Empirical Bayes method in the study of traffic safety accounting for spatial and temporal heterogeneity

The empirical Bayes (EB) approach is currently the standard method for estimating traffic safety. In general, expected crash frequencies across sites are estimated via the negative binomial (NB) model, assuming the homogeneous dispersion parameter and time invariant safety. The heterogeneous negative binomial (HNB) model has been recently used to estimate safety due to its flexibility for explaining the heterogeneous dispersion parameter. Hauer proposed a modified EB method since the time invariant safety assumption may be invalid. However, it is still unclear whether the HNB model increases the accuracy of the EB estimate materially. Furthermore, no attempts have been made to examine the generalizable form of the marginal distribution resulting from the modified EB framework. Because the hyper parameters have generally been unavailable from the resulting marginal distribution, an assessment is lacking on how accurately the modified EB method predicts safety in the presence of time variant safety and regression-to-the-mean (RTM) effects.;Recognizing these challenges, this study first examines the nature of the HNB model, and shows that the EB estimates obtained from the HNB model significantly minimize the prediction error when the gamma prior variances are truly heterogeneous among sites having similar traits. Thus, the study shows that the HNB model is an efficient tool for identifying a suitable reference group in obtaining the EB estimates. Secondly, this study derives the closed form marginal distribution in the modified EB method, and reveals that the marginal distribution is equivalent to the negative multinomial (NM) distribution, which is essentially the same as the likelihood function used in the random effects Poisson model. As a result, this study shows that the gamma posterior distribution from the multivariate Poisson-gamma mixture can be estimated using the NM model or the random effects Poisson model. This study also shows that the prediction errors from the modified EB method are systematically smaller than those from the comparison group method by simultaneously accounting for RTM and time variant safety effects. Hence, the modified EB method via the NM model is a generalizable method for estimating safety in the presence of time variant safety and RTM effects.