A hierarchical Bayesian approach to negative binomial regression

There is a growing interest in establishing the relationship between the count data y and numerous covariates x through a generalized linear model (GLM), such as explaining the road crash counts from the geometry and environmental factors. This paper proposes a hierarchical Bayesian method to deal with the negative binomial GLM. The Negative Binomial distribution is preferred for modeling nonnegative overdispersed data. The Bayesian inference is chosen to account for prior expert knowledge on regression coefficients in a small sample size setting and the hierarchical structure allows to consider the dependence among the subsets. A Metropolis-Hastings-within-Gibbs algorithm is used to compute the posterior distribution of the parameters of interest through a data augmentation process. The Bayesian approach highly over-performs the classical maximum likelihood estimation in terms of goodness of fit, especially when the sample size decreases and the model complexity increases. Their respective performances have been examined in both the simulated and real-life case studies.