Effects of low sample mean values and small sample size on the estimation of the fixed dispersion parameter of Poisson-gamma models: A Bayesian Perspective

There has been considerable research conducted on the development of statistical models for predicting motor vehicle crashes on highway facilities. Many of these developments were performed for the likelihood-based or frequentist modeling approach. Over the last few years, there has been a significant increase in the application hierarchical Bayes method for modeling motor vehicle crashes. Whether the inferences are estimates using the likelihood-based or the Bayesian method, the most common probabilistic structure used for modeling this type of data remains the traditional Poisson-gamma (or Negative Binomial) distribution. Crash data collected for highway safety studies often have the unusual attributes of being characterized by low sample mean values and, due to the prohibitive costs of collecting data, small sample sizes. Previous studies have shown that the dispersion parameter of Poisson-gamma models can be seriously mis-estimated when the coefficients are estimated using the maximum likelihood method (MLE) for these extreme conditions. Despite important work done on this topic for the MLE, nobody has so far examined how low sample mean values and small sample sizes affect the posterior mean of the dispersion parameter of Poisson-gamma models estimated using the hierarchical Bayes method. The inverse dispersion parameter (posterior mean) plays an important role in various types of highway safety studies, such as building confidence intervals for comparing the safety performance of different highway design alternatives and the application of the empirical Bayes (EB) method for refining the long-term mean of a highway entity, and is particularly important for practitioners who are not familiar with Bayesian methods. It is therefore vital to determine the conditions in which the inverse dispersion parameter may be mis-estimated for this category of models. To accomplish the objectives of this study, a series of Poisson-gamma distributions are simulated using different values describing the mean, the dispersion parameter, the sample size, and the prior specification. Non-informative and informative prior specifications are tested for determining the magnitude of the biases introduced by low sample mean values and small sample sizes. A series of Poisson-lognormal distributions are also simulated, in the light of recent work done by statisticians on this mixed distribution. The study shows that a dataset characterized by a low sample mean combined with a small sample size can seriously affect the estimation of the posterior mean of the dispersion parameter when a non-informative prior specification is used to characterize the gamma hyper-parameter. The risk of a mis-specified posterior mean can be greatly minimized when an appropriate informative prior distribution is used. Finally, the study shows that Poisson-lognormal models are recommended over Poisson-gamma models whenever crash data characterized by low sample mean values are used for developing crash prediction models.