A generalized Poisson-gamma model for spatially overdispersed data.

Modern disease mapping commonly uses hierarchical Bayesian methods to model overdispersion and spatial correlation. Classical random-effects based solutions include the Poisson-gamma model, which uses the conjugacy between the Poisson and gamma distributions, but which does not model spatial correlation, on the one hand, and the more advanced CAR model, which also introduces a spatial autocorrelation term but without a closed-form posterior distribution on the other. In this paper, a combined model is proposed: an alternative convolution model accounting for both overdispersion and spatial correlation in the data by combining the Poisson-gamma model with a spatially-structured normal CAR random effect. The Limburg Cancer Registry data on kidney and prostate cancer in Limburg were used to compare the conventional and new models. A simulation study confirmed results and interpretations coming from the real datasets. Relative risk maps showed that the combined model provides an intermediate between the non-patterned negative binomial and the sometimes oversmoothed CAR convolution model.