Metropolis approximation to assess dependence between fixed and random effects in a count model for overdispersed data

This analysis will provide inference on extra-Poisson variation in annual tornado counts from 12 US states, under the assumption that the counts arise from a Poisson distribution. Parameterizing the rate of occurrence as a function of time varying covariates including Sea Surface Temperatures yields a fixed effects Poisson model. We use such a model as a vehicle to test counts for equal mean and variance-a condition called equidispersion. The rejection of equidispersion prompts the introduction of state-specific random variability in the Poisson regression model to get a negative binomial model. We find that in the presence of the random effect, Sea Surface Temperatures becomes insignificant while Year remains significant. Assessing the relation between Year and the state-specific random effects proceeds by estimating the joint posterior full conditional of the regression parameters for different levels of the parameter representing state heterogeneity.