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MODELING THE INTERACTIVE EFFECTS OF SAFETY COUNTERMEASURES ON PENNSYLVANIA HIGHWAYS
ABSTRACT
Crash Modification Factors (CMFs) are typically used to quantify the effects of road safety countermeasures by estimating the expected safety impact with the installation of certain treatments. When multiple treatments are implemented together, multiplying individual CMFs is a common way to estimate the overall reduction on crashes quantitatively, carrying with an assumption of statistical independence of the treatments. The validity of the assumption of treatment independence has not been studied extensively.
When interaction among treatments exists, it is inappropriate to assume independence of the treatments. The inappropriate use of CMF multiplication is likely to cause errors in the estimation of the magnitude of the safety effects of the treatment (i.e. CMF). This research develops a methodology to identify and quantify the presence of interaction between countermeasures. Differences in safety effectiveness estimates are developed so as to better understand the implications of treatment interaction on safety.
The crosssectional modeling approach using negative binomial models was adopted to estimate the effect on CMFs of countermeasure interaction. This research analyzes the interaction using Generalized Linear Models (GLM) with an underlying negative binomial distribution. After careful selection and screening of predictor variables, a series of models with independent CMFs are compared to a comparable series without the assumption of treatment independence (i.e. with twoway interaction terms).
This dissertation consists of three papers. For the first paper in Chapter 2, the methodology is applied to rural multilane highways by studying the potential interactions among lane width, right should width and median type. These roadway countermeasures interact with each other throughout the crosssection of rural multilane highways in terms of their association with safety performance. Chapter 3 studies a more homogeneous type of road environment, rural twolane highways, with three different treatment combinations that were selected from the countermeasures throughout roadway, roadside, and alignment elements, including horizontal curve density and access density. Two types of interactions, categorical by categorical and categorical by continuous interactions, were developed and analyzed in the research. All treatment combinations have demonstrated the existence of interactions with the significance on interaction terms. For the last paper in Chapter 4, the interactive effects on limited access facilities (i.e. interstate freeways) were explored to see how the interaction of a same pair of treatments (e.g. lane width and shoulder width) vary across different road facilities (i.e. rural multilane and interstate freeways). The treatment combinations that are tested for independence include lane width, right shoulder width, and median type. The comparison of the GLM models demonstrates the existence of significant interactive effects.
The findings of all papers have achieved the research objectives to develop a feasible method for examining treatment independence and to demonstrate the influences of interaction in CMFs by effectively estimating the combined safety effectiveness with interactions. The differences in CMF with and without interaction are shown to lead to the divergences of conclusions for predicting and assessing the combined treatment effects. It is necessary to be aware and examine the independence of multiple treatments when potential interactions exist; neglecting interactions should be avoided to prevent the combined treatment effects from being incorrectly estimated. The findings of this dissertation are expected to bring contributions to the field by enhancing the quality of safety estimation with the application of CMFs, and therefore provide innovative yet reliable tools and methods for roadway safety improvement.
TABLE OF CONTENTS
List of Figures ………………………………………………………………………………………………………….. viii
List of Tables …………………………………………………………………………………………………………… xi
Acknowledgements …………………………………………………………………………………………………… xv
Chapter 1 Introduction ………………………………………………………………………………………………. 1
1.1 Research Background……………………………………………………………………………………. 11.2 Methodology ……………………………………………………………………………………………….. 9
1.3 Data Description ………………………………………………………………………………………….. 201.4 Research Questions and Objectives ………………………………………………………………… 22
1.5 Research Contributions …………………………………………………………………………………. 25
1.6 References …………………………………………………………………………………………………… 25
Chapter 2
Crash Modification Factors with Interaction: Modeling the Interactive Effects of
Roadway Countermeasures on Rural Multilane Highways ……………………………………… 29
Abstract ……………………………………………………………………………………………………………. 29
2.1 Introduction …………………………………………………………………………………………………. 30
2.1.1 Safety Effectiveness Evaluation of Multiple Treatments by CMFs ……………. 32
2.1.2 The Interaction Problem between Multiple Treatments ……………………………. 34
2.1.3 Research Objectives ……………………………………………………………………………. 37
2.2 Methodology ……………………………………………………………………………………………….. 37
2.2.1 Crosssectional Method ……………………………………………………………………….. 37
2.2.2 Modeling Interactive Effects ………………………………………………………………… 39
2.2.3 Estimation of the Combined Safety Effects with Interaction by CMF………… 44
2.3 The Data and the Selected Treatment Combinations …………………………………………. 48
2.3.1 Data Description …………………………………………………………………………………. 48
2.3.2 The Selected Treatment Combinations ………………………………………………….. 50
2.4 Modeling Results …………………………………………………………………………………………. 55
2.4.1 Treatment Combination 1: Crosssectional Estimation of Lane Width and
Right Shoulder Width …………………………………………………………………………… 56
2.4.2 Treatment Combination 2: Crosssectional Estimation of Median Type
(Group 1, Traversable/Nontraversable Medians) and Right Shoulder
Width …………………………………………………………………………………………………. 65
2.4.3 Treatment Combination 3: Crosssectional Estimation of Median Type
(Group 2, Traversable Medians) and Right Shoulder Width ………………………. 74
2.5 Discussion …………………………………………………………………………………………………… 83
2.6 Conclusions …………………………………………………………………………………………………. 87
2.7 References …………………………………………………………………………………………………… 89
Chapter 3 The Interactive Effects of Safety Effectiveness Evaluation with Combined
Treatments on Rural Twolane Highways …………………………………………………………….. 93Abstract ……………………………………………………………………………………………………………. 93
3.1 Introduction …………………………………………………………………………………………………. 94
3.1.1 Safety Effectiveness Evaluation of Multiple Treatments by CMFs ……………. 96
3.1.2 The Interaction Problem between Multiple Treatments ……………………………. 98
3.1.3 Research Objectives ……………………………………………………………………………. 101
3.2 Methodology ……………………………………………………………………………………………….. 102
3.2.1 Crosssectional Method ……………………………………………………………………….. 102
3.2.2 Modeling Interactive Effects ………………………………………………………………… 103
3.2.3 Estimation of the Combined Safety Effects with Interaction by CMF………… 109
3.3 The Data and Selected Treatment Combinations ………………………………………………. 113
3.3.1 Data Description …………………………………………………………………………………. 113
3.3.2 The Selected Treatment Combinations ………………………………………………….. 116
3.4 Modeling Results …………………………………………………………………………………………. 126
3.4.1 Crosssectional Estimation of Horizontal Curve Density and Travelway
Width …………………………………………………………………………………………………. 127
3.4.2 Crosssectional Estimation of Access Density and Number of Horizontal
Curves ………………………………………………………………………………………………… 147
3.4.3 Crosssectional Estimation of Roadside Hazard Rating and Right
Shoulder Width ……………………………………………………………………………………. 167
3.5 Discussion …………………………………………………………………………………………………… 180
3.6 Conclusions …………………………………………………………………………………………………. 185
3.7 References …………………………………………………………………………………………………… 187
Chapter 4 Screening, Identifying, and Estimating the Interactions of Roadway
Countermeasures on Highways Safety for Limited Access Facilities ……………………….. 192
Abstract ……………………………………………………………………………………………………………. 192
4.1 Introduction …………………………………………………………………………………………………. 193
4.1.1 Safety Effectiveness Evaluation of Multiple Treatments by CMFs ……………. 195
4.1.2 The Interaction Problem between Multiple Treatments ……………………………. 197
4.1.3 Research Objectives ……………………………………………………………………………. 200
4.2 Methodology ……………………………………………………………………………………………….. 201
4.2.1 Crosssectional Method ……………………………………………………………………….. 201
4.2.2 Modeling Interactive Effects ………………………………………………………………… 203
4.2.3 Estimation of the Combined Safety Effects with Interaction by CMF………… 208
4.3 The Data and Selected Treatment Combinations ………………………………………………. 212
4.3.1 Data Description …………………………………………………………………………………. 212
4.3.2 The Selected Treatment Combinations ………………………………………………….. 214
4.4 Data Analysis ………………………………………………………………………………………………. 221
4.4.1 Modeling Results of Lane Width and Right Shoulder Width ……………………. 2224.4.2 Modeling Results of Median Type and Right Shoulder Width ………………….. 232
4.4.3 Modeling Results of Median Type and Lane Width ………………………………… 242
4.5 Discussion …………………………………………………………………………………………………… 2524.6 Conclusions …………………………………………………………………………………………………. 256
4.7 References …………………………………………………………………………………………………… 259
Chapter 5 Summary, Conclusions, and Recommendations …………………………………………….. 263
5.1 Summary …………………………………………………………………………………………………….. 263
5.2 Conclusions …………………………………………………………………………………………………. 267
5.3 Recommendations for Future Research …………………………………………………………… 273
Appendix A The Distribution Plots of Continuous Variables …………………………………………. 276
A.1 The Distribution Plots of Paper 1 on Rural Multilane Highways (Chapter 2) ………. 276A.2 The Distribution Plots of Paper 2 on Rural Twolane Highways (Chapter 3) ………. 278
A.3 The Distribution Plots of Paper 3 on Pennsylvania Interstate Freeways (Chapter 4) ……………………………………………………………………………………………………………… 280
Appendix BThe Time Series Plots of Crash Variables ………………………………………………….. 281
B.1 The Time Series Plots of Paper 1 on Rural Multilane Highways (Chapter 2) ………. 281
B.2 The Time Series Plots of Paper 2 on Rural Twolane Highways (Chapter 3) ……….. 283
B.3 The Time Series Plots of Paper 3 on Pennsylvania Interstate Freeways (Chapter 4) ……………………………………………………………………………………………………………… 286
Appendix C CMF Comparisons with the CMFs in Highway Safety Manual (AASHTO, 2010) ……………………………………………………………………………………………………………….. 287
C.1 CMF Comparisons of Paper 1 on Rural Multilane Highways (Chapter 2) …………… 287
CMFs for Lane Width (LW) ………………………………………………………………………… 287
CMFs for Right Shoulder Width (RSW) ………………………………………………………… 288
CMFs for Median Type ……………………………………………………………………………….. 288
C.2 CMF Comparisons of Paper 2 on Rural Twolane Highways (Chapter 3) ……………. 289
CMFs for Roadside Hazard Rating (RHR) …………………………………………………….. 289
CMFs for Right Shoulder Width (RSW) ………………………………………………………… 290
References ……………………………………………………………………………………………………………….. 292
Chapter 1 Introduction
1.1 Research Background
How to reasonably and effectively quantify safety effects in response to any changes in road geometry has been an essential and popular topic in the road safety research literature. Generally, the expected number of crashes that occur within specific study units (e.g. section of roadway, unit of length, time period, driving mileage) is considered to be the most direct performance measurement in terms of “safety” (Hauer, 1997). Therefore, the methods to quantify safety basically revolve around the modeling of the number of crashes with characteristics of study interests, including geometric elements.
A specific area of interest is the effectiveness of safety countermeasures, where safety countermeasure represents the treatment or changes made to a road segment aiming at reducing the frequency and/or severity of crashes. Crash Modification Factors (CMFs) are typically used to quantify the effect of countermeasures. Currently, CMFs are used as a common tool to compute the change of crashes quantitatively by estimating the expected safety impact with the installation of certain treatments. The CMF is defined as the ratio of E(N_{with})/E(N_{without}), where E(N_{with}) represents the expected number of crashes per unit of time with a countermeasure is implemented, while E(N_{without}) stands for the expected number of crashes per unit time without a countermeasure (Bonneson & Pratt, 2008; AASHTO, 2010). The general form of the application of CMF can be illustrated as:
_{ }= _{ } _{1 }… … . _{ } (1.1)
where:
N_{pred }=the predicted number of crashes per study unit with countermeasure;
N_{baseline }=the predicted number of crashes per study unit from the baseline model, which can also be represented as the number of crashes without countermeasure;
CMF_{i}= the crash modification factor for specific safety countermeasure i (AASHTO, 2010). As shown in Equation 1.1, CMFs work as multipliers that indicate the expected change in crashes after implementing a given treatment (Gross, Jovanis, & Eccles, 2009). CMFs are generally represented as a single value or a function containing explanatory variables and may also be applied to different crash types and severities based on their usage instructions. In the form of a single value, CMF represents an index relative to the value of one and quantitatively indicates how much crash experience is expected to change with the treatment. For example, CMF greater than 1 represents the treatment contributes to higher number of crashes, whereas CMF less than 1 means the treatment decreases the expected number of crashes.
When the safety effect for a certain treatment is not suitable to be assumed equal for all conditions, a CMF in the form of function can be better used to capture the differences among locations, such as traffic volume or roadway functional class. In other words, functionstyle CMFs enable one to compute the estimated safety effects of treatments which change over the range of a variable or a combination of variables according to the characteristics of each site (Gross, Persaud, & Lyon, 2010). When multiple treatments are implemented together on a given site, CMF is a common tool to estimate the reduction on crashes quantitatively by the combined treatments. By multiplying CMFs that correspond to each individual countermeasure, the expected number of crashes with multiple treatments can be therefore estimated by applying
Equation 1.1 with CMFs shown above.
It is common that multiple treatments are combined and implemented together on a given site to deal with a specific safety problem, and CMF is used to estimate how much the safety can be improved quantitatively by the combined treatments. Equation 1.1 is adopted by both Highway Safety Manual (AASHTO, 2010) and CMF Clearinghouse (Federal Highway Administration, 2014) to estimate the combined safety effects when more than one treatment is implemented, either simultaneously or sequentially, at a given location. However, this multiplicative relationship of multiple CMFs targeting each individual safety countermeasures is based on an assumption that the effects of these treatments are independent of each other. That is, the prerequisite of applying the multiplicative equation in Equation 1.1 is that there is no interaction among multiple treatments and these treatments have independent effects on safety. Therefore, whether the assumption of independence is valid for the given treatments determines the applicability and feasibility of multiplicative relationship of multiple individual CMFs. When interaction among treatments exists, it is expected that each single treatment CMF may not accurately represent the full effect. The results of the interaction may be higher or lower safety effects than the product of the independent CMFs. However, there has been very limited research investigating the existence and characteristics of interaction and the correct way to estimate the combined effects of multiple treatments when the assumption of independence is violated.
In statistics, interaction means that the effects of predictor variables on the response variable are not additive but differ depending on the levels of the other predictor variables (Kutner, Nachtsheim, Neter, & Li, 2005). That is, an interaction effect exists when the effect of a predictor variable on the response variable depends on the value of another predictor variable. To generalize the idea of interaction in statistics to CMFs, interaction in this research is defined as the circumstance when the effects of a given countermeasure on safety are not constant but vary depending on the presence of other countermeasure or the levels of other countermeasures that are implemented, either simultaneously or sequentially, at the same location. Based on this definition, there are three types of treatments to be studied:
 The presence of a treatment with a discrete feature. That is, to distinguish sites with and without the presence of a particular treatment of interest, such as median type, roadway lighting and shoulder rumble strip.
 A treatment which is characterized by levels. That is, the study sites contain a treatment that is best characterized by various levels of a continuous variable, such as lane width and shoulder width.
 A treatment with continuous, quantitative features. That is, the study sites include a treatment that is by nature a continuous variable with numerical, decimal values, such as access density and horizontal curve density.
Therefore, the treatment interaction used in this research is not precisely the same as the interaction that is traditionally viewed in statistics, but depends on the nature of different countermeasures being studied. Because the prerequisite of using the multiplicative CMF prediction is based on the independence of these chosen countermeasures, it is important to detect in advance if any interactions exist among different treatments. In other words, there should not be any existing or known interaction among these multiple road treatments when the multiplicative CMF prediction method (i.e. Equation 1.1) is used.
Once the countermeasures are shown to be nonindependent and interact with each other, their combined safety effect should be estimated with the consideration of the interactive effects along with their consequential CMFs developed with interaction. Examples of CMFs estimated with single, independent treatment effect and CMFs developed with interaction considered are illustrated in the following Table 1.1 and Table 1.2 respectively.
Table 1.1 The Example of CMF Estimated with Single Treatment Effect without Interaction: Shoulder Width on Rural, TwoLane Highways with AADT>2000 veh/day (AASHTO, 2010)
CMF Estimated with Single Treatment Effect without Interaction  
Treatment (Shoulder Width)  CMF  CMF of Shoulder Width on Rural, TwoLane
Highways with AADT>2000 (veh./day) Shoulder Width 

0 ft.  1.50  
2 ft.  1.30  
4 ft.  1.15  
6 ft.  1.00
(baseline ) 

8 ft.  0.87  
Note: CMF are applied to singlevehicle runofftheroad, multiplevehicle headon, oppositedirection sideswipe, and samedirection sideswipe crashes. 
Table 1.2 The Example of CMF Developed with Interaction: Lane Width and Shoulder Width on Rural, TwoLane, Undivided Highways (Gross et al., 2009)
CMF Developed with Interaction  
Shoulder Width  
3 ft. 4 ft.  5 ft.  6 ft.  
Lane 10 ft.  1.13 1.20  1.22  1.00 (baseline)  
Width 11 ft.  1.19 1.14  1.06  0.84  
12 ft.  1.11 1.04

0.87  0.75  
Lane Width and Shoulder Width on Rural, Twolane Highways
10 ft 11 ft 12 ft 

A typical multiple regression model with more than one treatment predictor is a common practice to estimate the combined safety effects when multiple factors may be associated with crash occurrence, such as road geometry and traffic volume. By using a combined database including all populations (i.e. data gathered from all road segments with and without treatments), the changes or existence of treatments are represented as a predictor variable in the model indicating the effect of differences in condition (e.g. grade, speed, lane width, etc.) on crash frequency for estimating the combined treatment effects (Bonneson & Pratt, 2008). However, a naïve multiple regression model is a one that assumes all parameters are independent and constrained to have equal effects for all locations, and the effect of one treatment on safety is restricted to be equal across different levels or the presence of all the other treatments. In a case like this, the interactions between treatments are not presented in the model and are assumed to be zero; and therefore the differences in effects across different levels or the presence of all the other treatments cannot be tested and verified. To overcome the limitation of multiple regression models that assume independence of predictors, separate models corresponding to each group/level of a treatment can be commonly found in safety research (Harwood et al., 2003; Lord et al., 2008; Fitzpatrick et al., 2008; Stamatiadis et al., 2009). CMFs estimated by building individual models are appealing when the changes of road safety with the application of a certain treatment (e.g. widening lane width or shoulder width) are to be compared between different road patterns, especially when there are good theoretical reasons to believe that the treatment effect on crash vary between different levels or the presence of another treatment. Additionally, building separate models can also help to control some unknown factors other than known treatments that cannot be modeled or collected in database, and the precision on the prediction of treatment effectiveness by CMFs can therefore be improved. However, building individual models may lead the interaction effects to be due to the inherent differences of the samples gathered from different road types or environments, such as divided/undivided highways. That is, the resulting CMFs not only consist of pure treatment interaction but also the inherent influences of heterogeneity. Additionally, separate models cannot verify if the treatment effect on safety significantly varies between different groups or populations. This insufficiency may confuse and mix the true treatment effects with inherent differences due to statistical errors, different constant terms, or the nature differences between road types. As the result, an alternative approach that uses interaction models should be considered to overcome these limitations, including examining the existence of interactive effects by adding the interaction terms and further estimating the combined treatment effects with significant interactions.
When multiple countermeasures are implemented together on a site, it is important to detect in advance if any interactions exist among different treatments before using the multiplicative CMFs prediction. In literature, CMF Clearinghouse defines that treatments can be considered independent if the countermeasures are targeting different crash types (“CMF
Clearinghouse >> FAQs,” 2013). Expanding from this conceptual independence, the matrix table based on target crash types was proposed by Gross et al. for identifying the independence of countermeasures (Gross, Hamidi, & Yunk, 2012). Because some treatments don’t have specific target crash types and different treatments are still likely to have interactions even though they don’t target on the same crash type, comparing target crash types may be adequate as a screening
tool but not appropriate to precisely determine the effect of actual interactions.
In statistics, building a model with interaction terms is a scientific and solid method to verify if the effect of one treatment varies with another treatment when crash data are available for modeling. Generally, interaction models are developed for two purposes. First, the models work as the means to examine the actual existence of interactive effects between different predictor variables by including the interaction terms in the models. The interaction terms are developed by creating cross product of treatment variables, and statistical procedures are available to test the coefficients of interaction terms for differences from zero, such as Wald test (Kutner et al., 2005). Second, the combined treatment effects with significant interactions can be estimated by modeling interactive effects. When the effects of a given countermeasure on safety vary depending on the presence or the levels of other countermeasures, the change of the response variable with and without interactive effects provides important information on estimating safety at a site. When the assumption of zero interaction does not hold for the chosen treatments and interactive effects are ignored in safety estimation, modeling a multiple regression without interactions may lead to biased estimations of the true combined treatment effects along with its inappropriate use on the multiplication of individual CMFs. In a case like this, adding interaction terms into a regression model for developing CMFs with interaction should to be adopted to overcome the limitations of incorrect independent and constrained equal effects, and further to avoid the errors in the estimation of combined treatment effectiveness. The influences of wrong safety effectiveness estimations can be deep and wide, leading to incorrect implementation decisions with resulting increases in injuries or fatalities. This is the justification for further study of the existing method for estimating safety effectiveness based on multiplicative multiple CMFs. In conclusion, the current knowledge gaps on the issue of safety estimation for multiple treatments when interaction exists are summarized as follows:
 There is a need to develop methods to systematically check if the given treatments of interest are independent. That is, to identify if there is potential interaction among treatments.
 Once the assumption of the independence is shown to be not valid, there is a need to quantitatively estimate the combined effects by modeling interactions.
 There is a need to estimate the influence of the interactive effects on safety; that is, to quantify the consequences of ignoring the interaction.
1.2 Methodology
Focused on the crosssectional comparison of sites with and without treatments, this dissertation adopted the crosssectional method by generalized linear models (i.e. negative binomial models) for estimating the CMFs with interaction effects. The alternative before/after method was not considered in this dissertation due to a lack of readily available data, especially regarding the implementation dates of engineering changes. Even though before/after studies are conducted, they are comparatively rare because of the time required to collect sufficient crash data under these two required conditions (i.e. without the treatment followed by a period with the treatment at the same site). Crosssectional method is one of the most common tools in transportation safety research for estimating CMF by using observational data.
To capture the change in safety of some roadway segments with the installation of some treatments, crosssectional method compares the crash experience of locations with and without the treatment of interests and then attributes the difference in safety performance to that treatment (Gross et al., 2010). That is, to compare the expected crash frequencies or severities of one group of locations which shares some common characteristic to a different group of locations without having that characteristic to assess the differences in crash experience with/ without that characteristic (AASHTO, 2010). By fitting a regression model to observational data, the CMF is estimated as the ratio of the expected crash frequency for roadway segments with the treatment over the expected crash frequency without the treatment. Generally, the strengths of the crosssectional method is its superior applicability when beforeafter study is limited and not feasible while a large number of segments with and without the treatment is available to be obtained (Gross & Donnell, 2011). For example, when there are no before/after data available, such as crash or traffic volume data for the period before or after treatment implementation, crosssectional method is an useful alternative to overcome the difficulty of unavailable data.
The crosssectional method is reliable based on the ability to select sufficient segments that are as similar as possible to each other except for treatments of interest, which is sometimes not easy to fulfill in practice (Gross et al., 2010). To solve the problem of insufficient identical segments, controlling variables, such as traffic volume or geometric elements other than treatments of interest, will additionally be included in a multiple regression model to account for known differences across sites. Unlike exposure variables and treatment variables, a controlling variable is kept in the final model mostly when its coefficient is proven to be statistically significant by Wald test. In addition to the results of the Wald test, the inclusion of controlling variables also avoided any noticeable changes of signs or significance for the existing exposure and treatment variables in order to prevent potential correlation problem among variables. This key concept to avoid potential correlation is used as a principle for adding more predictors when building a model. Each time a controlling variable or interaction term is added, all other predictors are checked for changes in sign or magnitude.
However, the crosssectional method also has weaknesses that are of concern. The same drawbacks of the ordinary multiple regression modeling exist when the crosssectional method is used, such as inappropriate assumptions of data distributions (e.g. normal distribution), biases due to omitted variables and missing data, and potential correlations among variables. For these reasons, relevant statistical remedies are required to take actions when crosssectional design is adopted, such as the use of the concept of Generalized Linear Models (GLM).
Generalized linear models (GLM) are welldeveloped statistical tools to help solve many of the problems due to the limitations of ordinary linear models. Constituted by three different components (i.e. random component, structure component, and link function), GLM allows users to select a suitable distribution of the response variable identified by its random component and also to specify a function that connects the expected value (mean) of response variable with the predictors by its link function (Agresti, 2007). Because of the discrete, nonnegative, random and sporadic nature of the frequency of crash occurrence, a series of research completed in 1980s and 1990s has pointed out and demonstrated the necessity to improve the unsatisfactory statistical characteristics of the application of ordinary multiple linear model (Jovanis & Chang, 1986; E.
Hauer, Ng, & Lovell, 1988; Saccomanno & Buyco, 1988; Joshua & Garber, 1990; Miaou & Lum, 1993). According to the characteristics of crash occurrence, the crash data do not comply with the assumption of normality; that is, the expected crash frequency (i.e. the dependent variable, Y) is not normal distributed as well as its noise. Instead, the distribution of crash frequency tends to be very rightskewed and most observations concentrated at low frequencies (i.e. 0, 1, or 2) since crash occurrence is a rare events with random and sporadic nature. These unsatisfactory statistical assumptions of conventional linear regression were first pointed out by Jovanis and Chang in 1986 (Jovanis & Chang, 1986) by demonstrating Poisson regression is superior to linear regression for explaining the association between crash frequency and predictor variables. Meanwhile, the nonlinear relationship between crash frequency and traffic exposures was verified and frequently shown in literature, such as Highway Safety Manual (AASHTO, 2010), as illustrated in the following Figure 1.1. With the ability to design a suitable function that links the expected value of response variable and the predictors, GLM also enables the choice of nonlinear link function (e.g. g(µ)=log(µ) for Poisson models) for modeling the nonlinear nature of crash occurrence.
Figure 1.1 The Example of the Nonlinear Relationship between Crash Frequency and Traffic Exposures (AASHTO, 2010)
Overdispersion is defined as a situation when the observed variance of the dependent variable is higher than the variance estimated from a conditional model. For a model with Poisson distribution, overdispersion occurs when the observed variance exceeds the mean (Agresti, 2007), which is frequently found in count data with discrete feature . Because of the overdispersion problem of Poisson regression, negative binomial models (NB models) is considered as an advanced and extension version of Poisson models to better simulate crash occurrence. Negative binomial models for safety analysis was studied by a series of followup research by different groups from 1993 to 1996 for exploring the solution to overdispersion problem (Miaou & Lum,
1993; Shankar, Mannering, & Barfield, 1995; Poch & Mannering, 1996; Maher & Summersgill, 1996). From these studies, negative binomial regression was tested and reported as a successful enhancement to Poisson model with its ability to handle overdispersion and further became a common practice for modeling crash frequency as count data. In Highway Safety Manual (AASHTO, 2010), the use of negative binomial regression is also recommended for modeling crash frequency when safety effectiveness evaluation is conducted.
Using the basis of the GLM modeling framework, interaction among safety treatments is studied by using the following procedures with the comparison of CMFs which are illustrated in Figure 1.2 below:
 Step 1: To select treatment combinations for interaction modeling by building a reasonable pool of treatments to choose from. The pool of treatments is developed using two sources: a matrix of treatment target crash types (Gross et al., 2012) and the potential interaction treatments indicated by past studies.
 Step 2: To examine if the candidate combinations of treatments are independent and their interactive effects exist. The effects of each treatment are estimated firstly as if the treatments were independent by a main effect model, and then the interactive effects are examined with interaction terms included as an interaction model.
 Step 3: To estimate the combined safety effects of the candidate treatments with their interaction considered. The likelihood ratio test is conducted for main effects models and interaction models for assessing if the inclusion of interaction terms improves the overall goodness of fit.
 Step 4: To compare and analyze the change in safety performance by CMFs estimated with and without interactive effects, when the independence between treatments does not hold.
Figure 1.2 The Structure for the Modelling and Estimation of CMFs with Interactions
To derive the negative binomial models for examining interactive effects of treatments, let Y denote as count data representing the number of crashes occurred per year at a given road segment. The negative binomial models assume that Y follows Poisson distribution with its estimated mean (i.e. ̂) and the error term follows a gamma distribution. The general form of the negative binomial models is listed as follows:
~ ( ̂) (1.2)
̂ = exp ( _{ }+ _{ }) (1.3)
(1.4)
where:
̂= the estimated mean of the crash frequency per year,
_{ }=the error term that follows gamma distribution,
D=dispersion parameter, which can represent the heterogeneity in Poisson means.
The relationship of the expected value and variance for the negative binomial distribution is:
(1.5)
( ) = ̂ + ̂^{2} (1.6)
D is the dispersion parameter. The larger D is, a more serious overdispersion would be observed in the data. Estimated by the maximum likelihood method by using STATA (i.e. the statistics software developed by StataCorp), the link function of negative binomial model is g(µ)=log(µ) , and the model can be written together with the system component (i.e. the explanatory variables) in the following forms:
(1.7)
̂ = ^{ }^{0 }× ^{ }^{1 }× ℎ^{ }^{2 }× ^{∑}^{ =1 }^{ }^{ }^{ }^{ } (1.8) , where ̂ is the estimated mean of the crash frequency per year, AADT (the annual average daily traffic) and Length (segment length) are the exposure terms, and X_{j} is any of k explanatory variables which represent the main treatment effects, the interactive effects, and all the other controlling variables in the final model.
Only twoway interaction terms are explored in this dissertation, as it is the first study of its kind to systematically study whether and how safety treatment effectiveness can be modeled using interaction terms in a crosssectional modeling framework. Some quantitative, continuous types of treatments (e.g. shoulder width, travelway width) were categorized into levels, and their corresponding indicator variables were developed to represent each level of these quantitative variables, including the baseline level. The interaction terms were developed by creating cross product of treatment variables. Two types of interaction terms, categorical by categorical interaction and categorical by continuous interaction, were formed and studied in this research to reflect the treatment interactive effects. Twoway interactions are illustrated in the following Equation 1.9 for categorical by categorical interaction and in Equation 1.10 for categorical by continuous interaction:
Categorical by categorical interaction:
(1.9)
where:
X_{i}= the main effect term of Treatment 1 for each i level (except for the baseline),
X_{j}= the main effect term of Treatment 2 for each j level (except for the baseline),
X_{i} X_{j}=the twoway interaction terms for each ij level combination (except for the baseline). Categorical by continuous interaction
(1.10)
where:
X_{i}= the main effect term of Treatment 1 (i.e. categorical variable) for each i level (except for the baseline),
X= the main effect term of Treatment 2 (i.e. continuous variable),
XX_{i}=the twoway interaction terms for each i level combination (except for the baseline).
To estimate CMFs by crosssectional method, two sets of negative binomial models, one set with only main effects and one set with interaction terms, were built in the following forms for categorical by categorical interaction:
 Negative binomial model with only main effects (twotreatment combination):
(1.11)
 Negative binomial model with interaction terms (twotreatment combination):
(1.12)
where:
E(µ)=the expected crash frequency
X_{i}= the main effect variable of Treatment 1 (i.e. categorical variable) for each i level (except for the baseline),
X_{j}=the main effect variable of Treatment 2 (i.e. categorical variable) for each j level (except for the baseline),
X_{i}X_{j}=the interaction terms for each ij level combination (except for the baseline),
X_{k}= the controlling variables for each k level (except for the baseline).
The CMFs of categorical by categorical interaction are calculated as:
CMF= (  1 = , 2 = )/ (  1 = , 2 = ) (1.13)
where:
E(μtrt1 = i,trt2 = j)=the expected crash frequency when Treatment 1=i level and Treatment
2=j level when all the other variables hold constant;
E(μtrt1 = baseline,trt2 = baseline)= the expected crash frequency when Treatment
1=baseline level and Treatment 2=baseline level when all the other variables hold constant.
For categorical by continuous interaction, the two sets of negative binomial models, including one set with only main effects and one set with interaction terms, were built to estimate CMFs by crosssectional method in the following forms for categorical by continuous interaction:
 Negative binomial model with only main effects (twotreatment combination):
(1.14)
 Negative binomial model with interaction terms (twotreatment combination):
(1.15)
where:
E(µ)=the expected crash frequency,
X_{i}= the main effect variable of Treatment 1 (i.e. categorical variable) for each i level (except for the baseline),
X=the main effect variable of Treatment 2 (i.e. continuous variable),
XX_{i}=the interaction terms for each i level combination (except for the baseline),
X_{k}= the controlling variables for each k level (except for the baseline).
The baseline level of the continuous variable was set at zero, and the CMF Functions of categorical by continuous interaction are calculated as:
CMF Function= (  1 = , 2 = )/ (  1 = , 2 = 0) (1.16)
where:
E(μtrt1 = i,trt2 = X)=the expected crash frequency when Treatment 1=i level and Treatment
2=X when all the other variables hold constant;
E(μtrt1 = baseline,trt2 = 0)= the expected crash frequency when Treatment 1=baseline level and Treatment 2=0 when all the other variables hold constant.
To examine if the interactive effects between the selected treatments exist, this dissertation applies two kinds of significance tests, including:
 Testing the significance of interactive effects: every single coefficient of the interaction term is to be tested by Wald test with the null hypothesis as: H_{0}:β_{k}=0. The formulas of
Wald test are listed as follows:
(1.17)
(1.18)
− = ( >  _{⁄2}) (1.19)
where:
̂_{ }= the estimated coefficient of the treatment interaction term k,
= the estimated standard error of the estimated coefficient of the treatment interaction term k,
the estimated variance of the estimated coefficient of the treatment interaction
term k,
_{ }^{2 }_{=1}=the chisquare value with the degree of freedom of 1.
Based on the output of the negative binomial model, z statistic will be obtained with its corresponding twotailed pvalue. The choice of a more flexible significant level is set at α=0.15 which is supported by Hauer (Hauer, 2004) by the considerations of allowing more predictors to be assessed and recognizing small samples sizes that can occur in road safety. If the pvalue is smaller than α=0.15, the null hypothesis is rejected. That is, the interaction between treatments exists and the interactive effect is significant on the prediction of safety performance.
For example, for the single interaction term with its coefficient=0.294 and pvalue=0.140, its significance is tested by comparing its pvalue with α=0.15. Because the pvalue is smaller than α=0.15, the null hypothesis is rejected. That is, for this preliminary study as conducted, the interaction of these treatment levels exists and their interactive effect is significant on the prediction of safety performance.
 A likelihood ratio test used in assessing goodness of fit: to test if the inclusion of all interaction terms improves the overall goodness of fit of the model. The likelihood ratio
test is a test of two nested models, a “full model” and a “reduced model”, to determine whether the full model with interaction terms fits the data significantly better than the reduced model. The likelihood ratio test is conducted by comparing the likelihood values between the models with interaction terms (i.e. the full models) and the models without interaction terms (i.e. the reduced models). To test if all of the coefficients of the interaction terms are equal to zero, the null hypothesis to be tested is: H_{0}: β_{q}= β_{q+1}=…= β_{p1}=0, where the full model has p parameters while the reduced model containing q parameters. The test statistic for the likelihood ratio test is denoted by G^{2}, where L(F) is the likelihood value of the full model and L(R) is the likelihood value of the reduced model. G^{2} is tested by comparing with chisquare value (i.e. χ^{2}) when the significant level is set at α=0.15, and the decision rule for the test is:
^{2 }= −2 [log ( ) − log ( )] ~χ^{2}(1 − α; df = p − q) (1.20) If G^{2}≤ χ^{2}(1α; df=pq), conclude H_{0}. That is, the inclusion of interaction terms doesn’t actually improve the overall goodness of fit of the model.
If G^{2}> χ^{2}(1α; df=pq), conclude H_{a}. That is, the full model with interaction terms fits significantly better than the reduced model without interaction term.
For the likelihood ratio test, the test result may show that the inclusion of all interaction terms didn’t significantly improve the overall fit of the model because the null hypothesis cannot be rejected:
G^{2}=1.05< χ^{2}=3.79(0.85; df=2), conclude H_{0}.
That is, the interaction model actually wasn’t a significantly better fit than the main effect model.
1.3 Data Description
The dataset used in this research is segmentbased yearly crash data and corresponding roadway inventory information from Pennsylvania Department of Transportation (PennDOT). For the segmentbased crash data, all reported crashes were recorded yearly with a wide range of details of each crash occurrence, including the location information of crash (e.g. segment number and coordinates), weather, illumination, road condition, drivers, and vehicles. A reportable crash is defined in Pennsylvania’s Consolidated Statues as a crash occurs on a highway or trafficway that is open to the public and involves injury to or death of any person and/or damage to any other vehicle and therefore requires towing (“Consolidated Statutes,” 2014).
According to the yearly crash data, the total number of crashes occurring from 2005 to 2009 in Pennsylvania was 638,165. By merging the crash data with the roadway inventory files, the dataset contains details of each roadway segment tested in Pennsylvania for five continuous years from 2005 to 2009, including the count of crashes each year for all collision types and severity levels, roadway segment length, annual average daily traffic (AADT), speed limit, access control, highway functional classification, and elements of the geometric design of roads (e.g. lane width, shoulder width, median type, etc.). Exclusively for rural twolane highways, additional data elements of roadway and roadside features were available for eight continuous years from 2005 to 2012 for more varieties of safety analysis and modeling, including the roadside hazard rating (RHR), the presence of passing lanes, the presence of lowcost safety improvements (e.g. rumble strips and inlane pavement markings), access density, and horizontal curve radius and length.
To prepare the data for modeling, the data cleaning tasks were undertaken for the merged crash data and the roadway inventory data, including eliminating segments with incomplete traffic volume or geometric data and filtering data for the type of highways of research interest
(e.g. rural twolane or rural multilane). Based on the data availability of PennDOT’s yearly crash records and roadway inventory database, the mutual roadway geometry characteristics considered by the research for all roadway patterns are: paved roadway width, speed limit, right shoulder width (paved and total), right shoulder type, median type, lane width, and number of lanes in both directions, which are illustrated in Figure 1.3 for multilane highways and interstate freeway and in Figure 1.4 for twolane highways below. According to the roadway inventory file of 2013, the total mileage of roadway in PennDOT’s database is 52,856 miles with totally 126,396 segments of all road types. In average, a Pennsylvanian roadway segment is approximately 0.5 mile long. For the eightyear period from 2005 to 2012, totally 846,847 segmentlevel observations were recorded in the database.
Figure 1.3 The Illustration of Median, Shoulder Widths, Lane Width, and Roadway Width of Multilane Highways and Interstate Freeways
Figure 1.4 The Illustration of Shoulder Widths and Lane Width of Twolane Highways
1.4 Research Questions and Objectives
The main research questions of this dissertation are:
 How to reasonably estimate safety effectiveness when multiple treatments are implemented, especially when there is the interaction between treatments?
 What consequences occur if the interaction is ignored when the interaction between treatments exist? That is, how different will the estimated combined treatment effects be when the interaction is included?
 Among common elements of the road geometric design, what kinds of countermeasure combinations have interaction? That is, how the interactive effects vary among different safety treatments?
 How the interactive effects vary across different road facilities (e.g. rural multilane, rural twolane, and interstate freeways)? For example, how different
are the interactions of the treatment combination of lane width and shoulder width on rural multilane highways and on interstate freeways?
Expanded based on the research questions listed above, this dissertation consists of three papers with the following framework illustrated in Figure 1.5. The first paper (Paper 1) in Chapter 2 focuses on analyzing the interactive effects for the treatments of roadway elements on rural multilane highways, and two combinations with right shoulder width were selected for the comparison of safety effectiveness with/without the inclusion of interaction when multiple treatments are implemented together at the same time. The second paper (Paper 2) in Chapter 3 studies a more homogeneous type of road environment, rural twolane highways, together with the expansion on the selection of countermeasures in addition to roadway elements, such as access density and horizontal curve density. By using the recent data of roadside and alignment elements collected on rural twolane highways (Donnell, Gayah, & Jovanis, 2014), the interactive effects to be analyzed include the treatment combinations with roadside hazard rating (RHR) and horizontal curves. For the third paper (Paper 3) in Chapter 4, the interactive effects on limited access facilities consisting of interstate freeways are explored to answer the question on how the interactive effects of a same pair of treatments (e.g. lane width and shoulder width) vary across different road facilities (i.e. rural multilane and interstate freeways). With the expansions on the topic of modeling interactive effects both horizontally with various treatment combinations and vertically on different road facilities, this dissertation aims to acquire a comprehensive and deep understanding for the influence of interactions on the safety estimation of combined treatment effects. Upon the dissertation’s completion, the results and findings of these papers are expected to enhance the quality of safety estimation with the application of CMFs, and therefore provide innovative yet reliable tools and methods for roadway safety improvement.
Figure 1.5 The Framework of The Dissertation with Three Papers The research revolves the following specific objectives:
 To develop a feasible method to examine if these combinations of treatments are independent.
 To provide a solution and effective method for estimating combined safety effectiveness when there is interaction between treatments.
 To demonstrate the influences of interaction on safety effectiveness evaluation for roadway, roadside, and alignment countermeasures. That is, to demonstrate what the impacts will be if the interaction is ignored by CMFs estimated with/without interaction. The research scope is expected to expand both horizontally and vertically by exploring different treatment combinations on different types of road facilities.
 To investigate the applicability of multiplicative CMF prediction method when the interactions of combined treatments occur.
1.5 Research Contributions
With the goal to fill the current knowledge gaps of the interaction issue, the dissertation research is expected to bring the main contributions to the field as follows:
 Systematically study and quantify the effect of interactions between safety treatments. The goal is to broaden beyond simply treating the safety treatments as individual and in isolation, by considering how they may interact in their effect on safety.
 Develop and test the proposed methodology to check the independence of the given treatments and to quantify the safety effect when interaction exists.
 Explore different types of interaction terms which reflect the interactive effects of different types of treatment variables, including categorical by categorical interaction and categorical by continuous interaction.
 Compare and analyze the differences in safety effectiveness estimates with and without interactive effects. The purpose is to enhance the quality of safety estimation with the application of CMFs and to provide a better understanding for the implications of treatment interaction on safety.
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MODELING THE INTERACTIVE EFFECTS OF SAFETY COUNTERMEASURES ON PENNSYLVANIA HIGHWAYS