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17 CHAPTER 3 Data Analysis The first section of this chapter presents the methodological Negative binomial models are typically used in developing approach and related issues. The second section presents Crash Reduction Factors (CRFs) or Accident Modification the data used in the development of the prediction models Factors (AMFs). Even though these two terms are in general and AMFs. similar in concept, there are slight differences. A CRF is a value that represents the reduction of crashes due to a safety Methodology improvement at a roadway spot or section. Such values rep- resent the percent improvement on the roadway and most Over the past decades, interest has increased in estimating the often have a positive connotation--that is, the safety inter- safety implications from changes in various design elements. To vention will have a positive result. On the other hand, an AMF be able to determine these changes, models were developed is a constant that represents the safety change due to a change that could predict the crash-rate frequency or the number of in a value of the segment. These factors are typically the ratio crashes as a function of various traffic conditions and values of the expected values of crashes with and without the change. of geometric elements. A significant part of past research was AMFs are also used as multipliers for estimating the expected devoted to developing such models; in the past decade, most number of crashes, and values less than 1.0 indicate fewer researchers have used negative binomial models for modeling crashes as a result of the change. crashes. These models assume that unobserved crash variation The basic concept of the AMF is to capture the change in across roadway segments is gamma-distributed, while crashes crash frequency due to the change of a single element. How- within sites are Poisson-distributed (41). The Poisson, Poisson- ever, this is often not the case, and these factors have been Gamma (negative binomial), and other related models are developed using cross-sectional studies where multivariate collectively called "generalized linear models" (GLM). These models were developed and used in the determination of AMFs. models have the general form of Equation 1: The models typically include all contributing factors that could E [ N ] = EXPO e b +b X +b X + . . . + b X 0 1 1 2 2 n n (1) influence safety and then use them to estimate the change in crashes due to a change in one unit of the variable of concern. where This approach is typically completed with the assistance of an E[N ] = predicted number of crashes per year for a expert panel that evaluates the use of the prediction models roadway section, and estimates the potential effect for each variable of concern. EXPO = exposure to crashes, These evaluations could be further supported by the existing b0, . . . , bn = regression coefficients, and literature and current knowledge for the specific variable. X1, . . . , Xn = predictor variables. This approach was used in the two-lane rural roadway models as part of the IHSDM, where the models developed were used Models developed similar to Equation 1 will be capable of as the basis for the creation of the AMFs. AMFs may appear identifying the relationship of the number of crashes to the subjective in nature, but they represent a collective "wisdom" various elements to be considered. The measure of exposure based on expert panel knowledge, field observation, and used in these prediction models could be either the traditional findings in the research literature. The key limitation to this vehicle-miles (i.e., length Average Daily Traffic (ADT) vol- approach for AMF development is that there may not be ade- ume), or the length itself while the ADT becomes a predictor quate literature dealing with the identification of the safety variable. impacts from the elements of interest.