  In employing spatial regression models for counts, we usually meet two
issues. First, ignoring the inherent collinearity between covariates and the
spatial effect would lead to causal inferences. Second, real count data usually
reveal over or under-dispersion where the classical Poisson model is not
appropriate to use. We propose a flexible Bayesian hierarchical modeling
approach by joining non-confounding spatial methodology and a newly
reconsidered dispersed count modeling from the renewal theory to control the
issues. Specifically, 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 consequently, we can carry
out the fast computation using the integrated nested Laplace approximation
method. We also examine different popular approaches for handling spatial
confounding and compare their performances in the presence of dispersion. We
use the proposed methodology to analyze a clinical dataset related to stomach
cancer incidence in Slovenia and perform a simulation study to understand the
proposed approach's merits better.
