T7 T8

ln (β0) 2.477, 0.24 (2.02, 2.96)

2.441, 0.23 (1.99, 2.89)

2.37, 0.23 (1.93, 2.82)

2.450, 0.22 (2.00, 2.87)

ln (VKT) 0.231, 0.03 (0.17, 0.29)

0.199, 0.03 (0.15, 0.25)

0.21, 0.03 (0.16, 0.27)

0.206, 0.03 (0.15, 0.26)

Total Road Kilometers --- 0.027, 0.00 (0.02, 0.03)

0.02, 0.00 (0.02, 0.03)

0.027, 0.00 (0.02, 0.03)


0.127, 0.01 (0.10, 0.15)

0.11, 0.01 (0.09, 0.14)

0.118, 0.01 (0.09, 0.14)


0.151, 0.02 (0.11, 0.19)

0.15, 0.02 (0.11, 0.18)

0.148, 0.02 (0.11, 0.18)


0.155, 0.01 (0.13, 0.17)

--- --- ---

Total Rail Kilometers --- --- --- -0.012, 0.03 (-0.07, 0.05)

Number of Schools --- --- 0.02, 0.01

(0.003, 0.05) ---

α 0.072 0.068 0.118 0.095


(0.047, 0.302) 0.179, 0.053

(0.083, 0.288) 0.183, 0.060 (0.093, 0.33)

0.183, 0.065 (0.08, 0.34)


(0.020, 0.174) 0.139, 0.025

(0.086, 0.188) 0.026, 0.0299 (0.001, 0.11)

0.007, 0.008 (0.0007, 0.03)

Mean, Standard Deviation (95% Confidence Level)


- 153 -

Table 7-4 Regression Results Using FBSA Approach, Models S2-S4


S2 S3 S4

ln (β0) 2.201, 0.26

(1.67, 2.70)

1.164, 0.27

(0.65, 1.68)

1.073, 0.25

(0.56, 1.56)

ln (VKT) 0.058, 0.03

(0.001, 0.12)

0.218, 0.03

(0.15, 0.28)

0.222, 0.03

(0.16, 0.28)

Total Arterial Road Kilometers 0.250, 0.02

(0.22, 0.29) --- ---

Total Expressway Kilometers 0.114, 0.02

(0.07, 0.16) --- ---

Total Collector Kilometers 0.060, 0.02

(0.02, 0.10) --- ---

Total Laneway Kilometers 0.077, 0.03

(0.03, 0.13) --- ---

Total Local Road Kilometers -0.002, 0.006

(-0.01, 0.01) --- ---

Total Ramp Kilometers 0.065, 0.03

(0.01, 0.12) --- ---

Total Road Kilometers --- 0.044, 0.004

(0.04, 0.05) ---


0.148, 0.01

(0.12, 0.17)


0.176, 0.02

(0.13, 0.22)

α 0.138 0.219 0.069


(0.105, 0.308)

0.195, 0.058

(0.10, 0.32)

0.121, 0.055

(0.04, 0.26)


(0.0006, 0.021)

0.006, 0.007

(0.0006, 0.01)

0.10, 0.054

(0.010, 0.202)


- 154 -



S5 S6 S7 S8

ln (β0) 1.084, 0.24 (0.60, 1.57)

1.036, 0.23 (0.60, 1.51)

0.956, 0.24 (0.45, 1.42)

1.1016, 0.24 (0.54, 1.51)

ln (VKT) 0.222, 0.03 (0.16, 0.28)

0.199, 0.03 (0.14, 0.25)

0.206, 0.03 (0.15, 0.27)

0.200, 0.03 (0.14, 0.26)

Total Road Kilometers --- 0.028, 0.004 (0.02, 0.04)

0.026, 0.004 (0.02, 0.03)

0.028, 0.004 (0.02, 0.04)


0.116, 0.01 (0.09, 0.14)

0.116, 0.01 (0.09, 0.14)

0.119, 0.01 (0.09, 0.15)


0.158, 0.02 (0.12, 0.20)

0.156, 0.02 (0.12, 0.19)

0.157, 0.02 (0.12, 0.20)


0.154, 0.01 (0.13, 0.18)

--- --- ---

Total Rail Kilometers --- --- --- -0.013, 0.03 (-0.07, 0.05)

Number of Schools --- --- 0.023, 0.01 (<0.0, 0.05)

---

α 0.091 0.133 0.135 0.116

Variance of fstr 0.146, 0.060 (0.050, 0.28)

0.145, 0.05 (0.062, 0.25)

0.151, 0.05 (0.065, 0.27)

0.148, 0.049 (0.052, 0.266)


(0.001, 0.175) 0.026, 0.027

(0.001, 0.098) 0.019, 0.027

(0.0006, 0.09) 0.009, 0.010

(0.0009, 0.037) Mean, Standard Deviation (95% Confidence Level)


- 155 -

Similar to the GLM models, the FBSA models revealed that collision frequency increases with the increase in the following explanatory variables: total arterial road, collector, laneway, ramp, and road kilometers, number of signalized intersections and 4-legged and 3-legged signalized intersections, and number of schools in each TAZ. However, the total rail and total local road kilometers were found to be inversely related to collision frequency. A comparison of coefficients in the FBSA models with the corresponding coefficients in the GLM models indicated that they are roughly the same values with the same sign. As well, all of the variables that were significant in the GLM models are also significant in the FBSA models except for total local road kilometers. As highlighted by Aguero-Valverde and Jovanis (2006), the possible reason is that frequentist models, such as the GLM models developed in Chapter Five, do not take into consideration all sources of uncertainty for estimation of the variance-covariance matrix. Consequently, the standard errors in the GLM model are frequently underestimated. On the other hand, FBSA models consider all sources of uncertainty. Therefore, the standard errors for these models are higher. It has to be pointed out here that the variable for total rail kilometers is not significant in both the GLM and the FBSA models.

The dispersion parameters (α) for all the models are shown in Tables 7-2 to 7-5. The dispersion

parameters for both models (total and severe) are significantly different from 0 which confirms the appropriateness of an NB regression. The variances for structured and unstructured covariates for all the models are significant since their 95% confidence level do not include zero. Also, the variances due to structured covariates for all the models are higher than the variances due to unstructured covariates.

The spatial distribution of posterior means for structured, unstructured, and total spatial covariates for all the models in this section is shown in Appendix C.


- 156 -

7.4.3 FBSA Models with Land Use, Dwelling Unit and Employment Characteristics as Independent Variables

In this section, the relationship between collision frequency and land use, dwelling unit and employment is explored. Tables 7-6 and 7-7 present the posterior means, posterior standard deviation, and 95% confidence level for the total and severe collision models, respectively.

Models T9 and S9 explore the relationship between the number of collisions and land use types. It was found that all of the variables, except for the “other” land use variable, have positive effects on the number of collisions in each TAZ. The “other” land use variable includes all the land use types, such as parks, open space etc. Since these areas do not generate as many trips as the other land use types, the traffic exposure and consequently, number of collisions are relatively low. This can be a possible reason for the inverse relationship between the number of collisions and the “other” land use variable in the models. Another interesting observation from these models is that the coefficients that remain significant in the FBSA models are those which are also highly significant (high chi-square values) in the GLM models.

Models T10 and S10 explore the relationship between the number of collisions and dwelling units. For both total and severe collisions, these models show positive relationships between collisions and type of dwelling units. It must be noted here that the number of townhouses for Models 10 and 10S were found to be statistically insignificant since the 95% confidence level is between a negative and positive value.

For total collisions, Model T11 was calibrated using land use data and number of people living in each TAZ categorized by employment type. The coefficients for the variables that are significant in both FBSA and GLM models are very similar.

The variances for structured and unstructured covariates for all the models are significant since their 95% confidence level does not include zero. Also, the variances due to the structured covariate for all the models are higher than the variances due to the unstructured covariate. Again, this indicates that there is a spatial dependency in the data and structured covariates should be considered in the model.


- 157 -



T9 T10 T11

ln (β0) 2.378, 0.28

(1.84, 2.91)

2.249, 0.27

(1.72, 2.78)

2.389, 0.25

(1.90, 2.89)

ln (VKT) 0.267, 0.03

(0.20, 0.33)

0.304, 0.03

(0.24, 0.36)

0.262, 0.03

(0.20, 0.32)

Commercial ( 1000 m2) 0.001, 0.00

(>0.00, 0.002) ---

0.001, 0.0003

(>0.00, 0.002)

Residential (1000 m2) 0.0004, 6.33E-05

(>0.00, >0.00) --- ---

Resource & Industrial (1000 m2) 0.0004, 7.49E-05

(>0.00, >0.00) ---

0.0004, 6.73E-05

(>0.00, 0.0005)

Other (1000 m2) -7.89E-06, 4.90E-05

(<0.00, >0.00) ---

-2.47E-05, 4.61E-05

(<0.00, 6.75E-05)

Number of Houses × 10-3

---

0.055, 0.01

(0.027, 0.08) ---

Number of Apartments × 10-3 --- 0.040, 0.01

(0.02, 0.06) ---

Number of Townhouses × 10-3 --- 0.077, 0.09

(-0.09, 0.25) ---

Number of People Employed in Managements × 10-3 --- --- 0.134, 0.04

(0.06, 0.21)


0.300, 0.07

(0.18, 0.43)

α 0.229 0.250 0.226


(0.06, 0.30)

0.137, 0.047

(0.057, 0.255)

0.140, 0.0569

(0.05, 0.271)


(0.0006, 0.043)

0.074, 0.044

(0.008, 0.155)

0.0064, 0.008

(0.0004, 0.030)


- 158 -



S9 S10

ln (β0) 0.991, 0.28

(0.43, 1.53)

0.896, 0.27

(0.37, 1.41)

ln (VKT) 0.257, 0.03

(0.19, 0.32)

0.288, 0.03

(0.23, 0.35)

Commercial (1000 m2) 0.001, 0.0003

(0.0005, 0.001) ---

Residential (1000 m2) 0.0004, 6.32E-05

(0.0003, 0.0005) ---


(0.0002, 0.0005) ---


(1000 m2)

0.0002, 0.0001

(-3.47E-05, 0.0005) ---

Other (1000 m2) -3.454E-05, 5.98E-05

(-0.0002, 8.33) ---

Number of Houses × 10-3 --- 0.058, 0.015

(0.03, 0.09)

Number of Apartments × 10-3 --- 0.041, 0.01

(0.02, 0.06)

Number of Townhouses × 10-3 --- 0.112, 0.09

(-0.06, 0.28)

α 0.254 0.274


(0.033, 0.221)

0.138, 0.057

(0.056, 0.281)


(0.0004, 0.05)

0.025, 0.022

(0.0009, 0.082)


- 159 -

7.4.4 FBSA Comprehensive Models

Similar to GLM and GWPR approaches, a separate comprehensive model using land use, network, traffic intensity and socioeconomic and demographic variables for total and severe collisions was calibrated. The posterior summary of the model parameters for the comprehensive models for total (T12) and severe (S11) collisions are presented in Table 7-8.

All of the variables, with the exception of “other”, show positive and significant relationships with collision frequency at the 95% confidence level for both total and severe collisions. The “other” variable for both models is the total area of open space and parks. This variable was found to have a negative, but significant association to collision data. The dispersion parameter values for both models are different from 0, which confirms that the data are overdispersed relative to the Poisson distribution and that the NB error structure is justified.


- 160 -

Table 7-8 Regression Results Using FBSA Approach for Comprehensive Models


T12 S11

ln (β0) 2.446, 0.22

(2.05, 2.91)

0.986, 0.229

(0.54, 1.45)

ln (VKT) 0.195, 0.03

(0.14, 0.25)

0.198, 0.03

(0.14, 0.25)

Total Road Kilometers 0.023, 0.004

(0.01, 0.03)

0.025, 0.004

(0.02, 0.03)


0.111, 0.01

(0.09, 0.14)

0.108, 0.01

(0.09, 0.13)


0.143, 0.02

(0.11, 0.18)

0.149, 0.02

(0.11, 0.19)

Commercial (1000 m2) 0.0007, 0.0002

(0.0003, 0.001)

0.0005, 0.0002

(5.36E-05, 0.001)


(8.69E-05, 0.0003)

0.0002, 6.09E-05

(4.22E-05, 0.0003)


(1000 m2) ---

0.0002, 0.0001

(6.90E-05, 0.0004)

Other (1000 m2) -8.765E-05, 3.96E-05

(-0.000, -9.28E-06)

-0.0001, 5.3E-05

(-0.0002, -1.41E-05)

Population × 10-3 0.019, 0.007

(0.005, 0.03)

0.016, 0.008

(0.002, 0.03)

α 0.059 0.133


(0.063, 0.33)

0.155, 0.056

(0.07, 0.291)


(0.002, 0.11)

0.036, 0.031

(0.001, 0.11)


- 161 -

Figures 7-5 and 7-6 show the posterior means for structured, unstructured, and total spatial covariates for both total and severe collision models in each TAZ, respectively. Similar to Figures 7-2 and 7-3 for the traffic intensity models, the posterior means for the structured and total spatial covariates display a clear regional pattern. This spatial structure indicates rather higher prevalence of collision occurrences in TAZs of CBD and those of northwest and mid-east areas as opposed to the TAZs in other parts of the city. Also, posterior means for the structured covariates have greater values than corresponding unstructured covariates. From this result, it can be concluded that the number of collision occurrences in each TAZ is spatially correlated.


- 162 -


c) Total

Figure 7-5 Posterior Means for Spatial Covariates of Comprehensive Model, T12


- 163 -


C) Total

Figure 7-6 Posterior Means for Spatial Covariates of Comprehensive Model, S11


- 164 -

CURE plots were produced for Model T12 for total collisions and Model S11 for severe collisions and are shown in Figure 7-7. The figures show that both models tend to underestimate the collision counts for higher range of VKTs, suggesting that there may be a need to introduce additional explanatory variables which must be weighed against the disadvantages of over-fitting.

a) Total b) Severe

Figure 7-7 CURE Plot for Comprehensive Models Based on FBSA Approach

7.5 Discussion of Results of the Spatial Safety Planning Models

This chapter has conducted a spatial data analysis of collisions in TAZs within the City of Toronto by developing a series of spatial safety planning models. Twenty-three safety planning models are developed using land use, traffic intensity, road network, and socioeconomic and demographic zonal characteristics as explanatory variables. A FBSA approach is employed in which NB regression models are developed separately for total and severe (fatal and non fatal injuries) collisions. The flexibility of the full Bayesian method allows the construction of sophisticated models. The application of the MCMC technique makes it possible for estimation of the posterior inference which is intangible for complex models. The FBSA models allow analyzing the effects of explanatory various and spatial effects in much more detail than traditional GLM approaches. As will be explained in the next chapter, FBSA models perform better than the traditional GLM models and in some cases, even better than the GWPR models.

The results of developed FBSA models in this chapter indicate that FBSA and GLM models are mostly consistent in the direction and magnitude of the effects of the covariates for predicting number of collisions in a TAZ. In general, highly significant variables in the GLM models are also significant in the FBSA models. On the other hand, variables just marginally significant in the GLM


- 165 -

models (lower chi-square values) normally are non-significant in the FBSA models. A possible reason is that the GLM models do not take into consideration all sources of uncertainty for estimation of the variance-covariance matrix. Consequently, the standard errors in the GLM model are frequently underestimated.

The results of the posterior means for structured, unstructured, and total spatial covariates indicate that the number of collision occurrences in each TAZ is spatially correlated. Spatial correlation is significant in all of the models since the variance of structured covariate is different from zero. The variance of the spatially correlated term is significant, which implies that some of Poisson extra variation in the data can be explained by spatial correlation. Also, posterior means for the structured covariates have greater values than corresponding unstructured covariates. This supports the above discussion.

- 166 -

CCHHAAPPTTEERR EEIIGGHHTT

8.0 COMPARISON OF PERFORMANCE OF SAFETY PLANNING

MODELS

GOF statistics provide an ability to objectively assess the fit of a model to data. With the various model specifications determined, measures for evaluating the performance of the GLM, GWPR and FBSA models have to be defined. These measures were used to compare the accuracy of the GWPR and FBSA models to that of GLM models and each other for predicting the number of collisions for each TAZ. A number of such measures for assessing the performance of different model forms have been formulated in a number of papers. These measures are similar to those used by Oh et al. (2003). The following definitions draw considerably from their work.

8.1 Goodness-of-Fit Measures for Comparative Analysis

8.1.1 Pearson Product- Moment Correlation Coefficient

As indicated by Oh et al. (2003), the Pearson product-moment correlation coefficient is a measure of the linear relationship between two variables. A coefficient of exactly 1 for a model indicates that

Use of Advanced Techniques to Estimate Zonal Level Safety Planning Models and Examine Their Temporal Transferability Chapter 8: Comparison of Performance of Safety Planning Models

- 167 -

the model predicts the observed data perfectly. Conversely, a linear correlation coefficient of 0 suggests a complete lack of linear association between observed and predicted variables. It is defined as:

( )( )

( ) ( )[ ] 5.022 ˆˆ

ˆˆ

∑ ∑∑

−−

−−=

avgiavgi

avgiavgi

YYYY

YYYYr (8.1)

=iY observed number of collisions at TAZ i;

=iY predicted number of collisions at TAZ i;

=avgY mean predicted number of collisions per TAZ; and

=avgY mean observed number of collisions per TAZ.

It is worth noting that the random sampling error, which is to be expected, will not reduce the correlation coefficient significantly.

8.1.2 Mean Absolute Deviation (MAD)

Mean absolute deviation (MAD) is the sum of the absolute value of predicted number of collisions by the models for each TAZ minus corresponding observed number of collisions divided by the data sample size. This measure provides the average “misprediction” of the model. A value close to zero suggests that the model on average predicts the observed data well. It is defined as:

nYYMAD i

n

ii −=∑

=1

ˆ( (8.2)

n = data sample size

The MAD gives a measure of the average magnitude of variability of prediction. Smaller values are preferred to larger values.


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8.1.3 Mean Squared Prediction Error (MSPE) and Mean Squared Error (MSE)

The mean squared prediction error (MSPE) is the sum of the squared differences between observed and predicted collisions divided by the sample size. It is defined as:

nYYMSPEn

iii∑

=

−=1

2)ˆ( (8.3)

The mean squared error (MSE) is the sum of the squared differences between observed and predicted collisions divided by the sample size minus the number of model parameters. MSE is typically a measure of model error associated with the calibration or estimation data, and so,

degrees of freedom are lost (p) as a result of producing iY , the predicted response. It is defined

as:

)()ˆ(1

2 pnYYMSEn

iii −−=∑

=

(8.4)

where

p=number of model parameters

A comparison of MSPE and MSE shows potential over or under fitting of the models to the estimation data. If the MSE of a model is higher than the MSPE of another, this indicates that the first model may have been over fitted to the estimation data and some of the observed relationships may have not been correctly shown in the model.

8.2 Comparative Analysis of GLM, GWPR and FBSA Models

Tables 8-1 and 8-2 present the relative GOF measures for predicting the number of collisions using FBSA, GWPR, and GLM NB and Poisson regression models for total and severe collisions, respectively. Large differences in GOF measures of subject models are indicative of their significant differences in performance in terms of predicting the number of collisions.


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The results of Pearson product-moment correlation coefficient coefficients (r) show the improved performance of all GWPR and FBSA models over the GLM models. Mixed results were obtained when the linear correlation coefficients of the GWPR and FBSA models were compared to each other. For the network characteristics based models in total collisions, the FBSA models perform much better than the GWPR models.

Similar conclusions can be derived when the MAD values from the GWPR and FBSA models are compared to those obtained from the GLM regression models. For some of the models, such as Model T5, the improvement achieved with the GWPR is especially substantial. Again, mixed results were obtained when the MAD values of the GWPR and FBSA models were compared to each other. It seems that the GWPR models in most of the cases are performing better than the FBSA model. However, the difference between MAD values from these models is very marginal to come up with a conclusive result.

As indicated in the previous section, a comparison of the MSPE and MSE shows potential for over or under fitting of the models to the estimation data. If the MSE of a model is higher than the MSPE of another, this indicates that the first model may have been over fitted to the estimation data. The MSE values for predicting the 2001 total and severe yearly TAZ collisions using the GWPR and FBSA models are lower than the MSPE values for the corresponding NB and Poisson models. This indicates that the variability in the 2001 number of collisions is better captured by the GWPR and FBSA models than the GLM models. For most of the cases, the GWPR marginally performs better than the FBSA models. However, no firm conclusions can be achieved when the MSE and MSPE values for FBSA and GWPR models are compared to each other.

In conclusion, the results of the comparison analysis show that the GWPR and FBSA models perform much better than the conventional generalized linear models. Therefore, it can be concluded that the local and spatial model estimation techniques of GWPR and FBSA can improve the safety analysis of transportation networks.


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Table 8-1 Goodness of Fit Measures for Total Collisions

Model Numbers Model Types r MAD MSE MSPE

GLM - NB 0.48 76.09 11,695 11,548 GLM - Poisson 0.48 76.14 11,649 11,504 GWPR 0.77 56.40 6,441 6,360 T1

FBSA 0.69 65.05 8,685 8,576 GLM - NB 0.51 68.70 18,400 18,170 GLM - Poisson 0.60 65.61 9,771 9,649 GWPR 0.89 36.03 3,171 3,132 T2





FBSA ≈1.00 7.46 158 156 GLM - NB 0.64 60.24 9,432 9,314 GLM - Poisson 0.66 59.24 8,500 8,394 GWPR 0.90 34.18 3,004 2,967 T7






FBSA 0.89 34.44 3,259 3,219


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Table 8-2 Goodness of Fit Measures for Severe Collisions

Model Numbers Model Types r MAD MSE MSPE

GLM - NB 0.44 18.10 624 617 GLM - Poisson 0.44 18.07 623 615 GWPR 0.87 9.80 195 193 S1

FBSA 0.66 15.68 471 465 GLM - NB 0.52 16.33 898 887 GLM - Poisson 0.60 15.50 510 504 GWPR 0.81 12.03 265 261 S2









FBSA 0.83 9.96 249 246 GLM - NB 0.63 14.29 513 506 GLM - Poisson 0.65 14.23 450 444 GWPR 0.90 8.26 147 145

S11

FBSA 0.86 9.08 198 195


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8.3 Discussion of the Results

The results reported in this chapter show that the GWPR models perform much better than the conventional generalized linear models. Therefore, it is concluded that the local model estimation technique of GWPR can improve safety analysis of transportation networks. However, there is an issue with the GWPR models developed in this study. For some of the TAZs, the GWPR model produces counterintuitive coefficient signs for some the variables. Existence of multi-collinearity in the data set and insignificant effects of some variables in a model for some TAZs are two reasons for such unexpected results. Indeed, the results of t-values indicate that most of TAZs with unexpected coefficient signs are insignificant at the 90% confidence level.

Moreover, these models have the potential to be temporally transferred. However, these models are not spatially transferable, since the developed GWPR models produce the local coefficient for each TAZ. This is a relatively minor difficulty in that most jurisdictions tend to develop their own models.

To overcome the issues related to the GLM and GWPR techniques, a series of spatial safety planning models are developed using the FBSA regression technique. While such models are generally not considered as local models, they recognize the local nature of spatial data by including a spatial covariate in the GLM models.

The results of the FBSA models confirm the existence of spatial correlation on collision data at the TAZ level. The spatial correlation plays a major role for measuring the safety of a TAZ. The results of the developed FBSA models indicate that the FBSA and GLM models are mostly consistent in the direction and magnitude of the effects of the covariates for predicting number of collisions in a TAZ. In general, highly significant variables in the GLM models are also significant in the FBSA models. The results of the comparative analysis indicate that the FBSA models perform much better than the conventional generalized linear models. However, mixed results are encountered when the results of FBSA models are compared to the results of GWPR models.

In conclusion, the results of the comparative analysis show that the GWPR and FBSA models perform much better than the conventional generalized linear models. Therefore, it can be


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concluded that the local and spatial model estimation techniques of GWPR and FBSA can improve safety analysis of transportation networks.

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CCHHAAPPTTEERR NNIINNEE

9.0 TEMPORAL TRANSFERABILITY AND UPDATING OF SAFETY

PLANNING MODELS

9.1 Background

As explained previously, collision prediction models are commonly used to predict the number of collisions at intersections, arterial road sections, highways or transportation networks. Of late, there has been considerable interest in transportation planning applications, for which these models (usually developed at the TAZ level) are aimed at establishing a relationship between collision frequency and a number of explanatory variables, such as traffic volume, road network characteristics and socioeconomic and demographic features. It is important for safety planning models to enable transportation planners to assess the safety implications of alternative transport planning options that are considered for medium to long-range implementation. This assessment is in a way, similar to how road designers make use of tools, such as the Interactive Highway Safety Design Model (IHSDM), to assess the safety implications of design decisions. For this reason and others, it is desirable to obtain reasonably accurate safety planning models that are transferable across time and have good predictive performance.

Use of Advanced Techniques to Estimate Zonal Level Safety Planning Models and Examine Their Temporal Transferability Chapter 9: Temporal Transferability and Updating of Safety Planning Models

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In general, the importance of collision prediction models makes it crucial that they be properly calibrated (Persaud et al., 2002a). The quantity and quality of the data used to calibrate collision prediction models is very important to the success or failure of these models. Models calibrated with limited data will not perform adequately in terms of predictive accuracy. Therefore, having a reasonable dataset is very important in the development of collision prediction models. Moreover, the specification of an appropriate functional model form for calibration of collision prediction models is a complex task. It gets even more complicated for safety planning models because of the dearth of these models, a situation necessitated by scarcity of relevant transportation planning data.

Data for transportation planning variables are usually obtained from surveys held periodically. For example, the TTS is conducted once every 5 years in the GTA. These surveys usually cover a comprehensive area and are therefore, expensive in terms of the time and cost associated with data collection and analysis. As such, once a model is well specified to capture the decision process in one context, it is cost effective to transfer such a model to other contexts that have limited calibration data, as long as the basic nature of the decision-making process remains the same (Badoe and Miller 1995a).

There are two aspects to model transferability; temporal and spatial. The first results from application of a model estimated at one point of time for prediction at another point of time for the same spatial environment. This occurs when estimated models are applied in a forecasting framework. The second aspect; spatial transferability, involves the application of a model estimated in a specific geographic region for prediction in a different one for the same time period. Of course, there can be both temporal and spatial transfers.

As highlighted in the first chapter, the second objective of this research study is to examine the temporal transferability of the safety planning models using appropriate evaluation measures of predictive performance to assess whether the relationship between the dependent and independent variables holds reasonably well across time. The two temporal contexts that are used for this study are 1996 and 2001, with updated 1996 models being used to predict 2001 collisions in each TAZ in the City of Toronto. The chapter also examines alternative updating methods for temporal transfer by assuming that only a sample of 2001 data is available. The sensitivity of the


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performance of the updated models to the 2001 sample size is explored. The updating procedures examined include the Bayesian updating approach and application of calibration factors to the 1996 models. The results presented in the following sections have also been documented in Hadayeghi et al. (2006).

9.2 1996 Model Specification

A series of macro-level collision prediction models that estimate the number of zonal collisions in the City of Toronto for 1996 was developed in the work described in Hadayeghi (2002). A GLM approach was employed, in which NB regression models were developed separately for total and severe (fatal and non fatal injuries) collisions as a function of socioeconomic/demographic, traffic demand and network data variables. The following model form is used for the development of the 1996 safety planning models:

( ) i

n

ii XVKTYE ∑

=

β β∗∗β=1

0 exp)( 1 (9.1)

E(Y) = predicted collision frequency per TAZ;

VKT = vehicle kilometers travelled;

Xj = jth explanatory variables (j=2,…..,p); and

βj=jth model parameters (j=0,…..,p).

Separate models were calibrated for total and severe (fatal plus non-fatal injuries) collisions per year and for each, 1996 models were estimated for the entire day and the morning peak period (6:00 to 9:00 a.m.) to which the VKT and other traffic variable estimates from EMME/2 (INRO, 1998) pertain. Four models were developed: 1) total yearly collisions (Model I); 2) yearly severe collisions (Model II); 3) total yearly collisions for the morning peak period (6:00-9:00 a.m.) (Model III); and 4) yearly severe collisions for the morning peak period (6:00-9:00 a.m.) (Model IV).

The developed 1996 models included major arterial and minor road data in each TAZ, but excluded freeway data.


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9.3 Measures of Model Transferability

The transferability of a model relates to its application that is estimated in one context based on adequate data, to another context with limited data. Model transferability is based on the idea that the estimated model from a previous study in a different context may provide useful information for estimating parameters appropriate for the application context. “Local” or “estimation” context refers to the context for which the adequate data set was used in the model estimation. The “transfer” or “application” context refers to the context for which predictions are required, and to which the model from the local context is transferred for application.

Transportation planning models are not perfectly transferable between contexts since they do not capture the behavior of individuals completely and the data available for the estimation of the unknown parameters are not error free due to a variety of sources, such as sampling errors (Lerman, 1981). As Lerman (1981) and Ben-Akiva (1981) argued, it is appropriate to describe the measure of transfer effectiveness on a continuous scale rather than describe transferability as a dichotomous property. Transferability is a property that may exist to a greater or lesser degree depending on the formation of the model and characteristics of the estimation and application contexts (Lerman, 1981 and Ben-Akiva, 1981).

Measures of model transferability were used to assess the effectiveness of models when they are transferred for use in an application context (2001 in this study) that is different from their calibration context (1996). The indices and test-statistics are all based on log-likelihood measures. These measures, defined below, are largely similar to those used by Koppelman and Rose (1983) and Badoe (1994).

9.3.1 Nested Likelihood Ratio Test (LR)

This measure indicates the statistical similarity of the estimation and application model coefficients. In this test, the null hypothesis is that the application and estimation context can be presented by a single model. Basically, this test examines whether the model parameters have remained stable over time and is given by:

[ ])ˆ()ˆ()ˆ(2 jjiijiji LLLLLLLR θ−θ−θ−= ∪∪ (9.2)


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The above equation is chi-squared distributed with degrees of freedom equal to the number of model parameters. In the event of rejection of the null hypothesis, an asymptotic t-test can be performed to test which parameter(s) were responsible for the rejection. For this, the t-statistic is given as:

))ˆvar()ˆ(var()ˆˆ( kjkikjkit θ+θθ−θ= (9.3)

where

=kiθ estimate of parameter k in context i; and

=)ˆvar( kiθ variance of the estimate of parameter k in context i

9.3.2 Transfer Index (TI)

Transfer index (TI) is a relative measure that indicates how well a transferred model performs in predicting the application data set relative to a locally estimated model in the application context and with similar specification. For this particular research, this index compares the performance of the transferred and application context models relative to the performance of a 2001 “constant” model that has no independent variables. The index has an upper bound of 1.0, which is attained when the transferred model performs as well as an estimated model on the application data. It has no lower bound. Negative values imply that the transferred model is worse than the local constant model. It is expressed as:

)]c(LL)ˆ(LL[)]c(LL)ˆ(LL[TI jjjjij −θ−θ= (9.4)

where

=θ )ˆ( ijLL log-likelihood value in context j using model parameters estimated from year i;

=)ˆ( jjLL θ log-likelihood value given by application context model j; and

=)ˆ(cLL j log-likelihood value given by constant model estimated in application context j.


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9.3.3 Cumulative Residual (CURE) Plots

CURE plots are also used as measures for assessing transferability of safety planning models in this study. As highlighted in Chapter Five, the CURE plot shows how well the developed model fits the data with respect to each individual covariate. For measuring transferability, these plots were used to explore the similarity of the estimation and application models. It is expected that if the estimation and application models are compatible, they will have similar patterns in terms of their cumulative residuals.

9.4 Updating Safety Planning Models

This section provides an approach for updating local models into an application context in a situation when only limited data are available for calibrating a meaningful model. The updating technique is helpful in applying an “old” estimated model to an application context to predict present conditions with minimum data requirements, consequently cutting the cost of data collection and time. Also, it seems desirable, when transferring a model, to update it by making adjustments based on locally available information. To illustrate such a situation, and investigate the feasibility of this approach, random subsets of zonal information were selected from the 2001 database. To explore the impact of sample size on updating performance, 10 samples of each of 10, 20, 25, and 35 TAZs and their corresponding information were randomly selected from the 2001 data set. However before using updating procedures, a set of models was estimated using different sample sizes. These models were used to examine whether the coefficients in the estimation and application contexts are similar when limited data are available.

9.4.1 Updating Using a Calibration Factor

The updating procedure was recommended by Harwood et al. (2000) and Oh et al. (2003), for application in the IHSDM for transferring collision prediction models calibrated for one geographic region to another region. In this procedure, Harwood et al. (2000) in essence, obtained a calibration factor as the total number of collisions observed in a sample from one jurisdiction divided by the sum of the predicted number of collisions in the sample using the model from another jurisdiction. The calibration factor is calculated as follows:


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∑∑==

=n

ii

n

ii YYFactornCalibratio

11

ˆ (9.5)

Yi = the number of observed collisions in TAZ i in 2001 for a given sample size; and

iY = the predicted number of collisions in TAZ i by applying values of the 2001 independent variables (for the same sample size of 2001) to the 1996 model.

Then, the calibration factors estimated for each sample size are simply multiplied by the base models (1996 models) in order to “transfer” the 1996 models for application to 2001 conditions.

9.4.2 Bayesian Updating

Bayesian updating, which was introduced by Atherton and Ben-Akiva (1976), is another approach that was adopted for updating the 1996 models using the 2001 sample data. This methodology combines sample information with prior information in order to achieve more accurate updated information. Atherton and Ben-Akiva (1976) assumed both prior and posterior (updated) distributions of the parameters to be normally distributed. The updated parameters are expressed as:

)]/1()/1/[()/()/[( 201

296

20101

29696 samplesamplesampleupdated −−− σ+σσβ+σβ=β (9.6)

5.02sample01

296updated )]/1()/1[(1 −σ+σ=σ (9.7)

where

updatedβ = 2001 updated coefficient;

96β = coefficient estimated from 1996 model;

sample01−β = coefficient estimated from 2001 sample model;

updatedσ = standard deviation of 2001 updated coefficient;

96σ = standard deviation of coefficient estimated from 1996 model; and

sample01−σ = standard deviation of coefficient estimated from 2001 sample model.


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9.4.3 Measures of Updating Effectiveness

Measures of GOF from Chapter Eight were used to assess the performance of the updated models relative to the 1996 models. These measures are MAD, MSPE and MSE.

9.5 Results and Discussions

9.5.1 Transferability Results

The dependent variable of each developed model is the number of TAZ collisions per year. The regression parameters were estimated based on the maximum likelihood method for NB regression and using PROCGENMOD in SAS statistical software (SAS Institute Inc.).

It is worth mentioning that the developed 2001 safety planning models in this chapter are not the same as the models presented in Chapters Five, Six and Seven. In order to examine temporal transferability of safety planning models, the 2001 models in this chapter are developed based on the variables used for the 1996 safety planning models from the previous study (Hadayeghi, 2002).

9.5.1.1 Transferability of Yearly Model for Total Collisions

Table 9-1 presents the parameter estimates and corresponding chi-squares for the 1996 and 2001 models for total yearly collisions (Model I).


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Table 9-1 Results of Negative Binomial Regression for Yearly Total Collision Model (Model I)

Parameter 1996 Model 2001 Model

Intercept (Log of β0) 4.1767

(0.2957, 199.47) 2.8065

(0.3572, 61.75)

Log of VKT 0.2656

(0.0347, 58.76) 0.2845

(0.0540, 27.81)

Total Major Road in Kilometer 0.1175

(0.0186, 40.05) 0.1960

(0.0197, 93.98)

Number of Households × 10-3 0.0864

(0.015, 33.27) 0.0582

(0.0156, 13.89)

Speed -0.0431

(0.0071, 36.61) -0.0124

(0.0065, 3.64)

Volume/Capacity (V/C) -1.3678

(0.2766,24.45) -0.6472

(0.3112, 4.33)

Intersection Density 0.0052

(0.0012, 19.87) 0.0024

(0.0012, 3.98)

Goodness of Fit Measures

Mean Pearson Chi-Square /DF 1.0313 1.1559

Pearson R-Square 37.48% 37.24%

α 0.1941 0.2355

The dependent variable is the natural logarithm of yearly total collisions. Parameter estimate (standard error, Chi-square)

A comparison of the parameter estimates in Table 9-1 indicates that the speed becomes insignificant at the 95% confidence level in the 2001 model. The parameter estimates of each

variable have identical signs in both models. The dispersion parameters (α) are in the range of

0.19 and 0.23 for the 1996 and 2001 models, respectively, which indicates that the data are quite overdispersed relative to the Poisson distribution and the NB error structure is therefore, justified. The GOF measures for each model, also presented in Table 9-1, show the mean Pearson chi-


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squared ratios to be close to 1, an indication that the NB regression model fits the data very well. The Pearson r-squared values for the 1996 and 2001 models are 37.48% and 37.24% respectively, which indicate that the developed models explain the variation in the dependent variable to a reasonable degree. In addition, a test was done to examine whether keeping the dispersion parameter in the 2001 model, which is the same as the one in the 1996 model, would considerably affect its performance. It was found that the model parameters and corresponding parameter signs are almost the same as those of Model I. The total predicted number of collisions from this model for the whole study area is different by less than one percent from the total in Model I, which is quite insignificant in comparison. Therefore, no further investigation was conducted. Results of tests of parameter equality, which were done by the nested likelihood ratio test, are shown in Table 9-2.

Table 9-2 Results from Transferring 1996 Models to 2001 Data Set for Model I

a) Absolute and Relative Transfer Measures

Transfer Measures Model I

Nested Likelihood Ratio 128.6

LL01(θ96)×105 * 3.087

Transfer Index -0.966

b) Statistical Test of Parameter Equality in Two Context

Parameter Model I

Log of VKT -0.29

Total Major Road in Kilometer -2.71


Speed -3.19


Intersection Density 1.65 Note: * LL01(θ96) is the 2001 context Log-likelihood using θ96 from year 1996


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These results indicate that 1996 and 2001 model parameters are not statistically equal. Therefore, the null hypothesis can be rejected. The errors in the modelling calibration procedures are most likely to affect transferability of the explanatory variables and estimated parameters. To investigate whether the rejection of the null hypothesis is because of an individual coefficient or sub-set of coefficients, the asymptotic t-test was used to test the equality of 1996 and 2001 variable coefficient estimates. Part b of Table 9-2 presents the results of this test for the yearly model in total collisions. As evident, only two explanatory variables, major road kilometers and speed, can be rejected at the 95% confidence level under the null hypothesis of equality. Theoretically, it can be concluded that the roadway parameters, such as major road kilometers and speed, do not remain stable over time. The value of TI, which describes the degree to which the log likelihood of the transferred model exceeds the reference model relative to improvement provided by a model developed in the application context, is found to be –0.966. The negative sign implies that the transferred model is worse than the local reference model.

Figure 9-1 illustrates the cumulative residual plots for the variable VKT for applying the 1996 and 2001 models to fit 2001 data in total collisions. The results of the CURE plots show that indeed, the 1996 and 2001 collision models are not compatible with each other despite the fact that the coefficient of VKTs (presented in Table 9-1) for the two models are very similar.


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Figure 9-1 Comparison of CURE Plots for Applying 1996 and 2001 Safety Planning Models to 2001 Data for Total Collisions

9.5.1.2 Transferability of Yearly Model for Severe Collisions

The model parameters and corresponding chi-square values for the yearly model in severe collisions (Model II) are shown in Table 9-3. All of the coefficients are close in magnitude and share the same sign for the 1996 and 2001 models. The intersection density and V/C variables are insignificant at the 95% confidence level in the 2001 model, but significant in the 1996 model.


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Table 9-3 Results of Negative Binomial Regression for Yearly Severe Collision Model (Model II)

Parameter 1996 Model 2001 Model


(0.3239, 67.55) 1.9593

(0.3736, 27.50)

Log of VKT 0.2415

(0.039, 38.41) 0.1739

(0.0561, 9.60)

Total Major Road in Kilometer 0.1225

(0.0202, 36.77) 0.2423

(0.0208, 136.31)


(0.0164, 27.38) 0.0650

(0.0162, 16.03)

Speed -0.0312

(0.0078, 16.05) -0.0096

(0.0067, 2.07)


(0.3042, 16.18) -0.3963

(0.3193, 1.54)

Intersection Density 0.0038

(0.0013, 9.14) 0.0013

(0.0012, 1.03)


Mean Pearson Chi-Square /DF 1.0346 1.0110

Pearson R-Square 31.90% 43.06%

α 0.2092 0.2296

The dependent variable is the natural logarithm of yearly total collisions. Parameter estimate (standard error, chi-square)

The dispersion parameters (α) of 0.2092, and 0.2226 for the 1996 and 2001 models, respectively,

again confirm the appropriateness of using NB as opposed to Poisson regression. The mean Pearson chi-square ratios for the two models are within the acceptable range of 0.8 to 1.2, another indication that the NB error structure appears to be appropriate. The Pearson r-square values for


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the 1996 and 2001 models are 31.90 % and 43.3% respectively, which indicate that the developed models explain the variation in the dependent variable to a reasonable degree.

Results of the test of parameter equality in Table 9-4 indicate that the 1996 and 2001 model parameters are not statistically equal. V/C, speed, and major road kilometers are the parameters most responsible for the statistical inequality, as shown in Part b of Table 9-4. The value of TI of 0.528 suggests that the transferred model provides some portion of the information obtained from the 2001 model, irrespective of model specification.

Table 9-4 Results from Transferring 1996 Models to 2001 Data Set for Model II


Transfer Measures Model II

Nested Likelihood Ratio 52.4

LL01(θ96)×105 * 0.460

Transfer Index 0.528


Parameter Model II

Log of VKT 0.99

Total Major Road in Kilometer -4.13


Speed -2.10


Intersection Density 1.41 Note: * LL01(θ96) is the 2001 context Log-likelihood using θ96 from year 1996

Figure 9-2 illustrates the cumulative residual plots for the variable VKT for applying the 1996 and 2001 models to fit 2001 data for severe collisions. The results of CURE plots also confirm the results of Table 9-4 in that the 1996 and 2001 model parameters are not statistically equal. As


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evident from Figure 9-2, the 1996 and 2001 collision models are not compatible with each other. Therefore, it can be concluded that the 1996 safety planning models are not temporally transferable.

Figure 9-2 Comparison of CURE Plots for Applying 1996 and 2001 Safety Planning Models to 2001 Data for Severe Collisions

9.5.1.3 Transferability of Morning Peak Period Collision Models

For the morning peak period models, collisions between 6:00 to 9:00 a.m. were extracted from the dataset by running a query in Microsoft Access. Table 9-5 shows the parameter estimates for the total collision models (Model III) and severe collision models (Model IV) in the morning peak period.


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Table 9-5 Results of Negative Binomial Regression for Models III & IV

Yearly Total Collision Model for Morning Period (Model III)

Yearly Severe Collision Model for Morning Period (Model IV) Parameter

1996 Model 2001 Model 1996 Model 2001 Model


(0.2909, 37.66) 0.4328

(0.3127, 1.92) 0.2079

(0.0403,2.7) 1.0527

(0.4562, 5.32)

Log of VKT 0.2225

(0.0430, 26.73) 0.3268

(0.0574, 32.47) 0.2171

(0.0606, 12.85) 0.3163

(0.0803, 15.53)

Total Major Road in Kilometer

0.1171 (0.0201, 34.07)

0.2127 (0.0202, 111.39)

0.1316 (0.0264, 24.8)

0.2528 (0.0267, 89.51)

Number of Households × 10-3 ---- ----

0.0892 (0.0220, 16.43)

0.0701 (0.0216, 10.51)

Number of Employments × 10-3

0.0692 (0.0162, 18.18)

0.0533 (0.0155, 11.80)

---- ----

Speed -0.0324

(0.0073, 19.59) -0.0139

(0.0063, 4.91) -0.0213

(0.0102, 4.34) -0.0141

(0.0087, 2.61)


(0.3296, 11.46) -0.7236

(0.3234, 5.01) -0.9074

(0.4500, 4.07) -0.8137

(0.4412, 3.40)

Total Minor Road in Kilometer

0.0108 (0.0051, 4.52)

0.0006 (0.0049, 0.02)

---- ----


Mean Pearson Chi-Square /DF 1.0805 1.0802 1.0791 0.9659

Pearson R-Square 36.56% 48.46% 31.57% 42.17%

α 0.1586 0.1959 0.1961 0.2290

The dependent variable is the natural logarithm of yearly total collisions. Parameter estimate (standard error, chi-square)


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For both Models III and IV, the signs of all variables and constant terms are the same for the 1996 and 2001 models. The coefficients in the 2001 models are largely different from the 1996 models, in particular, those with low levels of significance, such as total minor road kilometers in the 2001 model. For both Models III and IV, the coefficient of the total major road kilometers variable for the 2001 models is three times as large as that for the 1996 models.

The GOF statistics show that the mean Pearson chi-square values are close to one, again

indicating that the models fit the data reasonably well. The dispersion parameters (α) for all the

models are significantly different from zero, again confirming the appropriateness of NB regression. The Pearson r-square values are reasonable for all the models.

Results of tests of parameter equality, which were done by the nested likelihood ratio test, are shown in Table 9-6.


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Table 9-6 Results from Transferring 1996 Models to 2001 Data Set for Models III and IV


Transfer Measures Model III Model IV

Nested Likelihood Ratio 177.4 30.84

LL01(θ96)×105 * 0.179 0.0116

Transfer Index -0.881 0.640


Parameter Model III Model IV

Log of VKT -1.45 -0.99

Total Major Road in Kilometer -3.35 -3.23

Number of Households × 10-3 ---- 0.62

Number of Employments × 10-3 0.71 ----

Speed -0.06 -0.54

Volume/Capacity (V/C) -1.21 -0.15

Total Minor Road in Kilometer 1.44 ---- Note: * LL01(θ96) is the 2001 context Log-likelihood using θ96 from year 1996

The statistical tests of equality of model parameters for Models III and IV reject the null hypothesis of equality, indicating that some or all of the model parameters have not remained stable over time. Both Models III and IV exhibit statistically significant equality at the 95% confidence level in all of the parameters, except major road kilometers. The findings from Table 9-6 are also confirmed by the CURE plots for the variable VKT in both Models III and IV as shown in Figures 9-3 and 9-4. As evident, the 1996 and 2001 plots for Models III and IV are not closely following each other. Therefore, it can be concluded that the 1996 and 2001 models are not compatible with each other and they are not temporarily transferable.


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Figure 9-3 Comparison of CURE Plots for Applying 1996 and 2001 Safety Planning Models to 2001 Data for Total Collisions for Morning Peak Period

Figure 9-4 Comparison of CURE Plots for Applying 1996 and 2001 Safety Planning Models to 2001 Data for Severe Collisions for Morning Peak Period


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The values of TI are found to be -0.881 and 0.640 for Models III and IV respectively, suggesting that the transferred model provides some portion of the information obtained from the 2001 model, irrespective of model specification. However, the negative value, as explained previously, implies that the transferred model is worse than the local reference model.

9.5.2 Updating Results

9.5.2.1 Results of Updating for Yearly Model for Total Collisions

Table 9-7 presents the relative GOF measures for predicting the number of collisions in 2001 using four bases: 1996 models, 2001 models, models that apply a calibration factor to the 1996 model, and the Bayesian approach. For the latter two, various sample sizes are used to estimate the “updated” models that were applied.

As expected, the measures indicate that the “2001 model” fits the 2001 data better than other models since its MAD value is smaller than those for other models. When MSE values for the “2001 model” and MSPE values for other models are compared, the variability in 2001 collisions is not being captured well. It is captured better when using the data based on 2001 model.

Comparing the performance of the “updated” models among themselves for different sample sizes with the “1996 model” using MAD values suggests that the updated Bayesian models perform better than the recalibrated “updated” models and the “1996 model”, regardless of sample size.


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Table 9-7 Goodness of Fit Measures for Predicting 2001 Collisions from 1996, 2001 and Updated Models for Model I

Model Types MAD MSE MSPE

2001 Model 59.20 8067 ---

1996 Model 65.17 --- 9294

Recalibrated 1996 Model

71.62 (4.86)

--- 9504

(1690) Updating with Sample of 10 TAZs in 2001

Bayesian Update 61.95

(16.97) ---

8993 (5823)


69.73 (2.61)

--- 8956 (294) Updating with

Sample of 20 TAZs in 2001


(16.14) ---

9685 (3506)


71.51 (5.38)




(18.26) ---

8392 (4600)


71.33 (5.61)




(16.39) ---

8762 (4463)

MAD = Mean Absolute Deviation, MSPE = Mean Squared Prediction Error, MSE = Mean Squared Error, (Standard Deviation)

9.5.2.2 Results of Updating for Yearly Model in Severe Collisions

Results of GOF measures for predicting 2001 collisions from 1996, 2001 and updated models in yearly severe collision models are presented in Table 9-8. The GOF measures from Table 9-8 suggest that the “2001 model” generally predicts TAZ collisions for 2001 better than the other models. However, for some cases, the Bayesian updated model performs marginally better than the “2001 model”, especially when the larger sample sizes are used.


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Comparing performance in terms of MAD values of the “updated” models among themselves for different sample sizes and with the “1996 model” suggests that the updated Bayesian models generally perform better than the recalibrated “updated” models and the “1996 model”. However, mixed results are found.

Table 9-8 Goodness of Fit Measures for Predicting 2001 Collisions from 1996, 2001 and Updated Models for Model II

Model Types MAD MSE MSPE

2001 Model 14.89 449 ---

1996 Model 15.36 --- 491


17.68 (2.38)

--- 545

(123) Updating with Sample of 10 TAZs in 2001


(13.33) ---

808 (902)


16.97 (0.56)



Bayesian Update 15.69 (2.14)

--- 454

(127)


16.78 (0.46)




--- 513 (90)


16.85 (0.73)




--- 441

(180) MAD = Mean Absolute Deviation, MSPE = Mean Squared Prediction Error, MSE = Mean Squared Error, (Standard Deviation)


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9.5.2.3 Results of Updating for Morning Peak Period Collision Models

Table 9-9 presents the relative GOF measures for predicting the number of collisions in 2001 using the four bases: 1996 models, 2001 models, models that apply a calibration factor to the 1996 model, and the Bayesian approach for Models III and IV. The MAD values for both Models III and IV indicate that the “2001 model” fits the 2001 data better than other models since the 2001 values are smaller than the other MAD values in both cases. A comparison of MSE values for the “2001 model” and MSPE values for the other models suggests that the variability in 2001 collisions is not being captured well. It is captured better with the data that is based on the “2001 model”.

Comparing performance in terms of the MAD values of the “updated” models among themselves for different sample sizes and with the “1996 model” suggests that the updated Bayesian models perform better than the recalibrated “updated” models and the “1996 model”. The MAD values become smaller when the sample size increases for most cases.


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Table 9-9 Goodness of Fit Measures for Predicting 2001 Collisions from 1996, 2001 and Updated Models for Models III and IV

Model III Model IV Model Types

MAD MSE MSPE MAD MSE MSPE

2001 Model 8.11 132 --- 2.39 10.90 ---

1996 Model 8.51 --- 168 2.43 --- 10.90


8.47 (0.77)

--- 152 (12)

2.65 (0.16)

--- 12.58 (1.09) Updating

with Sample of 10 TAZs in 2001 Bayesian

Update 7.49

(3.21) ---

138 (155)

2.66 (0.62)

--- 12.89 (7.06)


8.24 (0.36)

--- 140 (4)

2.54 (0.18)

--- 12.02 (0.65) Updating


Update 8.74

(1.63) ---

143 (50)

2.54 (0.18)

--- 11.16 (7.35)


8.62 (0.63)

--- 142 (9)

2.53 (0.13)

--- 11.72 (0.42) Updating


Update 10.16 (1.57)

--- 278

(149) 2.50

(0.38) --- 10.46

(3.98)


8.48 (0.70)

--- 143 (11)

2.47 (0.10)

--- 11.73 (0.46) Updating


Update 8.21

(2.10) ---

178 (149)

2.23 (0.39)

--- 9.22 (3.79)

9.5.3 Discussion of Results

As expected, none of the models calibrated with limited data performs reasonably in terms of the magnitude of the coefficient parameters or their level of significance in the application context, regardless of the sample size. The magnitude of coefficient parameters for some of the models are the same as the 1996 models, but typically, the parameters are insignificant at the 95%


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confidence level or the GOF measures are not satisfactory. This is expected since the sample sizes selected for this study are too small (less than 8% of the total data) to result in a meaningful conclusion. Therefore, it is concluded that the updating procedure is the best approach when limited data are available and no further investigation is conducted regarding developing models with limited data.

As expected, the measures indicate that the 2001 model fits the 2001 data better than other models. The real value of these results is in the comparison of the performance of the “updated” models among themselves for different sample sizes and with the 1996 model. This comparison shows that in general, the Bayesian “updated” models perform better than the recalibrated updated and 1996 models. However, no clear-cut conclusions can be drawn regarding the performance of the updated models with respect to the sample size used for estimating them.

The results of the nested likelihood ratio test indicate that the model parameters from the two urban contexts are not statistically equal and have not remained stable over time. Therefore, from a strictly theoretical perspective, the temporal transfer of safety planning models is not achievable. However, relative measures of transferability indicate that the transferred models yield useful information in the application context. TI values show that transferred models capture at least 52 percent of the variance provided by estimating all the parameters in the 2001 data. The results of the t-test for all the models indicate that the VKT, and socioeconomic and demographic parameters are stable over time.

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CCHHAAPPTTEERR TTEENN

10.0 SUMMARY AND CONCLUSIONS

This chapter is divided into three sections. The first section summarizes the work performed in this research study. In the second section, the contributions of this study are discussed, including suggestions on how the research community and practitioners can greatly benefit. Recommendations for future research are explained in the last section.

10.1 Summary

Given the knowledge gaps created by the recognized need for safety planning tools, and the paucity of relevant research, several research efforts, including the author’s previous work (Hadayeghi, 2002), have been already undertaken to model the level of road safety of a transportation network in the transportation planning process. The general motivation underlying all of these studies is to provide a safety planning decision-support tool, in order to facilitate a proactive approach to assessing safety implications of alternative network planning initiatives and scenarios.

Use of Advanced Techniques to Estimate Zonal Level Safety Planning Models and Examine Their Temporal Transferability Chapter 10: Summary and Conclusions

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The first objective of this research study is to develop a series of safety planning models that are consistent with regional travel demand models commonly used for urban transportation planning. To investigate the spatial variations in the relationship between the number of zonal collisions and potential transportation planning predictors, the GWPR and FBSA modelling techniques, in addition to the traditional GLM, are used for development of safety planning models.

The second objective of this research study is to examine the temporal transferability of the safety planning models and alternative updating methods for temporal transfer of these models. The rational is that the importance of collision prediction models in general makes it crucial that they are properly calibrated. The quantity and quality of the data used to calibrate collision prediction models is very important to their success or failure. Models calibrated with limited data will not perform adequately in terms of predictive accuracy. The importance of available data for transportation planning variables is even more crucial since the data for transportation planning variables are usually obtained from surveys held periodically (for example, the TTS is conducted once every 5 years in the GTA). Therefore, once a model is well specified to capture the decision process in one context, it is cost effective to transfer such a model to other contexts that have limited calibration data as long as the basic nature of the decision-making process remains the same.

The literature review, presented in Chapter Two, surveys the numerous statistical collision prediction models, including previous ones developed for safety planning. The models are built to find the relationship between traffic collisions and a series of explanatory variables. In this chapter, the concept of safety management programs is reviewed. It is suggested that although these programs, which are based on traditional or reactive approaches, are very successful in addressing the safety of a location, they are expensive and require a significant collision data history before any action can be taken. Therefore, to reduce the number of collisions and consequently, improving safety, government agencies and researchers have begun to pursue more proactive engineering approaches.

The concept of proactive safety programs is also discussed in Chapter Two, emphasizing how these programs can be used to improve the safety at the planning level and preclude an unsafe


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situation in the first place. The state of development of current proactive empirical programs, such as RSA programs, sustainable road safety programs, and concepts of SCP, were reviewed.

In Chapter Three, the motivation of this research study, which is the need to develop safety planning decision-support tools, is discussed. These tools can be used to facilitate a proactive approach to assessing safety implications of alternative network planning initiatives.

Several types of data are used for the completion of this study. These include data on collisions, street network, land use, traffic demand, socioeconomic and demographic characteristics, dwelling unit, and employment in the City of Toronto’s 463 TAZs for 1996 and 481 TAZs for 2001. The characteristics of the data and their descriptive analysis are presented in Chapter Four.

An overview of the methodology used for the development of safety planning models based on the GLM, GWPR, and FBSA modelling techniques are presented and discussed in Chapters Five, Six and Seven respectively. The advantages and shortcomings of each method are also discussed in these chapters. In Chapter Eight, an analysis is presented to compare the accuracy of GWPR and FBSA models to GLM models and each other.

In Chapter Nine, the temporal transferability of safety planning models is assessed. To do that, the safety planning models from the author’s previous research (Hadayeghi, 2002) are used. These models are calibrated based on 1996 collision and transportation planning data. The 1996 safety planning models are used to predict number of collisions for 2001 and results are compared to the 2001 models in order to examine temporal transferability. Moreover, this chapter examines alternative updating methods for temporal transfer by imagining that only a sample of 2001 data is available. The sensitivity of the performance of the updated models to the 2001 sample size is explored. The updating procedures that are examined include the Bayesian updating approach and the application of calibration factors to the 1996 models.

10.2 Conclusions

The following sections describe the conclusions and main contributions of this research that are offered, pertaining to the previously identified knowledge gaps.


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10.2.1 Conclusions on of Safety Planning Models

Sixty-nine (twenty-three for each modelling technique) safety planning models are developed using land use, traffic intensity, road network, and socioeconomic and demographic zonal characteristics as explanatory variables. The developed models represent a substantial advancement in the development of macro-level collision prediction models at TAZ levels. The developed models estimate the expected number of collisions of an urban transportation network rather than just intersections or particular road segments. These models can be used for several purposes.

Travel forecasting is one of the responsibilities of transportation planners. Travel forecasting is used to predict future travel demand for analyzing long-range transportation system alternatives. Travel forecasting and collision prediction models can be used to predict future operational characteristics and safety of a transportation network. The developed models based on traffic intensity explanatory variables can be employed for estimating the road safety performance of a transportation network using the EMME/2 (INRO, 1998) model output. Also, these models can assist transportation planners in examining transportation planning issues and policies. For example, urban sprawl is a major planning issue that plagues many big cities, including the City of Toronto. The implication is that population and employment are spreading out to the suburbs with low-density and dispersed growth patterns. People are increasingly willing to live farther away from work and relying more on their cars. Thus, even if population size does not grow (which is not realistic), VKT (i.e. exposure) increases with urban sprawl. The above models can be used to examine the safety impacts of such an issue and policies considered to mitigate its effects.

The models based on socioeconomic and demographics can be used by planners as indicators for deciding the employment and population density of a TAZ in order to achieve a reasonable level of road safety. For instance, the models with different types of dwelling units can be employed as a proxy for deciding the population density of a TAZ to achieve a specific safety goal.

The models based on land use variables in this study can assist planners in creating a land use scenarios for improving the safety of transportation networks. It can be a useful tool in long term analysis, providing the planner with ideas about the safety benefits of different land use scenarios.


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For instance, the models can be used for calculating proportions of green areas (labelled as “other” in the models) to built-up areas in each TAZ to achieve a reasonable level of safety.

The developed models can help inform planning initiatives at the regional and neighbourhood levels of likely road safety impacts. For example, they can be used to assess the safety implications of alternative street network designs. Also, the road safety impacts of strategic land use developments can be analyzed using the developed models.

It must be emphasized here that incorporating safety in the transportation planning process may sometimes conflict with achieving other planning goals or objectives. For example, the goals of increased mobility and safety can often conflict.

For these reasons, the development of these safety planning models as part of this research has been proposed as a significant contribution towards addressing the previously identified research gap, which include providing reliable empirical tools for use in road safety planning.

10.2.2 Conclusions of Temporal Transferability of Safety Planning Models

This section of the thesis examines the temporal transferability of safety planning models for the City of Toronto. In doing so, model calibration and related transferability issues are also addressed. The coefficient estimates for the 2001 models are consistent with the 1996 models. Most of the coefficient parameters and constant terms are significant for the 1996 and 2001 models across all four collision type models (Models I to IV).

The results of the nested likelihood ratio test indicate that the model parameters from two urban contexts are not statistically equal and have not remained stable over time. Therefore, from a theoretical perspective, the transfer of safety planning models is undesirable. However, relative measures of transferability indicate that the transferred models yield useful information in the application context. TI values show that transferred models capture at least 52 percent of the variance provided by estimating all the parameters in the 2001 data. The results of the t-test for all of the models indicate that the VKT, and socioeconomic and demographic parameters are stable over time.


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In the absence of a complete dataset for calibrating the 2001 model, it appears that the “updated” models, which are developed by applying the Bayesian updated models, perform reasonably better for predicting the number of collisions in year 2001 than the 1996 models or updated 1996 models based on calibration factors regardless of the sample size used for estimating these factors.

10.3 Recommendations for Future Work

The outcome of this research is a set of innovative tools for assessing safety impacts of alternative transportation plans and policies at the strategic long-term planning level. The research outcomes help in advancing state of the art road safety research by enabling safety to be included among other traditional evaluation criteria in the strategic planning of transportation systems. For the research community, these outcomes represent a point of departure to develop next-generation road safety planning models. Hence, several topics are recommended for future road safety research related to safety planning model development and transferability. They are as follows:

One of the key transportation planning processes is the design of residential street pattern or neighbourhood layout. Recently, planning policies have increasingly focused on promoting neighbourhood designs that attempt to reduce substantially, the need for car use, in response to environmental and social concerns. The developed safety planning models in this research study can be used to conduct a comparative assessment of the safety impacts of different street network layouts.

Although the developed GWPR models seem to perform well in explaining the relationship among considered variables, there is still a need for further improving safety planning models. The developed GWPR models are excellent tools for predicting the number of zonal collisions for a future planning year. The GWPR model attempted here can be refined for aggregate road safety modelling by employing the same approach, but with an NB distribution for the error terms. The theory to facilitate that approach needs to be developed and built into the software.

This research examines only the temporal transferability of safety planning models. The logical following step for further research would be the investigation of spatial transferability and combination of temporal and spatial transferability. Such research


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should aim to identify the conditions that make two study areas “similar enough” to justify transferring a model from one to the other.

Additional work is needed regarding the estimation of traffic flows by EMME/2 (INRO, 1998). As explained in Chapter Four, EMME/2 (INRO, 1998) based models typically estimate flows in vehicles per hour for the morning period. However, the developed collision models are for the whole year. Therefore, it is necessary to have better 24-hour traffic flow estimates for each zone.

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APPENDIX A:

SPATIAL ANALYSIS OF INDEPENDENT VARIABLES

Use of Advanced Techniques to Estimate Zonal Level Safety Planning Models and Examine Their Temporal Transferability Appendix A: Spatial Analysis of Independent Variables

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Figure A-1 Number of Apartments per TAZ

Figure A-2 Number of Houses per TAZ


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Figure A-3 Number of Town Houses per TAZ

Figure A-4 Average 85th Percentile Speed per TAZ


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Figure A-5 Average Posted Speed per TAZ

Figure A-6 Commercial Area (km2) per TAZ


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Figure A-7 Government & Institutional (km2) per TAZ

Figure A-8 Resource & Industrial (km2) per TAZ


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Figure A-9 Open Area (km2) per TAZ

Figure A-10 Parks and Recreational (km2) per TAZ


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Figure A-11 Residential (km2) per TAZ

Figure A-12 Employed in: General Office per TAZ


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Figure A-13 Employed in: Management and Professional per TAZ

Figure A-14 Employed in: Manufacturing, Construction and Trades per TAZ


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Figure A-15 Employed in: Sales and Service per TAZ

Figure A-16 Number of Not Employed per TAZ


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Figure A-17 Number of Full time Employees per TAZ

Figure A-18 Number of Part time Employees per TAZ


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Figure A-19 Number of Rail Station Train per TAZ

Figure A-20 Number of 3-Legged Signalized Intersections per TAZ


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Figure A-21 Number of 3-Legged Signalized Intersections per TAZ

Figure A-22 Total Arterial Road Kilometers per TAZ


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Figure A-23 Total Collector Kilometers per TAZ

Figure A-24 Total Expressway Kilometers per TAZ


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Figure A-25 Total Laneway Kilometers per TAZ

Figure A-26 Total Local Road Kilometers per TAZ


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Figure A-27 Total Rail Kilometer per TAZ

Figure A-28 Total Ramp Kilometers per TAZ


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Figure A-29 Number of Schools per TAZ

Figure A-30 Number of People Possessing a Driver License per TAZ


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Figure A-31 Number of Females per TAZ

Figure A-32 Number of Males per TAZ


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Figure A-33 Number of People Not Possessing a Driver License per TAZ

Figure A-34 Number of People Holding Transit Pass per TAZ


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Figure A-35 Number of People Older Than 65 Year Olds per TAZ

Figure A-36 Number of People per TAZ


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Figure A-38 Number of People Not Holding Transit Pass per TAZ

Figure A-39 Number of People Younger than 17 Year Olds per TAZ

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APPENDIX B:

CUMULATIVE RESIDUAL (CURE) PLOTS

Use of Advanced Techniques to Estimate Zonal Level Safety Planning Models and Examine Their Temporal Transferability Appendix B: Cumulative Residual (CURE) Plots

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CURE PLOT MODELS FOR TOTAL COLLISIONS BASED ON GLM APPROACH

T2 Model-GLM

T3 Model-GLM

T4 Model-GLM

T5 Model-GLM

T6 Model-GLM

T7 Model-GLM


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T8 Model-GLM

T9 Model-GLM

T10 Model-GLM

T11 Model-GLM


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CURE PLOT MODELS FOR SEVERE COLLISIONS BASED ON GLM APPROACH

S2 Model-GLM

S3 Model-GLM

S4 Model-GLM

S5 Model-GLM

S6 Model-GLM

S7 Model-GLM


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S8 Model-GLM

S9 Model-GLM

S10 Model-GLM


- 243 -

CURE PLOT MODELS FOR TOTAL COLLISIONS BASED ON GWPR APPROACH

T2 Model-GWPR

T3 Model-GWPR

T4 Model-GWPR

T5 Model-GWPR

T6 Model-GWPR

T7 Model-GWPR


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T8 Model-GWPR

T9 Model-GWPR

T10 Model-GWPR

T11 Model-GWPR


- 245 -

CURE PLOT MODELS FOR SEVERE COLLISIONS BASED ON GWPR APPROACH

S2 Model-GWPR

S3 Model-GWPR

S4 Model-GWPR

S5 Model-GWPR

S6 Model-GWPR

S7 Model-GWPR


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S8 Model-GWPR

S9 Model-GWPR

S10 Model-GWPR


- 247 -

CURE PLOT MODELS FOR TOTAL COLLISIONS BASED ON FBSA APPROACH

T2 Model-FBSA

T3 Model-FBSA

T4 Model-FBSA

T5 Model-FBSA

T6 Model-FBSA

T7 Model-FBSA


- 248 -

T8 Model-FBSA

T9 Model-FBSA

T10 Model-FBSA

T11 Model-FBSA


- 249 -

CURE PLOT MODELS FOR SEVERE COLLISIONS BASED ON FBSA APPROACH

S2 Model-FBSA

S3 Model-FBSA

S4 Model-FBSA

S5 Model-FBSA

S6 Model-FBSA

S7 Model-FBSA


- 250 -

S8 Model-FBSA

S9 Model-FBSA

S10 Model-FBSA

- 251 -

APPENDIX C:

THEMATHIC MAPS OF POSTERIOR MEAN FOR SPATIAL COVARITATES

Use of Advanced Techniques to Estimate Zonal Level Safety Planning Models and Examine Their Temporal Transferability Appendix C: Thematic Maps of Posterior Mean for Spatial Covariates

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TOTAL MODELS


c) Total

Figure C-1 Posterior Mean for Spatial Covariates of Intensity Based Model, T2


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c) Total



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c) Total



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c) Total



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c) Total



- 257 -


c) Total



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c) Total



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c) Total



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c) Total



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c) Total



- 262 -

SEVERE MODELS


c) Total

Figure C-11 Posterior Mean for Spatial Covariates of Intensity Based Model, S2


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c) Total



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c) Total



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c) Total



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c) Total



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c) Total



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c) Total



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c) Total



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c) Total


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