Empirical Bayes analysis for a hierarchical negative binomial generalized linear model

The negative binomial (NB) distribution has provided a representation for count data in many areas of research. We propose a parametric hierarchical empirical Bayes (EB) approach for estimating the parameters of negative binomial (NB) generalized linear regression models (GLiM). We re-parametrize the NB in terms of dispersion and proportion parameters and put a beta prior on the latter parameter. We link the covariate information to the prior mean of the proportions wia a parametric family of link functions. We estimate the hyperparameters and the dispersion parameter from the beta-negative binomial marginal likelihood and we dene an EB estimator of the proportion parameters as well as the mean responses (counts) at each value (or combination of values) of the covariates. We construct EB condence intervals for the proportion as well as the mean response parameters by using the denition of Morris (1983). We apply our procedure to the environmental toxicity data of Paul et al. (1997) to emphasis its utility. A simulation study is then carried out to investigate the performance of the proposed point and interval EB estimators as compared to the usual NB GLiM estimates in terms of biases and MSEs.