Modelling of Overdispersed Count Rates

This paper revisits the common problem of analysing counts recorded over time through the modelling of the underlying rate, motivated by the analysis of a cancer treatment related study. The baseline Poisson model is simply implemented though the inclusion of an offset for the different exposure/recording times and the underlying Poisson process gives other nice well-known properties. We consider how this approach can be extended to models for overdispersed data. The use of a simple offset with a negative binomial model is common practice and we consider the appropriateness of this and the resulting implications. We discuss how these ideas extend to more general mixed Poisson models, including ZIP, to handle zero-inflation, and ZINB for zero-inflation and overdispersion. The simple offset approach does not extend to other extended count models such as the COM-Poisson and general weighted Poisson distributions.