FACTORS ASSOCIATED WITH MEDIAN RELATED CRASH FREQUENCY AND SEVERITY

Transcription

1 The Pennsylvania State University The Graduate School Department of Civil and Environmental Engineering FACTORS ASSOCIATED WITH MEDIAN RELATED CRASH FREQUENCY AND SEVERITY A Dissertation in Civil Engineering by Wen Hu 2010 Wen Hu Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2010

2 The dissertation of Wen Hu was reviewed and approved* by the following: Eric T. Donnell Associate Professor of Civil and Environmental Engineering Dissertation Advisor Chair of Committee Paul P. Jovanis Professor of Civil and Environmental Engineering Martin T. Pietrucha Professor of Civil and Environmental Engineering Aleksandra B. Slavković Associate Professor of Statistics Eugene J. Lengerich Professor of Public Health Sciences, Family and Community Medicine, and Health Policy and Administration Peggy A. Johnson Professor and Head of the Department of Civil and Environmental Engineering *Signatures are on file in the Graduate School

3 iii ABSTRACT This study analyzed the frequency and severity of three median-related crash types, including: cross-median and rollover crashes on rural highways with earth-divided, traversable medians and no median barrier, and median barrier crashes on rural highways with longitudinal median barriers. Data from multi-lane, limited-access, rural highway segments in Pennsylvania and North Carolina over a five-year period were used for the analysis. This study included analysis employing data regarding median barrier offset from the edge of the traveled way, median barrier type, and the cross-slopes of the median, which traditionally have not been available for consideration in past medianrelated crash frequency and severity models. Both five-year aggregated and yearly crash frequencies were analyzed. To account for road segment random effects (unobserved heterogeneity) and serial correlation in the yearly crash frequency data, random-effects Poisson and negative binomial (NB) regression models were estimated, as were pooled Poisson and NB models. Also, Gaussian process-based spatial generalized linear models (SGLM) were estimated to account for the potential spatially dependent random effects in modeling crash frequencies along adjacent roadway segments. Crash severities were estimated using binary, multinomial, and nested logit models. The modeling results confirmed the importance of median design characteristics, such as median foreslopes, median backslopes, median width, and median barrier placement and type on median-related crash frequency and severity. While flatter and narrower medians were associated with an increase in cross-median crash frequency and

4 iv severity, steeper median side-slopes significantly increased rollover and median barrier crash frequency and severity. Although the presence of median barriers resulted in more median-related crashes, it likely prevented more severe crashes from occurring, such as cross-median and rollovers crashes. The modeling results also showed that curved road segments and higher daily traffic volumes led to more median-related crashes. Left shoulder rumble strips were associated with fewer median-related crashes. For median-related crash severity, the use of seatbelts significantly decreased the probability of high-severity outcomes given a crash. The presence of interchange entrance ramps and horizontal curves were associated with increased probabilities of higher-severity outcomes. Simulations were performed to compare the predictive capabilities of the estimated SGLMs and the classical/panel count models. The simulation results showed that SGLMs generally predicted crash frequencies that were closer to the observed values with narrower prediction intervals than the classical count models, by borrowing information from neighboring road segments. It was found that SGLMs could be utilized to detect road sites with crash frequencies significantly higher than adjacent road segments. Transferability tests were performed to test the transferability of the estimated crash frequency models between the two states. The test results show that if calibrated, the Pennsylvania cross-median crash frequency models could be transferred to the North Carolina data. However, the Pennsylvania rollover crash frequency models were not transferable to North Carolina. Similarly, the Pennsylvania and North Carolina median barrier crash frequency models were not transferable.

5 v TABLE OF CONTENTS LIST OF FIGURES... viii LIST OF TABLES... x ACKNOWLEDGEMENTS... xiii Chapter 1 Introduction Research Objectives Analysis Methodology Significance of Research Organization of Dissertation... 9 Chapter 2 Literature Review Crash Frequency Studies Models Predicting Median-Related Crash Frequencies Models Incorporating Spatial Dependency for Count Outcomes Crash Severity Studies Summary of Literature Chapter 3 Description of Data Pennsylvania Data Data for Median-Related Crash Frequency Analysis Data for Median-Related Crash Severity Analysis North Carolina Data Data for Median-Related Crash Frequency Analysis Data for Median-Related Crash Severity Analysis Chapter 4 Methodology Crash Frequency Modeling and Model Assessment Crash Frequency Models Classical Count Outcome Models Random-Effects Poisson and Random-Effects Negative Binomial Models Spatial Generalized Linear Model for Count Outcomes Crash Frequency Model Simulation and Assessment Simulations from Classical Count Outcome Models Simulations from Random-Effects Poisson and NB models Simulations from Spatial Generalized Linear Model Transferability and Recalibration of Crash Frequency Models Crash Severity Modeling... 89

6 vi Chapter 5 Crash Frequency Analysis Results Pennsylvania Cross-Median Crash Frequency Models Pennsylvania Five-Year Aggregated Cross-Median Crash Frequency Models and Model Assessment Pennsylvania Five-year Aggregated Cross-Median Crash Frequency Models Transferability of Pennsylvania Five-Year Aggregated Cross- Median Crash Frequency Models Pennsylvania Yearly Cross-Median Crash Frequency Models and Model Assessment Pennsylvania Yearly Cross-Median Crash Frequency Models Transferability of Pennsylvania Yearly Cross-Median Crash Frequency Models Pennsylvania Rollover Crash Frequency Models Pennsylvania Five-Year Aggregated Rollover Crash Frequency Models and Model Assessment Pennsylvania Five-Year Aggregated Rollover Crash Frequency Models Transferability of Pennsylvania Five-Year Aggregated Rollover Crash Frequency Models Pennsylvania Yearly Rollover Crash Frequency Models and Model Assessment Pennsylvania Yearly Rollover Crash Frequency Models Transferability of Pennsylvania Yearly Rollover Crash Frequency Models Pennsylvania Median Barrier Crash Frequency Models Pennsylvania Five-Year Aggregated Median Barrier Crash Frequency Models and Model Assessment Pennsylvania Five-Year Aggregated Median Barrier Crash Frequency Models Pennsylvania Five-Year Aggregated Median Barrier Crash Frequency Model Assessment Pennsylvania Yearly Median Barrier Crash Frequency Models and Model Assessment Pennsylvania Yearly Median Barrier Crash Frequency Models Pennsylvania Yearly Median Barrier Crash Frequency Model Assessment North Carolina Median Barrier Crash Frequency Models North Carolina Five-Year Aggregated Median Barrier Crash Frequency Models and Model Assessment North Carolina Five-Year Aggregated Median Barrier Crash Frequency Models North Carolina Five-Year Aggregated Median Barrier Crash Frequency Model Assessment North Carolina Yearly Median Barrier Crash Frequency Models and Model Assessment North Carolina Yearly Median Barrier Crash Frequency Models

7 North Carolina Yearly Median Barrier Crash Frequency Model Assessment Transferability of Pennsylvania and North Carolina Median Barrier Crash Frequency Models Transferability of Five-Year Aggregated Median Barrier Crash Frequency Models Transferability of Yearly Median Barrier Crash Frequency Models Chapter 6 Crash Severity Analysis Results Pennsylvania Cross-Median Crash Severity Model Pennsylvania Rollover Crash Severity Model Pennsylvania Median Barrier Crash Severity Model North Carolina Median Barrier Crash Severity Model No Injury and Class C Injury Results No Evident Injury, Class B Injury, and Class A/Fatality Summary Chapter 7 Conclusions Cross-Median Crashes Rollover Crashes Median Barrier Crashes Engineering Application Methodological Issues Recommendations for Future Research References Appendix Simulation Results vii

8 viii LIST OF FIGURES Figure 2.1. Anatomy of Traffic Crashes Involving the Median: An Illustration (Bligh et al., 2006) Figure 3.1. Roads for CMC Crash Frequency Analysis and CMC Crash Distribution in Pennsylvania Figure 3.2. Roads for Rollover Crash Frequency Analysis and Rollover Crash Distribution in Pennsylvania Figure 3.3. Roads for HMB Crash Frequency Analysis and HMB Crash Distribution in Pennsylvania Figure 4.1. Crash Frequency Analysis Process Figure 4.2. Semivariogram of a Second-Order Stationary Process with and without Nugget Effect (Schabenberger and Pierce, 2002) Figure 5.1. Regular and Robust Empirical Semivariograms from Residuals of a Poisson Base Model for Pennsylvania Five-Year Aggregated Cross-Median Crash Frequency.. 96 Figure 5.2. Regular and Robust Empirical Semivariograms from Residuals of a Poisson Base Model for Pennsylvania Yearly Cross-Median Crash Frequency Figure 5.3. Regular and Robust Empirical Semivariograms from Residuals of a Poisson Base Model for Pennsylvania Five-Year Aggregated Rollover Crash Frequency Figure 5.4. Regular and Robust Empirical Semivariograms from Residuals of a Poisson Base Model for Pennsylvania Yearly Rollover Crash Frequency Figure 5.5. Regular and Robust Empirical Semivariograms from Residuals of a Poisson Base Model for Pennsylvania Five-Year Aggregated Median Barrier Crash Frequency Figure 5.6. Simulation Results based on Pennsylvania Five-Year Aggregated Median Barrier Crash Frequency Models Figure 5.7. Regular and Robust Empirical Semivariograms from Residuals of a Poisson Base Model for Pennsylvania Yearly Median Barrier Crash Frequency Figure 5.8. Simulation Results based on Pennsylvania Yearly Median Barrier Crash Frequency Models, Year Figure 5.9. Simulation Results based on Pennsylvania Yearly Median Barrier Crash Frequency Models, Year

9 Figure Simulation Results based on Pennsylvania Yearly Median Barrier Crash Frequency Models, Year Figure Simulation Results based on Pennsylvania Yearly Median Barrier Crash Frequency Models, Year Figure Simulation Results based on Pennsylvania Yearly Median Barrier Crash Frequency Models, Year Figure Regular and Robust Empirical Semivariograms from Residuals of A Poisson Base Model for North Carolina Five-Year Aggregated Median Barrier Crash Frequency Figure Simulation Results based on North Carolina Five-Year Aggregated Median Barrier Crash Frequency Models Figure Regular and Robust Empirical Semivariograms from Residuals of A Poisson Base Model for North Carolina Yearly Median Barrier Crash Frequency Figure Simulation Results based on North Carolina Median Barrier Crash Frequency Models, Year Figure Simulation Results based on North Carolina Yearly Median Barrier Crash Frequency Models, Year Figure Simulation Results based on North Carolina Yearly Median Barrier Crash Frequency Models, Year Figure Simulation Results based on North Carolina Yearly Median Barrier Crash Frequency Models, Year Figure Simulation Results based on North Carolina Yearly Median Barrier Crash Frequency Models, Year Figure 6.1. Nesting structure of median barrier crashes ix

10 x LIST OF TABLES Table 1.1. Factors Expected to Be Associated With Median-Related Crash Frequency... 4 Table 1.2. Factors Expected to Be Associated With Median-Related Crash Severity... 5 Table 2.1. NB, RENB and NM Modeling Results and Comparison (Ulfarsson and Shankar, 2004) Table 2.2. Estimated Median-Related Crash Frequency Models by Using Full Bayes Framework (Bligh et al., 2006) Table 2.3. Crash Frequency Models for Depressed Medians without Barriers (Tarko et al., 2008) Table 3.1. Descriptive Statistics of Continuous Variables for Pennsylvania Median Barrier Crash Frequency Analysis Table 3.2. Descriptive Statistics of Categorical Variables for Pennsylvania Median Barrier Crash Frequency Analysis Table 3.3. Descriptive Statistics of Continuous Variables for Pennsylvania CMC/Rollover Frequency Analysis Table 3.4. Descriptive Statistics of Categorical Variables for Pennsylvania CMC/Rollover Frequency Analysis Table 3.5. Severity Distribution of CMC, Rollover and HMB in Pennsylvania Table 3.6. Pennsylvania CMC Crash Severity Distribution by Variables Table 3.6. Pennsylvania CMC Crash Severity Distribution by Variables (Continued) Table 3.7. Descriptive Statistics of Continuous Variables for Pennsylvania CMC Severity Model Estimation Table 3.8. Descriptive Statistics of Categorical Variables for Pennsylvania CMC Severity Model Estimation Table 3.9. Pennsylvania Rollover Crash Severity Distribution by Variables Table 3.9. Pennsylvania Rollover Crash Severity Distribution by Variables (Continued) Table Descriptive Statistics of Continuous Variables for Pennsylvania Rollover Severity Model Estimation... 50

11 xi Table Descriptive Statistics of Categorical Variables for Pennsylvania Rollover Severity Model Estimation Table Pennsylvania HMB Crash Severity Distribution by Variables Table Pennsylvania HMB Crash Severity Distribution by Variables (continued) Table Descriptive Statistics of Continuous Variables for Pennsylvania HMB Severity Model Estimation Table Descriptive Statistics of Categorical Variables for Pennsylvania HMB Severity Model Estimation Table Descriptive Statistics of Continuous Variables for North Carolina HMB Frequency Model Estimation Table Descriptive Statistics of Categorical Variables for North Carolina HMB Frequency Model Estimation Table North Carolina HMB Crash Severity Distribution by Variables Table Descriptive Statistics of Continuous Variables for North Carolina HMB Severity Model Estimation Table Descriptive Statistics of Categorical Variables for North Carolina HMB Severity Model Estimation Table 5.1. Poisson Models for Pennsylvania Five-Year Aggregated Cross-Median Crash Frequency Table 5.2. Models for Pennsylvania Yearly Cross-Median Crash Frequency Table 5.3. NB Models for Pennsylvania Five-Year Aggregated Rollover Crash Frequency Table 5.4. Models for Pennsylvania Yearly Rollover Crash Frequency Table 5.5. Models for Pennsylvania Five-Year Aggregated Median Barrier Crash Frequency Table 5.6. Simulation Results for Pennsylvania Five-Year Aggregated Median Barrier Crash Frequency Table 5.7. Models for Pennsylvania Yearly Median Barrier Crash Frequency Table 5.7. Models for Pennsylvania Yearly Median Barrier Crash Frequency (continued) Table 5.8. Simulation Results for Pennsylvania Yearly Median Barrier Crash Frequency Table 5.9. Models for North Carolina Five-Year Aggregated Median Barrier Crash Frequency

12 Table Simulation Results for North Carolina Five-Year Aggregated Median Barrier Crash Frequency Table Models for North Carolina Yearly Median Barrier Crash Frequency Table Models for North Carolina Yearly Median Barrier Crash Frequency (continued) Table Simulation Results for North Carolina Yearly Median Barrier Crash Frequency Table Five-Year Aggregated Median Barrier Crash Frequency Models for Transferability Test Table Yearly Median Barrier Crash Frequency NB Models for Transferability Test Table 6.1. Binary Logit Model of Pennsylvania Cross-Median Crash Severity Table 6.2. Multinomial Logit Model of Pennsylvania Rollover Crash Severity Table 6.3. Multinomial Logit Model of Pennsylvania Rollover Crash Severity Odds Ratio Table 6.4. Multinomial Logit Model of Pennsylvania Median Barrier Crash Severity Table 6.5. Multinomial Logit Model of Pennsylvania Median Barrier Crash Severity Odds Ratios Table 6.6. Estimation Results for the No Evident Injury Conditional Model (Lower Nest) Table 6.7. Estimation Results for the No Evident Jnjury, Class B, and Class A/Fatal model (Upper Nest) Table 6.8. Estimation Results for the No Evident Injury, Class B, and Class A/Fatal model (Upper Nest) Odds Ratios Table 7.1. Effects of Factors Associated With Cross-Median Crash Frequency Table 7.2. Effects of Factors Associated With Cross-Median Crash Severity Table 7.3. Effects of Factors Associated With Rollover Crash Frequency Table 7.4. Effects of Factors Associated With Rollover Crash Severity Table 7.5. Effects of Factors Associated With Median Barrier Crash Frequency Table 7.6. Effects of Factors Associated With Median Barrier Crash Severity xii

13 xiii ACKNOWLEDGEMENTS To my parents, Qingai Ou and Jiancheng Hu, as well as my boyfriend, Chengsheng Jiang, for their love and support. To my advisor, Dr. Donnell, for his guidance, encouragement, and support for which I will be eternally grateful. To the members of my dissertation committee, Dr. Jovanis, Dr. Pietrucha, Dr. Slavković and Dr. Lengerich, for their valuable advice. collection. To my fellow graduate students, especially Scott Himes, for helping me with data

14 Chapter 1 Introduction Improving traffic safety is stated generally as one of the primary goals of transportation agencies in the United States (U.S.). Approximately 40,000 motorists are killed and another 3 million are injured annually in crashes on the nation s highway and street network (NHTSA, 2001). These traffic crashes lead to enormous social and economic loss. Although freeways are designed using the most uniform and forgiving geometric design criteria, Neuman et al. (2008) cite a Federal Highway Administration statistic that indicates that there is one cross-median crash (CMC) fatality annually for every 200 freeway miles; approximately 250 fatalities occur annually on freeways as a result of CMC events; and median-related crashes are more than three times severe as other crash types on freeways. The authors also indicate that approximately 44 percent of CMC events occur on rural freeways and nearly two-thirds of fatal CMC crashes involve males. Additionally, more than 50 percent of fatal CMC events occur at night. To mitigate CMC events on divided highways, the installation of longitudinal median barrier is one treatment that has generally proven to be cost-effective (Nystrom, et al., 1997; HNTB, 1999; Macedo, 1999; Glad, et al., 2002; Bane, 2003; Powers, 2003; Miaou et al., 2005; Bligh et al., 2006; Donnell and Mason, 2006; Sicking et al., 2009). The American Association of State Highway and Transportation Officials (AASHTO) Roadside Design Guide (2006) contains median barrier warrant criteria, as well as median barrier type and placement guidelines. In the policy, median barrier is

15 recommended when the median width is equal to or less than 30 ft (10 m) and the 2 average daily traffic volume (ADT) exceeds 20,000 vehicles per day. Median barrier is recommended for consideration when the median width is between 30 and 50 ft (10 and 15 m) if the ADT exceeds 20,000 vehicles per day. For all other median width and ADT combinations, median barrier is considered optional. A variety of median barriers can be used to prevent CMC events. These are commonly categorized as flexible, semi-rigid, and rigid systems. AASHTO policy (AASHTO, 2006) indicates that flexible or semi-rigid median barriers can be used in wide medians with relatively flat cross-slopes as long as the dynamic deflection of the barrier is less than one-half the median width. Narrow medians, particularly along roadways with high traffic volumes, normally require a rigid median barrier system with a nominal or no dynamic deflection. Median barrier placement is generally recommended near the edge of the inside shoulder on divided highways when the median cross-slope is steeper than 10H:1V. When the cross-slope is 10H:1V or flatter, longitudinal barrier systems may be placed either on the slope or at the center of the median. The only exception to this is for cable median barrier, which has exhibited effective performance on slopes as steep as 6H:1V during crash tests (AASHTO, 2006). A variety of research studies have shown that the frequency of median-related crashes increases, while the severity of these events is reduced, after installation of median barrier (McNally and Yaksich, 1992; Elvik, 1995; Nystrom, et al., 1997; Sposito and Johnston, 1998; HNTB, 1999; Hunter, et. al., 2001; Bane, 2003; Monsere, et al., 2003; Miaou et al., 2005; Bligh et al., 2006; Donnell and Mason, 2006; Tarko et al., 2008; Sicking et al., 2009). Most statistical models related to the frequency and severity

16 3 of median-related crashes include only roadway geometric design and traffic variables in the model specification (Miaou et al., 2005; Bligh et al., 2006; Donnell and Mason, 2006; Tarko et al., 2008). Little is known about the effects of median (roadside) design features, such as median foreslopes and backslopes, median barrier type, and barrier offset from the edge of the traveled way, on median-related crash frequency and severity. 1.1 Research Objectives This study analyzed the frequency and severity of three types of the following median-related crashes: cross-median crashes on rural highways with earth-divided, traversable medians and no median barrier; rollover crashes on rural highways with earthdivided, traversable medians and no median barrier; and median barrier (HMB) crashes on rural divided highways with longitudinal median barriers. In this study, median-related crashes on multi-lane, rural, limited-access highway segments in Pennsylvania and North Carolina over a five-year period ( for Pennsylvania, for North Carolina) were used for the analysis. A novel data collection system was developed to collect median cross-section design data using a laser-scanning device. These data were appended to electronic roadway inventory and crash data to form a database that could be used to provide important insights concerning the frequency and severity of median-related collisions on limited-access, rural divided highways. As such, the main objective of this study was to perform an analysis of the roadway, roadside, environmental, and driver-related factors that are associated with median-related crash frequency and severity. The effects of median barrier offset from

17 4 the edge of the traveled way, median barrier type, and the cross-slopes of the median are factors included in the models which traditionally have not been available for consideration in past median-related crash frequency and severity models. Based on the published literature and engineering intuition, the factors that were expected to be associated with median-related crash frequency and their expected effects are summarized in Table 1.1. Similarly, those factors expected to be associated with median-related crash severity and their expected effects are summarized in Table 1.2. Table 1.1. Factors Expected to Be Associated With Median-Related Crash Frequency Expected Outcome No. Factors Expected effects on medianrelated crash frequency E1 Curved road segments 1 Increase E2 Presence of interchange entrance ramps 1 Increase E3 Presence of left shoulder rumble strips 1 Decrease E4 Higher ADT 2 Increase E5 Higher posted speed limits 1 Increase E6 Narrower medians 1 Increase in CMC frequency E7 Flatter median cross-slopes 1 Increase in CMC frequency E8 Wider medians 1 Increase in rollover frequency E9 Steeper median cross-slopes 1 Increase in rollover frequency E10 Increase in median barrier offset 1 Decrease in median barrier crash frequency E11 Flatter median foreslopes 1 Decrease in median barrier crash frequency Notes: 1. Roadway characteristics; 2. Traffic characteristics.

18 Table 1.2. Factors Expected to Be Associated With Median-Related Crash Severity Expected outcome No. Factors Expected effects on medianrelated crash severity E12 Curved road segments 1 More severe E13 Presence of left shoulder rumble strips 1 Less severe E14 Higher posted speed limits 1 More severe E15 Older or female drivers 3 More severe E16 Unbelted drivers 3 More severe E17 Impaired or aggressive drivers 3 More severe E18 Water/ice/snow/slush covered road 2 Less severe E19 Night 2 More severe E20 Presence of interchange entrance ramps 1 More severe E21 Heavy vehicle involvement 4 More severe E22 Overturning vehicle involvement 4 More severe E23 Multi-vehicle involvement 4 More severe E24 Narrower medians 1 More severe CMCs and rollovers E25 Flatter median cross-slopes 1 More severe CMCs E26 Steeper median cross-slopes 1 More severe rollovers E27 Increased median barrier offset 1 Less severe median barrier crashes E28 Cable median barrier 1 (compared to guardrail median barrier) Less severe median barrier crashes E29 E30 Notes: 1. Roadway characteristics; 2. Crash characteristics; 3. Driver characteristics; 4. Vehicle characteristics. Guardrail median barrier 1 (compared to concrete median barrier) Flatter median foreslopes 1 Less severe median barrier crashes Less severe median barrier crashes 5

19 1.2 Analysis Methodology 6 This study estimated econometric models of median-related crash frequency and severity. In the former, both five-year aggregated and yearly crash frequencies were analyzed. Aggregating crash frequencies over years could eliminate issues related to temporal correlation. However, aggregating data over years leads to information loss in time-varying explanatory variables, such as ADT. Ignoring the variation across temporal intervals in these time-varying variables may lead to the loss of significant explanatory information (Lord and Mannering, 2010). This issue was addressed in the present study by analyzing disaggregated yearly crash frequencies. To date most crash frequency models (Kniuman et al., 1993; Miaou, 1994; Bauer and Harwood, 1997; Milton and Mannering, 1998; Donnell et al., 2002; Bligh et al., 2006; Donnell and Mason, 2006; Tarko et al., 2008) use either Poisson or Negative Binomial (NB) regression to analyze crash occurrence. In yearly crash frequency datasets, there are multiple observations over several years on the same road segment (referred to as panel data), which introduces serial correlation. Random-effects Poisson/NB models can account for road segment random effects and serial correlation in panel count data (Hausman et al., 1984). In this study, random-effects Poisson/NB models were estimated using yearly crash frequencies, as were pooled Poisson/NB models, which ignored potential serial correlations and treated multiple observations over time on the same road segment as independent of each other. Another problem with the Poisson and NB models is that the derivation of these two models by maximum likelihood assumes that the events (crash frequency on adjacent road segments) are independent. However, it is possible that there is spatial dependency

20 7 among crash frequencies along adjacent road segments, which could affect the efficiency of the model estimation (Aguero-Valverde, 2008; Aguero-Valverde and Jovanis, 2008, 2010; El-Basyouny and Sayed, 2009). In this study, to account for this dependency, the following methods were used and compared: adding route indicators to Poisson or NB models and Gaussian process (GP)-based spatial generalized linear models (SGLM) (Breslow and Clayton, 1993; Gotway and Stroup, 1997; Diggle et al., 1998; Schabenberger and Pierce, 2002; Dalthorp, 2004; Schabenberger and Gotway, 2004; Lambert et al., 2006; Braimoh and Onishi, 2007; Diggle and Ribeiro, 2007; Haran, 2010). Classical Poisson and NB models, ignoring the possible effects of spatial correlation, were also estimated for the purpose of comparison. The estimated median-related crash frequency models were tested for transferability between the two states (NC and PA). Simulations were performed to compare predictive capabilities among the SGLMs and the other estimated models. With regard to crash severity, this study sought to provide new insights related to the injury severity of median-related crash events. A binary logit model of fatal and major injury crashes was considered when a comparatively small sample of medianrelated crash events was reported over the analysis period. For larger samples of reported crashes, a multinomial logit model was estimated if the independence of irrelevant alternatives assumption (IIA) was not violated, and a nested logit model was estimated if the IIA assumption was violated. Spatial dependency was not considered in this study because past research has shown that severity is most significantly associated with driver characteristics and, to a lesser extent, roadway characteristics (Donnell and Mason, 2005).

21 1.3 Significance of Research 8 From a methodological perspective, this study estimated random-effects count models to account for unobserved road segment heterogeneity and serial correlation in panel count data. This study also provided insights related to the spatial dependency among crashes on adjacent road segments. Although spatial dependency in traffic safety research is currently being investigated by several researchers, most have opted to employ Gaussian Markov random fields (GMRF)-based SGLMs. GMRF-based models provide conditional spatial dependency structures, while GP-based models provide direct measures of the spatial dependency as a function of spatial distances (Aguero-Valverde, 2008; Aguero-Valverde and Jovanis, 2010). Based on the advantages of GP-based models in terms of straightforward presentations of spatial dependency, and on the recommendations made by Aguero-Valverde (2008) and Aguero-Valverde and Jovanis (2010), this study estimated GP-based SGLMs to account for spatial dependency in models of median-related crash frequency. To assess the predictive capabilities of the estimated SGLMs, random-effects count models and classical count models, predicted crash frequencies were simulated and compared to the observed crash frequencies. Such an approach is seldom used in traffic safety research for model assessment. Transferability tests were also performed to determine whether the estimated crash frequency models were transferable between the two states. The inclusion of median cross-section design variables that have not typically been included in past crash frequency and severity models (e.g., foreslope and backslope dimensions, median barrier type, median barrier offset, etc.) is another major contribution

22 9 to the literature and will, perhaps, offer some new insights regarding median designsafety tradeoffs on rural divided highways. 1.4 Organization of Dissertation This dissertation is organized into six subsequent chapters. Chapter 2 is a literature review. It contains information related to previously estimated models predicting median-related crash frequencies on limited-access, divided highways, as well as models incorporating spatial dependency. Statistical models of median-related crash severity are also described and summarized in Chapter 2. Chapter 3 describes the Pennsylvania and North Carolina data used in this study to estimate the crash frequency and severity models. Chapter 4 contains a description of the crash frequency and severity modeling methodology used in this study. Additionally, transferability tests and model assessment methods using simulation are described in Chapter 4. Chapter 5 presents estimated crash frequency models and associated model explanations. Also, model assessment results using simulation and transferability test results are presented in Chapter 5. Chapter 6 presents estimated crash severity models and associated model explanations. Chapter 7 contains conclusions from this study and recommendations for future research.

23 Chapter 2 Literature Review This chapter of the Dissertation is organized into three sections. The first section describes various crash frequency model estimation methods used in highway safety research, with a focus on median-related crash frequencies. The second section contains a synthesis of various crash severity outcome model estimation methods used in highway safety research, with a focus on median-related crash severities. The third section in this chapter is a summary of the literature that highlights issues where the literature appears to offer converging evidence related to statistical associations between roadway, driver, and environmental factors and crash frequency or severity outcomes. Median-related design and safety issues that have either not been addressed, or where convergence related to specific issues have not been reached, are described in the last section of this chapter. 2.1 Crash Frequency Studies This section is organized into two subsections. The first describes published median-related crash frequency research, while the second describes various statistical methods used to account for spatial correlation in crash frequency models.

24 2.1.1 Models Predicting Median-Related Crash Frequencies 11 There have been several published research papers that estimated the expected frequencies of median-related crashes as a function of geometric and traffic characteristics. Donnell et al. (2002) used NB regression to model CMC crash frequencies on earth-divided, traversable highways with limited-access in Pennsylvania. The authors noted that CMC crashes were a safety concern because of the huge economic and social loss associated with these crashes. While CMC crashes are rare events, nearly 15 percent of them involve fatalities and 72 percent result in non-fatal injuries. CMC crashes occur where median barriers are not warranted by roadside design policies (AASHTO, 2006). A total of 267 CMC crashes on Interstate highways and expressways in Pennsylvania during a five-year (1994 to 1998) period were used in this study to analyze CMC crash frequencies. The resulting NB regression model is shown below: N CMC MW = e 5 L ADT e 2.1 where: N CMC = expected number of CMC crashes per year in one travel direction; L = road segment length (miles); ADT = one-way average daily traffic volume (vehicles/day); and MW = median width (ft). The results indicated that the expected CMC crash frequency decreased as median width increased and CMC crash frequency increased as the one-way ADT increased. The study also suggested that CMC crashes were more likely to occur immediately downstream of interchange entrance ramps, based on a limited sample of CMC locations.

25 12 Donnell and Mason (2006) applied NB regression to predict median barrier crash (HMB) frequencies on Interstate highways in Pennsylvania using five years of crash data (1994 to 1998). Separate NB regression models were estimated for the Pennsylvania Turnpike and all other Interstates. The resulting models are shown below: For Interstate highways: N IntBarrier e L where: = ( X ) exp( 0.066X 2 )exp( 0.481X 3)exp( 0.173X 4 )exp(0.092x 5 )exp( 0.035X 6 ) 2.2 N IntBarrier = expected HMB crash frequency per year for Interstate highway segments; L = segment length (miles); X 1 = ADT for one travel direction (vehicles/day); X 2 = interchange entrance ramp indicator (1 if no ramp present, 0 otherwise); X 3 = posted speed limit indicator (1if 55 mph, 0 otherwise); X 4 = horizontal curve indicator (1 if tangent section, 0 otherwise); X 5 = horizontal curve direction indicator (1 if curved to right, 0 otherwise); and X 6 = barrier offset distance from edge of travel way (ft). For the Turnpike: N TurnpikeBarrier = e L ( X ) 0.241X )exp( 0.105X )exp( 0.104X )exp(0.188x ) 1 exp( where: N TurnpikeBarrier = expected HMB crash frequency per year for Turnpike segments; L = section length (miles);

26 X 1 = ADT for one travel direction (vehicles/day); 13 X 2 = location indicator (1 if rural, 0 otherwise); X 3 = horizontal alignment indicator (1 if all curve radii in section >514m or 1685 ft, 0 otherwise); X 4 = terrain indicator (1 if level; 0 otherwise); and X 5 = interchange entrance ramp indicator (1 if ramp not present, 0 otherwise). The modeling results suggested that for Interstate highways, the HMB crash frequency increased as the ADT increased. The absence of an interchange entrance ramp, lower posted speed limits and traveling along tangent sections were associated with lower HMB crash frequencies when compared to the baseline of no interchange entrance ramp, higher posted speed limits, and curved horizontal alignments, respectively. The presence of curves to the right was shown to increase the expected HMB crash frequency when compared to curves to the left and tangent segments. The expected HMB crash frequency decreased as the offset to a longitudinal barrier increased. For the Turnpike, the expected HMB crash frequency increased as the ADT increased. Traveling in a rural environment decreased HMB crash frequency when compared to the baseline of an urban area, and traveling in a section containing all horizontal curves with adequate radii (greater than 514m) decreased HMB crash frequency when compared to segments that had less-than-adequate horizontal curve radii. Level terrain decreased HMB crash frequency when compared to the baseline of rolling terrain.

27 14 Ulfarsson and Shankar (2004) used negative multinomial (NM) models to analyze CMC crash frequencies on road sections without median barriers using a multi-year database. NM models could account for roadway section-specific serial correlation across time and the unobserved section-specific heterogeneity in the panel data structure. The panel data used for the analysis was balanced and consisted of annual CMC crash counts along 275 road sections in Washington State between the years 1990 and The modeling results suggested that higher ADT and narrower medians increased the expected CMC crash frequency. As the number of horizontal curves per kilometer increased, the expected CMC crash frequency decreased. As the length of the road section increased, the expected CMC crash frequency increased. Other explanatory variables included in the final model specification were related to differences between maximum and minimum shoulder widths along roadway segments, road friction factors, and state route indicators. The authors compared the NM modeling results with NB and random-effects negative binomial (RENB) regression results using the same data. The comparison is presented in Table 2.1. The NB models assumed independent observations and thus did not account for any roadway section-specific serial correlation across time and unobserved section-specific heterogeneity. The RENB models could account for serial correlation and unobserved section-specific heterogeneity in panel count data. In a study by Shankar et al. (1998), it was found that the RENB model outperformed the NB model when spatial and temporal indicators (fixed effects) were not included in the NB model, based on the log-likelihood at convergence. In the study by Ulfarsson and Shankar (2004), a likelihood ratio test indicated that the NM model outperformed the NB model. Collectively, these studies concluded that the NM model

28 was a superior method to analyze crash counts in the presence of panel data, when 15 compared to RENB and NB regression. Table 2.1. NB, RENB and NM Modeling Results and Comparison (Ulfarsson and Shankar, 2004) Bligh et al. (2006) estimated separate crash frequency models for the following four median-related crash types: CMC crashes on roadway sections without median barriers;

29 Other median-related crashes on roadway sections without median barriers; 16 HMB crashes on roadway sections with median barriers; All median-related crashes on roadway sections with median barriers. An illustration of these median-related crash types is presented in Figure 2.1. Figure 2.1. Anatomy of Traffic Crashes Involving the Median: An Illustration (Bligh et al., 2006)

30 17 A full Bayes approach was used for model estimation, which could account for and estimate the uncertainty of the model parameter estimates. The resulting models are presented in Table 2.2. That the median width variable was statistically significant in all of four models indicated that the expected crash frequency of the four median-related crash types decreased as the median width increased. The explanatory variable for number of lanes was statistically significant in the CMC crash frequency model, but not statistically significant in any of the three remaining models. The expected CMC crash frequency decreased as the number of lanes increased. The indicator variables related to posted speed limits were statistically significant in all four models. The expected crash frequencies of all median-related crash types generally increased as the posted speed limits increased. Table 2.2. Estimated Median-Related Crash Frequency Models by Using Full Bayes Framework (Bligh et al., 2006) Tarko et al. (2008) pointed out that median barriers were installed to prevent drivers from crossing medians and colliding with vehicles traveling in the opposing direction. However, the presence of median barrier could possibly increase crash

31 frequencies of other types, such as median barrier crashes. The impacts of median 18 designs on crash frequencies were analyzed using NB regression. Three types of crashes were investigated: single-vehicle (SV), multiple-vehicle same direction (SD), and multiple-vehicle opposite direction (OD). Crash, geometric, traffic, and construction data were collected from the following eight states: Indiana, Colorado, Illinois, Missouri, New York, Ohio, Oregon, and Washington. More than 30,000 crashes along 1,127 miles of roads were collected. Crash frequency models for two types of depressed medians without barriers (D2N, D3N) are presented in Table 2.3. Crash frequency models for medians with concrete barriers were not estimated due to small sample sizes. The modeling results suggested that frequencies of the three types of crashes increased as ADT increased, and the increase was the greatest in OD crashes and the lowest in SV crashes. The average horizontal curvature and frequency of horizontal curves were only significant in SV crash frequency models. The expected SV crash frequency on roadways with D2N medians increased as the average horizontal curvature increased. The expected SV crash frequency on divided highways with D3N medians decreased as the frequency of horizontal curves increased. The expected SV crash frequency on divided highways with D2N medians increased as the frequency of interchange off-ramps increased. The expected SD crash frequency on highways with D3N medians increased as the frequency of interchange ramps increased. The expected SV crash frequency on highways with D2N medians decreased as the inside shoulder width increased. The expected SD crash frequency on divided highways with D2N medians increased as the frequency of bridges increased. The expected OD crash frequency on divided highways with depressed

32 19 medians and no median barrier increased as posted speed limits increased. The expected OD crash frequency on divided highways with depressed medians and no median barrier was higher for narrower medians relative to wider medians. Table 2.3. Crash Frequency Models for Depressed Medians without Barriers (Tarko et al., 2008) D2N: 30 ft depressed median width 50 ft; D3N: depressed median width > 50 ft SV: Single vehicle crash; SD: multiple-vehicle same direction; OD: multiple-vehicle opposite direction Davis and Morris (2009) applied a simulation model to predict the frequency of CMC crashes on divided highways. The simulation model was used as a method to determine the mechanisms underlying crash occurrence (Davis and Morris, 2009). The fundamental equation for the simulation model is shown as follows: E[C] = E[N]P[U N]P[X U]P[C X] 2.4 where: E[C] = expected number of CMC crashes per year; E[N] = expected number of median encroachments per year;

33 P[U N] = probability an encroachment is uncontrolled; 20 P[X U] = probability an uncontrolled encroachment crosses the median; and P[C X] = probability a crossing encroachment collides with an opposing vehicle. A median encroachment occurs when a vehicle leaves the traveled way to the left and enters the median of a divided highway (Davis and Morris, 2009). The expected number of median encroachments per year was calculated as follows: E[N] =η(mvmt) 2.5 where: MVMT = million vehicle-miles of travel on the road section; and η= encroachments/mvmt. By using the data collected by Hutchinson and Kennedy (1966), η was estimated to be 0.77 with a standard error of 0.12 for March to November, 1.20 with a standard error of 0.25 for December to February, and P[U N] was estimated to be P[X U] was determined by a median traversal model with initial inputs including initial vehicle speed, trajectory angle, deceleration, a braking factor, the median cross-slope, median lateral width, and friction coefficient for the median slope s surface. The initial speed, direction of encroachment, and braking factor were randomly generated from distributions consistent with the data collected by Hutchinson and Kennedy (1966). If a simulated vehicle reached the far side of the median without stopping, its trajectory angle, braking factor, and speed upon exiting the median became the inputs for a collision model. P[C X] was determined by the collision model with randomly-generated opposing vehicles. The initial speeds of opposing vehicles were generated from a normal

34 21 distribution specified by the user. A probability was assigned to each combination of the initial conditions according to their distributions, and the probability of a CMC occurring was the probability assigned to all combinations of initial conditions that led to a crash. The simulations used to model CMC crash frequency required users to input geometric and traffic characteristics, such as median cross-section elements, ADT, and the mean and standard error of opposing traffic speeds. Two published statistical models (Donnell et al., 2002; Bligh et al., 2006) were used to calibrate and validate the simulation model. It was concluded that the simulation model matched the statistical models well based on the non-linear association between CMC frequency and ADT Models Incorporating Spatial Dependency for Count Outcomes It has been recognized that standard econometric techniques often fail in the presence of spatial dependency (Anselin, 1988). Spatial econometric methods have been increasingly applied to a wide range of fields such as regional science, health, and urban and real estate economics (Anselin, 1988, 1999). Modeling techniques that incorporate spatial dependency for count outcome data are quite complex (Anselin, 1988, 1999; Brock and Durlauf, 1995). Anselin (1988, 1999) pointed out that accounting for spatial dependency in count outcome models (e.g. Poisson, NB) was not trivial since the multivariate versions of discrete distributions were more difficult to tract than multivariate normality. There has been a significant amount of recent published literature in which spatial dependency in count data was accounted for. Besag (1974) introduced spatial auto-

35 22 models for lattice systems, which were arrangements of spatial units. The auto-poisson model was used to model count data exhibiting spatial dependency. However, a drawback of the auto-poisson model was that only negative spatial dependency was accommodated, while in many real world applications, most geo-referenced data exhibited positive spatial dependencies. Positive spatial dependency means that features are surrounded by features with similar values; while negative spatial dependency means that features are surrounded by features with dissimilar values. Griffith (2002, 2005) proposed a spatial filtering specification for the auto- Poisson model, where spatial autocorrelation could be accounted for in the mean response specification of an auto-poisson model (Griffith, 2002). The proposed specification converted variables that were spatially auto-correlated into spatially independent variables by partitioning the original geo-referenced attribute variables into two synthetic variables: a spatial filter variable capturing latent spatial dependency that otherwise would remain in the response residuals and a non-spatial variable that is free of spatial dependency in an auto-poisson regression framework (Griffith, 2005). The geographically weighted regression (GWR) was used by Lambert et al (2006) to analyze county-level manufacturing plant announcement count data from 2000 to GWR was a local spatial regression technique used to identify non-stationarity of regression coefficients across space. The marginal effect of an explanatory variable could be dependent on local, unobserved factors. Spatial correlation between local GWR estimates was incorporated in the modeling framework. However, since the subsamples constructed for each local unit for GWR estimation were not independent, the statistical

36 inference drawn from GWR estimates was unreliable. As a result, the application of 23 GWR was limited to exploratory diagnostics to uncover relationships across space. Spatial generalized linear models (SGLM) were introduced as generalized linear models with spatially-structured error terms, to account for spatial dependency in discrete data, such as count data (Breslow and Clayton, 1993; Gotway and Stroup, 1997; Diggle et al., 1998; Schabenberger and Pierce, 2002; Dalthorp, 2004; Schabenberger and Gotway, 2004; Lambert et al., 2006; Braimoh and Onishi, 2007; Diggle and Ribeiro, 2007; Haran, 2010). The spatial dependency in SGLMs can be modeled as Gaussian processes (GP) for continuous and fixed spatial domain (geostatistical data) (Diggle et al., 1998; Schabenberger and Pierce, 2002; Dalthorp, 2004; Schabenberger and Gotway, 2004; Lambert et al., 2006; Braimoh and Onishi, 2007; Diggle and Ribeiro, 2007; Haran, 2010). For discrete and fixed spatial domain (lattice data), the spatial dependency in SGLMs can be modeled as Gaussian Markov random fields (GMRF) (Diggle et al., 1998; Yang et al., 2009; Haran, 2010). The boundary between the geostatistical data and the lattice data is not clear and some spatial data could be treated as either of these two categories. As will be discussed in section , GMRF models use a sparse precision matrix and have massive computational benefits (Rue, 2001; Rue and Held, 2005). GMRF models have computational advantages over GP models, which make the GMRF models especially useful for large datasets and complicated problems. Although GP models have difficult computational issues due to massive matrix operations, especially when employed for large datasets, Gaussian processes are very flexible and widely used in geostatistics (Diggle and Ribeiro, 2007). The precision matrices used in GMRF models

37 24 provide information about conditional dependence, while the covariance matrices in GP models provide information about marginal dependence. As a result, GP models provide a clear interpretation and presentation of the spatial dependency structure, while it is difficult to interpret the spatial dependency structure from estimated GMRF models. In transportation safety, there is a growing body of published literature which employed SGLM specifications to incorporate spatial dependency into crash frequency models. Most of these studies modeled spatial random effects via GMRFs. Miaou et al. (2003) analyzed county-level crash counts using a generalized linear model specification with spatial and temporal random effects. The random spatial effect components were modeled as GMRFs with the conditional autoregressive (CAR) specification, which was a conditional specification of spatial dependency on neighbors. Annual fatal, incapacitating, and nonincapacitating injury crashes on rural, two-lane and low volume (fewer than 2000 vehicles per day) roads in Texas were used for analysis. The modeling results showed the existence of spatial effects in the data this was assessed based on a significant decrease in the Deviance Information Criterion (DIC) in the model with a spatial component when compared to a similar model without the spatial component. Aguero-Valverde and Jovanis (2006) analyzed county-level injury and fatal crash data in Pennsylvania using a generalized linear model specification with spatial, temporal, and space-time interaction random effects, which was referred to as the full Bayes hierarchical model in the article. Random spatial effect components were modeled as GMRFs with the CAR specification. The modeling results were compared to classical NB regression model estimates. The comparison showed that the full Bayes hierarchical model estimates were generally consistent with NB model estimates. The estimated

38 models showed the existence of spatial correlation in the county-level crash data and 25 provided a mechanism to quantify and reduce the effect of this correlation by including a random component representing spatial correlation in the model specification. Quddus (2008) pointed out that spatial dependency existed among area-level crash data and the application of a method that could take into account both spatial dependency and uncorrelated heterogeneity was essential. The research analyzed wardlevel total traffic casualties by severity levels in 633 census wards from the Greater London metropolitan area from 2000 to Before modeling the data, a global Moran s I was used to test for the presence of spatial dependency, and the test results showed that there was positive spatial correlation in the traffic casualty data. The relationships between different traffic casualties and three categories of contributing factors (traffic characteristics, road characteristics and socio-demographic factors) were estimated by using both non-spatial and spatial models. It was stated that non-spatial models such as NB regression models could account for unobserved heterogeneity but could not account for spatial dependency, while spatial models could control for both spatial dependency and unobserved heterogeneity. A generalized linear model specification with spatial random effects was used and referred to as the Bayesian hierarchical model in this study. Random spatial effect components were modeled as GMRFs with the CAR specification. The modeling results showed that the Bayesian hierarchical models were quite similar to NB models for most explanatory variables; however, for some of the explanatory variables included in the model specification, the Bayesian hierarchical model produced parameter estimates more consistent with engineering intuition.

39 Aguero-Valverde and Jovanis (2008) analyzed segment-level crash frequency 26 data using a SGLM specification with CAR spatial error terms. Crash, traffic, and roadway data for the rural two-lane undivided highway network in Centre County, Pennsylvania, were used for the analysis. Three adjacency-based neighboring structures (first-order, second-order and third-order neighbors) were tested to compare which structure best fit the data. It was found that, based on calculated DIC values, models with spatial random effects had a better fit than the models with only heterogeneity and no spatially dependent random effects; and that the model with the first-order neighboring structure had better fit than the models with the other neighboring structures. Aguero-Valverde (2008) and Aguero-Valverde and Jovanis (2010) analyzed segment-level crash frequency data using a multilevel SGLM specification with CAR spatial error terms. The first level was road segments and the second level was functional classes of the road segments. The multilevel models were implemented using a full Bayes hierarchical approach. Crash, traffic, and roadway data for primarily rural engineering districts in Pennsylvania from 2003 to 2006, and in Washington from 2003 to 2005, were used for the analysis. Four different segment neighboring structures (adjacency-based models, distance-order models, distance-exponential decay models, and adjacency-route information models) were used in these studies. It was found that models with spatial random effects had the same or better fit than the models with only heterogeneity and no spatially dependent random effects, and adjacency-based and distance-order models performed better than the others, based on the DIC values. The modeling results showed that the inclusion of spatial correlation significantly improved the estimation of crash

40 frequency models by explaining additional extra-poisson variability existing in the 27 data. Miaou and Song (2005) used multivariate SGLMs with CAR spatial effects to analyze crash data at four-legged signalized intersections in Toronto, and county-level crash data on rural two-lane roads in Texas. The exponential distance-decay function was used to model spatial effects. For the Texas dataset, crash frequencies of three crash severity levels (fatality, incapacitating injury, and non-incapacitating injury) were analyzed simultaneously at the county level. It was concluded based on DIC values that including spatial random effects significantly improved model fit over the models without spatial random effects. It was also found that the degree of overdispersion, which was measured by estimated inverse dispersion parameters, was significantly reduced by adding spatial random effects. Song et al. (2006) offered a detailed discussion of several multivariate spatial models, including models with multivariate CAR and correlated CAR (through the scale parameters in the CAR model) spatial random effects. County-level crash frequencies of four crash types (intersection crash, intersection-related crash, driveway-access crash and non-intersection crash) in Texas were analyzed. It was found that the model with correlated spatial random effects among the four crash types fitted the data better than the model with uncorrelated random effects. To summarize the literature review, there have been several studies that analyzed median-related crash frequency data. Among these studies, many used non-spatial count outcome models, such as Poisson or NB regression. Until recently, the existence of spatial dependency in crash count data has not been considered. There are three major

41 problems when applying spatial econometric methods to model crash frequency data, 28 including: 1) spatial methods are well-developed for continuous data, while for nonnegative count data, the modeling techniques are complex (Anselin, 1988, 1999; Brock and Durlauf, 1995); 2) most current applications of spatial econometric techniques to non-negative count data are focused on area-level data, while in transportation safety research, crashes occur on road networks and it would be more appropriate to apply spatial econometric methods at the link or segment-level; and, 3) most of the applied spatial econometric techniques for crash frequency data employed conditional spatial dependency structures (CAR) instead of marginal spatial dependency structures to model the unobserved spatial effects. In this study, SGLMs based on straightforward measures of spatial dependency in the random spatial error term as a function of road network distance were estimated and compared to classical crash frequency models such as Poisson and NB models. 2.2 Crash Severity Studies Published research related to median barrier and other median-related crash severities is limited. Donnell and Mason (2005, 2006) estimated the severity of CMC and median barrier crashes. Both an ordinal and nominal response was considered. An ordinal response is considered when there is a clear ordering of the discrete outcomes and the absolute distances between categories are unknown. In ordered logistic regression, a linear function of the independent variables is specified for each observation, serving as a basis for modeling the ordinal ranking of data. The probability of falling into outcome

42 category i is the probability that the estimated linear function plus a logistically 29 distributed random error is within the range of the estimated thresholds (Washington et al., 2003). The ordered logit model assumes that the ratio of the odds of being in the chosen category (or a higher category) to the odds of being in a lower category is the same no matter which category is chosen. If the proportional odds assumption for the ordered logit model is violated, the severity outcomes can be treated as nominal responses and a multinomial logit model can be estimated. Donnell and Mason (2005) estimated crash severity probabilities for 138 CMC and 4,416 median barrier crashes on Interstate highways in Pennsylvania that occurred over a five-year period between 1994 and The data showed that cross-median crashes resulted in a higher proportion of fatal outcomes than median barrier crashes. The crash severity outcomes considered were fatal, injury, and property-damage only (PDO). A score test was used to test the proportional odds assumption. The results indicated that the assumption was not violated in the case of cross-median crashes; however, the proportional odds assumption was violated for median barrier crashes. As such, an ordered logit model was used to estimate CMC severity probabilities and a multinomial logit model was used to estimate median barrier crash severity probabilities. The final modeling results indicated that driver impairment (i.e., drug/alcohol use), presence of a horizontal curve, and curve direction were statistically significant in the CMC severity model, while pavement surface condition, driver impairment, the presence of an interchange entrance ramp, and ADT were statistically significant in the median barrier crash severity model.

43 Miaou et al. (2005) and Bligh et al. (2006) estimated crash severities for four 30 median-related crash types, including: CMC on sections of divided highways with no median barrier, other median-related crashes on sections with no median barrier, all median-related crashes on sections with a median barrier, and HMB crashes on divided highway sections with a median barrier. Crashes that occurred in the median sections of Interstate highways, urban freeways, and rural arterial roads in 52 selected Texas counties over a two year period ( ) were used for analysis. Five crash severity outcomes were considered: fatal, incapacitating injury (A), nonincapacitating injury (B), possible injury (C), and PDO. An ordered multinomial logit modeling framework was used. Median width, ADT, number of lanes, and the posted speed limits were considered as explanatory variables for inclusion in the models. The modeling results showed that none of these variables were statistically significant in the CMC severity model. Median width and the posted speed limits were statistically significant in the severity model of other median-related crashes on divided highway sections with no median barrier. Median width, ADT, number of lanes, and the posted speed limits were statistically significant in severity models of all median-related crashes and HMB crashes on divided highway sections with a median barrier. Noyce et al. (2006) analyzed CMC severities using 631 reported CMCs occurring on divided freeways and expressways in Wisconsin over a three-year period between 2001 and Three crash severity outcomes were considered: PDO, injury, and fatality. Ordinal logistic regression models were estimated based on roadway, traffic, driver, crash, incident management, temporal, and environmental data. The final modeling results showed that traffic volume, median width, driver age, time of year, road

44 31 condition, weather condition, and emergency response time significantly affected CMC severities. Tarko et al. (2008) studied crash severities for three types of median-related crashes, including: single-vehicle (SV), multiple-vehicle same direction (SD), and multiple-vehicle opposite direction (OD). Crash, road geometric, traffic, and construction data were collected from the following eight states: Indiana, Colorado, Illinois, Missouri, New York, Ohio, Oregon, and Washington. More than 30,000 crashes along 1,127-miles of divided highways were collected. Two crash severity outcomes were considered: injury/fatal and non-injury. Three binary logit models were estimated for the three crash types, respectively. The modeling results showed that median width, median barrier type, number of lanes, presence of an roadside barrier, ADT, the posted speed limits, percentage of trucks, and the frequency of bridges, interchange entrance and exit ramps, vertical curves, and horizontal curves significantly affected the severity outcomes of median-related crashes. In addition to the median-related crash severity models described above, the published literature is vast with respect to statistical and econometric models of crash severity. Among them, several researchers have used multinomial logit models to estimate crash severity (e.g., Shankar and Mannering, 1996; Carson and Mannering, 2001). There have also been several published studies that estimated crash severity outcomes using nested logit models (e.g., Shankar et al., 1996; Lee and Mannering, 2002; Holdridge et al., 2005; Savolainen and Mannering, 2007). A nested logit model divides crash severities into a hierarchy of levels and takes into account the shared unobserved effects among severity levels. By assuming that the error terms are generalized extreme

45 value (GEV) distributed, the crash severity levels form a nested structure, where the 32 severity levels sharing unobserved effects form the same level. Ordinal logistic regression, multinomial logit and nested logit regressions are three commonly used statistical models to estimate crash severities. The ordered nature of crash severity levels suggests that the ordinal logistic regression model is a logical choice; however, the proportional odds assumption of the model places restrictions on the effects of the explanatory variables. Under this assumption, the explanatory variables either increase the probability of a higher crash severity level and decrease the probability of a lower crash severity level, or increase the probability of a lower crash severity level and decrease the probability of a higher severity level. It is possible, however, that an explanatory variable could increase (decrease) the probability of both the highest and lowest severity levels, and decrease (increase) the probability of middle severity levels (Washington et al., 2003; Holdridge et al., 2005). A limitation of the multinomial logit model is that the disturbance terms among the severity outcomes are assumed to be independent. Previous research (Shankar et al., 1996; Lee and Mannering, 2002; Holdridge et al., 2005; Savolainen and Mannering, 2007) has shown that there were shared unobserved effects at lower crash severity levels. As such, the analysis in this study involved estimating nested logit models of median-related crash severity if the IIA assumption was violated and multinomial logit models if the IIA assumption was not violated.

46 2.3 Summary of Literature 33 Based on the literature review, ADT, road segment length, posted speed limits, presence of horizontal curves and interchange entrance ramps, median width (for CMC frequency), and median barrier offset from the edge of travel way (for HMB crash frequency) were factors generally found to be associated with median-related crash frequencies. ADT, truck involvement, posted speed limits, median width, median barrier type, presence of horizontal curves and interchange entrance ramps, road surface conditions, driver impairment, and driver age were generally found to be associated with median-related crash severity. Median cross-section characteristics, such as foreslopes and backslopes, and the interaction between median barrier types and placement locations, have not been included in previous median-related crash frequency and severity studies. Furthermore, issues of vehicles rolling over in the median of highspeed, divided highways have not been studied in detail, particularly when examining the trade-offs between various median cross-section design elements and single- versus multi-vehicle crash propensities. Poisson and NB regression models are most frequently used to estimate crash frequencies. The estimation of Poisson and NB models assumes that the events are independent. However, potential spatial dependency could exist among crash frequencies along adjacent road segments. Recently, there have been several studies which incorporated spatial dependency in crash frequency models, and most of these studies estimated the models using conditional spatial dependency structures instead of direct spatial dependency structures to model the unobserved spatial effects. In this study, SGLMs based on straightforward measures of spatial dependency as a function of road

47 network distance in the random spatial error term were estimated and compared to 34 classical crash frequency models such as Poisson and NB models. Ordinal logistic regression, multinomial and nested logit models are most frequently used to estimate crash severity outcomes, given that a crash has occurred. The proportional odds assumption of the ordinal logistic regression model places restrictions on the effects of the explanatory variables, and the restrictions can be unreasonable. One limitation of the multinomial logit model is the IIA assumption, which has not been met in several previous research studies (Shankar et al., 1996; Lee and Mannering, 2002; Holdridge et al., 2005; Savolainen and Mannering, 2007). The nested logit model accommodates the shared unobserved effects among severity levels.

48 Chapter 3 Description of Data Data from two states, Pennsylvania and North Carolina, were used for the median-related crash frequency and severity analysis in this study. A unique aspect of this study is the availability of median cross-section data that are not normally available in electronic roadway inventory data. Researchers at the Thomas D. Larson Pennsylvania Transportation Institute developed a laser-scanning system that could be used to collect median barrier offset and median cross-slope data while traveling at highway speeds. A full description of the system is contained in Vemulapalli and Brennan (2009). 3.1 Pennsylvania Data For the median-related crash frequency and severity analysis in Pennsylvania, road information was obtained from PennDOT s Roadway Management System (RMS), and crash information was extracted from the PennDOT Crash Reporting System (CRS). Median-related crashes occurring on multi-lane, rural, limited-access highways in Pennsylvania between 2002 and 2006 were used for the analysis. Information on crash severity, median barrier type and offsets, median cross-slopes, roadway and weather characteristics, vehicle type, driver characteristics, and crash contributory factors were included in the electronic data. A total of 1,927 miles of road segments were included in the data. Approximately 55 percent of the roadway segments are functionally classified as Interstate highways and 45 percent are classified as expressways or principal arterials.

49 36 A total of 1,041 miles of road segments contain a fixed longitudinal median barrier, and a total of 886 miles of road segments do not contain longitudinal median barrier. Of the road segments with median barriers, 715 miles contain guardrail median barrier (a semirigid barrier); and 326 miles contain a concrete median barrier (a rigid barrier). Over the five-year analysis period, a total of 720 median barrier crashes were identified on rural divided highways with longitudinal median barrier. A total of 424 rollovers and 79 CMCs were identified on rural divided highways with no longitudinal median barrier Data for Median-Related Crash Frequency Analysis In this study, median barrier crash frequencies on roads with longitudinal median barrier, and CMC and rollover frequencies on roads without longitudinal median barriers were analyzed. Tables 3.1 and 3.2 show descriptive statistics of continuous and categorical variables in the Pennsylvania median barrier crash frequency analysis dataset. Tables 3.3 and 3.4 show descriptive statistics of continuous and categorical variables in the Pennsylvania CMC and rollover frequency analysis datasets. Figures 3.1, 3.2 and 3.3 show maps of Pennsylvania road segments with and without median barrier CMC, rollover, and HMB crash locations are also shown in Figures 3.1, 3.2, and 3.3, respectively. The geo-spatial information of Pennsylvania road segments was obtained from the Pennsylvania Spatial Data Access portal (PASDA, The Pennsylvania State University) and mapped in ArcGIS (ESRI Inc.). It is clear from these three figures that for most analysis segments, no CMC or rollover occurred during the five-year analysis period. As illustrated in Figures 3.1, 3.2 and 3.3, the road segments are not all connected.

50 37 This is due to the median type constraints placed on the selection of the study segments (e.g., CMC and rollovers can only occur on road segments with earth-divided, traversable medians). Table 3.1. Descriptive Statistics of Continuous Variables for Pennsylvania Median Barrier Crash Frequency Analysis Variable Obs Mean Std. Dev. Min Max Total HMB crashes per segment over five years Yearly HMB crashes per segment (2002) Yearly HMB crashes per segment (2003) Yearly HMB crashes per segment (2004) Yearly HMB crashes per segment (2005) Yearly HMB crashes per segment (2006) Road segment length (mile) Median barrier offset (ft) Median foreslope (H:V) : :1 3.55:1 Median backslope (H:V) : :1 3.55:1 Left shoulder paved width (ft) Total left shoulder width (ft) Median width (ft) Average daily traffic (veh/day) over five years Average daily traffic (veh/day, 2002) Average daily traffic (veh/day, 2003) Average daily traffic (veh/day, 2004) Average daily traffic (veh/day, 2005) Average daily traffic (veh/day, 2006)

51 Table 3.2. Descriptive Statistics of Categorical Variables for Pennsylvania Median Barrier Crash Frequency Analysis Variable Proportion in Sample Roadway Characteristics Proportion of road segments with two lanes per direction 99% Proportion of road segments with three lanes per direction 1% Proportion of left-curved segments 22% Proportion of right-curved segments 22% Proportion of segments with shoulder rumble strips 73% Proportion of median foreslopes steeper than 6H:1V 27% Proportion of median foreslopes 7H:1V 21% Proportion of median foreslopes 8H:1V 10% Proportion of median foreslopes 9H:1V 8% Proportion of median foreslopes flatter than 10H:1V 34% Proportion of median backslopes steeper than 6H:1V 27% Proportion of median backslopes 7H:1V 20% Proportion of median backslopes 8H:1V 11% Proportion of median backslopes 9H:1V 8% Proportion of median backslopes flatter than 10H:1V 33% Proportion of segments with interchange entrance ramps 21% Proportion of segments with interchange exit ramps 18% Proportion of segments with posted speed limit = 55 mph 15% Proportion of segments with posted speed limit = 65 mph 85% Route Indicators Proportion of segments along route 15 1% Proportion of segments along route 22 11% Proportion of segments along route 33 4% Proportion of segments along route 70 4% Proportion of segments along route 78 14% Proportion of segments along route 79 16% Proportion of segments along route 80 7% Proportion of segments along route 81 14% Proportion of segments along route 84 2% Proportion of segments along route 90 9% Proportion of segments along route 99 7% Proportion of segments along route 180 1% Proportion of segments along route 220 2% Proportion of segments along route 322 2% Proportion of segments along route 380 2% Proportion of segments along route 422 3% 38

52 Table 3.3. Descriptive Statistics of Continuous Variables for Pennsylvania CMC/Rollover Frequency Analysis Variable Obs Mean Std. Dev. Min Max Total CMC crashes per segment over five years Yearly CMC crashes per segment (2002) Yearly CMC crashes per segment (2003) Yearly CMC crashes per segment (2004) Yearly CMC crashes per segment (2005) Yearly CMC crashes per segment (2006) Total rollover crashes per segment over five years Yearly rollover crashes per segment (2002) Yearly rollover crashes per segment (2003) Yearly rollover crashes per segment (2004) Yearly rollover crashes per segment (2005) Yearly rollover crashes per segment (2006) Road segment length (mile) Median foreslope (H:V) : :1 3.42:1 Median backslope (H:V) : :1 3.42:1 Left shoulder paved width (ft) Total left shoulder width (ft) Median width (ft) Average daily traffic (veh/day) Average daily traffic (veh/day, 2002) Average daily traffic (veh/day, 2003) Average daily traffic (veh/day, 2004) Average daily traffic (veh/day, 2005) Average daily traffic (veh/day, 2006)

53 Table 3.4. Descriptive Statistics of Categorical Variables for Pennsylvania CMC/Rollover Frequency Analysis Variable Proportion in Sample Roadway Characteristics Proportion of road segments with two lanes per direction 99% Proportion of road segments with three lanes per direction 1% Proportion of left-curved segments 13% Proportion of right-curved segments 13% Proportion of segments with shoulder rumble strips 87% Proportion of median foreslopes steeper than 6H:1V 42% Proportion of median foreslopes 7H:1V 22% Proportion of median foreslopes 8H:1V 13% Proportion of median foreslopes 9H:1V 10% Proportion of median foreslopes flatter than 10H:1V 13% Proportion of median backslopes steeper than 6H:1V 42% Proportion of median backslopes 7H:1V 22% Proportion of median backslopes 8H:1V 13% Proportion of median backslopes 9H:1V 10% Proportion of median backslopes flatter than 10H:1V 13% Proportion of segments with posted speed limit = 55 mph 21% Proportion of segments with posted speed limit = 65 mph 79% Route Indicators Proportion of segments along route 15 2% Proportion of segments along route 22 2% Proportion of segments along route 33 4% Proportion of segments along route 70 7% Proportion of segments along route 79 17% Proportion of segments along route 80 4% Proportion of segments along route 81 26% Proportion of segments along route 84 3% Proportion of segments along route 90 10% Proportion of segments along route 99 10% Proportion of segments along route 322 1% Proportion of segments along route 380 5% Proportion of segments along route 422 9% 40

54 Figure 3.1. Roads for CMC Crash Frequency Analysis and CMC Crash Distribution in Pennsylvania 41

55 Figure 3.2. Roads for Rollover Crash Frequency Analysis and Rollover Crash Distribution in Pennsylvania 42

56 43 Figure 3.3. Roads for HMB Crash Frequency Analysis and HMB Crash Distribution in Pennsylvania Data for Median-Related Crash Severity Analysis For Pennsylvania median-related crash severity analysis, a total of 79 CMCs, 424 rollovers and 720 HMBs were identified. Crash severity outcomes were categorized into six levels: no injury, possible injury, minor injury, moderate injury, major injury and fatality. Table 3.5 shows the severity distribution of the three median-related crash types in Pennsylvania. Table 3.6 shows the distribution of CMC crash severity based on categorical variables available in the Pennsylvania analysis dataset. Descriptive statistics

57 of the data used in the Pennsylvania CMC crash severity modeling are shown in Table (continuous variables) and Table 3.8 (categorical variables). Table 3.5. Severity Distribution of CMC, Rollover and HMB in Pennsylvania Severity (%) No Injury Possible Injury Minor Injury Moderate Injury Major Injury Fatality CMC 16(20.3) 7(8.9) 21(26.6) 16(20.3) 8(10.1) 11(13.9) 79 Rollover 147(34.7) 15(3.5) 151(35.6) 71(16.7) 20(4.7) 20(4.7) 424 HMB 403(56.0) 13(1.8) 194(26.9) 77(10.7) 25(3.5) 8(1.1) 720 Table 3.6. Pennsylvania CMC Crash Severity Distribution by Variables Crash or Segment Characteristic No Injury Possible Injury Severity Frequency (%) Minor Moderate Injury Injury Major Injury Fatality Total CMC 16(20.3) 7(8.9) 21(26.6) 16(20.3) 8(10.1) 11(13.9) 79 Daytime crashes (excluding dawn and dusk) 11(18.6) 6(10.2) 17(28.8) 13(22.0) 3(5.1) 9(15.3) 59 Nighttime crashes (excluding dawn and dusk) 5(25.0) 1(5.0) 4(20.0) 3(15.0) 5(25.0) 2(10.0) 20 Crashes with shoulder rumble strips present 12(21.1) 6(10.5) 14(24.6) 14(24.6) 4(7.0) 7(12.3) 57 Crashes without shoulder rumble strips 3(15.8) 1(5.3) 6(31.6) 2(10.5) 4(21.1) 3(15.8) 19 Crashes with median foreslope steeper than 6H:1V 4(36.4) 0(0) 3(27.3) 2(18.2) 0(0) 2(18.2) 11 Crashes with median foreslope between 6H:1V and 10H:1V 7(18.9) 4(10.8) 13(35.1) 10(27.0) 3(8.1) 0(0) 37 Crashes with median foreslope flatter than 10H:1V 5(17.2) 3(10.3) 6(20.7) 4(13.8) 5(17.2) 6(20.7) 29 Crashes with median backslope steeper than 6H:1V 2(16.7) 2(16.7) 6(50.0) 0(0) 1(8.3) 1(8.3) 12 Crashes with median backslope between 6H:1V and 10H:1V 8(22.9) 2(5.7) 10(28.6) 12(34.3) 2(5.7) 1(2.9) 35 Crashes with median backslope flatter than 10H:1V 5(17.2) 3(10.3) 4(13.8) 4(13.8) 5(17.2) 8(27.6) 29 Crashes on a tangent segment 10(23.3) 4(9.3) 13(30.2) 6(14.0) 6(14.0) 4(9.3) 43 Crashes on a curved segment 5(15.2) 3(9.1) 7(21.2) 10(30.3) 2(6.1) 6(18.2) 33 Total Total

58 45 Table 3.6. Pennsylvania CMC Crash Severity Distribution by Variables (Continued) Severity Frequency (%) Crash or Segment Characteristic No Injury Possible Injury Minor Injury Moderate Injury Major Injury Fatality Total Year (22.2) 0(0) 3(33.3) 1(11.1) 2(22.2) 1(11.1) 9 Year (18.2) 0(0) 3(27.3) 2(18.2) 1(9.1) 3(27.3) 11 Year (30.0) 0(0) 6(30.0) 4(20.0) 2(10.0) 2(10.0) 20 Year (14.3) 2(9.5) 5(23.8) 6(28.6) 2(9.5) 3(14.3) 21 Year (16.7) 5(27.8) 4(22.2) 3(16.7) 1(5.6) 2(11.1) 18 Driver speeding 8(22.2) 3(8.3) 14(38.9) 3(8.3) 2(5.6) 6(16.7) 36 Driver not speeding 8(18.6) 4(9.3) 7(16.3) 13(30.2) 6(14.0) 5(11.6) 43 Driver driving aggressively 10(21.7) 3(6.5) 15(32.6) 8(17.4) 4(8.7) 6(13.0) 46 Driver not driving aggressively 6(18.2) 4(12.1) 6(18.2) 8(24.2) 4(12.1) 5(15.2) 33 Crashes involving an unbelted driver 0(0) 0(0) 4(28.6) 3(21.4) 2(14.3) 5(35.7) 14 Crashes involving a belted driver 16(24.6) 7(10.8) 17(26.2) 13(20.0) 6(9.2) 6(9.2) 65 Male driver 15(21.4) 7(10.0) 19(27.1) 11(15.7) 8(11.4) 10(14.3) 70 Female driver 1(11.1) 0(0) 2(22.2) 5(55.6) 0(0) 1(11.1) 9 Crashes when roadway surface was covered with water/snow/ice/slush 7(26.9) 4(15.4) 12(46.2) 1(3.8) 1(3.8) 1(3.8) 26 Crashes when roadway surface was dry 9(17.0) 3(5.7) 9(17.0) 15(28.3) 7(13.2) 10(18.9) 53 Crashes involving heavy trucks 9(31.0) 2(6.9) 8(27.6) 4(13.8) 4(13.8) 2(6.9) 29 Crashes involving no heavy trucks 7(14.0) 5(10.0) 13(26.0) 12(24.0) 4(8.0) 9(18.0) 50 The data presented in Table 3.6 provides some preliminary insights regarding the relationship between CMC crash severities and several roadway, driver, and vehicle variables. Nighttime crashes appear to have higher probability of fatality and major injury outcomes than daytime crashes. The presence of shoulder rumble strips appears to decrease the probability of fatality and major injury outcomes. Median foreslopes and backslopes flatter than 10H:1V appear to increase the probability of fatality and major injury outcomes, when compared to steeper median foreslopes and backslopes. The presence of curved roadway segments appears to increase the probability of fatality,

59 when compared to tangent segments. Crashes occurring in 2003 appear to have the 46 highest fatality outcome probability among the analysis years, while crashes occurring in 2004 appear to have the lowest fatality probability among the analysis years. Male and unbelted drivers appear to represent a relatively large proportion of severe and major injury crash outcomes. Crash severities appear to be lower in the presence of water/snow/ice/slush when compared to a dry roadway surface. Crashes involving heavy trucks appear to have lower probability of fatality and higher probability of no injury outcome. Other variables shown in Table 3.6 (speeding, aggressive driving) do not appear to significantly change the crash severity distribution across categories of the variables. All variables shown in Table 3.6 were considered in severity model estimations. Table 3.7. Descriptive Statistics of Continuous Variables for Pennsylvania CMC Severity Model Estimation Variable Obs Mean Std. Dev. Min Max Road segment length (ft) Average daily traffic (veh/day) Left shoulder paved width (ft) Total left shoulder width (ft) Median width (ft) Median foreslope (H:V) : :1 3.51:1 Median backslope (H:V) : :1 3.85:1

60 47 Table 3.8. Descriptive Statistics of Categorical Variables for Pennsylvania CMC Severity Model Estimation Categorical Variable Proportion in Sample Severity Outcome Proportion of severe crashes (fatal or major injury) 24% Proportion of non-severe crashes (moderate injury/minor injury/pdo/no 76% injury) Crash Characteristics Proportion at night 26% Proportion wet/icy/snow covered road 32% Roadway Characteristics Proportion on curved road segment 43% Proportion on segments with shoulder rumble strips 75% Proportion with median foreslope steeper than 6H:1V 14% Proportion with median foreslope between 6H:1V and 10H:1V 49% Proportion with median foreslope flatter than 10H:1V 38% Proportion with median backslope steeper than 6H:1V 16% Proportion with median backslope between 6H:1V and 10H:1V 46% Proportion with median backslope flatter than 10H:1V 38% Proportion with posted speed limit = 55 mph 26% Proportion with posted speed limit = 65 mph 74% Driver Characteristics Proportion with driver speeding contributory factor 45% Proportion with aggressive driving contributory factor 58% Proportion with unbelted driver 18% Proportion with female driver 12% Proportion with year old driver 22% Proportion with year old driver 21% Proportion with year old driver 18% Proportion with 60+ year old driver 39% Vehicle Characteristics Proportion involving heavy vehicles 37% Table 3.9 shows the distribution of rollover crash severity based on categorical variables available in the Pennsylvania analysis dataset. Descriptive statistics of the data used in the Pennsylvania rollover crash severity modeling are shown in Table 3.10 (continuous variables) and Table 3.11 (categorical variables).

61 Table 3.9. Pennsylvania Rollover Crash Severity Distribution by Variables Crash or Segment Characteristic No Injury Possible Injury Severity Frequency (%) Minor Moderate Injury Injury Major Injury Fatality Total Rollover 147(34.7) 15(3.5) 151(35.6) 71(16.7) 20(4.7) 20(4.7) 424 Daytime crashes (excluding dawn and dusk) 72(40.0) 5(2.8) 56(31.1) 26(14.4) 11(6.1) 10(5.6) 180 Nighttime crashes (excluding dawn and dusk) 71(30.6) 9(3.9) 89(38.4) 44(19.0) 9(3.9) 10(4.3) 232 Crashes with shoulder rumble strips present 66(36.7) 7(3.9) 59(32.8) 29(16.1) 7(3.9) 12(6.7) 180 Crashes without shoulder rumble strips 16(30.2) 4(7.5) 17(32.1) 11(20.8) 3(5.7) 2(3.8) 53 Crashes with median foreslope steeper than 6H:1V 25(31.6) 5(6.3) 25(31.6) 15(19.0) 2(2.5) 7(8.9) 79 Crashes with median foreslope between 6H:1V and 10H:1V 47(35.1) 4(3.0) 47(35.1) 23(17.2) 8(6.0) 5(3.7) 134 Crashes with median foreslope flatter than 10H:1V 11(47.8) 0(0) 9(39.1) 2(8.7) 0(0) 1(4.3) 23 Crashes with median backslope steeper than 6H:1V 23(29.9) 2(2.6) 29(37.7) 13(16.9) 4(5.2) 6(7.8) 77 Crashes with median backslope between 6H:1V and 10H:1V 48(35.8) 7(5.2) 43(32.1) 24(17.9) 6(4.5) 6(4.5) 134 Crashes with median backslope flatter than 10H:1V 11(45.8) 2(8.3) 5(20.8) 4(16.7) 0(0) 2(8.3) 24 Crashes on a tangent segment 113(34.3) 12(3.6) 119(36.2) 57(17.3) 10(3.0) 18(5.5) 329 Crashes on a curved segment 34(35.8) 3(3.2) 32(33.7) 14(14.7) 10(10.5) 2(2.1) 95 Year (31.0) 0(0.0) 23(39.7) 8(13.8) 3(5.2) 6(10.3) 58 Year (28.7) 1(1.1) 37(42.5) 15(17.2) 4(4.6) 5(5.7) 87 Year (38.5) 6(5.8) 37(35.6) 18(17.3) 2(1.9) 1(1.0) 104 Year (35.5) 4(3.6) 36(32.7) 19(17.3) 6(5.5) 6(5.5) 110 Year (38.5) 4(6.2) 18(27.7) 11(16.9) 5(7.7) 2(3.1) 65 Driver speeding 80(37.6) 7(3.3) 76(35.7) 29(13.6) 8(3.8) 13(6.1) 213 Driver not speeding 67(31.8) 8(3.8) 75(35.5) 42(19.9) 12(5.7) 7(3.3) 211 Driver driving aggressively 85(36.2) 9(3.8) 82(34.9) 36(15.3) 9(3.8) 14(6.0) 235 Driver not driving aggressively 62(32.8) 6(3.2) 69(36.5) 35(18.5) 11(5.8) 6(3.2) Total

62 49 Table 3.9. Pennsylvania Rollover Crash Severity Distribution by Variables (Continued) Severity Frequency (%) Crash or Segment Characteristic No Injury Possible Injury Minor Injury Moderate Injury Major Injury Fatality Total Crashes involving an unbelted driver 7(9.0) 4(5.1) 22(28.2) 19(24.4) 12(15.4) 14(17.9) 78 Crashes involving a belted driver 140(40.5) 11(3.2) 129(37.3) 52(15.0) 8(2.3) 6(1.7) 346 Male driver 115(39.7) 8(2.8) 90(31.0) 47(16.2) 16(5.5) 14(4.8) 290 Female driver 31(23.5) 7(5.3) 61(46.2) 24(18.2) 4(3.0) 5(3.8) 132 Crashes when roadway surface was covered with water/snow/ice/slush 73(47.4) 7(4.5) 50(32.5) 17(11.0) 2(1.3) 5(3.2) 154 Crashes when roadway surface was dry 74(27.4) 8(3.0) 101(37.4) 54(20.0) 18(6.7) 15(5.6) 270 Crashes involving heavy trucks 18(36.0) 1(2.0) 22(44.0) 7(14.0) 1(2.0) 1(2.0) 50 Crashes involving no heavy trucks 129(34.5) 14(3.7) 129(34.5) 64(17.1) 19(5.1) 19(5.1) 374 The data presented in Table 3.9 provides some preliminary insights regarding the relationship between rollover crash severities and several roadway, driver, and vehicle variables. Median foreslopes steeper than 6H:1V appear to increase the probability of fatality and decrease the probability of no injury outcome in a rollover crash, when compared to flatter median foreslopes. Median backslopes steeper than 6H:1V appear to decrease the probability of less severe injuries in a rollover crash, when compared to flatter median backslopes. The presence of curved roadway segments appears to significantly increase the probability of major injury, when compared to tangent segments. Crashes occurring in 2002 appear to have the highest fatality probability among the analysis years, while crashes occurring in 2004 appear to have the lowest fatality probability among the analysis years. Speeding and aggressive driving appear to increase the probability of fatality. Unbelted drivers appear to represent a relatively large proportion of severe crash outcomes. Male drivers appear to have higher no injury

63 50 probabilities than female drivers. Crash severities appear to be lower in the presence of water/snow/ice/slush when compared to a dry roadway surface. Other variables shown in Table 3.9 (day and night, shoulder rumble strips, heavy vehicle-involvement) do not appear to significantly change the crash severity distribution across categories of the variables. All variables shown in Table 3.9 were considered in severity model estimations. Table Descriptive Statistics of Continuous Variables for Pennsylvania Rollover Severity Model Estimation Variable Obs Mean Std. Dev. Min Max Road segment length (ft) Average daily traffic (veh/day) Left shoulder paved width (ft) Total left shoulder width (ft) Median width (ft) Median foreslope (H:V) : :1 3.55:1 Median backslope (H:V) : :1 3:1 Driver age

64 51 Table Descriptive Statistics of Categorical Variables for Pennsylvania Rollover Severity Model Estimation Categorical Variable Proportion in Sample Severity Outcome Proportion of severe crashes (fatal or major injury) 10% Proportion of moderate-injury crashes 17% Proportion of minor-injury crashes 33% Proportion of PDO or no-injury crashes 40% Crash Characteristics Proportion at night 41% Proportion wet/icy/snow covered road 32% Roadway Characteristics Proportion on curved road segment 45% Proportion on segments with shoulder rumble strips 77% Proportion with median foreslope steeper than 6H:1V 33% Proportion with median foreslope between 6H:1V and 10H:1V 57% Proportion with median foreslope flatter than 10H:1V 10% Proportion with median backslope steeper than 6H:1V 33% Proportion with median backslope between 6H:1V and 10H:1V 57% Proportion with median backslope flatter than 10H:1V 10% Proportion with posted speed limit = 55 mph 7% Proportion with posted speed limit = 65 mph 93% Driver Characteristics Proportion with driver speeding contributory factor 47% Proportion with aggressive driving contributory factor 54% Proportion with unbelted driver 19% Proportion with female driver 28% Proportion with year old driver 16% Proportion with year old driver 21% Proportion with year old driver 50% Proportion with 60+ year old driver 13% Vehicle Characteristics Proportion involving heavy vehicles 12% Table 3.12 shows the distribution of median barrier crash severities based on categorical variables available in the Pennsylvania analysis dataset. Descriptive statistics of the data used in the Pennsylvania median barrier crash severity modeling are shown in Table 3.13 (continuous variables) and Table 3.14 (categorical variables).

65 Table Pennsylvania HMB Crash Severity Distribution by Variables Crash or Segment Characteristic No Injury Possible Injury Severity Frequency (%) Minor Moderate Injury Injury Major Injury Fatality Total median barrier crash 403(56.0) 13(1.8) 194(26.9) 77(10.7) 25(3.5) 8(1.1) 720 Daytime crashes (excluding dawn and dusk) 208(53.6) 8(2.1) 111(28.6) 45(11.6) 14(3.6) 2(0.5) 388 Nighttime crashes (excluding dawn and dusk) 172(58.1) 5(1.7) 75(25.3) 28(9.5) 10(3.4) 6(2.0) 296 Collisions with guardrail barrier 146(60.1) 5(2.1) 58(23.9) 24(9.9) 9(3.7) 1(0.4) 243 Collisions with concrete barrier 199(53.8) 6(1.6) 104(28.1) 43(11.6) 14(3.8) 4(1.1) 370 Crashes with shoulder rumble strips present 208(60.3) 4(1.2) 86(24.9) 36(10.4) 9(2.6) 2(0.1) 345 Crashes without shoulder rumble strips 141(50.4) 8(2.9) 79(28.4) 33(11.9) 14(5.0) 3(1.1) 278 Crashes with median foreslope steeper than 6H:1V 32(61.5) 1(1.9) 13(25.0) 3(5.8) 2(3.8) 1(1.9) 52 Crashes with median foreslope between 6H:1V and 10H:1V 26(49.1) 2(3.8) 13(24.5) 12(22.6) 0(0.0) 0(0.0) 53 Crashes with median foreslope flatter than 10H:1V 34(40.0) 3(3.5) 33(38.8) 11(12.9) 3(3.5) 1(1.2) 85 Crashes with ramps present 135(58.7) 4(1.7) 59(25.7) 20(8.7) 10(4.3) 2(0.9) 230 Crashes without ramps present 207(54.6) 7(1.8) 103(27.2) 47(12.4) 13(3.4) 2(0.5) 379 Crashes on a tangent segment 311(55.8) 11(2.0) 149(26.8) 66(11.8) 17(3.1) 3(0.5) 557 Crashes on a curved segment 89(56.0) 2(1.3) 45(28.3) 11(6.9) 8(5.0) 4(2.5) 159 Year (54.9) 0(0.0) 28(23.0) 13(10.7) 9(7.4) 5(4.1) 122 Year (58.3) 2(1.3) 46(30.5) 13(8.6) 2(1.3) 0(0.0) 151 Year (53.4) 3(2.1) 45(30.8) 15(10.3) 4(2.7) 1(0.7) 146 Year (56.9) 5(2.9) 41(23.6) 21(12.1) 6(3.4) 2(1.1) 174 Year (55.9) 3(2.4) 34(26.8) 15(11.8) 4(3.1) 0(0.0) 127 Driver speeding 194(56.7) 9(2.6) 90(26.3) 34(9.9) 11(3.2) 4(1.2) 342 Driver not speeding 209(55.3) 4(1.1) 104(27.5) 43(11.4) 14(3.7) 4(1.1) 378 Driver driving aggressively 228(55.3) 10(2.4) 116(28.2) 39(9.5) 15(3.6) 4(1.0) 412 Driver not driving aggressively 175(56.8) 3(1.0) 78(25.3) 38(12.3) 10(3.2) 4(1.3) 308 Crashes involving an unbelted driver 24(28.6) 1(1.2) 22(26.2) 22(26.2) 11(13.1) 4(4.8) 84 Crashes involving a belted driver 379(59.6) 12(1.9) 172(27.0) 55(8.6) 14(2.2) 4(0.6) Total

66 53 Table Pennsylvania HMB Crash Severity Distribution by Variables (continued) Severity Frequency (%) Crash or Segment Characteristic No Injury Possible Injury Minor Injury Moderate Injury Major Injury Fatality Total Male driver 306(58.5) 8(1.5) 116(22.2) 62(11.9) 24(4.6) 7(1.3) 523 Female driver 93(48.9) 5(2.6) 78(41.1) 13(6.8) 1(0.5) 0(0.0) 190 Crashes when roadway surface was covered with water/snow/ice/slush 131(67.9) 2(1.0) 39(20.2) 15(7.8) 4(2.1) 2(1.0) 193 Crashes when roadway surface was dry 272(51.6) 11(2.1) 155(29.4) 62(11.8) 21(4.0) 6(1.1) 527 Crashes involving heavy trucks 58(46.8) 2(1.6) 29(23.4) 22(17.7) 8(6.5) 5(4.0) 124 Crashes involving no heavy trucks 345(57.9) 11(1.8) 165(27.7) 55(9.2) 17(2.9) 3(0.5) 596 Vehicle overturned 13(18.8) 3(4.3) 23(33.3) 19(27.5) 6(8.7) 5(7.2) 69 No vehicle overturned 390(59.9) 10(1.5) 171(26.3) 58(8.9) 19(2.9) 3(0.5) 651 The data presented in Table 3.12 provide some preliminary insights regarding the relationship between median barrier crash severities and various roadway, driver, and vehicle characteristics. A collision with a concrete median barrier appears to increase the probability of more severe injuries in a median barrier crash than a collision with a guardrail median barrier. The presence of shoulder rumble strips appears to decrease the probability of severe crash outcomes given a median barrier crash. Steeper median foreslopes appear to increase the probability of less severe injuries in a median barrier crash. The presence of ramps appears to slightly increase the probability of more severe injuries in a median barrier crash. The presence of a curved roadway segment appears to increase the probability of more severe crashes, when compared to tangent roadway segments. Crashes occurring in 2002 appear to have the highest fatality probability among the analysis years. Unbelted drivers appear to be involved in a greater proportion of high severity crashes when compared to belted drivers. Male drivers appear to be associated with higher severity crash outcomes than female drivers. Crash severities

67 54 appear to be lower when the roadway surface is covered with water/snow/ice/slush when compared to dry roadway surface conditions. The involvement of heavy trucks appears to increase the probability of severe crashes when compared to crashes involving passenger cars. Overturned vehicles appear to significantly increase the severity level in a median barrier crash. The remaining variables in Table 3.12 (day and night, speeding, aggressive driving) do not appear to be associated with crash severity outcomes. All of the variables shown in Table 3.12 were considered in the severity model estimations. Table Descriptive Statistics of Continuous Variables for Pennsylvania HMB Severity Model Estimation Variable Obs Mean Std. Dev. Min Max Road segment length (mile) ADT (veh/day) Left shoulder pavement width (ft) Left shoulder width (ft) Median width (ft) Median foreslope (H:V) : :1 3.55:1 Median barrier offset (ft) Driver age

68 Table Descriptive Statistics of Categorical Variables for Pennsylvania HMB Severity Model Estimation Categorical Variable Proportion in Sample Severity Outcome Proportion of severe crashes (fatal or major injury) 3% Proportion of moderate-injury crashes 13% Proportion of minor-injury crashes 32% Proportion of PDO or no-injury crashes 52% Crash Characteristics Proportion at night 42% Proportion wet/icy/snow covered road 25% Roadway Characteristics Proportion on curved road segment 22% Proportion on segments with shoulder rumble strips 55% Proportion with median foreslope steeper than 6H:1V 27% Proportion with median foreslope between 6H:1V and 10H:1V 28% Proportion with median foreslope flatter than 10H:1V 45% Proportion on segments with interchange entrance ramps 38% Proportion with concrete median barriers 60% Proportion with guardrail median barriers 40% Proportion with posted speed limit = 55 mph 15% Proportion with posted speed limit = 65 mph 85% Driver Characteristics Proportion with driver speeding contributory factor 45% Proportion with aggressive driving contributory factor 53% Proportion with unbelted driver 12% Proportion with female driver 23% Proportion with year old driver 22% Proportion with year old driver 23% Proportion with year old driver 46% Proportion with 60+ year old driver 9% Vehicle Characteristics Proportion involving heavy vehicles 18% Proportion involving overturned vehicles 15% North Carolina Data Electronic roadway and crash data were acquired from the North Carolina Highway Safety Information System (HSIS) dataset (Council et al., 2006), and supplemented with data collected using the laser-scanning system described earlier. Median-related crashes occurring between 2000 and 2004 on rural Interstate highways and expressways were used for the analysis. Information on crash severity, median

69 56 barrier type, median barrier offset from the edge of the traveled way, median crossslopes, roadway and weather characteristics, vehicle type, driver characteristics, and crash contributory factors were included in the electronic data. A total of 506 miles of road segments were included in the analysis. Of these road segments, 468 miles contain longitudinal median barriers, and 38 miles have earth-divided traversable medians with no barrier. Of the road segments with median barriers, 264 miles contain cable median barrier; 180 miles contain guardrail median barrier; and 24 miles contain concrete median barrier. Over the five-year analysis period, a total of 3,691 median barrier crashes were identified on rural divided highways with longitudinal median barrier; and a total of 34 rollover and 11 CMCs were identified on rural divided highways with earth-divided, traversable medians and no barrier Data for Median-Related Crash Frequency Analysis For the median-related crash frequency analysis, the small sample of road segments without longitudinal median barrier prohibited model estimation of CMC and rollover crash frequencies; therefore, only median barrier crash frequency on roads with longitudinal median barrier were modeled in this study. Tables 3.15 and 3.16 show descriptive statistics for key continuous and categorical variables, respectively, in the North Carolina median barrier crash frequency analysis dataset.

70 57 Table Descriptive Statistics of Continuous Variables for North Carolina HMB Frequency Model Estimation Std. Dev. Min Max Variable Obs Mean Total HMB crashes per segment over five years Yearly HMB crashes per segment (2000) Yearly HMB crashes per segment (2001) Yearly HMB crashes per segment (2002) Yearly HMB crashes per segment (2003) Yearly HMB crashes per segment (2004) Road segment length (ft) Median barrier offset (ft) Median foreslope (H:V) : :1 4.71:1 Median backslope (H:V) : :1 3.59:1 Left shoulder width (ft) Median width (ft) Average daily traffic (veh/day) over five years Average daily traffic (veh/day, 2000) Average daily traffic (veh/day, 2001) Average daily traffic (veh/day, 2002) Average daily traffic (veh/day, 2003) Average daily traffic (veh/day, 2004)

71 58 Table Descriptive Statistics of Categorical Variables for North Carolina HMB Frequency Model Estimation Variable Proportion in Sample Roadway Characteristics Proportion of road segments with two lanes per direction 87% Proportion of road segments with three lanes per direction 4% Proportion of left-curved segments 6% Proportion of right-curved segments 5% Proportion of segments with shoulder rumble strips 74% Proportion of median foreslopes steeper than 6H:1V 3% Proportion of median foreslopes 7H:1V 5% Proportion of median foreslopes 8H:1V 9% Proportion of median foreslopes 9H:1V 8% Proportion of median foreslopes flatter than 10H:1V 75% Proportion of median backslopes steeper than 6H:1V 26% Proportion of median backslopes 7H:1V 18% Proportion of median backslopes 8H:1V 12% Proportion of median backslopes 9H:1V 6% Proportion of median backslopes flatter than 10H:1V 38% Proportion of concrete median barriers 6% Proportion of guardrail median barriers 37% Proportion of cable median barriers 57% Proportion of segments with interchange entrance ramps 22% Proportion of segments with interchange exit ramps 22% Proportion of segments with posted speed limit = 55 mph 15% Proportion of segments with posted speed limit = 60 mph 3% Proportion of segments with posted speed limit = 65 mph 42% Proportion of segments with posted speed limit = 70 mph 40% Route Indicators Proportion of segments along route 1 5% Proportion of segments along route 26 5% Proportion of segments along route 40 38% Proportion of segments along route 64 7% Proportion of segments along route 74 3% Proportion of segments along route 77 3% Proportion of segments along route 85 14% Proportion of segments along route 95 22% Proportion of segments along route 321 3%

72 3.2.2 Data for Median-Related Crash Severity Analysis 59 The small sample of CMC and rollover crashes on earth-divided, traversable medians in North Carolina did not permit crash severity models to be estimated. As such, only the severity of North Carolina median barrier crashes on rural, divided highways was estimated in this study. The primary reason for observing a small number of CMC and rollover crashes in North Carolina is because, in 1998, the North Carolina Department of Transportation funded a systematic median barrier installation program on high-speed, divided highways with medians less than 70 feet (23 m) wide (Lynch, 1998). A total of 3,691 median barrier crashes over the five-year analysis period on 453 miles of roadway were included in the crash severity analysis. Of the 453 roadway miles included in the analysis database, 452 miles had median foreslopes flatter than or equal to 6H:1V. A 1-mile segment had a 4.7:1 foreslope and a 3.7:1 backslope. This segment was included in the 6H:1V category for the purposes of this analysis. Of the median barrier crashes that were identified, 25 were fatal crashes, 50 were class A crashes, 272 were class B crashes, 635 were class C crashes, and 2709 were crashes without injury (PDO). Table 3.17 shows the distribution of median barrier crash severities based on categorical variables available in the analysis dataset. Descriptive statistics of the data to be used in the North Carolina median barrier crash severity model are shown in Table 3.18 (continuous variables) and Table 3.19 (categorical variables).

73 Table North Carolina HMB Crash Severity Distribution by Variables Crash or Segment Characteristic Severity Frequency (%) No Injury Class C Class B Class A Fatal Total median barrier crashes 2709(73.4) 635(17.2) 272(7.4) 50(1.4) 25(0.7) 3691 Daytime crashes (excluding dawn and dusk) 1719(73.8) 392(16.8) 168(7.2) 37(1.6) 13(0.6) 2329 Nighttime crashes (excluding dawn and dusk) 867(72.2) 218(18.2) 91(7.6) 13(1.1) 12(1.0) 1201 Collisions with cable barrier 1660(79.1) 290(13.8) 114(5.4) 20(1.0) 15(0.7) 2099 Collisions with guardrail barrier 945(66.7) 299(21.1) 139(9.8) 24(1.7) 10(0.7) 1417 Collisions with concrete barrier 104(59.4) 46(26.3) 19(10.9) 6(3.4) 0(0.0) 175 Crashes with shoulder rumble strips present 1782(72.2) 440(17.8) 195(8.9) 32(1.3) 20(0.8) 2469 Crashes without shoulder rumble strips 632(76.1) 134(16.1) 50(6.0) 11(1.3) 3(0.4) 830 Crashes on a tangent segment 2448(74.1) 568(17.2) 230(7.0) 36(1.1) 23(0.7) 3305 Crashes on a curved segment 250(67.2) 64(17.2) 42(11.3) 14(3.8) 2(0.5) 372 Crashes on road with concrete pavement 621(79.0) 120(15.3) 30(3.8) 9(1.1) 6(0.8) 786 Crashes on road with asphalt pavement 2074(71.8) 512(17.7) 241(8.3) 41(1.4) 19(0.7) 2887 Crashes on road with on-ramps 592(75.0) 126(16.0) 58(7.4) 11(1.4) 2(0.3) 789 Crashes on road with off-ramps 599(75.3) 131(16.5) 50(6.3) 12(1.5) 3(0.4) 795 Crashes on road without ramps 1116(73.2) 266(17.5) 107(7.0) 22(1.4) 13(0.9) 1524 Crashes involving a belted driver 2660(74.7) 612(17.2) 238(6.7) 36(1.0) 13(0.4) 3559 Crashes involving an unbelted driver 25(34.2) 10(13.7) 22(30.1) 9(12.3) 7(9.6) 73 Male driver 1784(75.7) 350(14.8) 174(7.4) 31(1.3) 18(0.8) 2357 Female driver 915(69.3) 285(21.6) 96(7.3) 18(1.4) 7(0.5) 1321 Crashes involving impaired driver 418(68.5) 114(18.7) 63(10.3) 9(1.5) 6(1.0) 610 Crashes where vehicle overturned 63(24.9) 61(24.1) 84(33.2) 28(11.1) 17(6.7) 253 Single-vehicle crashes 2440(77.7) 459(14.6) 200(6.4) 27(0.9) 16(0.5) 3142 Multi-vehicle crashes 269(49.0) 176(32.1) 72(13.1) 23(4.2) 9(1.6) 549 Crashes involving a heavy vehicle 190(58.5) 74(22.8) 46(14.2) 6(1.8) 9(2.8) 325 Crashes when roadway surface was covered with water/snow/ice/slush 1216(80.7) 215(14.3) 58(3.9) 11(0.7) 6(0.4) 1506 Crashes when roadway surface was dry 1490(68.3) 419(19.2) 214(9.8) 39(1.8) 19(0.9) Total The data presented in Table 3.17 provides some preliminary insights regarding the relationship between median barrier crash severities and various roadway, driver, and vehicle characteristics. Among the three median barrier types, collisions with concrete

74 61 barriers appear associated with higher crash severities when compared to crashes with a guardrail or cable barrier. Collisions with guardrail appear associated with higher crash severities when compared to collisions with a cable barrier. Crash severities appear to be higher in the presence of a curved roadway segment and on asphalt pavement surfaces, when compared to tangent roadway segments and concrete pavement surfaces, respectively. Female and physically-impaired drivers appear to suffer from higher severities when compared to male and unimpaired drivers. Unbelted drivers, overturning vehicles, multi-vehicle crashes, and median barrier crashes involving heavy vehicles appear to increase crash severity outcomes. Crash severities appear to be lower in the presence of water/snow/ice/slush. No apparent trends exist between crash severity and other explanatory variables shown in Table 3.17 (day and night, shoulder rumble strips, ramps). Table Descriptive Statistics of Continuous Variables for North Carolina HMB Severity Model Estimation Variable Mean Std. Dev. Min Max Median foreslope (H:V) 11.0: :1 4.7:1 Median backslope(h:v) 7.7: :1 3.7:1 Median barrier offset (ft) Left shoulder width (ft) Median width (ft) ADT (veh/day) Vehicle traveling speed (mph) Driver age

75 62 Table Descriptive Statistics of Categorical Variables for North Carolina HMB Severity Model Estimation Categorical Variable Proportion in Sample Severity Outcome Proportion of severe crashes (fatal or class A injury) 2% Proportion of class B crashes 7% Proportion of class C crashes 17% Proportion of no-injury crashes 74% Crash Characteristics Proportion at night 33% Proportion wet/icy/snow covered road 59% Roadway Characteristics Proportion on curved road segment 10% Proportion on road segment with concrete pavement 19% Proportion on segments with shoulder rumble strips 75% Proportion with median foreslope flatter than 10H:1V 70% Proportion on segments with interchange entrance ramps 22% Proportion with concrete median barriers 5% Proportion with guardrail median barriers 40% Proportion with cable median barriers 55% Proportion with posted speed limit = 55 mph 5% Proportion with posted speed limit = 60 mph 5% Proportion with posted speed limit = 65 mph 53% Proportion with posted speed limit = 70 mph 37% Driver Characteristics Proportion with physically impaired driver contributory factor 17% Proportion with unbelted driver 2% Proportion with female driver 36% Vehicle Characteristics Proportion involving heavy vehicles 9% Proportion involving overturned vehicles 6% Proportion involving multiple vehicles 15%

76 Chapter 4 Methodology This chapter consists of two sections. The first section describes the process and methodology to estimate and assess crash frequency models with and without the presence of spatial dependency. The second section describes the modeling techniques used to estimate crash severity outcomes. 4.1 Crash Frequency Modeling and Model Assessment The process to estimate and assess median-related crash frequency models in this study is presented in Figure 4.1. The present study did not only estimate crash frequency models, but also simulated the predicted crash frequencies from the estimated models and compared the simulated crash frequencies with the observed ones to evaluate the predictive capabilities of the estimated models. For each analysis dataset, 90 percent of the sample was randomly selected for model estimation and the remaining 10 percent of the sample was used for model assessment using simulation. Also, transferability tests were performed to test whether the estimated regression parameters from the crash frequency models could be transferred between two states. This study estimated five-year aggregated median-related crash frequency, as well as the yearly median-related crash frequency. The crash frequencies aggregated over five years may result in information loss when compared to yearly crash frequencies. The information loss could introduce unobserved heterogeneity, as well as bias in model

77 64 estimation (Washington et al., 2003, 2010; Lord and Mannering, 2010). At the beginning of model estimation, Poison or Negative Binomial (NB) models were estimated as base models. For the five-year aggregated crash frequency model estimation, route indicators were then added to the base models to account for route-specific effects, such as weather and driver population effects. The addition of route indicators was expected to account for some spatially-dependent effects since road segments along the same route were connected to each other. Then, spatial generalized linear models (SGLM) were estimated if there were significant unobserved spatially-dependent effects. For the yearly crash frequency model estimation, route and year indicators were added to the estimated base models to account for spatial and temporal effects. Then, random-effects Poisson or NB models were estimated to account for road segment-specific random effects and serial correlation in the panel count data. SGLMs were also estimated if there were significant unobserved spatially-dependent effects. A point that is worth explanation is that the significance of route-specific effects does not necessarily mean that there are significant unobserved spatially-dependent effects. Route-specific effects are estimated by treating all road segments along the same route as one group and the average crash frequencies are compared among different route groups to determine whether the average crash frequencies of several groups are significantly higher or lower than those of other groups. The road segments in the same group are assumed to be independent of each other and are given the same weight regardless of the distances among these road segments and directions. While in the spatial generalized linear models, road segments along the same route are assumed to be dependent upon each other if they are in the same direction, and the dependence is related

78 to the distance along road segments. The closer the road segments are, the higher the 65 dependence is assumed to be between them.

79 Figure 4.1. Crash Frequency Analysis Process 66

80 4.1.1 Crash Frequency Models 67 This section consists of three sections. The first section briefly describes the most commonly used count outcome models. The second section describes the random-effects Poisson and NB models. The third section describes spatial generalized linear models for count outcomes Classical Count Outcome Models Poisson regression provides a starting point for non-negative count data analysis. By assuming the response variable Y has a Poisson distribution, the probability of Y i equaling a particular count y i is specified in equation 4.1. Pr( Y i λ e i i i = yi ) = λ 4.1 y! i y ln λ = x β 4.2 i i where: λ i = expected count for analysis unit i; x i = vector of explanatory variables for analysis unit i; and β = vector of coefficients of explanatory variables. A characteristic of the Poisson distribution is that its mean is equal to its variance, which is often violated in crash frequency data. In the case of overdispersion, a NB specification could be used (Cameron and Trivedi, 1998; Washington et al., 2003). The expected count for the NB model is specified in equation 4.3. The conditional variance σ i of yi given xi

81 68 is generally specified as in equation 4.4. The probability of Y i equaling a particular count y i is specified in equation 4.5. ln λ + i = xiβ ε i 4.3 σ = λ 1+ αλ ) 4.4 i i ( i Γ((1/ α) + y ) 1/ α λ Pr( Y i i ) Γ(1/ α) y! (1/ α) + λ (1/ α) + λ i 1/ α i y = y ) = ( ) ( i 4.5 i i i where: ε i = disturbance term specific to each observation (exp( ε i ) is gamma-distributed with mean one and variance α ); σ i = conditional variance of y i given x i (if σ i is equal to 0, the NB model reduces to a Poisson model); and Γ (.) = a gamma function. Both Poisson and NB models are estimated by maximum likelihood procedures Random-Effects Poisson and Random-Effects Negative Binomial Models In the yearly crash frequency datasets, there are multiple observations over several years on the same road segment (referred to as panel data), which causes serial correlation problems. Hausman (1984) proposed random- and fixed-effects approaches for analyzing panel count data to account for serial correlation. The proposed randomeffects count models account for road segment-specific effects which vary randomly across road segments, in addition to serial correlation. Fixed-effects count models do not account for the unobserved heterogeneity, and were not estimated in this study. The

82 69 random-effects Poisson and NB models are specified as follows (Hausman et al., 1984; Cameron and Trivedi, 1998; Shankar et al., 1998): ln λ + it = xit β ε i 4.6 where: λ it = expected count for analysis unit i in year t; x it = vector of explanatory variables for analysis unit i in year t; and ε i = disturbance term specific to analysis unit i, In the case of random-effects Poisson models, exp( ε i ) is gamma-distributed with mean one and variance α.the joint probability of the counts Y i1,, Y ini equaling y i1,, y ini for the i th unit group is specified in equation 4.7. Pr( Y = y 1,..., Y = y i i in i in 1 i ) = n i t= 1 i y Γ((1/ α) + it λ it yit! Γ(1/ α) n t= 1 y it ) 1/ α ni (1/ α) + λit t= 1 1/ α 1 (1/ α) + n i t= 1 λit n i yit t= where: n i = number of years of observations on the i th unit group. In the case of random-effects NB models, for analysis unit group i, the dispersion (variance divided by the mean) is equal to (1+δ i ), which indicates that dispersion is constant within the group. Across groups, δ i varies randomly, and 1/(1+ δ i ) is assumed to be beta distributed with parameters r and s. The joint probability of the counts Y i1,, Y ini equaling y i1,, y ini for the i th unit group is specified in equation 4.8.

83 Pr( Y = y 1,..., Y = y i i in i in 1 i ) 70 i Γ( r + s) Γ( r + λit ) Γ( s + t= 1 = n Γ( r) Γ( s) Γ( r + s + λ + n i t= 1 it n i t= 1 n i t= 1 y y it it ) ) n i t= 1 Γ( λ + y it it it it ) Γ( λ ) y! 4.8 Both random-effects Poisson and negative binomial models are estimated by maximum likelihood procedures Spatial Generalized Linear Model for Count Outcomes Both Poisson and NB models assume independent observations for the estimation to remain efficient. However, it is possible that there is spatial dependency among crash frequencies along adjacent road segments. In the presence of spatial dependency, modeling techniques should be improved to incorporate spatial dependency for better inference and superior predictions (Haran, 2010). A spatial generalized linear model (SGLM) for count outcome Y i of spatial unit i (i=1,2,,n) is specified in equation 4.9 (Breslow and Clayton, 1993; Gotway and Stroup, 1997; Diggle et al., 1998; Schabenberger and Pierce, 2002; Dalthorp, 2004; Schabenberger and Gotway, 2004; Lambert et al., 2006; Braimoh and Onishi, 2007; Diggle and Ribeiro, 2007; Haran, 2010). log Y = X β + w + Z 4.9 i i i i where: X i = explanatory variable vector of unit i, i=1,, n; β = coefficient vector of explanatory variables; w i = spatial random effect at unit i; and

84 Z i = nugget, non-spatially structured random effect at unit i, mutually 71 independently and normally distributed with mean 0 and varianceτ 2. In a spatial generalized linear model as specified in equation 4.9, a spatial random effect w i is added to the linear predictor to account for the unobserved spatially dependent effects. In a spatial Poisson model without a nugget effect Z, the conditional variance of Y i given w i is constrained to be equal to the conditional expectation of Y i. However, additional variability in the data may be encountered in practice, which is not spatially structured. The nugget effect Z is added to the linear predictor to account for additional i non-spatial variation in the data (Diggle et al., 1998; Diggle and Ribeiro, 2007). In a spatial generalized linear model, spatial dependency is imposed on the process by modeling the spatial random effects w = ( w 1,, i w n ) T as a Gaussian random field. The choice of the spatial structure of w is dependent on the characteristics of the spatial domain (Haran, 2010). When the spatial domain is continuous and fixed, it is geostatistical data and w could be modeled as a zero mean stationary Gaussian process (GP) (Diggle and Ribeiro, 2007; Haran, 2010), which implies that w follows a n- dimensional multivariate normal distribution with mean zero and a n*n covariance matrix Σ ( θ ). The components of covariance matrix ( θ ) Σ depend on the distance between two spatial locations and θ are the parameters controlling the range and scale of spatial dependency. When the spatial domain is discrete and fixed, it is lattice data and w is modeled as combinations of contributions from neighboring sites, typically as Gaussian Markov

85 72 random fields (GMRF) (Besag, 1974). GMRF models employ a conditional specification of spatial dependence, assuming that the value of a spatial unit depends primarily on its neighbors. For wi at spatial unit i, the conditional distribution of w i given w J, the spatial random effect vector associated with all the other spatial units J (any j in J i) in the spatial domain, is a multivariate normal distribution as specified in equation 4.10 (Besag, 1974; Haran, 2010). w i w J ~ n N d j= 1 ij 1 w j, pi 4.10 where: d ij = spatial proximity parameter between spatial unit i and j, non-zero only when i and j are neighbors; and p i = precision parameter in the precision matrix, which is the inverse of the covariance matrix and is a sparse matrix. The boundary between these geostatistical data and lattice data is not clear and some spatial data could be treated as either of these two categories. In this study, the spatial units were road segments and crash counts on these road segments were modeled. These road segments could be treated as lattice data by treating each road segment as a spatial entity where the crash frequencies were recorded. Or, the road segments could be treated as geostatistical data, since crashes could occur at any location and there is a crash frequency associated with any location along these road segments. As summarized in Chapter 2, most of the previous research on spatial models of crash frequency employed the lattice data category and modeled spatial random effects w as GMRFs. GMRF models use a sparse precision matrix and have massive computational

86 73 benefits (Rue, 2001; Rue and Held, 2005). GMRF models have computational advantages over GP models, which make the GMRF models especially useful for large datasets and complicated problems. Although GP models have difficult computational issues due to massive matrix operations, especially when employed for large datasets, Gaussian processes are very flexible and widely used in geostatistics (Diggle and Ribeiro, 2007). To specify a Gaussian process, only a valid mean and covariance structure are needed, which automatically determine the joint distribution. The precision matrices used in GMRF models provide conditional specifications of dependence, while the covariance matrices Σ( θ ) in GP models provide marginal specifications of dependence (Haran, 2010). As a result, GP models provide a clear interpretation and presentation of the spatial dependency structure, while it is difficult to interpret the spatial dependency from estimated GMRF models. In this study, the geostatistical data category was employed, and the spatial random effects w were modeled as zero-mean second-order stationary GPs. In the presence of panel count data (yearly crash frequency data sets), Miaou et al. (2003) stated that when the data was quite rich spatially but rather limited temporally (251 counties and 8 years of data in their case), simple temporal effects were considered. The modeling results showed that models with fixed temporal effects performed better than those with random temporal effects (Miaou et al., 2003). The present study assumed a constant GP model over years and fixed temporal effects to analyze the yearly crash frequencies. The following provides a detailed description of the methods and concepts in GP-based SGLM modeling, based on the description provided by Schabenberger and Pierce (2002) and Diggle and Ribeiro (2007).

87 Variogram and Covariogram 74 Consider a random field W(s) as a function of spatial coordinates s in a spatial domain. W(s) is second-order stationary if the following two conditions are satisfied: E ( W ( s)) = μ 4.11 Cov [ W ( s), W ( s + h)] = C( h) 4.12 The condition expressed in equation 4.11 indicates that the mean of the random field is constant and independent of spatial locations. The condition expressed in equation 4.12 indicates that the covariance between two observations is only determined by their spatial distances h. C (h) is known as a covariogram of the spatial process. If covariogram C (h) is not changing for different orientations of h, it is isotropic and could be specified as in equation Cov [ W ( s), W ( s + h)] = C( h ) 4.13 where: h = Euclidean distance between two spatial observations. Another form of stationarity, intrinsic stationarity, assumes two conditions. The first condition is specified in equation The other condition is that the variance of the increments W ( s) W ( s + h) is only dependent on spatial distances h and is stated as follows: 1 Var[ W ( s) W ( s + h)] = γ ( h) γ (h) in equation 4.14 is defined as a semivariogram of W (s). If the semivariogram is a function of h, the random field is furthrtmore isotropic (Schabenberger and Pierce,

88 2002). If a random field is second-order stationary, the relationship between γ (h) and C (h) is as follows: γ ( h) = C(0) C( h) 4.15 The semivariogram is regarded as a significant tool to represent spatial dependency structure in random fields. Semivariogram functions are considered valid only when they are positive and increase as the spatial distance h increases, and 75 approach a constant value when spatial distance h approaches certain values. The general shape of a semivariogram of a second-order stationary random field is presented in Figure 4.2. There are three important elements of a semivariogram: sill, range and nugget. Sill is the upper asymptote in a semivariogram s shape (Schabenberger and Pierce, 2002). When the semivariogram meets its upper asymptote, the covariance C (h) is zero and as a result, sill is equal to C (0). The spatial distance h where the semivariogram meets its upper asymptote is known as the range of the semivariogram (Schabenberger and Pierce, 2002). In Figure 4.2, the practical range is defined as the distance where the semivariogram reaches 95% of the sill (Schabenberger and Pierce, 2002). Nugget effect, θ 0, results when γ (h) is not approaching 0 and when h approaches 0. The partial sill of the semivariogram, θ s, is defined as the difference between the sill and nugget, C ( 0) θ 0. In principle, there are two possible explanations to the existence of a nugget effect. One is measurement error, and the other is small-scale spatial variation, which means that a spatial process works at distances shorter than the

89 smallest distance in the analysis data set (Schabenberger and Pierce, 2002; Diggle and 76 Ribeiro, 2007). There are numerous statistical semivariogram models, and this study employed the two most frequently used ones, Spherical and Exponential, which are specified in equations 4.16 and 4.17, respectively (Schabenberger and Pierce, 2002). r r 0 h = h 1 h ; = θ 0 + θ s 0 < h α α 2 α θ 0 + θ s h > α ( h θ ) 0 h = 0 θ = h θ 0 + θ 1 exp h s α ( h; ) where: α = range. The two main characteristics of the Spherical model are the linear behavior near the origin and the fact that at distance the semivariogram meets the sill and remains flat thereafter (Schabenberger and Pierce, 2002). The second characteristic raises some concern that the covariance may not be exactly 0 when spatial distances are longer than a certain distance (Stein, 1999). The Exponential model makes the semivariogram approach sill asymptotically as the spatial distance approaches infinity (Schabenberger and Pierce, 2002). SGLM results from the two semivariogram models were compared in this study, as will be presented in Chapter 5.

90 77 Range Figure 4.2. Semivariogram of a Second-Order Stationary Process with and without Nugget Effect (Schabenberger and Pierce, 2002) Estimation of Spatial Generalized Linear Model for Count Data SGLMs could be estimated using Bayesian or maximum likelihood-based inference. For SGLMs, the marginal distributions are not in closed forms for any parameter to be estimated, and both approaches are computationally expensive (Breslow and Clayton, 1993; Diggle and Ribeiro, 2007; Liu et al., 2008; Haran, 2010). Schabenberger and Pierce (2002) proposed an estimation approach for GP-based SGLMs, which avoids repeated inversion of large matrices. This estimation approach is an iterative process which updates spatial dependency and model parameters iteratively. Spatial dependency parameters are estimated based on the estimates of mean parameter coefficients (βs), and given the estimated spatial dependency parameters, the mean

91 parameter coefficients (βs) are re-estimated. This process continues until some 78 convergence criterion is met. The process is described in detail as follows (Schabenberger and Pierce, 2002), where all the parameters are as previously defined: Step #1: fit a classical generalized linear model assuming independent data; Step#2: construct n*(n-1)/2 unique pairwise differences D ij = w w, given the i j zero-mean second-order stationary GP assumption, D ij is a Gaussian random variable with mean 0 and variance 2γ ( h, θ ) ; Step #3: apply a composite likelihood (CL) approach where the composite score function of D ij turns out to be the nonlinear weighted least squares objective function of the model specified in equation Then, the CL estimates are calculated as the weighted nonlinear least square estimates of θ with a Gauss-Newton algorithm by using the PROC NLIN procedure in SAS (SAS Institute Inc., ); ij D = 2γ ( ; θ ) + δ 2 ij h ij ij 4.18 where: δ ij = independently distributed with means 0 and variance 8γ 2 ij ( h ij ; θ ) ; Step #4: construct a marginal variance-covariance matrix of Y, by inserting the estimated spatial dependency parametersθˆ into the matrix; Step #5: re-estimate the model parameters β by using a marginal quasi-likelihood approach with approximations involved, using the estimated marginal variance-covariance matrix;

92 79 Step #6: repeat the process from steps #2 to #5 until the largest relative change in the estimated model parameters is less than some critical value. The CL estimation avoids the inversion of a large matrix by inverting smaller matrices instead. The largest matrix to be inverted for the estimation of spatial dependency parameters θ is a 3*3 matrix if there is nugget effect, or a 2*2 matrix if there is no nugget effect (Schabenberger and Pierce, 2002). This GP-based SGLM estimation process was programmed in a SAS macro by Schabenberger and Pierce (2002). This macro was employed by Schabenberger and Pierce (2002) and Lambert et al. (2006). The authors concluded that this produced reasonable modeling results. One point that is worth discussion is that when there is no spatial dependency, or the theoretical range is very close to 0, this process is not able to estimate the range parameter. Because when the theoretical range is close to 0, almost all of the pairwise distances in the analysis dataset are greater than the theoretical range. Under this situation, according to the two semivariogram models specified in equations 4.16 and 4.17, the range parameter is not contained in the models and only sill and nugget are in the models for estimation Crash Frequency Model Simulation and Assessment This section describes how the estimated crash count models were assessed. For the estimation of the classical count models (Poisson or NB models), the maximum likelihood estimator was used, and the fit of the estimated Poisson and NB models could be judged by the calculated Akaike information criterion (AIC) and Bayesian information

93 criterion (BIC) values. However, as discussed earlier in this chapter, for SGLMs, the 80 marginal distributions are not in closed forms for any parameter to be estimated and the likelihood function is intractable. Approximations to the likelihood function were used for SGLM estimation in this study. As a result, AIC or BIC is not a good measure of the SGLM s fit. In order to compare the fit of the estimated SGLM models and the classical count models in the present study, predicted crash frequencies were simulated from the estimated models and compared with the observed crash frequencies on the same road segments to assess the predictive capabilities of different models. As stated earlier, ten percent of the total road segment samples were randomly selected for simulation and the remaining ninety percent were used to estimate crash frequency models. For the road segments randomly selected for simulation, predicted crash frequencies on these road segments were simulated based on the estimated SGLMs and the classical count model with the best fit (lowest AIC and BIC values). Simulations were performed in R (R Development Core Team, 2010) in the present study. The following describes how to simulate the predicted crash frequencies from the estimated models. In the present study, 1000 simulations of predicted crash frequency were generated for each road segment. The mean value of the 1000 simulated crash frequencies was calculated for each road segment, as well as the 10th percentile and 90th percentile of the 1000 simulated crash frequencies to measure the predictive capabilities of these models. The model with mean simulated crash frequencies that were the closest to the observed crash frequencies, with the highest coverage of the prediction intervals (the 10th percentile and 90th percentile of the 1000 simulated crash frequencies) on the

94 81 observed crash frequencies, and the narrowest prediction intervals was considered as the best model with the best predictive capability. This section consists of three subsections. The first subsection briefly describes simulations from estimated Poisson or NB models. The second subsection describes simulations from random-effects NB models. The third subsection describes simulations from GP-based SGLMs. All the parameters appearing in this section share the same explanation as in section Simulations from Classical Count Outcome Models Simulations from the estimated Poisson or NB models are straight forward. Given the estimated model parameters βˆ, expected crash frequency λi on road segment i is calculated as specified in equation λ = exp( x ˆ) β i i 4.19 If the crash frequency is estimated by a Poisson model, predicted crash frequency ŷ i on road segment i is Poisson-distributed with λ i. If crash frequency is estimated by a NB model, predicted crash frequency ŷ i on road segment i is NB-distributed with λi and overdispersion parameterα. Simulations could then be performed based on the calculated expected crash frequency and the Poisson or NB distributions. In this study, 1000 simulations of predicted crash frequencies were generated for each road segment. The mean value of the 1000 simulated crash frequencies was calculated for each road

95 82 segment, as well as the 10th percentile and 90th percentile of the 1000 simulated crash frequencies to measure the predictive capabilities of these models Simulations from Random-Effects Poisson and NB models In the case of random-effects Poisson models, given the estimated model parameters βˆ, expected crash frequency λit on road segment i at year t is calculated as specified in equation λ exp( ˆ it = xit β + ε i ) 4.20 As described in section , exp( ε i ) is gamma-distributed with mean one and variance α. Given the estimatedα, a road segment-specific random effect could be simulated for each road segment group. Then the predicted crash frequency for road segment i at year t ŷ it follows a Poisson distribution with expected value λ it. In the present study, 1000 simulations of predicted crash frequency were performed based on λit and Poisson distribution for each road segment each year. In the case of RENB models, predicted crash frequency ŷ it follows a Poisson distribution with expected frequency λ it. Then λ it follows a gamma distribution with two parameters, γ ij = exp( xit βˆ ) and δ i. As described in section , 1/(1+ δ i ) is beta distributed with parameters r and s (Hausman et al., 1984; Cameron and Trivedi, 1998). Given the estimated beta distribution parameters, a value of δ i could be simulated for each road segment group. With the simulated δ i and calculatedγ ij based on the estimated

96 83 RENB model parameters, the expected crash frequency λ it could be simulated based on the gamma distribution. In the present study, 1000 simulations were generated for each λit and a mean of the 1000 simulated values was calculated for each λ it. Then, 1000 simulations of predicted crash frequencies were generated for each road segment at each year based on the mean of simulated λ it and the Poisson distribution. The mean value of the 1000 simulated crash frequencies was calculated for each road segment, as well as the 10th percentile and 90th percentile of the 1000 simulated crash frequencies to measure the predictive capabilities of these models Simulations from Spatial Generalized Linear Model From the estimated GP-based SGLMs, simulations of the predicted crash frequencies on the road segments randomly selected for model assessment are conditional on the random sample for model estimation (Diggle and Ribeiro, 2007; Haran, 2010). Since the spatial random effects are added to the linear predictor, the natural logarithm of expected crash frequency is simulated first. Suppose Z 1 is a vector of natural logarithm of observed crash frequencies in the sample for model estimation, Z 2 is a vector of natural logarithm of expected crash frequencies in the sample for model assessment. Then Z 2 given Z 1 is calculated as shown in equation ˆ 1 Z 2 Z1 = X 2β + Σ 21Σ11 ( Z1 X 1 ˆ) β 4.21 where: X 1, X 2 = explanatory variable vector in sample for model estimation and sample for model assessment, respectively;

97 Σ 11= a n 1 *n 1 covariance matrix with ( Σ 11) ij =C(h ij, θ ), n 1 is the number of road segments in the sample for model estimation, h ij is the spatial distance between road segment i and j in the sample for model estimation, i, j=1,,n 1 ; and Σ 21= a n 2 *n 1 covariance matrix with ( Σ 21 ) kl =C(h kl, θ ), n 2 is the number of road segments in sample for model assessment, h kl is the spatial distance between road segment k in the sample for model assessment and road segment l in the sample for model estimation, k =1,,n 2, l=1, n 1. The covariogram, C(h, θ ), is calculated as specified in equations 4.22 and 4.23 for Spherical and Exponential semivariogram models, respectively, based on the discussion in section C( h; θ ) = θ s θ + θ 0 s h 1 h + α 2 α 0 3 h = 0 0 < h α h > α 4.22 h θ exp( ) h > 0 C( h; θ ) = s α 4.23 θ 0 + θ s h = 0 In the present study, after calculation of the conditional expectations of natural logarithm of expected crash frequencies in the model assessment sample Z 2 Z1, a GP was simulated for 1000 times based the estimated GP parameters (sill, nugget and range) and calculated mean vector Z 2 Z1. Then, for every road segment in the sample for model assessment, there were 1000 simulated values of natural logarithm of the expected crash frequency. Then, the exponential for each of these simulated values were calculated and this gave 1000 simulated values of the expected crash frequency for every road segment

98 85 in the sample for model assessment. The predicted crash frequency ŷ k on road segment k in the model assessment sample was simulated based on the simulated expected crash frequency and the Poisson distribution simulated predicted crash frequencies were generated for each road segment. The mean value of the 1000 simulated crash frequencies was calculated for each road segment, as well as the 10th percentile and 90th percentile of the simulated crash frequencies to measure the predictive capabilities of these models Transferability and Recalibration of Crash Frequency Models Measures of model transferability were used to assess the effectiveness of models when they were transferred for use in different time periods and/or space. Two measures were used in this study as described below. The first measure is a likelihood ratio test (Hadayeghi et al., 2006; Washington et al., 2003). This test examines whether the model parameters are stable spatially and/or temporally. The null hypothesis of this test is that the model could be transferred for use in different application scenarios (different time periods and/or areas). Suppose the transferability of model parameters between two application scenarios, i and j, is tested, the test could be conducted as shown in equation Variables common to each dataset should be used in each model. X 2 = 2 [ LL( β ) LL( β ) LL( β )] c i j 4.24 where: 2 X = χ 2 distributed with degrees of freedom equal to the number of model parameters;

99 LL( β c ) = log likelihood at convergence of the model estimated with the 86 combined data with both scenarios, i and j; and LL ( β ), LL( β ) i j = log likelihood at convergence of the model estimated with the scenario i and j data, respectively. When the null hypothesis is rejected, an asymptotic t-test could be performed to test which variables are responsible for the rejection (Hadayeghi et al., 2006). The t-statistic is calculated as in equation ( ˆ β ˆ β )/ var( ˆ β ) + var( ˆ β ) t = 4.25 ai aj ai aj where: βˆ ai, βˆ aj = coefficient estimate for variable a in application scenarios i and j, respectively; and var ( βˆai ), var ( ) βˆaj scenarios i and j, respectively. = variance of estimated coefficient for variable a in application The other measure of model transferability is based on the measurement of the goodness of fit of a crash frequency model to any crash dataset. It could test whether a model estimated from data collected in application scenario i could make reliable predictions for application scenario j (Vogt and Bared, 1998; Sawalha and Sayed, 2006). This measure is applied for the situation where limited data are available in application scenario j. By using this measure, no additional models are needed other than the model estimated from application scenario i. This could save modeling efforts for highway engineers. A z-score is calculated to measure how far the calculated Pearson χ 2 is from its expected value, as shown in equation 4.26.

100 χ E( χ ) z = sd( χ ) N k = 1 ( y E( y )) 2 k k χ =, E ( y k ) var( y ) 2 where: χ =Pearson χ 2 statistic; k 2 2 λ, var( y k ) = λ i + αλ i 4.27 = i y k = observed crash count on road segment k (k=1,, N); λ i = expected crash count for road segment k; var( y k ) = variance of y k ; and α = scalar parameter, when α is not significantly different from 0, the NB model degenerates to a Poisson model. It is reported (Vogt and Bared, 1998) that if the tested crash frequency model is correct and the observations in the new dataset to which the crash frequency model is transferred are independent, the expected Pearson χ 2 value and its standard deviation could be calculated as shown in equations 4.28 and ( ) N E χ = 4.28 N 2 1 sd( χ ) = 2N(1 + 3α ) k = 1 λi + ( 1 αλ ) where: N= number of observations in the dataset. i A value of z-score near zero indicates that the model could be transferred to the new dataset. However, Vogt and Bared (1998) did not indicate the limit of z-score away from zero beyond which the calculated z-score indicates that the model is not transferable to the new dataset. The possible reason for this is that the distribution of z-score is

101 88 unclear and thus the limit for acceptance or rejection of the null hypothesis that a model is transferable for a new dataset could not be set. If the second transferability measure is used, Sawalha and Sayed (2006) stated that there were two transfer alternatives. One is to apply the model to the new dataset as is without any change to the parameter estimates. The other is to multiply the estimates from the model to be transferred by a calibration factor (Harwood et al. (2000)). This is equivalent to recalibrating the constant of the model to be transferred by multiplying the constant by a scale factor, leaving the remaining parameter estimates unchanged. The calibration factor is calculated by dividing the sum of observed crash counts in application scenario j by the sum of estimated crash counts in application scenario j by using the model estimated using application scenario i. Sawalha and Sayed (2006) pointed out that no matter which transfer alternative is chosen, the overdispersion parameter should be recalibrated for NB models before a transferability test, since it is not reasonable to assume that the overdispersion parameters of the two different datasets are the same. The overdispersion parameter could be estimated by using equation 4.30 (Kulmala, 1995; Sawalha and Sayed, 2006). N 2 [ ( yi μi ) μi ] 1 N i= 1 α = 4.30 N 1 2 μi N i= 1

102 4.2 Crash Severity Modeling 89 This section describes the binary logit, multinomial logit and nested logit models (Washington et al., 2003). A linear function to determine the severity outcome i for crash n, S in, is defined as: S in = β X + ε 4.31 i in in where: X in = a vector of explanatory variables used to determine the severity outcome i for a crash n; β i = a vector of estimable coefficients for severity outcome i; and ε in = an error term to account for the unobserved factors associated with severity outcome i and crash n. In the case of a binary logit model, there are only two severity outcomes, i =1 or 0, and the cumulative density of ε is a logistic function (Kennedy, 2003). The probability of crash n experiencing severity outcome i =1, P n (i=1) is shown in equation 4.32, and the probability of crash n experiencing severity outcome i =0, P n (i=0) is shown in equation P P n n ( i 1) = P( β X + ε > 0) = P( ε > β X ) ( β1x 1n ) ( β X ) exp = 1 1n 1n 1n 1 1n = exp 1 1n 1 i = 0 = exp β X ( ) ( ) 1 1n In the case of the multinomial logit model, there are more than two severity outcomes and ε in is independently and identically generalized-extreme-value distributed. The probability that crash n falls in severity level i, P n (i) is expressed as follows:

103 90 P ( i) = P S S for any i j in A 4.34 n ( ) in jn where j includes all possible severity outcomes in the severity set A. Substituting equation 4.31 into equation 4.34 yields the following: Pn ( i) = P( β i X in + ε in β j X jn + ε jn ) or P( β i X in β j X jn ε jn ε in ) for any i j in A 4.35 For the probability of crash n experiencing severity outcome i, the multinomial logit model takes the form shown in equation 4.36 (Washington et al., 2003): P n () i = exp i A ( β i X in ) ( β X ) exp i in 4.36 where exp ( β i X in ) is equal to 1 ( β i = 0) if the severity category is the base category. Several previous studies (Shankar et al., 1996; Lee and Mannering, 2002; Holdridge et al., 2005; Savolainen and Mannering, 2007) showed that there were shared unobserved effects across lower crash severity levels and thus the IIA assumption of the multinomial logit model may not be satisfied. Violating this assumption could lead to inconsistent coefficient estimates and erroneous severity probabilities. A Hausman test (Hausman and McFadden, 1984; Kennedy, 2003) could be used to test the IIA assumption before the adoption of multinomial logit models. This test is based on the idea that if the IIA assumption is true, dropping a category does not change the coefficient estimates significantly (Hausman and McFadden, 1984; Kennedy, 2003). When the IIA assumption is violated, the nested logit model is used to account for the shared unobserved effects across severity levels. The nested logit model has the following functional form (McFadden, 1981; Washington et al., 2003):

104 P P L n n in () i ( j i) ( β i X in + θ i Lin ) ( β X θ L ) exp = exp i in + = i I exp i J ( β X jn ) j i ( β X ) exp j i jn [ exp( β X jn )] j J i in = ln 4.39 j i where P n () i is the unconditional probability of crash n falling in severity outcome i, P n ( j i) is the conditional probability of crash n falling in severity outcome j (lower level) given that the crash falls in severity category i (higher level). I is the unconditional severity set (higher level), and J is the conditional severity set (lower level) on I. Lin is the inclusive value representing the maximum value of the attributes that determine the probability of crash n falling in severity i. θi is the coefficient of inclusive value L in and its value has to be greater than 0 and less than 1 to be consistent with the nested logit derivation (McFadden, 1981). If the estimated value of θ i is not significantly different from 1 and the nested structure has only 2 levels, the nested logit model is reduced to a simple multinomial logit model. The multinomial logit model is estimated by maximum likelihood. Full information maximum likelihood (FIML) is used to estimate the nested logit model. FIML estimates the parameters in all levels simultaneously which leads to consistent and efficient parameter estimates (Holdridge et al, 2005). 91

105 Chapter 5 Crash Frequency Analysis Results This chapter presents the crash frequency analysis results. The five-year aggregated crash frequency and yearly crash frequency data were analyzed for Pennsylvania and North Carolina. Estimated models of cross-median, rollover and median barrier crash frequencies are presented and explained. This chapter also presents the transferability test results of the estimated models between the two states, and simulation results of predicted crash frequencies from the estimated models. The predictive capabilities of the estimated models were compared based on the simulation results. 5.1 Pennsylvania Cross-Median Crash Frequency Models This section is divided into two parts. The first part presents estimated models for five-year aggregated cross-median crash frequency in Pennsylvania and model assessment results. The second part presents estimated models for yearly cross-median crash frequency in Pennsylvania and model assessment results.

106 Pennsylvania Five-Year Aggregated Cross-Median Crash Frequency Models and Model Assessment Pennsylvania Five-year Aggregated Cross-Median Crash Frequency Models This section presents the estimated models for Pennsylvania five-year aggregated cross-median crash frequencies. Two crash frequency models were estimated, one without route indicators as the base model and one with a route indicator. The purpose of adding route indicators was to incorporate route-specific effects such as weather and driver population into the models. The addition of route indicators was expected to account for some spatially-dependent effects since road segments along the same route are connected to each other. Negative Binomial (NB) models were first estimated to account for possible overdispersion in the crash frequency distribution. However, the overdispersion parameters were not statistically significant in both models (p-values > 0.33). As such, two Poisson models were estimated for the five-year aggregated crossmedian crash frequency data in Pennsylvania and the results are shown in Table 5.1. The natural logarithm of the road segment length was treated as an offset in both models since its estimated coefficients in both models were very close to 1 and its 95% confidence intervals contained 1. As discussed in chapter 4, the statistical significance of route-specific effects does not necessarily mean that there were significant unobserved spatially dependent effects. Efforts were made to estimate spatial generalized linear models (SGLM) for Pennsylvania five-year aggregated cross-median crash frequencies. However, the range parameter could not be estimated. As explained in section , a possible reason for

107 94 the failure was that theoretically there were no significant unobserved spatially-dependent effects in the cross-median crash frequency model. This was not unexpected since crossmedian crashes were relatively rare events and their occurrences were significantly affected by median width and median foreslopes and backslopes, which were already included in the estimated models. No other unobserved spatially dependent factors were expected to significantly affect cross-median crash occurrences. Two empirical semivariograms were plotted from the Poisson base model residuals as presented in Figure a regular semivariogram and a robust one to account for outliers in the data. In Figure 5.1, it is clear that the semivariograms did not increase as the spatial distance increased. According to equation 4.15, this indicates that the covariance C (h) was not significantly different from zero, which means that no significant spatial dependency was found in the residuals.

108 Table 5.1. Poisson Models for Pennsylvania Five-Year Aggregated Cross-Median Crash Frequency Five-year aggregated cross-median crash frequency Poisson Poisson with Route Indicator Constant (4.120) (4.351) Leftcurve indicator : if left curved segment;0: otherwise (0.555) (0.573) Rightcurve indicator : if right curved segment;0: otherwise (0.505) (0.527) Natural log adt (0.421) (0.445) Shoulder rumble strip indicator : if left shoulder rumble strip is present; 0: otherwise (0.449) (0.455) Median foreslope 10H:1V or flatter indicator : if median foreslope 10H:1V or flatter; 0:otherwise (0.786) (0.802) Median foreslope between 7H:1V and 9H:1V indicator 1: if median foreslope between 7H:1V and 9H:1V; 0: otherwise (0.699) (0.702) Median backslope 10H:1V or flatter indicator : if median backslope 10H:1V or flatter; 0: otherwise (0.628) (0.651) Median width (ft) Median width (ft) if Median foreslope and backslope both 6H:1V or steeper, 0 otherwise Median width (ft) if Median foreslope and backslope both 10H:1V or flatter, 0 otherwise (0.010) (0.010) (0.009) (0.009) (0.015) (0.015) Route 90 indicator 1: if route 90; 0: otherwise (0.632) Natural log road segment length (mile) offset Log likelihood for constant-only model Log likelihood at convergence AIC BIC Notes: standard errors are given in parentheses; No. of observations:

109 Lag Di st ance t ype regul ar robust Figure 5.1. Regular and Robust Empirical Semivariograms from Residuals of a Poisson Base Model for Pennsylvania Five-Year Aggregated Cross-Median Crash Frequency In the two models presented in Table 5.1, the estimated coefficients of the variables in both models are similar. In the model with a route indicator, the route indicator was statistically significant at the 85% confidence level. However, the overall model fit was not improved by adding the route indicator, as indicated by the calculated values of Akaike information criterion (AIC) and Bayesian information criterion (BIC). The AIC and BIC values of the Poisson model without the route indicator are slightly lower than those of the model with the route indicator.

110 97 Based on the results shown in Table 5.1, elasticities were computed to determine the marginal effects of the independent variables on the expected crash frequency (Washington et al., 2003). For continuous variables, elasticity of the expected crash frequency λ i on road segment i with respect to a continuous variable x im, I E λ x im, is calculated by using equation 5.1(Washington et al., 2003). The calculated elasticity can be explained as the percent change in the expected crash frequency due to one-percent change in the independent variable. For indicator variables, a pseudo-elasticity calculated to estimate an approximate elasticity of the expected crash frequency λ i on road segment i with respect to an indicator variable x in. Washington et al. (2003) proposed calculating I E λ x in I E λ x in using equation 5.2, which is defined as the incremental change in the expected crash frequency λ i on road segment i caused by changes in the indicator variables. However, this study calculated I E λ x in by using equation 5.3, which is defined as the percent change in expected crash frequency λ i on road segment i given a change in the value of the indicator variables from zero to unity. is λ Ex I im = β x 5.1 m im E λ x I in exp( β n ) 1 = 5.2 exp( β ) n E λ x I in = n exp( β ) where: β m = estimated coefficient for continuous variable x im ; and β n = estimated coefficient for indicator variable x in. Elasticities were computed for each analysis unit (road segment) and in this dissertation, a single elasticity is reported for each independent variable as the average

111 elasticity over all analysis units (Washington et al., 2003). Elasticities with absolute 98 values greater than one are interpreted as being elastic, while the elasticities with absolute values smaller than one are interpreted as being inelastic (Milton and Mannering, 1998). Expected crash frequencies have a relatively low sensitivity to the changes in inelastic explanatory variables, while the opposite is true for explanatory variables that are elastic (Shankar, et al., 1995). The pseudo-elasticities of the expected five-year aggregated cross-median crash frequency in Pennsylvania with respect to left curve and right curve indicators are 1.61 and 2.57, respectively. This indicates that the presence of left curves and right curves increases the expected cross-median crash frequency by 161% and 257%, respectively, when compared to tangent road segments. These findings support the expected outcome E1 stated in Chapter 1 and are consistent with statistics by McGee and Hanscom (2006), which showed that crash rates were three times higher on horizontal curves than on tangent roadway segments. Right-curved road segments lead to a greater increase in cross-median crash frequency than left-curved road segments. All these findings were expected because vehicles traveling on curved road segments, especially those curved right, are more likely to leave the roadway and enter into the median. The pseudo-elasticity of the expected five-year aggregated cross-median crash frequency in Pennsylvania with respect to the left shoulder rumble strip indicator is -0.70, which means that the presence of left shoulder rumble strips decreases the expected cross-median crash frequency by 70%. This finding supports the expected outcome E3 stated in Chapter 1. It is consistent with a study by Torbic et al.(2009), who reported that shoulder rumble strips were expected to reduce the frequency of single-vehicle run-off-

112 99 road crashes on divided highways by 11 to 22%, and that centerline rumble strips have been shown to reduce the expected number of opposite-direction crashes on undivided highways by 30 to 40%. Shoulder rumble strips produce auditory and tactile vibrations when the tire of the vehicle makes contact with the shoulder rumble strip. This noise and vibration alert drivers after leaving the roadway to the left and thus likely prevent median-related crashes. The average elasticity of the expected five-year aggregated cross-median crash frequency in Pennsylvania with respect to ADT is This means that a 1% increase in ADT is associated with a 1.16% increase in the expected cross-median crash frequency. This finding supports the expected outcome E4 stated in Chapter 1 that higher traffic volumes lead to higher cross-median crash frequencies. Higher traffic volumes not only increase the probability that a vehicle will leave the roadway to the left and enter the median, but also increase the probability that a vehicle crossing the median will collide with a vehicle traveling in the opposing direction. The pseudo-elasticities of the expected five-year aggregated cross-median crash frequency in Pennsylvania with respect to median foreslope equal to or flatter than 10V:1H and median foreslopes between 7H:1V and 9H:1V indicator are 1.26 and 0.30, respectively, when compared to median foreslopes steeper than 6H:1V. This indicates that the presence of median foreslopes equal to or flatter than 10V:1H increases the expected cross-median crash frequency by 126% relative to median foreslopes steeper than 7H:1V, and the presence of median foreslopes between 7H:1V and 9H:1V increases the expected cross-median crash frequency by 30% relative to median foreslopes steeper than 7H:1V. Although the median foreslopes between 7H:1V and 9H:1V indicator is not

113 statistically significant at the 50% confidence level in either model, it was kept in the 100 model since the comparison between the two median foreslope indicators clearly showed that flatter median foreslopes led to greater increases in cross-median crash frequency. Median foreslopes equal to or flatter than 10V:1H lead to a greater increase in crossmedian crash frequency than median foreslopes between 7H:1V and 9H:1V. The pseudoelasticity of the expected five-year aggregated cross-median crash frequency with respect to median backslopes equal to or flatter than 10H:1V indicator is This indicates that median backslopes equal to or flatter than 10V:1H increases the expected cross-median crash frequency by 54% relative to steeper median backslopes. This finding was expected since flat median slopes permit vehicles entering the median to cross easily when compared to steeper median slopes. The median cross-slope findings support the expected outcome E7 stated in Chapter 1 and are consistent with a recent research by Stine et al. (2010), which indicated that flatter median slopes increases the likelihood that vehicles encroaching into the median will then enter into the opposing travel lanes.. The average elasticity of the expected five-year aggregated cross-median crash frequency in Pennsylvania with respect to median width is This means that a 1% increase in median width decreases the expected cross-median crash frequency by 0.8%. This finding is consistent with all past cross-median crash research (e.g., Donnell et al., 2002), and supports the expected outcome E6 in Chapter 1 that wider medians lead to fewer cross-median crashes. Wider medians offer drivers more space and time to stop or recover their vehicles in the median prior to entering the opposing traveling lanes. The interaction between median width and median foreslope and backslope both equal to 10H:1V or flatter was not statistically significant at the 50% confidence level in

114 101 either model, but was retained in the models for the purpose of comparison with the other interaction term (the interaction between median width and median foreslope and backslope both equal to 6H:1V or steeper). The positive estimated coefficient of the interaction between median width and median foreslopes and backslopes both equal to 6H:1V or steeper indicates that the decrease in cross-median crash frequency due to a 1% increase in median width is reduced in magnitude (the average elasticity of crash frequency with respect to median width was reduced to -0.23) when both the median foreslope and backslope are equal to 6H:1V or steeper relative to flatter median crossslopes, although cross-median crash frequency still decreases slightly as the median width increases (a 1% increase in median width reduces the expected cross-median crash frequency by 0.23%). This was expected because if both the median foreslope and backslope are steep, there is an increased probability of vehicles rolling over in the median instead of crossing the median. The negative estimated coefficient of the interaction between median width and median foreslope and backslope both equal to 10H:1V or flatter indicates that the decrease in cross-median crash frequency due to a 1% increase in median width is inflated in magnitude, if both the median foreslope and backslope are equal to 10H:1V or flatter, relative to steeper median cross-slopes. The average elasticity of crash frequency with respect to median width is increased to -1.23, which means a 1% increase in median width decreases the expected cross-median crash frequency by 1.23%. This was expected because if the median foreslope and backslope are very flat, it is easier for errant vehicles in the median to recover before they enter the opposing travel lanes.

115 102 The pseudo-elasticity of the expected five-year aggregated cross-median crash frequency in Pennsylvania with respect to the Route 90 indicator is This indicates that the expected cross-median crash frequency on Route 90 is 153% more than on other rural divided highways in the sample. A possible reason for this finding is that Interstate 90 is located in northwest PA along lake Erie and the Route 90 indicator may be a surrogate for an increase in weather-related crashes (i.e., snow). All other possible route indicators were not statistically significant in the estimated model. To summarize, for the analysis of five-year aggregated cross-median crash frequency in Pennsylvania, two Poisson models were estimated and no spatial models were estimated. No simulation was performed for the purpose of model assessment, since no SGLMs were estimated, and the estimated two crash frequency models are similar to each other. The fit of the two models could be clearly judged from AICs and BICs. The base Poisson model without route indicators has slightly lower AIC and BIC and is thus considered a better fit than the model with a route indicator Transferability of Pennsylvania Five-Year Aggregated Cross-Median Crash Frequency Models The base Poisson model without route indicator shown in Table 5.1 was tested for transferability. As noted in Chapter 4, a likelihood ratio test or a goodness-of-fit test (zscore) can be used to test the transferability of the Pennsylvania cross-median crash model to North Carolina. The goodness-of-fit based test was chosen to test the transferability of the five-year aggregated cross-median crash frequency model estimated from Pennsylvania data, due to the small sample of North Carolina cross-median crash

116 103 data. The likelihood ratio test requires estimation of cross-median frequency models from the North Carolina dataset, but the small sample of road segments without longitudinal median barrier in North Carolina prohibited model estimation of cross-median crash frequency. The two transfer alternatives (Sawalha and Sayed, 2006) noted in section were tested and compared. Since the transferred model was a Poisson model, there was no need to recalibrate the overdispersion parameter. First, the Pennsylvania cross-median crash frequency model was transferred to North Carolina cross-median crash data as is without any change to the parameter estimates. A z-score was calculated and its value was , which was not close to zero. The other transfer alternative was to first estimate North Carolina cross-median crash frequency by applying the Pennsylvania cross-median frequency model, and then to multiply the estimates by a calibration factor. The calibration factor was calculated by dividing the sum of observed cross-median crash frequencies in the North Carolina dataset by the sum of estimated cross-median crash frequencies in North Carolina by using the Pennsylvania cross-median crash frequency model. The calculated calibration factor was A z-score was again calculated and its value was , which is much closer to 0 than the z-score of the first transfer alternative (-0.837). This indicates that recalibration of the model to be transferred by using a calibration factor prior to the transferability test works better than transferring the model as is, since the calibration factor updated the transferred model by making adjustment based on locally available information.

117 5.1.2 Pennsylvania Yearly Cross-Median Crash Frequency Models and Model Assessment Pennsylvania Yearly Cross-Median Crash Frequency Models This section presents the estimated models for Pennsylvania yearly cross-median crash frequency. Two specifications of crash frequency models were employed for the yearly crash frequencies, Poisson models and random-effects Poisson models. For each model specification, three models were estimated, one without year or route indicators as the base model, one with year indicators, and one with both year and route indicators. The same sets of variables were included in the model specifications. Also, the variables in the yearly crash frequency models were the same as those variables in the five-year aggregated crash frequency models shown in Table 5.1. NB models were estimated first to account for possible overdispersion. However, the overdispersion parameters were not statistically significant (p-values > 0.25). As such, Poisson models were estimated for the yearly cross-median crash frequencies in Pennsylvania, as presented in Table 5.2. A Likelihood ratio test was conducted for each of the three random-effects Poisson models by comparing the panel estimator with the pooled Poisson estimator, and the test results showed that alpha was not statistically different from 0 in the three random-effects Poisson models (p-values>0.35). This indicates that the road segment random effects were not statistically significant. The estimated random-effects Poisson models are presented for the purpose of comparison with Poisson models. In these models, the natural logarithm of road segment length (miles) was treated as an offset, since its

118 estimated coefficients in all the models were very close to 1 and its 95% confidence 105 intervals contained 1. As discussed in chapter 4, the significance of route-specific effects does not necessarily mean that there were significant unobserved spatially dependent effects. Efforts were made to estimate SGLMs for Pennsylvania yearly cross-median crash frequencies. However, the range parameters could not be estimated. As explained in section , a possible reason for the failure is that theoretically there were no significant unobserved spatially-dependent effects in the cross-median crash frequency model. Empirical semivariograms were plotted from the Poisson base model residuals as presented in Figure a regular semivariogram and a robust one to account for outliers in the data. In Figure 5.2, it is clear that the semivariograms did not increase as the spatial distance increased. According to equation 4.15, this indicates that the covariance C (h) was not significantly different from zero, which means that there was no significant spatial dependency in the residuals.

119 Table 5.2. Models for Pennsylvania Yearly Cross-Median Crash Frequency Yealy cross-median crash frequency Poisson RE Poisson Poisson with year indicator RE Poisson with year indicator Poisson with year/route indicator 106 RE Poisson with year/route indicator Constant (4.184) (4.265) (4.188) (4.271) (4.419) (4.491) Leftcurve indicator (0.555) (0.595) (0.556) (0.595) (0.574) (0.604) Rightcurve indicator (0.505) (0.515) (0.505) (0.515) (0.527) (0.538) Natural log adt (0.426) (0.434) (0.426) (0.435) (0.449) (0.457) Shoulder rumble strip indicator Median foreslope 10H:1V or flatter indicator Median foreslope between 7H:1V and 9H:1V indicator Median backslope 10H:1V or flatter indicator (0.453) (0.462) (0.453) (0.463) (0.459) (0.471) (0.786) (0.796) (0.786) (0.796) (0.801) (0.814) (0.696) (0.702) (0.696) (0.702) (0.699) (0.706) (0.626) (0.645) (0.626) (0.645) (0.648) (0.671) Median width (ft) (0.010) (0.010) (0.010) (0.010) (0.010) (0.011) Median width (ft) if Median foreslope and backslope both 6H:1V or steeper, 0 otherwise (0.009) (0.009) (0.009) (0.009) (0.009) (0.009) Median width (ft) if Median foreslope and backslope both 10H:1V or flatter, 0 otherwise (0.015) (0.015) (0.015) (0.015) (0.015) (0.016) Route 90 indicator (0.632) (0.643) Year 2002 indicator (0.618) (0.618) (0.618) (0.618) Natural log road Offset length (mile) Alpha (0.793) (0.789) (0.820) LL_ LL_ convergence AIC BIC NOTES: 1.Standard errors are given in parentheses; an indicator variable is 1 if the condition holds and 0 otherwise; 2. No. of observations: 3099; 3. Likelihood ratio test of alpha=0: χ 2 (1) = 0.07, Prob>= χ 2 = 0.396; 4. Likelihood ratio test of alpha=0: χ 2 (1) = 0.07, Prob>= χ 2 = 0.397; 5. Likelihood ratio test of alpha=0: χ 2 (1) = 0.11, Prob>= χ 2 = 0.370; 6. log likelihood for constant-only model; 7. log likelihood at convergence.

120 Lag Di st ance t ype regul ar robust Figure 5.2. Regular and Robust Empirical Semivariograms from Residuals of a Poisson Base Model for Pennsylvania Yearly Cross-Median Crash Frequency The estimated coefficients of variables in the models in Table 5.2 are similar to those in the models presented in Table 5.1. As a result, interpretation of the model parameters is similar and is not repeated here. The only variable that was not included in the models presented in Table 5.1 is the year 2002 indicator. The pseudo-elasticity of the expected yearly cross-median crash frequency in Pennsylvania with respect to year 2002 is This means that cross-median crash frequency was 42% lower in year 2002, when compared to the years 2003 through 2006 (inclusive). One possible reason for this finding is that the crash reporting system in Pennsylvania was changed in 2002, and the

121 108 new system was easier to use for crash reporting. As a result, the crash frequencies might be under reported in year 2002 before the system change in From the comparison between the estimated Poisson and random-effects Poisson models, it was found that the coefficient estimates remained stable across the two model specifications, while the standard errors were consistently higher for the random-effects Poisson models. The random-effects model has more noise due to its distribution assumption. The overall model fit was not improved by adding the year and route indicators, as indicated by the calculated values of AIC and BIC. Also, the randomeffects specification did not improve the overall model fit, since the road segment random effects were not statistically significant. The AIC and BIC values of the Poisson model without the year and route indicators are the lowest among the six estimated models shown in Table 5.2 and is thus considered the best fit. Since no SGLMs were estimated, and the model fit of the estimated Poisson and random-effects Poisson models could be clearly judged from the calculated AIC and BIC values, no simulation was performed for model assessment Transferability of Pennsylvania Yearly Cross-Median Crash Frequency Models The base Poisson model without route/year indicators shown in Table 5.2 was tested for transferability. As noted previously, the goodness-of-fit based test was chosen to test the transferability of the yearly cross-median crash frequency model estimated from Pennsylvania data due to the small sample of North Carolina cross-median crash data.

122 109 The two transfer alternatives (Sawalha and Sayed, 2006) noted in section were tested and compared. Since the transferred model was a Poisson model, it was not necessary to recalibrate the overdispersion parameter. First, the Pennsylvania yearly cross-median crash frequency model was transferred to North Carolina yearly crossmedian crash data as is without any change to the parameter estimates. A z-score was calculated and its value was The other transfer alternative was to first estimate North Carolina yearly crossmedian crash frequencies by applying the Pennsylvania yearly cross-median crash frequency model, and then to multiply the estimates by a calibration factor. The calibration factor was calculated by dividing the sum of observed yearly cross-median crash frequencies in North Carolina by the sum of estimated yearly cross-median crash frequencies in North Carolina using the Pennsylvania yearly cross-median crash frequency model. The calculated value of the calibration factor was A z-score was again calculated and its value was , which was much closer to 0 than the z-score of the first transfer alternative (-2.717). This indicates that recalibration of the model to be transferred by using a calibration factor before the transferability test works better than transferring the model as is, since the calibration factor updated the model to be transferred by making adjustments based on locally available information. To summarize the transferability test results of the estimated Pennsylvania crossmedian crash frequency models, for both the five-year aggregated crash frequencies and yearly crash frequencies, Poisson models were estimated. The transferability test results show that if the estimated Poisson models using Pennsylvania data were calibrated by using a calibration factor based on the available information in North Carolina, the

123 Pennsylvanian cross-median crash frequency models could be transferred to estimate 110 North Carolina cross-median crash frequencies. This finding was expected because crossmedian crash frequencies were significantly affected by median cross-section design characteristics, which were included in the estimated Pennsylvania models. Furthermore, there is likely little variability in median cross-section design features on rural divided highways without median barrier throughout the U.S. (i.e., flat slopes, wide, etc.). As a result, the Pennsylvania models were expected to be able to predict cross-median crash frequencies in other states relatively well when calibrated Pennsylvania Rollover Crash Frequency Models This section is divided into two parts. The first part presents estimated models for five-year aggregated rollover crash frequencies in Pennsylvania and model assessment results. The second part presents estimated models for yearly rollover crash frequency in Pennsylvania and model assessment results. Variables that were statistically significant at the 50% confidence level were included in the models Pennsylvania Five-Year Aggregated Rollover Crash Frequency Models and Model Assessment Pennsylvania Five-Year Aggregated Rollover Crash Frequency Models This section presents the estimated models for Pennsylvania five-year aggregated rollover crash frequencies. Two crash frequency models were estimated, one without

124 111 route indicators as the base model and one with route indicators. The purpose of adding route indicators was to incorporate route-specific effects into the models. The addition of route indicators was expected to account for some spatially-dependent effects since road segments along the same route were connected to each other. Two NB models were estimated to account for overdispersion and the overdispersion parameter was statistically significant in both models (p-values < 0.15), as presented in Table 5.3. In the two models, the natural logarithm of road segment length (miles) was treated as an offset, since its estimated coefficients in both models were close to 1 and the 95% confidence intervals contained 1. Efforts were made to estimate SGLMs for Pennsylvania five-year aggregated rollover crash frequency. However, the range parameters could not be estimtated. As explained in section , a possible reason for the failure was that theoretically there were no significant unobserved spatially-dependent effects in the rollover crash frequency model. This was not unexpected, since rollover crashes are relatively rare events and their occurrences are significantly affected by median width and median foreslopes and backslopes, which were already included in the estimated models. No other unobserved spatially dependent factors were expected to significantly affect rollover crash occurrence. Two empirical semivariograms were plotted as shown in Figure 5.3 from residuals of a Poisson base model without route indicators -- a regular semivariogram and a robust one to account for outliers in the data. In Figure 5.3, it is clear that the semivariograms did not increase significantly as the spatial distance increased. Although it appears that the semivariograms started to increase when the spatial distances between road segments were longer than 70 miles, this increase was

125 112 expected to be random variation due to the limited number of road segment pairs with distances between them longer than 70 miles. According to equation 4.15, this indicates that the covariance C (h) was not significantly different from zero, which means that there was no significant spatial dependency in the residuals. Table 5.3. NB Models for Pennsylvania Five-Year Aggregated Rollover Crash Frequency Five-year aggregated cross-median crash frequency NB NB with Route Indicator Constant (2.200) (2.079) Leftcurve indicator : if left curved segment;0: otherwise (0.278) (0.276) Rightcurve indicator : if right curved segment;0: otherwise (0.291) (0.289) Natural log adt (0.228) (0.218) Shoulder rumble strip indicator : if left shoulder rumble strip is present; 0: otherwise (0.264) (0.266) Median foreslope 10H:1V or flatter indicator 1: if median foreslope 10H:1V or flatter; 0:otherwise Median foreslope between 8H:1V and 9H:1V indicator 1: if median foreslope between 8H:1V and 9H:1V; 0: otherwise (0.448) (0.445) (0.332) (0.331) Median foreslope between 6H:1V and 7H:1V indicator : if median backslope 10H:1V or flatter; 0: otherwise (0.262) (0.262) Median width (ft) (0.005) (0.005) Median width (ft) if Median foreslope and backslope both H:1V or steeper, 0 otherwise (0.003) (0.003) Median width (ft) if Median foreslope and backslope both H:1V or flatter, 0 otherwise (0.019) (0.019) Route 79 indicator 1: if route 90; 0: otherwise (0.301) Route 90 indicator 1: if route 90; 0: otherwise (0.728) Alpha (0.391) (0.354) Natural log road length (mile) offset Log likelihood for constant-only model Log likelihood at convergence AIC BIC NOTES: 1.standard errors are given in parentheses; 2. No. of observations: Likelihood ratio test of alpha=0: χ 2 (1) = 2.11, Prob>= χ 2 = 0.073; 4. Likelihood ratio test of alpha=0: χ 2 (1) = 1.34, Prob>= χ 2 = 0.123;

126 Lag Di st ance t ype regul ar robust Figure 5.3. Regular and Robust Empirical Semivariograms from Residuals of a Poisson Base Model for Pennsylvania Five-Year Aggregated Rollover Crash Frequency In the two models presented in Table 5.3, the estimated coefficients are similar. In the model with two route indicators, both route indicators were statistically significant at the 95% confidence level. The overall model fit was slightly improved by adding the route indicators, as indicated by the calculated AIC and BIC values. The NB model with route indicators has slightly lowered AIC and BIC values than the base model without route indicators. Based on the results shown in Table 5.3, elasticities were computed by using equations 5.1 and 5.3, to determine the marginal effects of the independent variables on

127 the expected five-year aggregated rollover crash frequencies in Pennsylvania 114 (Washington et al., 2003). The pseudo-elasticities of the expected five-year aggregated rollover crash frequency in Pennsylvania with respect to left curve and right curve indicators are 0.63 and 0.54, respectively. These indicate that the presence of a left curve and a right curve increases the expected rollover crash frequency by 63% and 54%, respectively, relative to tangent road segments. This finding supports expected outcome E1 in chapter 1 that curved road segments will experience more median-related crashes. As reported by McGee and Hansom (2006), the crash rate along curved roadway segments exceeded the crash rate along tangent segments. Vehicles traveling on curved road segments are more likely to leave the roadway and enter into the median. The pseudo-elasticity of the expected five-year aggregated rollover crash frequency in Pennsylvania with respect to left shoulder rumble strip indicator is This indicates that the presence of left (or inside) shoulder rumble strips decreases the expected rollover crash frequency by 39% when compared to roadway segments without inside or left shoulder rumble strips. This finding supports expected outcome E3 in chapter 1 and is consistent with a recent research reported by Torbic et al. (2009) as described earlier. There are auditory and tactile vibrations produced by rumble strips when a vehicle tire makes contact with them. This noise and vibration alerts drivers that they have left the travel lanes and thus could prevent median-related crashes. The average elasticity of the expected five-year aggregated rollover crash frequency in Pennsylvania with respect to ADT is This indicates that a 1% increase in ADT increases the expected rollover crash frequency by 0.38%. This finding supports

128 the expected outcome E4 in chapter 1 and is intuitive since higher traffic volumes 115 increase the probability of crash occurrence. Although the median foreslopes between 6H:1V and 7H:1V indicator was not statistically significant at the 50% confidence level in the NB model with route indicators, it was kept in the model since the comparison among the three median foreslope indicators clearly showed a trend in the expected rollover crash frequency as the median foreslope becomes steeper. The pseudo-elasticities of the expected five-year aggregated rollover crash frequency in Pennsylvania with respect to median foreslopes equal to or flatter than 10V:1H indicator, median foreslopes between 8H:1V and 9H:1V indicator, and median foreslopes between 6H:1V and 7H:1V indicator are -0.43, and -0.17, respectively. This indicates that the presence of a median foreslope equal to or flatter than 10V:1H decreases the expected rollover crash frequency by 43% relative to a median foreslope steeper than 6H:1V; the presence of a median foreslope between 8H:1V and 9H:1V decreases the expected rollover crash frequency by 29% relative to a median foreslope steeper than 6H:1V; and, the presence of a median foreslope between 6H:1V and 7H:1V decreases the expected rollover crash frequency by 17% relative to a median foreslope steeper than 6H:1V. Median foreslopes equal to or flatter than 10V:1H lead to a greater decrease in rollover crash frequency than median foreslopes between 8H:1V and 9H:1V, and median foreslopes between 8H:1V and 9H:1V lead to a greater decrease in rollover crash frequency than median foreslopes between 6H:1V and 7H:1V. These findings support the expected outcome E9 in chapter 1 and are consistent with Stine et al. (2010), who indicated that the probability of rollovers in the median of divided highways increased as the cross-slopes became steeper in vehicle dynamics simulations.

129 The average elasticity of the expected five-year aggregated rollover crash 116 frequency in Pennsylvania with respect to median width is 1.0. This means that a 1% increase in median width increases the expected rollover crash frequency by 1.0%. This supports the expected outcome E8 in chapter 1 that a wider median leads to more rollover crashes, because wider medians increase the probability of vehicles staying in the median area instead of entering into the opposing travel lanes. The two interaction terms between median width and median cross-slopes were included to test whether the effect of median width on rollover crash frequency changes as the median cross-slopes change. The positive estimated coefficient of the interaction between median width and median foreslope and backslope equal to 6H:1V or steeper indicates that the increase in rollover crash frequency due to a 1% increase in median width is slightly inflated in magnitude (the average elasticity of rollover crash frequency with respect to median width is increased to 1.1, which means that a 1% increase in median width increases the expected rollover crash frequency by 1.1% ), if both the median foreslope and backslope are equal to 6H:1V or steeper relative to flatter median cross-slopes. This was expected because if both the median foreslope and backslope are steep, there is an increased probability of vehicles rolling over in the median. This is consistent with Stine et al. (2010), who showed that, when using vehicle dynamics simulations, wider medians were associated with an increase in the probability of median rollovers when the cross-slopes were 6H:1V. When both the median foreslope and backslope are equal to 10H:1V or flatter, the average elasticity of rollover crash frequency with respect to median width was decreased to -0.78, which means a 1% increase in median width reduces the expected rollover crash

130 117 frequency by 0.78%. This was expected because if the median foreslope and backslope are flat, there is a decreased probability of rollover crashes when errant vehicles enter the median. These findings support the expected outcome E9 in chapter 1. The pseudo-elasticities of the expected five-year aggregated rollover crash frequency in Pennsylvania with respect to Routes 79 and 90 are and -0.83, respectively. This indicates that the expected rollover crash frequency on Routes 79 and 90 were 45% and 83% lower than other routes, respectively. Interstate 79 is located in western PA and Interstate 90 is located in northwest PA. Both of these routes are partially of fully located in heavy snowfall regions of the state. It is possible that winter maintenance operations store snow in the median, creating a nearly flat profile during several months, and reducing the frequency of rollover events. As noted in the CMC models earlier, a higher expected cross-median crash frequency is expected on Interstate 90, which too may be the result of the a flatter median profile during several months of the year from snow storage. All other possible route indicators were not statistically significant in the estimated model. To summarize, for the analysis of five-year aggregated rollover crash frequencies in Pennsylvania, two NB models were estimated and no spatial models were estimated. No simulation was performed for the purpose of model assessment, since no SGLMs were estimated, and the estimated two crash frequency models were similar to each other. The fit of the two NB models could be clearly judged from the AIC and BIC. The NB model with route indicators has slightly lower AIC and BIC and is thus considered a better fit than the base model without route indicators.

131 Transferability of Pennsylvania Five-Year Aggregated Rollover Crash Frequency Models The base NB model without route indicators shown in Table 5.3 was tested for transferability. The goodness-of-fit based test was chosen to test the transferability of the five-year rollover crash frequency model estimated from Pennsylvania data, due to the small sample of North Carolina rollover crash data. The two transfer alternatives (Sawalha and Sayed, 2006) noted in section were tested and compared. First, the Pennsylvania rollover frequency model was transferred to North Carolina rollover crash data as is without any change to the parameter estimates. A z-score was calculated and its value was However, since the transferred model was a NB model, as discussed in section 4.1.3, the overdispersion parameter should be recalibrated prior to transferability test. Then, the Pennsylvania rollover frequency model was transferred to the North Carolina rollover data as is without any change to the parameter estimates except the overdispersion parameter. The overdispersion parameter was calculated by using equation 4.30, and the resulting z-score was calculated to be 2.604, which was much closer to zero than the z-score with the overdispersion parameter uncalibrated. This indicates that accounting for different overdispersion parameters in different application datasets could enable model transferabiity outside the application scenario where the model was estimated. However, a z-score of was still significantly greater than 0. The second transfer alternative was to first estimate North Carolina rollover crash frequencies by applying the Pennsylvania rollover frequency model, and then to multiply the estimates by a calibration factor. The calibration factor was calculated by dividing the

132 sum of observed rollover crash counts in the North Carolina dataset by the sum of 119 estimated rollover crash counts in North Carolina using the Pennsylvania rollover model, and its value was The recalibrated overdispersion parameter was calculated to be The resulting z-score was calculated and its value was 3.629, which was significantly greater than 0. It was found that after the overdispersion parameter was recalibrated, the recalibration of the Pennsylvania model by using a calibration factor resulted in a z-score which was further away from zero than the z-score calculated when the Pennsylvania model was transferred to the North Carolina data as is. This finding is similar to those reported by Sawalha and Sayed (2006). From the calculated z-scores, it was found that even though the model was calibrated, the Pennsylvania five-year aggregated rollover crash frequency model was not transferable to the North Carolina data Pennsylvania Yearly Rollover Crash Frequency Models and Model Assessment Pennsylvania Yearly Rollover Crash Frequency Models This section presents the estimated models for Pennsylvania yearly rollover crash frequency. NB models were estimated to account for possible overdispersion, but the overdispersion parameter was not statistically significant in the estimated models (pvalue > 0.5). Two crash frequency model specifications were employed for the yearly rollover crash frequency, Poisson models and random-effects Poisson models. For each model specification, three models were estimated, one without year or route indicators as the base model, one with year indicators, and one with both year and route indicators.

133 The same sets of variables were included in the two model specifications. Also, the 120 variables in the yearly crash frequency models were the same as those variables in the five-year aggregated crash frequency models shown in section A Likelihood ratio test was conducted for the each of the three estimated random-effects Poisson models by comparing the panel estimator with the pooled Poisson estimator, and the test results showed that alpha was statistically different from 0 in the three estimated random-effects Poisson models. This indicates that the road segment random effects were statistically significant. In these models, the natural logarithm of the road segment length (mile) was treated as an offset, since its estimated coefficients in all of the models were very close to 1 and the 95% confidence intervals contained 1. As discussed in chapter 4, the significance of route-specific effects does not necessarily indicate that there are significant unobserved spatially dependent effects. Efforts were made to estimate spatial generalized linear models for Pennsylvania yearly rollover crash frequency. However, the range parameters could not be estimated. As explained in section , a possible reason for the failure is that theoretically there were no significant unobserved spatially-dependent effects in the estimated rollover crash frequency models. Two empirical semivariograms were plotted from the Poisson base model residuals as presented in Figure a regular semivariogram and a robust one to account for outliers in the data. In Figure 5.4, it is clear that the semivariograms did not increase as the spatial distance increased. According to equation 4.15, this indicates that the covariance C (h) was not significantly different from zero, which means that there was no significant spatial dependency in the residuals.

134 Table 5.4. Models for Pennsylvania Yearly Rollover Crash Frequency Yealy cross-median crash frequency Poisson RE Poisson Poisson with year indicator RE Poisson with year indicator Poisson with year/route indicator 121 RE Poisson with year/route indicator Constant (2.049) (2.163) (2.054) (2.166) (1.950) (2.042) Leftcurve indicator (0.261) (0.277) (0.261) (0.277) (0.263) (0.275) Rightcurve indicator (0. 275) (0.290) (0.275) (0.290) (0.277) (0.288) Natural log adt (0.211) (0.224) (0.212) (0.224) (0.204) (0.214) Shoulder rumble strip indicator Median foreslope 10H:1V or flatter indicator Median foreslope between 8H:1V and 9H:1V indicator Median foreslope between 6H:1V and 7H:1V indicator (0.248) (0.262) (0.248) (0.262) (0.253) (0.264) (0.431) (0.446) (0.431) (0.446) (0.432) (0.443) (0.314) (0.329) (0.314) (0.329) (0.316) (0.327) (0.248) (0.261) (0.247) (0.260) (0.249) (0.259) Median width (ft) (0.004) (0.005) (0.004) (0.005) (0.004) (0.004) Median width (ft) if Median foreslope and backslope both 6H:1V or steeper, 0 otherwise (0.003) (0.003) (0.003) (0.003) (0.003) (0.003) Median width (ft) if Median foreslope and backslope both 10H:1V or flatter, 0 otherwise (0.019) (0.019) (0.019) (0.019) (0.019) (0.019) Route 79 indicator (0.292) (0.299) Route 90 indicator (0.723) (0.727) Year 2002 indicator (0.318) (0.318) (0.318) (0.318) Natural log road length Offset (mile) Alpha (0.379) (0.377) (0.337) LL_ LL_convergence AIC BIC NOTES: 1. No. of observations: 3099; 2.Standard errors are given in parentheses; an indicator variable is 1 if the condition holds and 0 otherwise; 3. Likelihood ratio test of alpha=0: χ 2 (1) = 1.87, Prob>= χ 2 = 0.086; 4. Likelihood ratio test of alpha=0: χ 2 (1) = 1.82, Prob>= χ 2 = 0.088; 5. Likelihood ratio test of alpha=0: χ 2 (1) = 1.04, Prob>= χ 2 = log likelihood for constant-only model; 7. log likelihood at convergence.

135 Lag Di st ance t ype regul ar robust Figure 5.4. Regular and Robust Empirical Semivariograms from Residuals of a Poisson Base Model for Pennsylvania Yearly Rollover Crash Frequency The estimated coefficients for the variables in the models shown in Table 5.4 are similar to those in the models presented in Table 5.3. As such, interpretations of the coefficients are similar and are not repeated. The only variable that was not included in the models presented in Table 5.3 is the year 2002 indicator. The elasticity of the expected yearly rollover crash frequency in Pennsylvania with respect to year 2002 is This means that rollover crash frequency was 55% lower in year 2002, when compared to the years 2003 through 2006 (inclusive). One possible reason for this finding is that the crash reporting system in Pennsylvania was changed in 2002, and the new system was easier to use for crash reporting. As a result, the crash frequencies might be

136 under reported in year 2002 before the system change in From the comparison 123 between the estimated Poisson and random-effects Poisson models, it was found that the coefficient estimates remained stable across the two model specifications, while the standard errors were slightly higher for the random-effects Poisson models. The randomeffects model has more noise due to its distribution assumption as described in section The overall model fit was improved by adding the year and route indicators, as indicated by the calculated AIC and BIC values. Given the same set of explanatory variables, the random-effects Poisson specification did not seem to improve the overall model fit over the pooled Poisson specification. The AIC and BIC values of the randomeffects Poisson model with year and route indicators and the Poisson model with year and route indicators are the lowest among the six estimated models shown in Table 5.4, and are thus considered the best fits. The AIC and BIC values of these two models are very similar to one another. Since no SGLMs were estimated, and the model fit of the estimated Poisson and random-effects Poisson models could be clearly judged from the calculated AIC and BIC values, no simulation was performed for model assessment Transferability of Pennsylvania Yearly Rollover Crash Frequency Models The base Poisson model without year and route indicators shown in Table 5.4 was tested for transferability. As discussed previously, the goodness-of-fit based transferability test was chosen to test the transferability of the yearly rollover crash

137 124 frequency model estimated from the Pennsylvania data, due to the small sample of the North Carolina rollover crash data. The two transfer alternatives (Sawalha and Sayed, 2006) noted in section were tested and compared. Since the transferred model was a Poisson model, it was not necessary to recalibrate the overdispersion parameter. First, the Pennsylvania rollover frequency model was transferred to North Carolina rollover crash data as is without any change to the parameter estimates. A z-score was calculated and its value was The second transfer alternative was to first estimate North Carolina rollover crash frequencies by applying the Pennsylvania rollover frequency model, and then to multiply the estimates by a calibration factor. The calibration factor was calculated by dividing the sum of observed rollover crash counts in North Carolina by the sum of estimated rollover crash counts in North Carolina by using the Pennsylvania rollover model, and its value was The resulting z-score was calculated and its value was Although this z- score was much closer to 0 than the z-score of the first transfer alternative (20.00), it was significantly different from 0. As a result, the Pennsylvania yearly rollover crash frequency model was not transferable. In summary, the transferability test results showed that the estimated models of the Pennsylvania five-year aggregated and yearly rollover crash frequencies were not transferable to the North Carolina data. Although calibrations were performed prior to the transferability tests, the Pennsylvania rollover crash frequency models could not be transferred to the North Carolina data. There are two possible reasons for this finding. One is that the common variable(s) in the rollover crash frequency models produced different effect sizes. The other reason is that in the estimated Pennsylvania rollover crash

138 125 frequency models, no explanatory variables related to median backslopes were included, since they were not statistically significant. However, backslope information might significantly affect median rollover crash frequencies in North Carolina Pennsylvania Median Barrier Crash Frequency Models This section is divided into two parts. The first part presents estimated models for five-year aggregated median barrier crash frequency in Pennsylvania and model assessment results. The second part presents estimated models for yearly median barrier crash frequency in Pennsylvania and model assessment results. Variables that were statistically significant at the 50% confidence level were included in the models Pennsylvania Five-Year Aggregated Median Barrier Crash Frequency Models and Model Assessment Pennsylvania Five-Year Aggregated Median Barrier Crash Frequency Models This section presents the estimated models for Pennsylvania five-year aggregated median barrier crash frequencies. Five crash frequency models were estimated, a Poisson model without route indicators as the base model for the purpose of comparison, a NB model without route indicators, a NB model with a route indicator, and two SGLMs with exponential and spherical semivariogram models, respectively. The purpose of adding route indicators to the NB models was to incorporate route-specific effects into the models. The addition of route indicators was expected to account for some spatially-

139 dependent effects since road segments along the same route were connected to each 126 other. As indicated by the calculated AIC and BIC values shown in Table 5.5, the addition of route indicators to the Pennsylvania five-year aggregated median barrier crash frequency NB model did not improve the model fit significantly. In the estimated NB models, the overdispersion parameter was statistically significant in both models (pvalues=0.008). Two empirical semivariograms were plotted as presented in Figure 5.5 from the Poisson base model residuals -- a regular semivariogram and a robust one to account for outliers in data. From the two empirical semivariograms, it is not clear whether there was significant spatial dependency in the residuals. In the two estimated SGLMs, the estimates of partial sill, nugget and range showed the significant presence of spatial dependency. In the estimated SGLM using the exponential semivariogram model, the estimated partial sill was and the estimated nugget was This implies that spatial dependency accounted for 56 percent (1.825/( )) of the extra-poisson variance, while the non-spatial random effect accounted for 44 percent (1.431/( )) of the extra-poisson variance. In the estimated SGLM using the spherical semivariogram model, the estimated partial sill was and the estimated nugget was This implies that spatial dependency accounted for 52 percent (1.775/( )) of the extra-poisson variance, while the non-spatial random effect accounted for 48 percent (1.620/( )) of the extra-poisson variance. The estimated ranges in the two SGLMs were similar to each other, 9.03 miles in the exponential semivariogram model and 8.79 miles in the spherical semivariogram model. The estimated ranges indicate that the unobserved spatially dependent effects were

140 127 present within a distance of about 9 miles or less in the estimated Pennsylvania five-year aggregated median barrier crash frequency models. This finding can be associated with the distances between interchanges on rural divided highways in Pennsylvania. For example, Interstate 80 in Pennsylvania runs about 311 miles through the state and there are 59 interchanges along Interstate 80. The interchanges are about 5.2 miles apart on average. The estimated ranges from SGLMs indicated that the unobserved spatial effects in the five-year aggregated median barrier crash frequency models existed within three consecutive interchanges. The five estimated models are presented in Table 5.5. In the five models, the natural logarithm of road segment length (miles) was treated as an offset, since the estimated coefficients in all of the models were close to 1 and the 95% confidence intervals contained 1.

141 Table 5.5. Models for Pennsylvania Five-Year Aggregated Median Barrier Crash Frequency Variable 2 Poisson NB NB with route SGLM (Exponential) indicator Constant (2.398) (2.599) (2.602) (3.430) (3.526) Difference between Median barrier offset and paved left shoulder width 4 (ft) 128 SGLM (Spherical) (0.028) (0.030) (0.030) (0.033) (0.033) Paved left Shoulder Width (ft) (0.062) (0.070) (0.070) (0.101) (0.103) Presence of entrance ramp indicator (0.198) (0.225) (0.224) (0.247) (0.247) Leftcurve indicator (0.232) (0.265) (0.266) (0.287) (0.287) Rightcurve indicator (0.230) (0.260) (0.260) (0.287) (0.288) Natural log adt (0.265) (0.290) (0.288) (0.381) (0.390) Left shoulder rumble strips indicator (0.218) (0.244) (0.253) (0.342) (0.349) Median foreslope 6H:1V or steeper indicator Route 22 indicator (0.456) Natural log road length (0.224) (0.241) (0.241) (0.273) (0.275) offset Alpha (Overdispersion parameter) (0.248) (0.243) Nugget (0.431) (0.299) Partial sill (0.424) (0.331) Range (mile) (2.71) (2.81) Log likelihood for constant-only model Log likelihood at convergence AIC BIC NOTES: 1. No. of observations: 411; 2.standard errors are given in parentheses; 3.an indicator variable is 1 if the condition holds and 0 otherwise; 4. median barrier offset from the edge of the traveled way paved left shoulder width; 5. Likelihood ratio test of alpha=0: χ 2 (1) = 5.90, Prob>= χ 2 = 0.008; 6. Likelihood ratio test of alpha=0: χ 2 (1) = 5.77, Prob>= χ 2 = 0.008;

142 Lag Di st ance t ype regul ar robust Figure 5.5. Regular and Robust Empirical Semivariograms from Residuals of a Poisson Base Model for Pennsylvania Five-Year Aggregated Median Barrier Crash Frequency In the Poisson and NB models presented in Table 5.5, the estimated coefficients are very similar. Similarly, the estimated coefficients are very similar between the two SGLMs. From the comparison among the estimated Poisson, NB models and SGLMs, it was found that the coefficient estimates remained relatively stable across the classical count models (Poisson and NB) and SGLMs except the coefficient estimate for the left shoulder rumble strips indicator. The estimated coefficients for this indicator in the classical count models were greater in magnitude than in the two SGLMs. A possible reason for the difference is the correlation between the spatial random effects and the left

143 130 shoulder rumble strips indicator in the SGLMs. The standard errors were consistently the lowest in the Poisson model and the highest in the SGLMs. The Poisson model underestimates standard errors because of its distribution properties. The NB model and SGLMs have more noise due to their distribution assumptions on the random error terms. In the NB model with a route indicator, the route indicator was statistically significant at the 85% confidence level. But the overall model fit was not significantly improved by adding the route indicator to the NB model, as indicated by the calculated AIC and BIC values presented in Table 5.5. Based on the results shown in Table 5.5, the pseudo-elasticities of the expected five-year aggregated median barrier crash frequency in Pennsylvania with respect to the left curve and right curve indicators are 1.01 and 0.97, respectively. This indicates that the presence of left and right curves increases the expected median barrier crash frequency by 101% and 97%, respectively. These findings, again, support expected outcome E1 and are consistent with McGee and Hansom (2006). Vehicles traveling on curved road segments are more likely to leave the roadway and enter into the median. The pseudo-elasticity of the expected five-year aggregated median barrier crash frequency in Pennsylvania with respect to left shoulder rumble strip indicator is in the NB models and in SGLMs. This indicates that the presence of left shoulder rumble strips decreases the expected median barrier crash frequency by about 30%. This finding supports expected oucomee3 and is consistent with a recent research reported by Torbic et al. (2009) described earlier. Rumble strips produce auditory and tactile vibrations that alert drivers that their vehicles have left the traveled way. The noise and

144 131 vibration produced when vehicle tires contact rumble strips could prevent drivers from entering into the median by steering back onto the through travel lanes. The pseudo-elasticity of the expected five-year aggregated median barrier crash frequency in Pennsylvania with respect to the interchange entrance ramp indicator is This indicates that the presence of interchange entrance ramps increases the expected median barrier crash frequency by 16%. This finding supports expected outcome E2 in chapter 1 and is consistent with a study performed by Donnell and Mason (2006), who found that the presence of interchange entrance ramps increased the expected median barrier crash frequency on Pennsylvania Interstates by approximately 7%. When there are entrance ramps present along a freeway, vehicles traveling on the through lanes may be forced to change lanes as vehicles merge onto the through lane from the adjacent entrance ramps. This increases the probability of encroaching into the median due to merging traffic. The average elasticity of the expected five-year aggregated median barrier crash frequency in Pennsylvania with respect to ADT is This means that a 1% increase in ADT increases the expected median barrier crash frequency by 0.31%. This supports expected outcome E4 in chapter 1 that higher traffic volumes increase the probability of median-related crashes. The pseudo-elasticity of the expected five-year aggregated median barrier crash frequency in Pennsylvania with respect to median foreslopes equal to or steeper than 6V:1H indicator is This indicates that median foreslopes equal to or steeper than 6V:1H increase the expected median barrier crash frequency by 35% relative to flatter median foreslopes. This supports the expected outcome E11 in chapter 1 that steep

145 132 median foreslopes increase median barrier crash frequencies, which is intuitive because steep median foreslopes make it difficult for drivers who have entered the median to stop or recover before colliding with a longitudinal median barrier. The average elasticity of the expected five-year aggregated median barrier crash frequency in Pennsylvania with respect to the lateral offset between the median barrier and the inside edge of the paved left shoulder is This indicates that a 1% increase in the median barrier offset decreases the expected median barrier crash frequency by 0.44%. This supports the expected outcome E10 in chapter 1 that installing median barrier further away from travel lane decreases median barrier crash frequencies. This finding is also consistent with a study by Donnell and Mason (2006), who found that increased median barrier offset led to decreased median barrier crash frequency. The further the median barrier is away from left shoulder, more space is provided for errant vehicles encroaching into the median to stop or recover before colliding with a median barrier. The average elasticity of the expected five-year aggregated median barrier crash frequency in Pennsylvania with respect to the width of the paved left shoulder is This means that a 1% increase in the paved left shoulder width increases the expected median barrier crash frequency by 0.96%. A possible reason for this finding is that wider paved inside shoulder may increase driving comfort, resulting in higher travel speeds and increased median encroachments. The pseudo-elasticity of the expected five-year aggregated median barrier crash frequency in Pennsylvania with respect to the Route 22 indicator was This indicates that road segments along route 22 experienced fewer cross-median crashes on

146 133 average during the five-year analysis period than other routes. This indicator might be a surrogate for the driver population or some traffic characteristics which lead to less median barrier crashes. It is not clear what exactly led to this finding and further exploration is recommended. No other possible route indicators were statistically significant in the estimated model. Efforts were made to include interactions between median foreslopes and median barrier offsets in the models, but these interaction terms were not statistically significant at the 50% confidence level Pennsylvania Five-Year Aggregated Median Barrier Crash Frequency Model Assessment As discussed in section 4.1.2, AIC or BIC is not a good measure of the SGLM s fit. In order to compare the fit of the estimated SGLM models and the classical count models, simulations of the predicted crash frequencies from the estimated SGLMs and the classical count models with the best fit (lowest AIC and BIC values) were performed to compare the predictive capabilities of these models. In the case of Pennsylvania fiveyear aggregated median barrier crash frequencies, the AIC and BIC values of the three estimated classical count models were similar to each other. Simulations were performed in R (R Development Core Team, 2010) for all the five estimated models on the randomly selected 45 road segments for simulation, which contained 10 percent of the total sample as described in Chapter simulations of predicted crash frequencies were generated for each road segment based on each of the five estimated models. For each model, the mean value of

147 134 the 1000 simulated crash frequencies was calculated for each road segment, as well as the 10th percentile and 90th percentile of the simulated crash frequencies to measure the predictive capabilities of these models. The model with mean simulated crash frequencies that were the closest to the observed crash frequencies, the highest coverage of the prediction intervals (the 10th percentile and 90th percentile of the simulated crash frequency) on the observed crash frequencies, and the narrowest prediction intervals was considered the model with the best predictive capabilities. Figure 5.6 shows the observed crash frequencies on the randomly selected 45 road segments for simulation, as well as the mean values of simulated crash frequencies based on the five estimated models shown in Table 5.5. The prediction intervals of the five estimated models are shown in Figures A.1 to A.5 in Appendix A. Table 5.6 presents the root mean squared errors, prediction interval coverage on the observed crash frequencies, mean prediction interval width and median prediction interval width of the five estimated models. 4 Crash Count PA Median Barrier Crash Site No. Observed simulated_nb_mean simulated_sglm(exponential)_mean Simulated_Poisson_Mean simulated_nb_route_mean simulated_sglm(spherical)_mean Figure 5.6. Simulation Results based on Pennsylvania Five-Year Aggregated Median Barrier Crash Frequency Models

148 135 Table 5.6. Simulation Results for Pennsylvania Five-Year Aggregated Median Barrier Crash Frequency Poisson NB NB with route SGLM SGLM indicator (Exponential) (Spherical) Root Mean Squared Error Pred. Interval Coverage a,b (%) Avg. Pred. Interval Width Median Pred. Interval Width Note: a: Prediction Interval Coverage = number of sites whose observed crash counts fall in prediction interval / total number of sites; b: Prediction interval width = 90%th percentile of 1000 simulated predicted crash counts from an estimated model for a site - 10%th percentile of the 1000 simulated predicted crash counts. From Figure 5.6, the predictions from the three classical count models are similar while the predictions from the two SGLMs are similar. It is difficult to determine whether the classical count model predictions or the SGLM predictions are closer to the observed crash frequencies. In Table 5.6, the root mean squared errors of the five models are very similar. The prediction interval coverage of the three classical count models is higher than the coverage of the two SGLMs. The mean and median prediction interval widths of the two SGLMs are significantly smaller than those of the classical count models, which is also shown in Figures A.1 to A.5. In summary, the predictions from the classical count models do not appear accurate. The SGLMs seem to be able to predict crash frequencies more accurately with much narrower prediction intervals than the classical count models, except several outliers where the simulated predicted crash frequencies from SGLMs differed considerably from the observed crash frequencies. The outliers from the SGLM predictions (sites #30, 41 and 45 in Figure 5.1) were further investigated and it was found that crash frequencies on these sites were different from most of their neighbors. Site #30 had no median barrier crashes in the five-year analysis period, but its closest neighbors in

149 136 the analysis dataset had 1-3 median barrier crashes during the same period. Site #41 had only 1 median barrier crash in the five-year analysis period, but its closest neighbors in the analysis dataset had 2-4 median barrier crashes during the same period. Site #45 had 2 median barrier crashes in the five-year analysis period, but most of its neighbors in the analysis dataset experienced no median barrier crashes during the same period. The simulated predicted median barrier crash frequencies on these outlier sites from the estimated two SGLMs were closer to most of their neighbors crash frequencies. This finding indicates that predictions from SGLMs borrow values from neighbors. When the crash frequencies on adjacent road segments are similar to each other, predictions from SGLMs are likely improved and more accurate than predictions from the classical count models, by borrowing information from neighbors. However, when the crash frequency on a road segment is very different from crash frequencies on most of its neighboring road segments, predictions from SGLMs appear to result in less accuracy. This characteristic of SGLMs could be utilized to detect outliers of road sites with crash frequencies significantly different from their neighbors Pennsylvania Yearly Median Barrier Crash Frequency Models and Model Assessment Pennsylvania Yearly Median Barrier Crash Frequency Models This section presents the estimated models for Pennsylvania yearly median barrier crash frequency. Three specifications of crash frequency models were used for the yearly median barrier crash frequency modeling, NB models, random-effects NB models and

150 SGLMs. For NB and random-effects NB model specifications, three models were 137 estimated in each model specification, one without year or route indicators as the base model, one with year indicators, and one with both year and route indicators. The purpose of adding route indicators to NB models is to incorporate route-specific effects into the models. The addition of route indicators was expected to account for some spatiallydependent effects since road segments along the same route are connected to each other. The overdispersion parameters in the estimated NB models were statistically significant (p-values<=0.01). A Likelihood ratio test was conducted for the each of the three random-effects NB models by comparing the panel estimator with the pooled NB estimator, and the test results showed p-values smaller than This indicates that the road segment random effects were statistically significant at the 90% confidence level. Empirical semivariograms were plotted as shown in Figure 5.7 from the Poisson base model residuals -- a regular semivariogram and a robust one to account for outliers in the data. From the comparison between the empirical semivariograms of the Pennsylvania five-year aggregated median barrier crash frequency model (Figure 5.5) and the Pennsylvania yearly median barrier crash frequency model (Figure 5.7), it was found that the semivariograms of the yearly crash frequency model residuals were much smoother than the five-year aggregated crash frequency model. This was expected because in the yearly crash frequency dataset, there were replicates of the spatial structure over years and more information was available for spatial dependence estimation. Two SGLMs were estimated with exponential and spherical semivariogram models, respectively. In the estimated SGLMs, year indicators were included and a constant spatial dependence structure over years was assumed. In the two estimated SGLMs, the

151 138 estimates of sill and range showed the significant presence of spatial dependency. Nugget was not significantly different from 0 in both models, which indicates that there was no significant non-spatial extra-poisson variance in both models. The estimated ranges in the two SGLMs were similar to each other, 2.69 miles in the exponential semivariogram model and 2.11 miles in the spherical semivariogram model. The estimated ranges indicate that unobserved spatially dependent effects were present within a distance of about 3 miles or less in the estimated Pennsylvania yearly median barrier crash frequency models. Again, this finding can be associated with the distances between interchanges on rural divided highways in Pennsylvania. For example, Interstate 79 in Pennsylvania runs about 184 miles through the state and there are 51 interchanges along Interstate 79. These interchanges are about 3.6 miles apart on average. The estimated ranges from the SGLMs indicated that the unobserved spatial effects in the yearly median barrier crash frequency models existed between two consecutive interchanges. A total of eight models were estimated for the Pennsylvania yearly median barrier crash frequency, as presented in Table 5.7. The same sets of variables were included in the three model specifications. The variables included in these yearly crash frequency models are the same as those variables in the five-year aggregated crash frequency models shown in Table 5.5, except the year indicator. In the estimated models,,the natural logarithm of road segment length (miles) was treated as an offset, since its estimated coefficients in all the models were very close to 1 and the 95% confidence intervals contained 1.

152 139 Table 5.7. Models for Pennsylvania Yearly Median Barrier Crash Frequency Variable 2 NB RENB NB with year indicator RENB with year indicator NB with year/route indicator RENB with year/route indicator SGLM (Exponential) SGLM (Spherical) Constant (2.380) (2.603) (2.394) (2.670) (2.390) (2.700) (2.832) (2.768) Difference between Median barrier offset and paved left shoulder width 3 (ft) (0.029) (0.030) (0.029) (0.030) (0.029) (0.030) (0.030) (0.030) Paved left Shoulder Width (ft) (0.065) (0.067) (0.065) (0.068) (0.065) (0.068) (0.089) (0.087) Presence of interchange entrance ramp indicator (0.211) (0.220) (0.210) (0.221) (0.210) (0.220) (0.226) (0.219) Leftcurve indicator (0.248) (0.264) (0.248) (0.265) (0.249) (0.266) (0.266) (0.261) Rightcurve indicator (0.243) (0.251) (0.243) (0.252) (0.244) (0.252) (0.263) (0.257) Natural log adt (0.266) (0.274) (0.267) (0.276) (0.265) (0.275) (0.318) (0.312) Left shoulder rumble strips indicator (0.229) (0.238) (0.228) (0.238) (0.237) (0.248) (0.298) (0.293) Median foreslope 6H:1V or steeper indicator (0.232) (0.239) (0.232) (0.240) (0.233) (0.241) (0.248) (0.246) Year 2002 indicator (0.249) (0.241) (0.249) (0.241) (0.318) (0.310) Year 2003 indicator (0.245) (0.239) (0.246) (0.239) (0.337) (0.329) Route 22 indicator (0.444) (0.450) Natural log road length offset

153 140 Table 5.7. Models for Pennsylvania Yearly Median Barrier Crash Frequency (continued) Variable 2 NB RENB NB with year indicator RENB with year indicator NB with year/route indicator RENB with year/route indicator SGLM (Exponential) SGLM (Spherical) Constant (2.380) (2.603) (2.394) (2.670) (2.390) (2.700) (2.832) (2.768) Alpha (Overdispersion parameter) (0.550) (0.540) (0.536) r (54.560) (61.438) (68.000) s (3.098) (2.776) (2.746) Sill (0.426) (0.406) Range (mile) (0.51) (0.28) Log likelihood for constant-only model Log likelihood at convergence AIC BIC NOTES: 1.standard errors are given in parentheses; 2.an indicator variable is 1 if the condition holds and 0 otherwise; 3. median barrier offset from the edge of the traveled way paved left shoulder width; 4. No. of observations: Likelihood ratio test of alpha=0: χ 2 (1) = 5.76, Prob>= χ 2 = 0.008; 6. Likelihood-ratio test vs. pooled: χ 2 (1) = 2.00, Prob>= χ 2 = 0.078; 7. Likelihood ratio test of alpha=0: χ 2 (1) = 5.46, Prob>= χ 2 = 0.010; 8. Likelihood-ratio test vs. pooled: χ 2 (1) = 2.23, Prob>= χ 2 = 0.068; 9. Likelihood ratio test of alpha=0: χ 2 (1) = 5.45, Prob>= χ 2 = 0.010; 10. Likelihood-ratio test vs. pooled: χ 2 (1) = 2.27, Prob>= χ 2 =

154 Lag Di st ance t ype regul ar robust Figure 5.7. Regular and Robust Empirical Semivariograms from Residuals of a Poisson Base Model for Pennsylvania Yearly Median Barrier Crash Frequency The estimated coefficients of variables in the models in Table 5.7 are similar to those in the models presented in Table 5.5 except the natural logarithm of ADT, and the year 2002 and 2003 indicators which were not included in the models in Table 5.5. For the similar explanatory variables, interpretation is not repeated here. The estimated coefficients of the natural logarithm of ADT in the estimated NB models, random-effects NB models and two SGLMs were not significantly different from 0. The pseudoelasticities of the expected yearly median barrier crash frequency in Pennsylvania with respect to year 2002 and 2003 are and -0.28, respectively. This means that median

155 142 barrier crash frequency was 30% lower in year 2002 and 28% lower in year 2003, when compared to years 2004 through 2006 (inclusive). One possible reason for this finding is that the crash reporting system in Pennsylvania was changed in 2002, and the new system was easier to use for crash reporting. As a result, the crash frequencies might be under reported in year 2002 before the system change in From comparisons of the estimated models presented in Table 5.7, it was found that the estimated NB and random-effects NB models with the same set of explanatory variables had similar estimated coefficients except for the constant term. The estimated constants in the NB models are more negative than in the random-effects NB models. The estimated coefficients are very similar between the two SGLMs. Also, it was found that the coefficient estimates remained stable across the NB models, random-effects NB models and SGLMs, except the coefficient estimates for left shoulder rumble strip indicator. The estimated coefficients of the left shoulder rumble strips indicator in the estimated NB and random-effects NB models were greater in magnitude than in the two SGLMs. A possible reason for the difference is the correlation between the spatial random effects and the left shoulder rumble strips indicator in the SGLMs. The standard errors are consistently the lowest for the NB model and the highest for the SGLMs. The random- effects NB model and SGLMs has more noise due to their distribution assumptions on their random error terms. From the comparison of AIC and BIC values calculated for the estimated NB and random-effects NB models presented in Table 5.7, it was found that adding the year and route indicators did not improve the overall model fit. Also the random-effects NB specification did not improve the overall model fit over the pooled NB specification. The

156 143 NB model without year and route indicators has the lowest AIC and BIC values among the six estimated NB and random-effects NB models shown in Table 5.7 and is thus considered the best fit among the six models. The assessment of SGLMs is discussed in the next section Pennsylvania Yearly Median Barrier Crash Frequency Model Assessment As discussed in section 4.1.2, AIC or BIC is not a good measure of the SGLM s fit. In order to compare the fit of the estimated SGLM models, the classical count models and the panel count models, simulations of the predicted crash frequencies from the estimated SGLMs and the classical/panel count models with the best fit (lowest AIC and BIC values) were performed to compare the predictive capabilities of these models. In the case of Pennsylvania yearly median barrier crash count models, the NB model without year or route indicators has the lowest AIC and BIC values. Then, simulations of predicted crash frequencies were performed in R (R Development Core Team, 2010) based on the estimated NB model without year or route indicators and two SGLMs on the randomly selected 45 road segments over five years, which contains 10 percent of the total sample as described in Chapter simulations of predicted crash frequencies were generated for each road segment in each year based on each of the three estimated models. For each model, the mean value of the 1000 simulated crash frequencies was calculated for each road segment in each year, as well as the 10th percentile and 90th percentile of the simulated crash frequencies to measure the predictive capabilities of these models. The model with mean

157 simulated crash frequencies that are the closest to the observed crash frequencies, the 144 highest coverage of the prediction intervals (the 10th percentile and 90th percentile of the simulated crash frequency) on the observed crash frequencies, and the narrowest prediction intervals was considered the model with the best predictive capabilities. Figures 5.8 to 5.12 show, by year, the yearly observed crash frequencies on the randomly selected 45 road segments for simulation, as well as the mean values of simulated crash frequencies based on the three estimated models shown in Table 5.7. The prediction intervals of the three estimated models are shown by year in Figures A.6 to A.20 in Appendix A. Table 5.8 presents the root mean squared errors, prediction interval coverage on the observed crash frequencies, mean prediction interval width and median prediction interval width of the three estimated models. 4 Crash Count PA Median Barrier Crash Site No.,Year 2002 Observed_yearly simulated_sglm(exponential)_mean simulated_nb_mean simulated_sglm(spherical)_mean Figure 5.8. Simulation Results based on Pennsylvania Yearly Median Barrier Crash Frequency Models, Year 2002

158 3 145 Crash Count PA Median Barrier Crash Site No. Year 2003 Observed_yearly simulated_sglm(exponential)_mean simulated_nb_mean simulated_sglm(spherical)_mean Figure 5.9. Simulation Results based on Pennsylvania Yearly Median Barrier Crash Frequency Models, Year Axis Title PA Median Barrier Crash Site No. Year 2004 Observed_yearly simulated_sglm(exponential)_mean simulated_nb_mean simulated_sglm(spherical)_mean Figure Simulation Results based on Pennsylvania Yearly Median Barrier Crash Frequency Models, Year 2004

159 3 146 Axis Title PA Median Barrier Crash Site No. Year 2005 Observed_yearly simulated_sglm(exponential)_mean simulated_nb_mean simulated_sglm(spherical)_mean Figure Simulation Results based on Pennsylvania Yearly Median Barrier Crash Frequency Models, Year Crash Count PA Median Barrier Crash Site No. Year 2006 Observed_yearly simulated_sglm(exponential)_mean simulated_nb_mean simulated_sglm(spherical)_mean Figure Simulation Results based on Pennsylvania Yearly Median Barrier Crash Frequency Models, Year 2006

160 147 Table 5.8. Simulation Results for Pennsylvania Yearly Median Barrier Crash Frequency NB SGLM(exponential) SGLM(spherical) Root Mean Squared Error Pred. Interval Coverage a,b (%) Avg. Pred. Interval Width Median Pred. Interval Width Note: a: Prediction Interval Coverage = number of sites whose observed crash counts fall in prediction interval / total number of sites; b: Prediction interval width = 90%th percentile of 1000 simulated predicted crash counts from an estimated model for a site - 10%th percentile of the 1000 simulated predicted crash counts. From Figures 5.8 to 5.12, when the observed crash frequencies are very low (0), the NB model appears to produce predictions closer to the observed crash frequencies than the SGLMs. When the observed crash frequencies increase, predictions from the NB model get further away from the observed crash frequencies, while predictions from the SGLMs get closer to the observed crash frequencies for some road segments. In Table 5.8, the root mean squared error of the NB model is the lowest, and the root mean squared error of the SGLM with spherical semivariogram model is the highest. But the root mean squared errors of the three models are not very different from each other. The prediction interval coverage of the SGLM with spherical semivariogram model is the highest among the three models. The mean and median prediction interval width of the NB model is the smallest, which is also shown in Figures A.6 to A.20. In summary, in the case of yearly Pennsylvania median barrier crash frequency, the NB model seems to predict the yearly crash frequencies better than the SGLMs. Outliers where the simulated predicted crash frequencies from SGLMs differed considerably from the observed crash frequencies were further investigated and it was found that crash frequencies at these outlier sites were different from most of their

161 neighbors. For example, site #9 in year 2002 had no median barrier crashes, but its 148 closest neighbors in the analysis dataset had 1 median barrier crash during the same period. Site #33 in year 2006 had no median barrier crashes, but its closest neighbors in the analysis dataset had 1-2 median barrier crashes during the same period. The simulated predicted median barrier crash frequencies from the estimated two SGLMs on these outlier sites are closer to most of their neighbors crash frequencies. These findings are similar to those discussed in section , which indicate that predictions from SGLMs borrow information from neighbors. When the crash frequencies on adjacent road segments are similar to each other, predictions from the SGLMs are improved over the classical count models. When the crash frequency on a road segment is very different from crash frequencies on most of its neighboring road segments, predictions from SGLMs are less accurate than the classical count models. Again, this characteristic of SGLMs could be utilized to detect outliers of road sites with crash frequencies significantly higher or lower than their neighbors North Carolina Median Barrier Crash Frequency Models This section is divided into two parts. The first part presents estimated models for five-year aggregated median barrier crash frequency in North Carolina and model assessment results. The second part presents estimated models for yearly median barrier crash frequency in North Carolina and model assessment results. Variables that were statistically significant at the 50% confidence level were included in the models.

162 North Carolina Five-Year Aggregated Median Barrier Crash Frequency Models and Model Assessment North Carolina Five-Year Aggregated Median Barrier Crash Frequency Models This section presents the estimated models for North Carolina five-year aggregated median barrier crash frequencies. Five crash frequency models were estimated as presented in Table 5.9, a Poisson model without route indicators as the base model for the purpose of comparison, a NB model without route indicators, a NB model with a route indicator, and two SGLMs with exponential and spherical semivariogram models, respectively. In the estimated NB models, the overdispersion parameter was statistically significant in both models (p-values<0.001). Empirical semivariograms were plotted as shown in Figure 5.13 from the Poisson base model residuals -- a regular semivariogram and a robust one to account for outliers in data. From the two empirical semivariograms, it is clear that there was significant spatial dependency in the residuals, since the semivariograms increased as the spatial distance increased, and approached constant values when spatial distance approached certain values. Also, in the two estimated SGLMs, the estimates of partial sill, nugget and range show the significant presence of spatial dependency. In the estimated SGLM using the exponential semivariogram model, the estimated partial sill was and the estimated nugget was This implies that spatial dependency accounted for 77 percent (0.282/( )) of the extra-poisson variance, while the non-spatial random effect accounted for 23 percent (0.084/( )) of the extra-poisson variance. In the estimated SGLM using the

163 150 spherical semivariogram model, the estimated partial sill was and the estimated nugget was This implies that spatial dependency accounted for 62 percent (0.228/( )) of the extra-poisson variance, while the non-spatial random effect accounted for 38 percent (0.141/( )) of the extra-poisson variance. The estimated ranges in the two SGLMs were similar to each other, 12.7 miles in the exponential semivariogram model and 13.4 miles in the spherical semivariogram model. The estimated ranges indicate that the unobserved spatially dependent effects took effect within a distance of about 13 miles or less in the estimated North Carolina five-year aggregated median barrier crash frequency models. This finding can be associated with the distances between interchanges on rural divided highways in North Carolina. For example, Interstate 95 in North Carolina runs about 180 miles through the state and there are 59 interchanges along Interstate 95. These interchanges are about 3.1 miles apart on average. The estimated ranges from the SGLMs indicate that the unobserved spatial effects in the North Carolina five-year aggregated median barrier crash frequency models existed within five consecutive interchanges. The five estimated models are presented in Table 5.9. In the five models, the natural logarithm of road segment length (miles) was treated as an offset, since its estimated coefficients in all the models were close to 1 and the 95% confidence intervals contained 1.

164 151 Table 5.9. Models for North Carolina Five-Year Aggregated Median Barrier Crash Frequency Variable 2 Poisson NB NB with route indicator SGLM (Exponential) SGLM (Spherical) Constant (0.537) (1.047) (1.506) (1.603) (1.6) Median Barrier offset (ft) (0.017) (0.034) (0.035) (0.041) (0.041) Leftcurve indicator (0.092) (0.171) (0.164) (0.151) (0.153) Rightcurve indicator (0.07) (0.158) (0.155) (0.145) (0.146) Natural log adt (0.053) (0.104) (0.145) (0.158) (0.158) Left shoulder rumble strips indicator (0.045) (0.087) (0.089) (0.14) (0.14) Median foreslope 7H:1V or steeper indicator (0.299) (0.528) (0.526) (0.556) (0.558) Median foreslope between 7H:1V and 9H:1V indicator (0.189) (0.364) (0.35) (0.331) (0.334) Median barrier offset (ft) if Median foreslope is 7H:1V or steeper, 0 otherwise (0.063) (0.102) (0.102) (0.105) (0.106) Median barrier offset (ft) if Median foreslope is between 7H:1V and 9H:1V, 0 otherwise (0.037) (0.07) (0.067) (0.065) (0.065) Speed limit 70mph or higher indicator (0.062) (0.113) (0.16) (0.185) (0.185) Speed limit 65 mph indicator (0.059) (0.114) (0.14) (0.175) (0.174) Route 26 indicator 0.47 (0.238) Route 40 indicator (0.185) Route 64 indicator (0.254) Route 74 indicator (0.304) Route 85 indicator (0.214) Route 95 indicator (0.194) Route 321 indicator (0.275) Natural log road segment length (miles) offset Alpha (Overdispersion parameter) (0.037) (0.034) Nugget (0.021) (0.014) Partial sill (0.021) (0.015) Range (mile) (2.01) (1.57) Log likelihood for constant-only model Log likelihood at convergence AIC BIC NOTES: 1.standard errors are given in parentheses; 2.an indicator variable is 1 if the condition holds and 0 otherwise; obs.; 4. LR test of alpha=0:χ 2 (1) = , Prob>χ 2 = 0.000;5.LR test of alpha=0: χ 2 (1) = , Prob> χ 2 = 0.000

165 152 Figure Regular and Robust Empirical Semivariograms from Residuals of A Poisson Base Model for North Carolina Five-Year Aggregated Median Barrier Crash Frequency From the comparison of the three estimated Poisson and NB models presented in Table 5.9, it was found that the estimated coefficients in the Poisson and NB models without route indicators are similar except for the left shoulder rumble strips indicator. In both models, the left shoulder rumble strips indicator was not significantly different from 0 at the 50% confidence level and was included in the models for the purpose of comparison with other models. The estimated coefficients are similar between the two estimated SGLMs. The coefficients in the estimated NB model with route indicators differ in magnitude from the estimated Poisson model, NB model without route

166 153 indicators and the two SLGMs. A possible reason for the difference is the correlation between the route indicators and the other explanatory variables in the model, such as ADT. From the comparison between the estimated classical count models (Poisson and NB) and SGLMs, it was found that the coefficient estimates remained relatively stable across the classical count models and spatial count models. The standard errors were consistently the lowest for the Poisson model and the highest for the SGLMs. As discussed earlier, the Poisson model underestimates standard errors because of its distribution property. The NB model and SGLMs have more noise due to their distribution assumptions on the random error terms. In the NB model with route indicators, the route indicators were statistically significant at the 70% confidence level. As indicated by the calculated AIC and BIC values shown in Table 5.9, the addition of route indicators to the North Carolina five-year aggregated median barrier crash frequency NB model slightly improved the NB model fit. The use of NB model specification significantly improved the model fit over the Poisson model specification. Based on the results shown in Table 5.9, the explanation of the estimated models are provided below. The pseudo-elasticities of the expected five-year aggregated median barrier crash frequency in North Carolina with respect to the left curve and right curve indicators are 0.06 and 0.16, respectively. This indicates that the presence of left and right curves increases the expected median barrier crash frequency by 6% and 16%, respectively. These findings support expected outcome E1 and are consistent with McGee and Hanscom (2006). However, in the NB model with route indicators the left and right curve indicators were not statistically significant.

167 The estimated coefficient of the left shoulder rumble strip indicator was not 154 significantly different from 0 in the NB model but was significantly different from 0 in the SGLM with spherical semivariogram model. In this SGLM, the pseudo-elasticity of the expected five-year aggregated median barrier crash frequency in North Carolina with respect to the left shoulder rumble strip indicator is This indicates that the presence of left shoulder rumble strips decreases the expected median barrier crash frequency by 8%. This finding supports expected outcome E3 in chapter 1 and is consistent with previous research, as explained in previous sections. The average elasticity of the expected five-year aggregated median barrier crash frequency in North Carolina with respect to ADT is This finding indicates that a 1% increase in ADT increases the expected median barrier crash frequency by 0.63%. Again, this supports the expected outcome E4, as explained in previous sections. The pseudo-elasticities of the expected five-year aggregated median barrier crash frequency in North Carolina with respect to the speed limit 70 mph or higher indicator and the speed limit 65 mph indicator are 0.42 and 0.28, respectively. This finding indicates that a speed limit of 70 mph or higher and a speed limit of 65 mph increase the expected median barrier crash frequency by 42% and 28% respectively, relative to a speed limit lower than 60 mph. A speed limit of 70 mph leads to a higher increase in median barrier crash frequency than a speed limit of 65 mph. which supports the expected outcome E5. However, in the estimated NB model with route indicators, a speed limit of 70 mph leads to a lower increase in median barrier crash frequency than a speed limit of 65 mph. A possible reason for the disagreement in the NB model with route

168 155 indicators is the correlation between the route indicators and the speed limit indicators in the model. The pseudo-elasticities of the expected five-year aggregated median barrier crash frequency in North Carolina with respect to the median foreslopes equal to or steeper than 7V:1H indicator and the median foreslope between 7V:1H and 9V:1H indicator are 1.46 and 0.28, respectively. This finding indicates that the median foreslopes equal to or steeper than 7V:1H and between 7V:1H and 9V:1H increase the expected median barrier crash frequency by 146% and 28% respectively, relative to median foreslopes equal to or flatter than 10V:1H. This finding indicates that steeper median foreslopes increase median barrier crash frequencies, which supports the expected outcome E11 and is intuitive because steep median foreslopes make it difficult for drivers who have entered the median to stop or recover before colliding with a longitudinal median barrier. The average elasticity of the expected five-year aggregated median barrier crash frequency in North Carolina with respect to the median barrier offset from the edge of travel lane was not significantly different from 0 in the NB model with route indicators and the two SGLMs, but was significantly different from 0 in the Possion and NB models without route indicators. This, again, is likely due to the correlation between the route indicators or spatial random effects with median barrier offsets. In the Possion and NB models without route indicators, the average elasticity of median barrier crash frequency with respect to the median barrier offset is This finding indicates that a 1% increase in the median barrier offset decreases the expected median barrier crash frequency by 0.15%. This finding supports the expected outcome E10 and is again consistent with a study by Donnell and Mason (2006), who found that increased median

169 156 barrier offset led to decreased median barrier crash frequency. The further the median barrier is away from left shoulder, more space is provided for errant vehicles encroaching into the median to stop or recover before colliding with a median barrier. When the median foreslope is equal to 7H:1V or steeper, the average elasticity of the expected five-year aggregated median barrier crash frequency in North Carolina with respect to the median barrier offset is This indicates that when the median foreslope is equal to 7H:1V or steeper, a 1% increase in median barrier offset reduces the expected median barrier crash frequency by 2.04% relative to when the median foreslope is equal to 10H:1V or flatter. When the median foreslope is between 7H:1V and 9H:1V, the average elasticity of the expected five-year aggregated median barrier crash frequency in North Carolina with respect to the median barrier offset is This means that when the median foreslope is between 8H:1V and 9H:1V, a 1% increase in median barrier offset reduces the expected median barrier crash frequency by 0.52% relative to when the median foreslope is equal to 10H:1V or flatter. These findings were expected because when the median foreslope gets steeper, it is more difficult for drivers to recover errant vehicles in the median prior to colliding with median barriers. The effects of median barrier offset found in this study are again consistent with a study by Donnell and Mason (2006), as described earlier. The pseudo-elasticities of the expected five-year aggregated median barrier crash frequency in North Carolina with respect to routes 26, 40, 64, 74, 85, 95 and 321 are 0.60, 1.29, 0.83, 2.44, 0.26, 0.82 and 0.84, respectively. This indicates that road segments along routes 26, 40, 64, 74, 85, 95 and 321 experienced more median barrier crashes on average during the five-year analysis period than the remaining routes in the sample.

170 These indicators might be surrogates for the driver population or some traffic 157 characteristics which lead to more median barrier crashes. It is not clear what exactly led to this finding and further exploration is recommended. Efforts were made to include other variables such as an interchange entrance ramp indicator in the models, but these variables were not statistically significant at the 50% confidence level North Carolina Five-Year Aggregated Median Barrier Crash Frequency Model Assessment In order to compare the fit of the estimated SGLM models and the classical count models, in the case of North Carolina five-year aggregated median barrier crash frequencies, simulations of the predicted crash frequencies from the estimated SGLMs and the classical count models presented in Table 5.9 were performed on the randomly selected 45 road segments for simulation, which contains about 10 percent of the total sample as described in Chapter simulations of predicted crash frequencies were generated for each road segment based on each of the five estimated models. For each model, the mean value of the 1000 simulated crash frequencies was calculated for each road segment, as well as the 10th percentile and 90th percentile of the simulated crash frequencies to measure the predictive capabilities of these models. The model with mean simulated crash frequencies that are the closest to the observed crash frequencies, the highest coverage of the prediction intervals (the 10th percentile and 90th percentile of the simulated crash frequency) on the observed crash frequencies, and the narrowest prediction intervals was

171 considered the model with the best predictive capabilities. Figure 5.14 shows the 158 observed five-year median barrier crash frequencies on the randomly selected 45 road segments for simulation, as well as the mean values of simulated crash frequencies based on the five estimated models shown in Table The prediction intervals of the five estimated models are shown in Figures A.21 to A.25 in Appendix A. Table 5.10 presents the root mean squared errors, prediction interval coverage on the observed crash frequencies, mean prediction interval width and median prediction interval width of the five estimated models Crash Counts North Carolina Median Barrier Crash Site No. Observed simulated_sglm(spherical)_mean simulated_nb_mean simulated_sglm(exponential)_mean simulated_poisson_mean simulated_nb_route_mean Figure Simulation Results based on North Carolina Five-Year Aggregated Median Barrier Crash Frequency Models

172 159 Table Simulation Results for North Carolina Five-Year Aggregated Median Barrier Crash Frequency Poisson NB NB with route indicator SGLM (exponential) SGLM (Spherical) Root Mean Squared Error Pred. Interval Coverage a,b (%) Avg. Pred. Interval Width Median Pred. Interval Width Note: a: Prediction Interval Coverage = number of sites whose observed crash counts fall in prediction interval / total number of sites; b: Prediction interval width = 90%th percentile of 1000 simulated predicted crash counts from an estimated model for a site - 10%th percentile of the 1000 simulated predicted crash counts. From Figure 5.14, the predictions from the three classical count models are similar and predictions from the two SGLMs are similar. It seems that when the observed crash frequencies were very low or very high, simulated crash frequencies from the two SGLMs were closer to the observed values than the classical count models. In Table 5.10, the root mean squared errors of the two SGLMs are slightly lower than those of the three classical count models. The prediction interval coverage of the Poisson model is the lowest and the prediction interval coverage of the SGLM with exponential semivariogram model is the highest. The mean and median prediction interval widths of the Poisson model are the smallest among the five models, and the mean and median prediction interval widths of the two SGLMs are smaller than those of the two NB models, which is also shown in Figures A.21 to A.25. In summary, the SGLMs could predict crash frequencies better and more accurately than the Poisson and NB models, except several outliers where the simulated predicted crash frequencies from SGLMs differed considerably from the observed crash frequencies. These outliers from the SGLMs predictions (sites #26 and 45 in Figure 5.1) were further investigated and it was found that crash frequencies on these sites were

173 160 different from most of their neighbors. Similar findings were described in sections and Site #26 had 8 median barrier crashes in the five-year analysis period, but its closest neighbors in the analysis dataset had an average of 15 median barrier crashes during the same period, which was much higher than site #26. Site #45 had 22 median barrier crashes in the five-year analysis period, but its closest neighbors in the analysis dataset had an average of 4 median barrier crashes during the same period. The simulated predicted median barrier crash frequencies from the estimated two SGLMs for these outlier sites are closer to most of their neighbors crash frequencies. This finding indicates that predictions from SGLMs borrow information from their neighbors. When the crash frequencies on adjacent road segments are similar to each other, predictions from the SGLMs are improved over the classical count models. When the crash frequency on a road segment is very different from crash frequencies on most of its neighboring road segments, predictions from SGLMs are less accurate than the classical count models. Again, this characteristic of SGLMs could be utilized to detect outliers of road sites with crash frequencies significantly higher or lower than their neighbors North Carolina Yearly Median Barrier Crash Frequency Models and Model Assessment North Carolina Yearly Median Barrier Crash Frequency Models This section presents the estimated models for North Carolina yearly median barrier crash frequencies. Three specifications of crash frequency models were used for the yearly median barrier crash frequency modeling, NB models, random-effects NB

174 161 models and SGLMs. For NB and random-effects NB model specifications, three models were estimated in each model specification, one without year or route indicators as the base model, one with year indicators, and one with both year and route indicators. The overdispersion parameters in estimated NB models were statistically significant (pvalues<0.001). A Likelihood ratio test was conducted for the each of the three randomeffects NB models by comparing the panel estimator with the pooled NB estimator, and the test results showed p-values less than This indicates that the road segment random effects were statistically significant at the 95% confidence level. Empirical semivariograms were plotted as presented in Figure 5.15 from the Poisson base model residuals -- a regular semivariogram and a robust one to account for outliers in the data. From the two empirical semivariograms, it appears that there was spatial dependency in the residuals, since the semivariograms increased as the spatial distance increased, and approached constant values when the spatial distance approached certain values. From the comparison between the empirical semivariograms from the North Carolina five-year aggregated median barrier crash frequency model (Figure 5.13) and the North Carolina yearly median barrier crash frequency model (Figure 5.15), it was found that the semivariograms from the yearly crash frequency model were much smoother than from the five-year aggregated crash frequency model. As discussed in section , this was expected because in the yearly crash frequency dataset, there were replicates of the spatial structure over years and more information were available for spatial dependence estimation. Two SGLMs were estimated with exponential and spherical semivariogram models, respectively. In the estimated SGLMs, year indicators were included and a

175 constant spatial dependence structure over years was assumed. In the two estimated 162 SGLMs, the estimates of nugget, partial sill and range showed the significant presence of spatial dependency. In the estimated SGLM with the exponential semivariogram model, the estimated partial sill was and the estimated nugget was This implies that spatial dependency accounted for 71 percent (1.020/( )) of the extra-poisson variance, while the non-spatial random effect accounted for 29 percent (0.426/( )) of the extra-poisson variance. In the estimated SGLM with the spherial semivariogram model, the estimated partial sill was and the estimated nugget was This implies that spatial dependency accounted for 64 percent (0.858/( )) of the extra-poisson variance, while the non-spatial random effect accounted for 36 percent (0. 489/( )) of the extra-poisson variance. The estimated ranges in the two SGLMs were similar to each other, 81 miles in the exponential semivariogram model and 71 miles in the spherical semivariogram model. The estimated ranges indicate that the unobserved spatially dependent effects were present within a distance of about 80 miles or less in the estimated North Carolina yearly median barrier crash frequency models. A possible reason for the large estimated range parameters is that some spatial effects which were present in large areas were missing from the models, such as weather. A total of eight models were estimated for the North Carolina yearly median barrier crash frequencies, as shwon in Table The same sets of variables were included in the three model specifications. The variables included in these yearly crash frequency models are the same as those variables in the five-year aggregated crash frequency models shown in Table 5.9, except the year indicators. In the estimated

176 163 models, the natural logarithm of road segment length (miles) was treated as an offset, since its estimated coefficients in all the models were close to 1 and the 95% confidence intervals contained 1.

177 164 Table Models for North Carolina Yearly Median Barrier Crash Frequency Variable 2 NB RENB NB with RENB with NB with RENB with SGLM SGLM year info. year info. year/route info. year/route info. (Exponential) (Spherical) Constant (0.795) (0.952) (0.763) (0.970) (1.094) (1.307) (1.729) (1.641) Median Barrier offset (ft) (0.026) (0.032) (0.025) (0.033) (0.027) (0.034) (0.036) (0.035) Leftcurve indicator (0.133) (0.165) (0.127) (0.168) (0.126) (0.161) (0.137) (0.137) Rightcurve indicator (0.120) (0.151) (0.114) (0.154) (0.115) (0.149) (0.126) (0.125) Natural log adt (0.079) (0.095) (0.076) (0.096) (0.105) (0.126) (0.166) (0.158) Left shoulder rumble strips indicator (0.067) (0.083) (0.064) (0.085) (0.069) (0.087) (0.130) (0.122) Median foreslope 7H:1V or steeper indicator (0.425) (0.522) (0.412) (0.536) (0.437) (0.551) (0.491) (0.478) Median foreslope between 7H:1V and H:1V indicator (0.280) (0.361) (0.269) (0.365) (0.268) (0.349) (0.287) (0.285) Median barrier offset (ft) if Median foreslope is 7H:1V or steeper, 0 otherwise (0.084) (0.103) (0.082) (0.105) (0.087) (0.109) (0.096) (0.094) Median barrier offset (ft) if Median foreslope is between 7H:1V and 9H:1V, 0 otherwise (0.054) (0.070) (0.052) (0.070) (0.052) (0.068) (0.057) (0.056) Speed limit 70mph or higher indicator (0.088) (0.109) (0.085) (0.111) (0.119) (0.147) (0.213) (0.204) Speed limit 65 mph indicator (0.088) (0.107) (0.084) (0.109) (0.106) (0.133) (0.173) (0.166) Year 2000 indicator (0.082) (0.079) (0.081) (0.078) (0.280) (0.266) Year 2001 indicator (0.076) (0.066) (0.075) (0.067) (0.276) (0.263) Year 2002 indicator (0.074) (0.062) (0.073) (0.062) (0.276) (0.263) Route 26 indicator (0.187) (0.231) Route 40 indicator (0.146) (0.182)

178 165 Table Models for North Carolina Yearly Median Barrier Crash Frequency (continued) Variable 2 NB RENB NB with RENB with NB with RENB with SGLM SGLM year info. year info. year/route info. year/route info. (Exponential) (Spherical) Route 64 indicator (0.204) (0.251) Route 74 indicator (0.240) (0.308) Route 85 indicator (0.165) (0.208) Route 95 indicator (0.151) (0.188) Route 321 indicator (0.223) (0.292) Natural log road length offset Overdispersion parameter (0.056) (0.049) (0.047) r (1.554) (1.687) (2.128) s (1.194) (0.843) (1.188) Nugget (0.017) (0.014) Partial sill (0.024) (0.022) Range (mile) (5.42) (3.50) Log likelihood for constant-only model Log likelihood at convergence AIC BIC NOTES: 1.standard errors are given in parentheses; 2.an indicator variable is 1 if the condition holds and 0 otherwise;3.no. of observations: 1790; 4. Likelihood ratio test of alpha=0: χ 2 (1) = , Prob>= χ 2 = 0.000; 5. Likelihood ratio test of alpha=0: χ 2 (1) = , Prob>= χ 2 = 0.000; 6. Likelihood ratio test of alpha=0: χ 2 (1) = , Prob>= χ 2 = 0.000; 7. Likelihood-ratio test vs. pooled: χ 2 (1) = 81.79, Prob>= χ 2 = 0.000; 8. Likelihood-ratio test vs. pooled: χ 2 (1) = , Prob>= χ 2 = 0.000; 9. Likelihood-ratio test vs. pooled: χ 2 (1) = 86.23, Prob>= χ 2 =

179 166 Figure Regular and Robust Empirical Semivariograms from Residuals of A Poisson Base Model for North Carolina Yearly Median Barrier Crash Frequency From the models presented in Table 5.11, the elasticities of the expected yearly median barrier crash frequency in North Carolina with respect to years 2000, 2001 and 2002 are -0.62, and respectively. This indicates that median barrier crash frequency was 62% lower in year 2000, 45% lower in year 2001 and 39% lower in year 2002, when compared to the years 2003 and It is not clear what exactly led to this trend and further exploration is recommended. For the other explanatory variables, similar explanations are found in section From comparison of the estimated models presented in Table 5.11, it was found that the estimated NB and random-effects NB models with the same set of explanatory

180 167 variables had relatively similar estimated coefficients. The estimated coefficients were very similar between the two SGLMs. Also, it was found that the coefficient estimates remained relatively stable across the NB models, random-effects NB models and SGLMs, except the coefficient estimates of the left shoulder rumble strips indicator. The estimated coefficients of the left shoulder rumble strip indicator in the estimated NB and random effects NB models were mostly positive and not significantly different from 0, while in the two SGLMs they were negative and significantly different from 0 at the 70% confidence level. This supports expected outcome E3 that the presence of left shoulder rumble strips reduces median-related crashes. A possible reason for the difference is that the SGLMs captured the spatial structure in an appropriate way and led to more reasonable parameter estimates. The standard errors were consistently lower in the NB model and higher in the SGLMs and random-effects NB models. The random-effects NB models and SGLMs have more noise due to their distribution assumptions on the random error terms. From the comparison of AIC and BIC values calculated for the estimated NB and random-effects NB models presented in Table 5.11, it was found that adding the year and route indicators improved the overall fit. Also, the random-effects NB specification improved the overall model fit over the pooled NB specification. The AIC and BIC values of the random-effects NB model with year and route indicators were the lowest among the six estimated NB and random-effects NB models shown in Table 5.11 and was thus considered the best fit among the six models. The assessment of SGLMs is discussed in the next section.

181 North Carolina Yearly Median Barrier Crash Frequency Model Assessment 168 In order to compare the fit of the estimated SGLM models and the classical count models and the panel count models for North Carolina yearly median barrier crash frequencies, simulations of predicted crash frequencies were performed in R (R Development Core Team, 2010) based on the estimated NB model with year and route indicators, random-effects NB model with year and route indicators and two SGLMs on the randomly selected 45 road segments over five years, which contains about 10 percent of the total sample as described in Chapter simulations of predicted crash frequencies were generated for each road segment in each year based on each of the four estimated models. The mean value of the 1000 simulated crash frequencies was calculated for each road segment in each year, as well as the 10th percentile and 90th percentile of the simulated crash frequencies to measure the predictive capabilities of these models. The model with mean simulated crash frequencies that are the closest to the observed crash frequencies, the highest coverage of the prediction intervals (the 10th percentile and 90th percentile of the simulated crash frequency) on the observed crash frequencies, and the narrowest prediction intervals was considered the model with the best predictive capabilities. Figures 5.16 to 5.20 show by year the yearly observed crash frequencies on the randomly selected 45 road segments for simulation, as well as the mean values of simulated crash frequencies based on the four estimated models shown in Table The prediction intervals of the four estimated models are shown by year in Figures A.26 to A.45 in Appendix A. Table 5.12 presents the root mean squared errors, prediction interval

182 169 coverage on the observed crash frequencies, mean prediction interval width and median prediction interval width of the four estimated models. 7 6 Crash Count North Carolina Median Barrier Crash Site No. Year 2000 Observed simulated_nb_yearroute simulated_renb_yearroute simulated_sglm(exponential) simulated_sglm(spherical) Figure Simulation Results based on North Carolina Median Barrier Crash Frequency Models, Year Crash Count North Carolina Median Barrier Crash Site No. Year 2001 Observed simulated_nb_yearroute simulated_renb_yearroute simulated_sglm(exponential) simulated_sglm(spherical) Figure Simulation Results based on North Carolina Yearly Median Barrier Crash Frequency Models, Year 2001

183 Crash Count North Carolina Median Barrier Crash Site No. Year Observed simulated_nb_yearroute simulated_renb_yearroute simulated_sglm(exponential) simulated_sglm(spherical) Figure Simulation Results based on North Carolina Yearly Median Barrier Crash Frequency Models, Year 2002 Crash Count North Carolina Median Barrier Crash Site No. Year 2003 Observed simulated_nb_yearroute simulated_renb_yearroute simulated_sglm(exponential) simulated_sglm(spherical) Figure Simulation Results based on North Carolina Yearly Median Barrier Crash Frequency Models, Year 2003

184 Crash Count North Carolina Median Barrier Crash Site No. Year Observed simulated_nb_yearroute simulated_renb_yearroute simulated_sglm(exponential) simulated_sglm(spherical) Figure Simulation Results based on North Carolina Yearly Median Barrier Crash Frequency Models, Year 2004 Table Simulation Results for North Carolina Yearly Median Barrier Crash Frequency NB with year/route indicator RENB with year/route indicator SGLM (Exponential) SGLM (Spherical) Root Mean Squared Error Pred. Interval Coverage a,b (%) Avg. Pred. Interval Width b Median Pred. Interval Width Note: a: Prediction Interval Coverage = number of sites whose observed crash counts fall in prediction interval / total number of sites; b: Prediction interval width = 90%th percentile of 1000 simulated predicted crash counts from an estimated model for a site - 10%th percentile of the 1000 simulated predicted crash counts From Figures 5.16 to 5.20, the random-effects NB model significantly overestimated crash frequencies, which was not unexpected because the estimated constant in the random-effects NB model was significantly higher than the constant in the

185 172 NB model. Also, it was found that when the observed crash frequencies were very low or very high, simulated crash frequencies from the two SGLMs were closer to the observed values than the NB model. Similar trends were found in section In Table 5.12, the root mean squared errors of the NB and two SGLMs models were not significantly different, but the root mean squared error of the random-effects NB model was significantly higher than the others. The prediction interval coverage of the NB model was slightly higher than the two SGLMs. The mean and median prediction interval widths of the two SGLMs were significantly lower than the NB model, which is also shown in Figures A.26 to A.45. In summary, using the yearly North Carolina median barrier crash frequencies on the road segments for analysis, the SGLMs could predict the yearly crash frequencies more accurately than the NB and random-effects NB models. Outliers where the simulated predicted crash frequencies from SGLMs differed significantly from the observed crash frequencies were further investigated and it was found that crash frequencies on these outlier sites were different from most of their neighbors. For example, site #20 in year 2003 had 1 median barrier crash, but its closest neighbors in the analysis dataset had 4-5 median barrier crashes during the same period. Site #32 in year 2004 had 3 median barrier crashes, but its closest neighbors in the analysis dataset had an average of 6 median barrier crashes during the same period. The simulated predicted median barrier crash frequencies from the estimated two SGLMs for these outlier sites are closer to most of their neighbors crash frequencies. These findings are similar to those as discussed in sections , and , which indicate that predictions from SGLMs borrow information from neighbors.

186 5. 5 Transferability of Pennsylvania and North Carolina Median Barrier Crash Frequency Models 173 This section is divided into two parts. The first part presents transferability test results for five-year aggregated median barrier crash frequency models in Pennsylvania and North Carolina. The second part presents the transferability test results for yearly median barrier crash frequency models in Pennsylvania and North Carolina Transferability of Five-Year Aggregated Median Barrier Crash Frequency Models The likelihood ratio test was applied to test the transferability of the five-year aggregated median barrier crash frequency models between the two states. Models with different explanatory variables were estimated for each state as shown in Tables 5.5 and 5.9. In order to perform the test, variables common to each dataset should be used in each model. Then, three models were estimated to perform the transferability test with the same explanatory variables: a NB model for Pennsylvania, a NB model for North Carolina and a NB model estimated by combining the Pennsylvania and North Carolina five-year aggregated median barrier crash data. The three estimated models are presented in Table Based on the log likelihoods at convergence of the estimated models, a chisquare statistic was calculated to be with 8 degrees of freedom (number of estimated parameters). The p-value was less than and thus the hypothesis that the five-year aggregated median barrier crash frequency models in Pennsylvania and North Carolina were transferable was rejected. This finding was not unexpected because Pennsylvania and North Carolina have different median barrier cross-section

187 characteristics. The median barrier types most commonly installed in Pennsylvania 174 include strong-post w-beam guardrail and concrete barriers, while cable guardrail is the primary median barrier type used in North Carolina. Cable median barrier is a flexible system with a large dynamic deflection and thus is installed in medians that are wide when compared to medians that contain semi-rigid or rigid barriers such as w-beam guardrail and concrete median barriers. These differences could lead to different characteristics of median barrier crashes in the two states. As discussed in section 4.1.3, when the hypothesis that models are transferable is rejected, an asymptotic t-test could be performed to test which variables are responsible for the rejection. The t-test results of the five-year aggregated median barrier crash frequency models are also presented in Table It seems that the left shoulder rumble strip indicator was responsible for the rejection of transferability of the five-year aggregated median barrier crash frequency models.

188 Table Five-Year Aggregated Median Barrier Crash Frequency Models for Transferability Test Variable Pennsylvania Carolina 4 Carolina 5 t-statistic 2 3 North Pennsylvania & North Median barrier offset (ft) (0.029) (0.028) (0.010) Left curve indicator (0.260) (0.154) (0.128) Right curve indicator (0.254) (0.160) (0.130) Shoulder rumble strip indicator (0.220) (0.085) (0.078) Natural log adt (0.257) (0.086) (0.075) Speed limit 60 mph or lower indicator (0.315) (0.099) (0.097) Constant (2.388) (0.881) (0.792) Natural log segment length (mile) offset -- Alpha (overdispersion parameter) (0.291) (0.038) (0.040) Log likelihood Notes:1. standard errors are given in parentheses; 2.an indicator variable is 1 if the condition holds and 0 otherwise; 3. No. of observations: 456; 4. No. of observations: 403; 5. No. of observations: 859; 6. Likelihood ratio test of alpha=0: χ 2 (1) = 12.09, Prob>= χ 2 = 0.000; 7. Likelihood ratio test of alpha=0: χ 2 (1) = , Prob>= χ 2 = 0.000; 8. Likelihood ratio test of alpha=0: χ 2 (1) = , Prob>= χ 2 = Transferability of Yearly Median Barrier Crash Frequency Models The likelihood ratio test was applied to test the transferability of the yearly median barrier crash frequency models between the two states. Models with different explanatory variables were estimated for each state as shown in Tables 5.7 and In order to perform the test, three NB models were estimated to perform the transferability test with the same explanatory variables: a NB model for Pennsylvania, a NB model for North Carolina, and a NB model estimated by combining the Pennsylvania and North Carolina yearly median barrier crash data. The three estimated NB models were presented in Table Based on the log likelihoods at convergence of the estimated

189 models, a chi-square statistic was calculated to be with 8 degrees of freedom 176 (number of estimated parameters). The p-value was less than and thus the hypothesis that the yearly median barrier crash frequency models in Pennsylvania and North Carolina were transferable was rejected. The t-test statistics of the yearly median barrier crash frequency models are also presented in Table 5.14, to test which variables were responsible for the rejection. It is clear that the left shoulder rumble strip indicator was responsible for the rejection of transferability of the yearly median barrier crash frequency models. Similar explanations for these findings can be found in section Table Yearly Median Barrier Crash Frequency NB Models for Transferability Test Variable Pennsylvania Carolina 4 Carolina 5 t-statistic 2 3 North Pennsylvania & North Median barrier offset (ft) (0.028) (0.020) (0.009) Left curve indicator (0.241) (0.116) (0.102) Right curve indicator (0.236) (0.118) (0.102) Left shoulder rumble strip indicator (0.203) (0.064) (0.059) Natural log adt (0.230) (0.064) (0.057) Speed limit 60 mph or lower indicator (0.296) (0.076) (0.074) Constant (2.147) (0.661) (0.608) Natural log segment length (mile) offset -- Alpha (overdispersion parameter) (0.673) (0.055) (0.056) Log likelihood Notes:1. standard errors are given in parentheses; 2.an indicator variable is 1 if the condition holds and 0 otherwise; 3. No. of observations: 2186; 4. No. of observations: 2015; 5. No. of observations: 4201; 6. Likelihood ratio test of alpha=0: χ 2 (1) = 10.93, Prob>= χ 2 = 0.000; 7. Likelihood ratio test of alpha=0: χ 2 (1) = , Prob>= χ 2 = 0.000; 8. Likelihood ratio test of alpha=0: χ 2 1) = , Prob>= χ 2 = In summary, the transferability test results show that the estimated models of the median barrier crash frequency models were not transferable between the two states. The

190 t-test results show that the rejection of transferability was caused by the inconsistent 177 effects of the left shoulder rumble strip indicator on the median barrier crash frequencies between the two states.

191 Chapter 6 Crash Severity Analysis Results This chapter presents the estimated cross-median, rollover and median barrier crash severity models for Pennsylvania and North Carolina. A positive coefficient estimate in an injury category indicates an increased probability of falling into the subject category relative to the baseline category, while a negative coefficient estimate in an injury category indicates a decreased probability of falling into the subject category relative to the baseline category. In most cases, the property damage only (pdo) or no injury category was used as the baseline. 6.1 Pennsylvania Cross-Median Crash Severity Model This section presents the estimated crash severity model for Pennsylvania crossmedian crashes. A binary logit model was estimated instead of a multinomial logit model, since the sample size of 76 cross-median crashes was small. A binary variable Severe was created in a way that its value was 1 if the crash severity outcome was fatal or major injury, and 0 if the crash severity outcome was moderate/ minor/pdo/no injury. Severe was the dependent variable used in model estimation. The modeling results are shown in Table 6.1. Based on the modeling results, odds ratios were calculated and are shown in Table 6.1 for each explanatory variable as exp ( β i ), where β i is the estimated coefficient for variable x i. An odds ratio greater than 1 indicates increased odds of a fatal or major

192 179 injury versus a less severe injury in a cross-median crash, resulting from a change in the indicator variable from 0 to 1; while an odds ratio smaller than 1 indicates decreased odds of a fatal or major injury versus a less severe injury in a cross-median crash, resulting in a change in the indicator variable from 0 to 1. Table 6.1. Binary Logit Model of Pennsylvania Cross-Median Crash Severity Std. Err. z P> z Odds ratio 4,5 Fatal or major injury cross-median crash Coef. Unbelted indicator 1: if the driver is unbelted; 0: otherwise Median width and backslope interaction #1 1: if Median width is less than 60 ft and backslope is flatter than10h:1v;0: otherwise Median width and backslope interaction #2 1: if median width is wider than 60 ft and backslope is flatter than 10H:1V; 0: otherwise Median foreslope flatter than 10H:1V indicator 1: if foreslope is flatter than 10H:1V; 0: otherwise Driver age indicator 1: if driver is older than 55 years; 0: otherwise (1.70, 33.57) 6.13 (1.01, 37.31) 4.55 (0.75, 27.75) 2.58 (0.48, 13.90) 0.24 (0.05, 1.14) Constant Notes: 1. The estimated coefficients are specific to the fatal or major injury category, and are relative to the moderate, minor, pdo or no injury category; 2. Number of observations: 76; 3. Log likelihood = ; Likelihood Ratio χ 2 (5) = 21.57; Prob. > χ 2 =0.0006; 4. Odds of a fatal or major injury versus a less severe injury in a cross-median crash; 5. 95% confidence intervals are given in parentheses. A confidence interval containing 1 indicates that the change in the explanatory variables does not statistically significantly affect the odds of a fatal or major injury versus a less severe injury in a cross-median crash, at 95% confidence level. Based on the results shown in Table 6.1, a positive coefficient estimate and an odds ratio significantly greater than 1 for the seatbelt usage indicator suggest that not using a seat belt significantly increases the probability of fatal or major injury outcomes relative to lower levels of injury in a cross-median crash. This finding supports the expected outcome E16 in chapter 1 and is consistent with research performed by Islam and Mannering (2006), who found that not using seatbelts increased the probability of a fatality in a single-vehicle crash by over 200% and increased the probability of an injury

193 by over 100%. The negative coefficient estimate and an odds ratio less than 1 (not 180 statistically different from 1 at 95% confidence level) for driver age indicates that drivers older than 55 years experience a lower probability of fatal or major injury outcomes relative to lower levels of injury in cross-median crashes when compared to drivers younger than 55 years. This finding does not support the expected outcome E15 in chapter 1. A possible reason for this is that older drivers tend to travel at slower speeds and are thus at a lower risk of suffering from severe injuries or fatalities in cross-median crashes than younger drivers. The positive coefficient estimates and odds ratios significantly greater than 1 for the two median width and median backslope flatter than 10H:1V interaction variables indicate that a median backslope flatter than 10H:1V increases the probability of fatal or major injury outcomes relative to lower levels of injury in a cross-median crash. The magnitude of the coefficient estimate and the odd ratio of the interaction between the median width less than 60 ft indicator and the median backslopes flatter than 10H:1V indicator is larger than those of the interaction between the median width greater than 60 ft indicator and the median backslopes flatter than 10H:1V indicator. This indicates that, given median backslopes flatter than 10H:1V, a narrower median increases the probability of fatal or major injury outcomes relative to lower levels of injury in a crossmedian crash. The positive coefficient estimate and an odds ratio greater than 1 (not statistically different from 1 at 95% confidence level) for the median foreslope indicator indicate that median foreslopes flatter than 10H:1V increase the probabilities of fatal or major injury outcomes relative to lower levels of injury in a cross-median crash, when compared to

194 steeper median foreslopes. All of these findings have signs in the expected direction 181 based on expected outcome E24 and E25 in chapter 1 that flatter and narrower medians lead to more severe injury outcomes in cross-median crashes. This is likely because vehicles enter and cross narrow medians with flat cross-slopes at higher speeds when compared to wider medians or median with steeper cross-slopes. Efforts to include main effects for average daily traffic (ADT), heavy vehicle involvement indicator, driver gender, road alignment, road surface condition and driver impairment were made, but none were statistically significant at the 80% confidence level. 6.2 Pennsylvania Rollover Crash Severity Model This section presents the estimated crash severity model for Pennsylvania rollover crashes. A multinomial logit model was estimated and is presented in Table 6.2. Based on the modeling results, odds ratios were calculated and are shown in Tale 6.3 for each explanatory variable and each injury category as exp ( β ij ), where β ij is the estimated coefficient for variable x i in injury category j. An odds ratio greater than 1 indicates increased odds of a given injury level versus PDO or no injury in a median rollover crash, resulting in achange in the indicator variable from 0 to 1. An odds ratio smaller than 1 indicates decreased odds of a given injury level versus PDO or no injury in a median rollover crash, resulting in a change in the indicator variable from 0 to 1. A multilevel categorical variable Severity was created in a way that its value was 0 if the crash severity outcome was pdo or no injury, 1 if the crash severity outcome was

195 minor injury, 2 if the crash severity outcome was moderate injury, and 3 if the crash 182 severity outcome was fatal or major injury. Severity was used as the dependent variable for the multinomial logit model estimation. The assumption of independence of irrelevant alternatives (IIA) in the multinomial logit model was tested and the test results (p-values > 0.5) showed that the shared unobserved effects across crash severity levels were not statistically significant. Table 6.2. Multinomial Logit Model of Pennsylvania Rollover Crash Severity Variable Major Injury Minor Injury Moderate Injury or Fatality Coef. z Coef. z Coef. z Constant Median foreslope steeper than 7H:1V indicator 1: if foreslope is steeper than 7H:1V; : otherwise Median backslope steeper than 7H:1V indicator 1: if backslope is steeper than 7H:1V; 0: otherwise Leftcurved road segment indicator 1: if left curve is present in crash segment; 0: otherwise Unbelted indicator 1: if driver is not belted; 0: otherwise Driver age indicator 1: if drive is older than 55 years; 0: otherwise Male driver indicator 1: if driver is male; 0:otherwise Road with water/ice/snow/slush indicator 1: if there is water/ice/snow/slush present on roadway; 0: otherwise Median width less than 70 ft indicator 1: if median width is less than 70 ft; 0: otherwise Median width between 70 ft and 100 ft indicator 1: if median width is between 70 ft and 100 ft; 0: otherwise Notes: 1. PDO or no injury is the base severity level; 2. Blank cells are treated as the baseline severity category specific to the variable; 3. Number of observations: 231; 4. Log likelihood = ; Likelihood Ratio χ 2 (20) = 77.26; Prob > χ 2 =0.0000; 5. The coefficients for the driver gender indicator were restricted to be equal across moderate and major injury/fatality levels.

196 183 Table 6.3. Multinomial Logit Model of Pennsylvania Rollover Crash Severity Odds Ratio Variable Median foreslope steeper than 7H:1V indicator Median backslope steeper than 7H:1V indicator Leftcurved road segment indicator Unbelted indicator Minor Injury Moderate Injury Major Injury or Fatality Odds ratio 1,3 Odds ratio 1,3 Odds ratio 1, (0.81, 3.92) 2.60 (0.90, 7.47) 2.38 (0.84, 6.69) 5.32 (1.67, 17.02) 1.60 (0.60, 4.24) 0.52 (0.23, 1.19) 0.24 (0.09, 0.63) 2.30 (0.70, 7.49) 1.69 (0.73, 3.89) 5.85 (0.99, 34.50) 2.33 (0.54, 10.11) 2.35 (0.79, 7.01) (11.12, ) 2.51 (0.68, 9.31) 0.52 (0.23, 1.19) 0.15 (0.04, 0.60) 5.05 (1.04, 24.53) 3.25 (0.88, 11.97) Driver age indicator 0.53 Male driver indicator (0.26, 1.09) 0.54 Road with water/ice/snow/slush indicator (0.28, 1.04) 1.65 Median width less than 70 ft indicator 2 (0.69, 3.96) Median width between 70 ft and 100 ft indicator 2 Notes: 1. Odds of a given injury versus a PDO or no injury in a rollover crash; 2. Baseline category: median wider than 100 ft; 3. 95% confidence intervals are given in parentheses. A confidence interval containing 1 indicates that the change in the explanatory variables does not statistically significantly affect the odds of a given injury versus a PDO or no injury in a rollover crash, at 95% confidence level. When controlling for other factors, the positive coefficient estimates and odds ratios greater than 1 for the median foreslope steeper than 7H:1V indicator in the minor injury, moderate injury and major injury/fatality categories indicate that median foreslopes steeper than 7H:1V increase the probability of higher severity outcomes relative to PDO or no-evident injury in a median rollover crash, when compared to flatter median foreslopes. The odds ratio of the median foreslope steeper than 7H:1V indicator is smaller for the minor injury category than for the moderate injury category, and smaller for the moderate injury category than for the major injury/fatality category. This indicates that in the presence of median foreslopes steeper than 7H:1V, the probability of a major injury or fatality outcome in a rollover crash is higher than the probability of a

197 184 moderate injury outcome, and the probability of a moderate injury outcome is higher than the probability of a minor injury outcome. The positive coefficient estimate odds ratio greater than 1 (not statistically different from 1 at 95% confidence level) for the median backslope steeper than 7H:1V indicator in the major injury/fatality category indicate that median backslopes steeper than 7H:1V increase the probability of major injury/fatality outcomes relative to other lower levels of severity in a median rollover crash, when compared to flatter median backslopes. These findings were expected (E26) that steeper median cross-slopes lead to higher probabilities of more severe injury outcomes in rollover crashes. The positive coefficient estimate and an odds ratio greater than 1 (not statistically different from 1 at 95% confidence level) for the left curved road segment indicator indicate that the presence of a left curve in a road segment increases the probability of the major injury/fatality outcome relative to lower levels of severity in a rollover crash, when compared to right-curved or tangent road segments. This finding supports the expected outcome E12, and was expected because vehicles leaving the roadway at locations curved to the left enter the median at relatively high angles. In doing so, a driver s natural reaction may be to attempt to recover (i.e., steering back onto the through travel lanes), which leads to significant tire yaw on an earth-divided median, which may contribute to soil-tripped rollovers. Alternatively, high departure angles may lead to high impact angles with median backslopes, causing vehicles to become instable and rollover. The positive coefficient estimates and odds ratios significantly greater than 1 for the unbelted indicator in the minor injury, moderate injury and major injury/fatality outcome categories indicate that not using a seat belt increases the probability of higher

198 levels of injury outcome relative to pdo or no-evident injury outcomes in a median 185 rollover crash. The odds ratio of the unbelted indicator is smaller for the minor injury category than for the moderate injury category, and smaller for the moderate injury category than for the major injury/fatality category. This indicates that not using a seatbelt leads to significantly higher probabilities of major injury or fatality outcome than moderate injury outcome, and higher probabilities of moderate injury outcome than minor injury outcome. These findings are consistent with the earlier stated expected outcome E16 that driver seatbelt usage leads to lowers levels of injury in a median rollover crash. These findings are again consistent with research performed by Islam and Mannering (2006), as described earlier. The positive coefficient estimates and odds ratios greater than 1 (not statistically different from 1 at 95% confidence level) for the driver age indicator show that drivers older than 55 years experience higher probabilities of moderate injury or major injury/fatality outcomes relative to lower severity outcomes in a median rollover crash, when compared to drivers younger than 55 years. The odds ratio of the driver age indicator is smaller for the moderate injury category than for the major injury/fatality category. This finding is consistent with the expected outcome E15 in chapter 1 and a study performed by Islam and Mannering (2006). It was found that in crashes involving overturned vehicles, older drivers (65 and above) had much higher probabilities of fatalities than younger drivers (Islam and Mannering, 2006). The difference in reaction times, body structures and health conditions between older and younger drivers could possibly lead to the increased probabilities of higher injury outcomes experienced by older drivers.

199 186 The negative coefficient estimates and odds ratios smaller than 1 (not statistically different from 1 at 95% confidence level) for the male driver indicator in minor injury, moderate injury and major injury/fatality categories indicate that male drivers have lower probabilities of higher injury outcomes relative to PDO or no-evident injury in a median rollover crash, when compared to females drivers. This finding supports the expected outcome E15. The coefficients for the male driver indicator were similar in magnitude across moderate and major injury/fatality levels. In the study performed by Islam and Mannering (2006), it was found that in crashes involving overturned vehicles, older female drivers had much higher probabilities of fatalities than other driver population (Islam and Mannering, 2006). The difference in reaction times, body weights, body structures between males and females could possible lead to the increased probabilities of higher injury outcomes experienced by female drivers. The negative coefficient estimates and odds ratios smaller than 1 for the road with water/ice/snow/slush indicator in minor injury, moderate injury and major injury/fatality categories indicate that the presence of water/ice/snow/slush leads to lower probabilities of higher severity outcomes relative to PDO or no-evident injury in a rollover crash. The odds ratio of the road with water/ice/snow/slush indicator is closer to 1 for the minor injury category than for the moderate injury category, and closer to 1 for the moderate injury category than for the major injury/fatality category. These findings support the expected outcome E18 stated in chapter 1. A possible reason for this finding is that drivers travel more cautiously in adverse weather, resulting in lower vehicle operating speeds. Lower speeds result in lower impact severity outcomes.

200 187 The positive coefficient estimates odds ratios greater than 1 for the two median width indicators in the minor injury, moderate injury and major injury/fatality categories indicate that medians narrower than 100 ft increase the probability of higher injury outcomes relative to PDO or no-evident injury in a rollover crash, when compared to wider medians. With respect to the medians narrower than 70 ft wide indicator, its odds ratio is the smallest for the minor injury outcome and the largest for the major injury/fatality outcome. For the medians between 70 ft and 100 ft wide indicator, its odds ratio is larger for the major injury/fatality category than for the moderate injury category. For the same severity category, the magnitude of the odds ratio of the median width between 70 ft and 100 ft indicator is smaller than that of the median width narrower than 70 ft indicator. The median width categories were decided according to the median width distribution in the analysis data set. These findings support the expected outcome E26 in chapter 1 and were expected because narrower medians provide less recovery area for errant vehicles than wider medians. Efforts to include main effects for ADT, heavy vehicle involvement indicator and driver impairment were made, but none were statistically significant at the 80thpercentile confidence level. Additionally, several interaction terms were included in the model (e.g., median foreslope/backslope and median width, and road alignment and road surface condition indicator); however, none were statistically significant in the model.

201 6.3 Pennsylvania Median Barrier Crash Severity Model 188 This section presents the estimated crash severity model for Pennsylvania median barrier crash severity. A multinomial logit model was estimated as presented in Tables 6.4. Based on the modeling results, odds ratios were calculated and are shown in Tale 6.5 for each explanatory variable and each injury category. An odds ratio greater than 1 indicates increased odds of a given injury level versus PDO or no injury in a median barrier crash, resulting in a change in the indicator variable from 0 to 1. An odds ratio smaller than 1 indicates decreased odds of a given injury level versus PDO or no injury in a median barrier crash, resulting in a change in the indicator variable from 0 to 1. A multilevel categorical variable Severity was created as the dependent variable for model estimation using the same categories as those defined for the rollover severity model described in section 6.2. The assumption of independence of irrelevant alternatives (IIA) was tested and the test results (p-values > 0.6) showed that the shared unobserved effects across crash severity levels were not statistically significant.

202 189 Table 6.4. Multinomial Logit Model of Pennsylvania Median Barrier Crash Severity Minor Injury Moderate Injury Major Injury or Fatality Variable Coef. z Coef. z Coef. z Constant Ramp indicator 1: if ramp is present in crash segment; 0: otherwise Left shoulder rumble strip indicator 1: if left shoulder rumble strips are present I crash segment; 0: otherwise Concrete barrier indicator 1: if median barrier is concrete; 0: otherwise Curved road segment indicator 1: if curve is present in crash segment; 0: otherwise Unbelted indicator 1: if driver is not belted; 0: otherwise Driver age indicator 1: if drive is older than 55 years; 0: otherwise Female driver indicator 1: if driver is female; 0:otherwise Speeding indicator 1: if driver was speeding; 0: otherwise Fatigue indicator 1: if driver was fatigue; 0: otherwise Road with water/ice/snow/slush 1: if there is water/ice/snow/slush present on roadway; 0: otherwise Nighttime crash indicator 1: if a crash occurred during night time; 0: otherwise Overturning vehicle indicator 1: if vehicle overturned in a crash; 0: otherwise Heavy truck indicator 1: if heavy truck involved in a crash; 0: otherwise Notes: 1. PDO or no injury is the base severity level; 2. Blank cells are treated as the baseline severity category specific to the variable; 3. Number of observations is 609; 4. Log likelihood = ; Likelihood Ratio χ 2 (25) = ; Prob > χ 2 =0.000; 5. The coefficients for the driver age indicator were restricted to be equal across minor and moderate injury levels; 6. The coefficients for the speeding indicator, the road with water/ice/snow/slush indicator, the nighttime crash indicator and the overturning vehicle indicator were restricted to be equal across moderate and major injury/fatality levels.

203 190 Table 6.5. Multinomial Logit Model of Pennsylvania Median Barrier Crash Severity Odds Ratios Ramp indicator Variable Left shoulder rumble strip indicator Concrete barrier indicator Curved road segment indicator Unbelted indicator Driver age indicator Female driver indicator Speeding indicator Fatigue indicator Road with water/ice/snow/slush Nighttime crash indicator Overturning vehicle indicator Minor Injury Moderate Injury Major Injury or Fatality Odds ratio 1,2 Odds ratio 1,2 Odds ratio 1, (0.95, 2.11) 2.29 (1.16, 4.52) 1.34 (0.81, 2.22) 1.86 (0.69, 4.99) 0.72 (0.46, 1.12) 3.12 (1.48, 6.60) 1.60 (0.88, 2.92) 7.71 (3.58, 16.61) 1.34 (0.81, 2.22) 0.58 (0.27, 1.23) 3.65 (1.18, 11.34) 2.06 (0.79, 5.38) 0.47 (0.24, 0.94) 0.56 (0.33, 0.97) 9.77 (4.33, 22.04) 2.53 (1.31, 4.91) 1.98 (0.80, 4.93) 0.55 (0.20, 1.51) 1.81 (0.59, 5.56) 2.35 (0.89, 6.19) (6.06, 47.49) 2.93 (1.05, 8.16) 0.14 (0.02, 1.10) 3.65 (1.18, 11.34) 5.28 (1.63, 17.11) 0.47 (0.24, 0.94) 0.56 (0.33, 0.97) 9.77 (4.33, 22.04) 3.58 (1.39, 9.19) Heavy truck indicator Note: 1. Odds of a given injury versus a PDO or no injury in a median barrier crash; 2. 95% confidence intervals are given in parentheses. A confidence interval containing 1 indicates that the change in the explanatory variables does not statistically significantly affect the odds of a given injury versus a PDO or no injury in a median barrier crash, at 95% confidence level. Based on the modeling results presented in Tables 6.4 and 6.5, when controlling for other factors, the positive coefficient estimate and an odds ratio greater than 1 (not statistically different from 1 at 95% confidence level) for the ramp indicator in the major injury/fatality category indicate that the presence of interchange ramps increases the probability of major injury/fatality outcomes in a median barrier crash relative to lower severity levels. This finding supports the expected outcome E20 in chapter 1.One possible reason is that the presence of ramps may increase the likelihood that drivers in

204 the through travel lanes are forced to change lanes to allow merging traffic on the 191 interchange ramp to enter the through lanes. In doing so, the drivers changing lanes may encroach into the median at high speeds and at high angles, increasing the impact severity with longitudinal median barriers. The negative coefficient estimate and an odds ratio smaller than 1 (not statistically different from 1 at 95% confidence level) for the left shoulder rumble strip indicator in the major injury/fatality category indicate that the presence of left shoulder rumble strips decreases the probability of major injury/fatality outcomes in a median barrier crash relative to lower severity levels. This finding supports the expected outcome E13 that left shoulder rumble strips may alert a drowsy or fatigued driver when encroaching into the median, which can result in braking or steering away from a longitudinal median barrier. In doing so, the impact speed or angle of impact is reduced, resulting in a less-severe outcome. The positive coefficient estimates and odds ratios greater than 1 (not statistically different from 1 at 95% confidence level) for the concrete barrier indicator in minor injury, moderate injury and major injury/fatality categories indicate that colliding with a concrete median barrier increases the probability of minor injury, moderate injury and major injury/fatality outcomes relative to pdo or no-evident injury in a median barrier crash when compared to guardrail median barriers. The odds ratio of the concrete median barrier is the smallest in the minor injury category and the largest in the major injury/fatality outcome category. These findings support the expected outcome E29 that the presence of concrete median barrier leads to higher injury outcomes than guardrail barriers in a median barrier crash. These findings were expected because a concrete

205 192 median barrier has a lower dynamic deflection than either cable or semi-rigid guardrail median barriers, resulting in increased vehicle occupant decelerations in a collision. The positive coefficient estimate and an odds ratio greater than 1 (not statistically different from 1 at 95% confidence level) for the curved road indicator in the major injury/fatality category indicate that the presence of curves increases the probability of major injury/fatality outcomes in a median barrier crash relative to lower levels of severity outcomes, when compared to tangent road segments. This finding supports the expected outcome E12, and was expected because vehicles leaving the roadway and entering into the median on curved roadway segments likely impact a longitudinal barrier at higher impact angles than vehicles leaving the roadway on a tangent roadway segment. The positive coefficient estimates and odds ratios significantly greater than 1 for the unbelted indicator in minor injury, moderate injury and major injury/fatality categories indicate that not wearing a seatbelt increases the probability of higher injury levels relative to PDO or no-injury. The odds ratio of the unbelted indicator is the largest in the major injury/fatality category and the smallest in the minor injury outcome category. This finding is in agreement with the expected outcome E16 that unbelted drivers will experience higher severity outcomes in a median barrier crash than belted drivers. Again, this finding is consistent with Islam and Mannering (2006), as described earlier. The positive coefficient estimates and odds ratios greater than 1 for the driver age indicator in the minor injury, moderate injury and major injury/fatality outcome categories indicate that older drivers have higher probabilities of higher severity outcomes relative to PDO or no-injury severity outcomes when compared to younger

206 193 drivers. The coefficients for the driver age indicator were restricted to be equal across minor and moderate injury levels since the coefficients were similar in magnitude. This finding was intuitive and supports the expected outcome E15. The negative coefficient estimates and odds ratios smaller than 1 (not statistically different from 1 at 95% confidence level) for the female driver indicator in the moderate injury and major injury/fatality categories indicate that female drivers have lower probabilities of moderate injury or major injury/fatality outcomes relative to lower severity outcomes. This finding does not support the expected outcome E15 in chapter 1. A possible reason for this finding is that females drive more carefully and at lower speeds than males and thus the impact speeds with a median barrier are lower. The lower impact speeds lead to lower injury outcomes in a median barrier crash. The relative effect of the female driver indicator is larger in major injury/fatality category than in moderate injury category. The positive coefficient estimates and odds ratios greater than 1 for the speeding indicator in the minor injury, moderate injury and major injury/fatality categories indicate that speeding increases the probability of higher severity outcomes in a median barrier crash. The coefficient for the speeding indicator was restricted to be equal across moderate and major injury/fatality levels since the coefficients were similar in magnitude. The odds ratio of speeding is higher in the moderate injury and major injury/fatality categories than in the minor injury category. These findings support the expected outcome E17 in chapter 1. The positive coefficient estimates and odds ratios greater than 1 for the driver fatigue indicator in the moderate injury and major injury/fatality categories indicate that

207 fatigued drivers have higher probabilities of moderate injury/major injury/fatality 194 outcomes relative to lower severity levels in a median barrier crash. The odds ratio of the fatigue indicator is higher in the major injury/fatality category than in the moderate injury category. These findings support the expected outcome E17 in chapter 1. The negative coefficient estimates and odds ratios less than 1 for the road surface with water/ice/snow/slush indicator in the minor injury, moderate injury and major injury/fatality categories indicate that the presence of water/ice/snow/slush leads to lower probabilities of higher injury outcomes relative to pdo or no-evident injury in a median barrier crash. The coefficient for the road surface with water/ice/snow/slush indicator was restricted to be equal across moderate and major injury/fatality levels since they were similar in magnitude. The odds ratio of the road surface with water/ice/snow/slush indicator is closer to 1 for the minor injury category than for the moderate injury/major injury/fatality category. These findings support the expected outcome E18 in chapter 1. Drivers travel more cautiously and at slower speeds in adverse weather, resulting in lower impact severities in a median barrier crash. The coefficients for the nighttime crash indicator were restricted to be equal across the moderate and major injury/fatality outcome levels. The negative coefficient estimate and an odds ratio smaller than 1 indicate that a median barrier crash occurring at night has lower probabilities of moderate and major injury/fatality outcomes relative to lower injury outcomes. This finding does not support the expected outcome E19 stated in chapter 1. A possible reason for this is that drivers drive more cautiously and at lower speeds at night, resulting in lower impact severities in a median barrier crash.

208 The positive coefficient estimates and odds ratios greater than 1 for the 195 overturning vehicle indicator in the minor injury, moderate injury and major injury/fatality outcomes categories indicate that overturning vehicles in a median barrier crash lead to higher probabilities of more severe injuries than PDO or no-injury outcomes. The coefficient for the overturning vehicles indicator was restricted to be equal across moderate and major injury/fatality levels since they were similar in magnitude. The odds ratio of the overturning vehicle indicator is smaller in the minor injury category than in the moderate and major injury/fatality categories. These findings support the expected outcome E22 stated in chapter 1. The positive coefficient estimates and odds ratios greater than 1 for the heavy truck indicator in the moderate injury and major injury/fatality categories indicate that heavy truck involvement in a median barrier crash increases the probability of more severe injuries than minor injury or PDO/no-injury outcomes. The odds ratio of heavy truck involvement is smaller in the moderate injury category than in the major injury/fatality category. These findings support the expected outcome E21 stated in chapter 1. Efforts to include main effects for ADT, median foreslope and median barrier offset were made, but none were statistically significant at the 80th-percentile confidence level. Additionally, several interaction terms were included in the model (e.g., median foreslope and barrier offset, barrier type and barrier offset, and barrier type and median foreslope); however, none were statistically significant in the model.

209 6.4 North Carolina Median Barrierr Crash Severity Model 196 A nested logit model was estimated to analyze the effects of median, roadway, driver, environmental, and vehicle factors on median barrier crash severities in North Carolina. Four crash severity outcomes were modeled: No injury, Class C Injury, Class B Injury, Class A Injury/Fatality. The Class A injury and fatality categories were combined due to the small number of cases in these two categories. Numerous nesting structures were tested --- the one used in the present analysiss is shown in Figure This structure accounts for the correlation of unobserved effects across the no-injury and class C severity levels, is consistent with nesting structures used in previous studies (e.g., Shankar et al., 1996; Lee and Mannering, 2002; Holdridge et al., 2005; Savolainen and Mannering, 2007), and was the most appropriate for the present study based on tests of other possible nesting structures. Figure 6.1. Nesting structure of median barrier crashes