A Flexible Generalized Poisson Likelihood for Spatial Counts Constructed by Renewal Theory, Motivated by Groundwater Quality Assessment

In recent years, the availability of spatial count data has massively increased. Due to the ubiquity of over- or under-dispersion in count data, we propose a Bayesian hierarchical modeling approach based on the renewal theory that relates nonexponential waiting times between events and the distribution of the counts, relaxing the assumption of equi-dispersion at the cost of an additional parameter. Particularly, we extend the methodology for analyzing spatial count data based on the gamma distribution assumption for waiting times. The model can be formulated as a latent Gaussian model, and therefore, we can carry out fast computation using the integrated nested Laplace approximation method. The analysis of a groundwater quality dataset and a simulation study show a significant improvement over both Poisson and negative binomial models.Supplementary materials accompanying this paper appear on-line.