The appropriateness of the Poisson model is frequently challenged when examining spatial count data marked by unbalanced distributions, over-dispersion, or under-dispersion. Moreover, traditional parametric models may inadequately capture the relationships among variables when covariates display ambiguous functional forms or when spatial patterns are intricate and indeterminate. To tackle these issues, we propose an innovative Bayesian hierarchical modeling system. This method combines non-parametric techniques with an adapted dispersed count model based on renewal theory, facilitating the effective management of unequal dispersion, non-linear correlations, and complex geographic dependencies in count data. We illustrate the efficacy of our strategy by applying it to lung and bronchus cancer mortality data from Iowa, emphasizing environmental and demographic factors like ozone concentrations, PM2.5, green space, and asthma prevalence. Our analysis demonstrates considerable regional heterogeneity and non-linear relationships, providing important insights into the impact of environmental and health-related factors on cancer death rates. This application highlights the significance of our methodology in public health research, where precise modeling and forecasting are essential for guiding policy and intervention efforts. Additionally, we performed a simulation study to assess the resilience and accuracy of the suggested method, validating its superiority in managing dispersion and capturing intricate spatial patterns relative to conventional methods. The suggested framework presents a flexible and robust instrument for geographical count analysis, offering innovative insights for academics and practitioners in disciplines such as epidemiology, environmental science, and spatial statistics.

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