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 approachs merits better.
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