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years to more accurately determine the effectiveness of roadway safety measures.One newly released tool to aid in transportation safety analysis is

the Highway Safety Manual (HSM) published by AASHTO (1). The HSM was developed to incorporate the explicit role of highway safety in decisions on roadway planning, design, maintenance, construction, and operations. Currently, there are no such widely accepted tools available for agencies responsible for managing the safety of road-ways. As a result, safety considerations often carry little weight in decision making.The purpose of this paper is to summarize a statistical methodology

that can be used to analyze the effectiveness of roadway safety treatments. The paper analyzes the effectiveness of raised median installations in Utah by using a hierarchical Bayesian model to compare crash distributions before and after installation of a raised median. The paper first presents an overview of the background necessitating the research and then discusses the safety model devel-oped for use in the research. The raised median analysis site selection and data collection components are then addressed, followed by the analysis results for individual sites and a combined site analysis. Finally, conclusions and recommendations are presented.

Background

raised Medians

In this study, the safety data collection and analysis techniques devel-oped are applied to study sites at which raised medians have been installed throughout the state of Utah. A raised median is a physical barrier, such as a concrete or landscaped island, in the center portion of the roadway that separates opposing lanes of traffic and is designed so that it is not easily traversed. Raised medians are appropriate in some, although not all, locations and have been found to be most useful on high-volume, high-speed roadways (2).Raised medians have frequently been used as an access man-

agement technique to improve roadway safety in two primary ways: (a) raised medians reduce the number of conflict points by allowing turning movements to be made only at designated openings or at signalized intersections, and (b) raised medians provide a physical barrier separating opposing traffic in an effort to eliminate head-on collisions (3).Raised medians are not always used to mitigate one specific

type of crash or factor (2). Raised medians have been installed for beautification purposes, as an access management technique, as a traffic calming device, for pedestrian refuge, and for various other purposes (3). Regardless of the purpose, raised medians have had a

Analyzing Raised Median Safety Impacts Using Bayesian Methods

Grant G. Schultz, Daniel J. Thurgood, Andrew N. Olsen, and C. Shane Reese

Because traffic safety studies are not performed in a controlled envi-ronment such as a laboratory, but rather in an uncontrolled real-world setting, traditional analysis methods often lack the capability to evaluate the effectiveness of roadway safety measures adequately. In recent years, however, advanced statistical methods have been used in traffic safety studies to determine the effectiveness of such measures more accurately. These methods, especially Bayesian statistical techniques, can account for the shortcomings of traditional methods. Hierarchical Bayesian modeling is a powerful tool for expressing rich statistical models that more fully reflect a given problem than traditional safety evaluation methods could. This paper uses a hierarchical Bayesian model to ana-lyze the effectiveness of raised medians on overall and severe crash fre-quency in Utah by determining the effect each newly installed median has on crash frequency and frequency of severe crashes at study sites before and after installation of the medians. Several sites at which raised medians have been installed in the past 10 years were evaluated with available crash data. The results show that the installation of a raised median is an effective technique to reduce the overall crash frequency and frequency of severe crashes on Utah roadways, with results showing a reduction in overall crash frequency of 39% and frequency of severe crashes of 44% along corridors where raised medians were installed. The results also show that hierarchical Bayesian modeling is a useful method for evaluating effectiveness of roadway safety measures.

Because traffic safety studies are not performed in a controlled environment, such as a laboratory, but rather in an uncontrolled real-world setting, traditional analysis methods often lack the capa-bility to adequately evaluate the effectiveness of roadway safety measures. However, safety studies have historically had to rely on these traditional methods because of the complexity of more effec-tive models. Fortunately, the development of advanced statistical software and methodologies in recent years has helped overcome the complexity of the models and made advanced statistical meth-ods more attainable. These advanced statistical methods, especially Bayesian statistical methods, have the capabilities to account for the shortcomings of traditional methods and have thus been used in recent

G. G. Schultz, Department of Civil and Environmental Engineering, 368 Clyde Building, and A. N. Olsen, 223 TMCB, and C. S. Reese, 208 TMCB, Department of Statistics, Brigham Young University, Provo, UT 84602. D. J. Thurgood, Depart-ment of Civil and Environmental Engineering, Brigham Young University, 5865 West, 10100 North, Highland, UT 84003. Corresponding author: G. G. Schultz, gschultz@byu.edu.

Schultz, Thurgood, Olsen, and Reese 97

positive impact on safety that has generally been well documented in the literature. For example, researchers have shown that roadways with raised medians generally experience lower crash rates than roadways of similar use and annual average daily traffic (AADT) that are undivided or include a two-way left-turn lane (28). These analyses were generally conducted using traditional before-and-after approaches to analyze crash statistics. Over the past several years, more statistically rigorous methodologies for safety analysis have emerged.

Safety analysis

To determine the safety of a site during analysis, it is essential to first define what safety is and how it is measured. Roadway safety is usually defined and evaluated by recorded numbers of crashes or crash rates. Severity of crashes also plays an important role in the understanding of roadway safety. For example, one site may experi-ence considerably more crashes than another site. However, the second site may have a much larger proportion of severe, even fatal, crashes. Therefore, both frequency of crashes and severity of crashes are essential in determining the safety of a facility.One of the key reasons it is difficult to understand why crashes

occur is that crashes are usually not the result of one factor, but instead the combination of several events, circumstances, and factors. The combination of multiple events can significantly alter the amount of risk a driver may face. Even if it were possible to account for all possible factors that lead to a crash, the ability to predict a crash is not absolute. The key principle is that understanding the nature of crashes is a vastly complex and random process when only the known factors are considered. It is also important to remember that unknown factors also contribute to crashes. The complexity of known and unknown contributing factors can be overcome through the devel-opment and use of proper statistical tools that can correctly model crash characteristics and behavior. To properly analyze crash statis-tics, it is essential to analyze the characteristics of crash statistics to determine the proper statistical tools to use.

Characteristics of Crashes

One of the key concepts of a crash study is that although there are trends and factors that increase the likelihood of crashes, the occur-rence of a crash is not completely predictable. Crashes, by nature, are random events. The frequencies of crashes will naturally fluctuate from year to year. Fluctuations in crash frequency make it difficult to determine whether a reduction in the number of crashes is a result of a specific treatment, changes in site conditions over time, or a result of natural fluctuations from stochastic processes. These fluctuations present a phenomenon referred to as regression-to-the-mean (RTM) bias. The RTM phenomenon expects that a value that is determined to be extreme will tend to regress to the long-term average over time. This means that a period of high crash frequencies at a site is statistically probable to be followed by a period of low crash frequencies (9).Traditional methods have been used for many years to analyze

crash statistics. Traditional methods use relatively straightforward before-and-after approaches to compare the crash frequency or rate of an entity immediately before an improvement was made with the crash frequency or rate directly after the improvement to determine the effectiveness of the treatment (9). One of the problems with

many traditional analysis methods is that they do not account for the RTM bias. If the RTM bias is not accounted for, the effectiveness of a specific treatment is likely to be inaccurately reported.

Predicting Crashes

The RTM bias provides evidence of limitations when short-term data are used for analysis, which would lead to the assumption that using data for longer periods provides a better representation of crash behavior at a site. However, there are problems associated with this method as well. The characteristics of a site, such as traffic volume, weather, and pavement condition, change over time. Some of these characteristics, such as weather, continually fluctuate. Other factors, such as pavement condition and roadway markings, dete-riorate gradually over years of use. These latter factors create a legitimate limitation when using long-term crash statistics for site analysis.Factors contributing to crashes can never be completely controlled

or maintained, providing a level of difficulty in accurately predicting crashes. Crash data are statistically classified as count data and by nature are nonnegative integers; therefore generalized linear models are insufficient because the assumption that the dependent variable is continuous is not true for crash studies (10). Previous studies have suggested the use of Poisson models or negative-binomial models as more appropriate for count statistics (1113).

Empirical Bayesian Method

Several methods have been developed that more accurately determine the effectiveness of a safety measure than traditional methods by combining observed crash statistics with predicted values obtained by the use of safety performance functions (SPFs), crash modification factors (CMFs), and local calibration factors (1). In recent years, interest in the use of various Bayesian approaches in traffic safety studies has increased significantly. One of the common methods being used in safety studies is the use of the empirical Bayesian (EB) method of analysis. The EB method corrects for the RTM bias by determining the expected crash frequency of an entity. The EB method combines an estimation of the crash frequency of the study site with charac-teristics of similar sites using SPFs to estimate the predicted number of crashes. The EB approach is demonstrably better suited to estimate safety than traditional methods (9).The EB method does suffer from deficiencies of its own. Perhaps

most prominent is the need to spend time, resources, and effort on the estimation of SPFs required for implementation of the EB method. Another major disadvantage of the EB approach is that the SPF is estimated using aggregate crash data for more than a year. Therefore, to accurately apply this model, the units of crash frequencies per three years needs to be maintained (i.e., annual crash data cannot be used in place of three-year aggregated data) (14). The EB method is also applicable only when both predicted and observed crash frequencies are available for a roadway network. Additionally, the EB approach has been criticized for its inability to incorporate uncertainties in the model parameters. The EB approach assumes the parameters are error free and can be replaced simply by their estimates for the posterior analysis. These limitations can be overcome with the use of the flexible modeling associated with the hierarchical Bayesian method (15).

98 Transportation Research Record 2223

Hierarchical Bayesian Method

In recent years, a full or hierarchical Bayesian approach has been suggested as a useful alternative to the EB approach. Though more complex, the hierarchical Bayesian approach has several advantages over the EB approach in that the hierarchical approach is believed to require less data for untreated reference sites, it better accounts for uncertainty in data used, and it provides more detailed causal inferences and more flexibility in selecting crash count distributions (16).In a hierarchical Bayesian analysis, prior (before) information

and all available data are integrated into posterior (after) distributions from which inferences can be made; therefore, all uncertainties are accounted for in the analyses. Hierarchical Bayesian methods may well be less costly to implement and may result in safety estimates with more realistic standard errors. A study performed by Iowa State University argues that with the use of a hierarchical Bayesian approach, it is possible to improve on the prediction of the expected number of crashes at a site while at the same time avoiding the need to obtain estimates of SPFs or CMFs (17).One important difference between hierarchical Bayesian and

the EB approach is the manner in which the model parameters are determined. In the EB approach, model parameters are dependent on the data only. Model parameters are estimated using techniques involving the use of crash data such as the maximum likelihood technique. However, to produce results close to the true population parameters, maximum likelihood estimators frequently require large amounts of data. In the hierarchical Bayesian approach, the prior distributions are fixed by modelers and are combined with the data to create a joint posterior distribution. In situations in which fewer data are available, the prior distribution simply receives compara-tively more weight in the posterior distribution, and a valid result is produced. Hence, hierarchical Bayesian methods are still reliable with fewer data points when priors are chosen carefully. Priors also account for RTM bias if the data observations are extreme. There has been increased interest in this approach over the past few years because of the modeling flexibility associated with these methods (15).The hierarchical Bayesian method was applied to evaluate the

safety effect of conversion from stop to signalized control at rural intersections in California. The results were then compared with those from the EB method, and it was found that the hierarchical Bayesian method can provide results similar to, if not better than, those of the EB approach (18).

Safety Model developMent

Because of the limitations and published concerns of traditional analysis methods, and the benefit of using Bayesian techniques to account for RTM bias and therefore more accurately evaluate safety benefits, a hierarchical Bayesian model was constructed as the analysis tool for this study. The analysis provides an opportunity to estimate the safety impacts of the installation of raised medians in Utah. The model uses crash frequency (overall crashes and severe crashes) and AADT data for selected analysis sites as inputs (although the model is developed such that other covariates may also be included). The details of the model can be found in the literature (19, 20); the basic model development is included in this section to aid the reader in understanding the analysis performed.For the model development, it was assumed that yi is Poisson

distributed as outlined in Equation 1. The Poisson distribution is

used because of the nature of crash data classified as count data. The Poisson distribution also allows for an exposure parameter, which in this case is the length of the roadway segment. The number of crashes per mile (or severe crashes per mile), rather than the number of crashes solely, is then modeled for each roadway segment. Data from segments with varying segment length may thus be included in the same model appropriately.

yi i~ ( )Poisson ( ) 1where yi is the number of crashes per mile for each AADT segment and qi is the mean number of crashes per mile.The estimation of the mean number of crashes per mile is calculated

using Equation 2.

log ( ) i A i B i iA B( ) = + + 1 2AADTwhere

b = regression coefficient of the independent variable (A represents the after period, B represents the before period, and 1 corresponds to AADT),

Ai = 1 if the ith observation is from the after period and 0 otherwise,

Bi = 1 if the ith observation is from the before period and 0 otherwise, and

AADTi = AADT for the ith observation.

The analysis result of this model is the consideration of two intercepts, one for the before period and one for the after period, with the coefficient for AADT constrained to be the same for both periods. The log transformation was chosen as part of the standard Poisson regression procedures (21). The prior distribution for each bj where j {A, B, 1} is normally distributed as discussed in the literature (12) and specifically defined in Equation 3.

j ~ , ( )norm 0 1 3( )The priors, p(bj), were selected with mean zero because previous

analyses of this type have not been performed on the chosen road-way segments to suggest higher or lower mean values for the prior distributions. Zero mean values allow the posterior distributions of the coefficients to be positive or negative according to the data with-out the prior pulling the posterior results in one way or the other. The posterior distribution, p(bA, bB, b1y), for the b parameters, up to a constant, is expressed in Equation 4.

pi pi pi pi

y P y

e

A B

eXi

i

n

( ) ( ) ( ) ( ) ( ) =( )

=

1

1 ( ) + +( ) =

( ) e ey Xin i i A B 12

1

2 212

(44)

where

Xi = matrix with Ai, Bi, and AADTi as its columns;b = matrix of parameters bA, bB, and b1; and n = total number of observations.

Because of the complexity of the normalizing constant of the pos-terior distribution, rather than deriving the distribution theoretically, Markov chain Monte Carlo methodology is used to sample from the posterior distribution using the scaled distribution shown in Equation 4 (12, 13, 22, 23). The results of the algorithm are several

Schultz, Thurgood, Olsen, and Reese 99

thousand random draws from the posterior distribution for each of the bj parameters. Inference and conclusions may be made on the basis of these random draws because they represent the appropriate marginal posterior densities.In this study, frequencies of both overall and severe crashes are

analyzed. The ability to consider more than one type of crash without recalibrating SPFs or CMFs is one of the flexibility features of the hierarchical Bayesian approach, which can be immensely helpful if a safety project is targeted to reduce a specific type of crash. Each of the raised median sites is analyzed individually, which allows for insights as to why the project may be effective at one study site but not at another. The sites are also combined to perform an overall analysis of the effectiveness of raised medians.

raiSed Median analySiS

Site Selection

Before a detailed crash analysis is conducted, study sites need to be selected. The study sites that have been selected for analysis where raised medians have been installed are discussed in this section. For crash analysis purposes, the study sites will be roadway segments where raised medians have been installed, as outlined in previous research in Utah (24). The following study sites were selected for analysis:

1. University Parkway (SR-265), Mile Point 1.21 to 1.96;2. Alpine Highway (SR-74), Mile Point 2.40 to 4.29;3. 400/500 South (SR-186), Mile Point 5.48 to 7.53;4. St. George Boulevard (SR-34), Mile Point 0.00 to 1.74; and5. SR-36, Mile Point 59.29 to 60.82.

Listed mile points are current mile points at the time of this study. More detail on the study sites is available in the literature (25).

data collection

Raw crash statistics were provided by the Utah Department of Trans-portation (DOT) Traffic and Safety Division from the Utah DOT crash database. The database contains records and statistics obtained from police reports for crashes that occurred on Utah roadways. At the time of this study, consistent data were available from 1998 to 2008.AADT data are used to measure total volume of vehicle traffic of a

highway or road. Previous research has determined that a relationship exists between crashes and AADT. Although the exact relationship is still not entirely known, it is known that the relationship is gener-ally nonlinear (9). However, AADT is still an important parameter in predicting crash frequency and was used as a covariant in the development of the model.Locations of crashes are reported as the mile point at which the

crash occurred on the corresponding roadway. However, mile points on Utah highways have undergone several changes over the past 10 years. Shifts in mileposts are usually the result of either a realign-ment of the roadway or an extension added on to either end of the roadway. Although the segments of each roadway of interest were held constant through each analysis year, corresponding mile points have changed over the course of the study period. To ensure that data for the correct segment was used for each analysis year, correct mileposts were verified through the Utah DOT.

An overall analysis on frequency of all crashes and a severe crash analysis on frequency of severe crashes were performed for the raised median study sites. Crash severity refers to the severity correspond-ing to the most severe of injuries sustained as a result of a crash. According to the National Safety Council, There are five mutually exclusive categories of injury severity for classification of road vehicle (crashes) (26). The five categories are fatal, incapacitating injury, nonincapacitating evident injury, possible injury, and non-injury. A common abbreviation for these severity levels is the KABCO scale, with each letter representing fatal through noninjury levels of severity, respectively. The five severity classifications are mutually exclusive because a crash is classified according to the most severe injury (e.g., a crash with a fatality and a minor injury is classified as a fatal crash, not a fatal crash and a minor injury crash). In this study, severe crashes were determined to be crashes indicated on the report as fatal (K) or incapacitating injury (A).

analySiS reSultS

individual Site analysis results

A hierarchical Bayesian analysis was performed at selected study sites at which raised medians have been installed. An analysis was performed on overall crash frequency and severe crash frequency for each segment. The results of the hierarchical Bayesian model before and after the raised median installation were used to calculate a percent change in crash frequency. In addition to the percent change, the probability that the crash frequency (overall or severe) decreased was calculated. The probability of reduction helps to identify the statistical significance of the change.The analysis of the individual study sites at which raised medians

have been installed showed three of the five study sites experienced a statistically significant (greater than 95% probability) reduction (26% to 43%) in the overall crash frequency. The probability of difference for the remaining two sites (SR-265 and SR-74) was too low to confidently determine whether a reduction or increase occurred (although the results for SR-74 were practically significant at a 93% probability of decrease). In these situations the mean is increasing, but not at a statistically significant rate. Table 1 provides a summary of the impact of raised medians on all crashes.Similar to the overall crash analysis, several of the study sites

also showed a reduction in the frequency of severe crashes. Table 2 provides a summary of the impact of raised medians on severe crashes. The results indicate that three of the five study sites experienced a significant (greater than 95% probability) reduction (60% to 67%) in the frequency of severe crashes along the segment after raised medians were installed. The analysis indicated an increase (55%) may have occurred at one of the remaining sites (SR-74); however, the probability of a difference was too low to confidently determine if a reduction or increase occurred. The final site (SR-36) showed a 90% probability (practically significant) that a decrease of 43% occurred. More detailed analysis results for both overall and severe crash frequencies are provided in the literature (25).

results of combined Site analysis

The safety impacts of all study sites chosen for analysis combined are discussed in this section. Two types of plots are produced for each analysis performed. The first plot is a plot of the actual data.

100 Transportation Research Record 2223

The data plot displays the actual data points and the mean of the posterior predictive distribution, which is a representation of the mean regression line through the points from a Bayesian perspective. The reduction is calculated by taking the mean of the posterior distribution of differences between the two intercepts. The mathematical details are discussed in the literature (19, 20), but are conceptually equivalent to taking the after curve and dividing it by the before curve to obtain the percent reduction.The second plot produced for the overall analysis results is the

plot of the distribution of the differences between the before and after periods. The difference plots display the posterior distributions of differences between the before and after intercepts of the model. Negative values indicate that the after time period saw a reduction in crashes. Because the exact form of the posterior distributions is unknown, the model uses simulated draws from the posterior with the Markov chain; because those draws represent the actual poste-rior distribution, the proportion of the draws less than zero repre-sents the probability that there was a reduction in crashes from the before time period to the after time period.Figure 1 displays the overall crash frequency for the before and

after periods as a function of AADT. The overall analysis results indicate a 39% reduction in overall crash frequency after the raised medians were installed. Figure 2 shows the corresponding probability distribution of the differences between the before and after periods for overall crashes. The entire distribution of differences in Figure 2 is less than zero, indicating a 100% probability that a reduction in overall crash frequency occurred after raised medians were installed.The analysis results of severe crashes display an even greater

reduction. Figure 3 displays the results for all study sites at which raised medians were installed. The severe crash analysis results on all study sites show a 44% reduction in severe crash frequency after raised median installation. Figure 4 shows the corresponding prob-ability distribution of the differences between the before and after periods for severe crashes. As with the overall analysis, the entire distribution of differences shown in Figure 4 is less than zero, indi-

cating a 100% probability that a reduction in severe crash frequency occurred after raised medians were installed at the study sites.

concluSionS

The analysis in this report was performed with a hierarchical Bayesian model developed to analyze the effectiveness of various treatments for improving roadway safety. The model is a valuable tool with various applications to transportation safety. As part of this research, the model was applied to study sites at which raised medians had been installed throughout the state of Utah. This study analyzed the effectiveness of raised medians on roadway safety by determin-ing the effect each newly installed median has had on frequency of overall crashes and of severe crashes at the study sites. An analysis was performed on individual sites at which raised medians have been installed as well as an overall analysis on all study sites grouped together.The results of the raised median analysis indicated a significant

improvement in both crash frequency and frequency of severe crashes along corridors where raised medians were installed. Results from all study sites combined show that the overall crash frequency was reduced by 39% and the frequency of severe crashes was reduced by 44% after the installation of raised medians along the study sites. The reduction in frequency of severe crashes is anticipated to be a result in the change in types of collisions. This study provides fur-ther and more statistically significant evidence that installing raised medians is an effective technique to reduce crash frequency, especially severe crashes caused by sideswipes or head-on collisions.The model developed for this research can be expanded to

determine the impact of other safety measures on various crash types. Additionally, AADT was the only covariate used in this study. The model has been developed such that additional covariates may be included in the analysis. Selection of the appropriate covariates to be used depends on the scope of the study being performed. Finally,

TABLE 1 Summary for All Crashes

State Year of Probability of Percent Study Site Route County Installation Reduction (%) Change

University Parkway 265 Utah 2002 38 3Alpine Highway 74 Utah 2002 93 -19400/500 South 186 Salt Lake 2001 100 -29St. George Boulevard 34 Washington 2006 100 -26SR-36 36 Tooele 2005 100 -43

TABLE 2 Summary of Severe Crashes

State Year of Probability of Percent Study Site Route County Installation Reduction (%) Change

University Parkway 265 Utah 2002 100 -60Alpine Highway 74 Utah 2002 41 55400/500 South 186 Salt Lake 2001 100 -67St. George Boulevard 34 Washington 2006 99 -61SR-36 36 Tooele 2005 90 -49

Schultz, Thurgood, Olsen, and Reese 101

30,000 AADT

10,000 20,000 40,000 50,000

100

Cras

hes

per M

ile p

er Y

ear

0 50

15

0 20

0

FIGURE 1 Crash frequency for all raised median study sites.

FIGURE 2 Distribution of differences for crashes on all raised median study sites.

102 Transportation Research Record 2223

30,000 AADT

10,000 20,000 40,000 50,000

10

Seve

re C

rash

es p

er M

ile p

er Y

ear

0 5

15

FIGURE 3 Severe crash frequency for all raised median study sites.

FIGURE 4 Distribution of differences of severe crashes for all raised median study sites.

Schultz, Thurgood, Olsen, and Reese 103

one of the important elements of transportation safety planning is identifying locations that experience an unusually high crash fre-quency. The model outlined in this report can be used to identify outlier sites for various types of crashes. The raised median analysis revealed several outlier sites that experienced an unusually high crash frequency in either overall or severe crashes. Further exploration can be performed to identify any factors that contribute to the unusually high crash frequency that can be mitigated.

acknowledgMentS

This research was made possible with funding from the Utah DOT and Brigham Young University. Special thanks to Robert Hull, W. Scott Jones, Tim Taylor, David Stevens, Mitsuru Saito, Steven Dudley, and others at the Utah DOT and the university who played key roles as members of the technical advisory committee and provided input to the research.

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The contents of this paper reflect the views of the authors, who are responsible for the facts and accuracy of the information, and are not necessarily representative of the sponsoring agency.

The Access Management Committee peer-reviewed this paper.