The initial Poisson model (with the offset variable only) provided a deviance equal to 161.20, which was much greater than the degrees of freedom (equal to 40), suggesting a strong heterogeneity.
Akaike's Information Criterion is equal to the deviance of the maximum likelihood model, offset by a penalty for complexity equal to twice the number of parameters in the model.
To assess goodness of fit, we used the posterior mean residual deviance, whose expected value is approximately equal to the number of data points under the assumption that the model is true (35, 37, 38).
Under the null hypothesis that the model provides an adequate fit to the data, the residual deviance is expected to have a mean equal to the number of unconstrained data points.
We evaluate significance by computing the deviance statistic (minus 2 times the log-likelihood; change in deviance is distributed as chi-square, with degrees of freedom equal to the number of parameters deleted from the model) and with the help of 10,000 samples from the posterior distributions of the coefficients using Markov chain Monte Carlo sampling.
Thus, the effect of each variable was assessed by comparing differences in residual deviances with a χ distribution with degrees of freedom equal to the difference in residual degrees of freedom.
P-values were adjusted for overdispersion, when present, by using an inflation factor equal to the square root of the model deviance divided by the degrees of freedom.
The terms were judged by their explanatory contribution to the model, using the reduction in deviance due to individual variables: under the null hypothesis of no effect each deviance reduction has approximately a chi-squared distribution, with expectation equal to the number of degrees of freedom (d.f).f
In order to adjust for over-dispersion, we adjusted all standard errors and test statistics by a scale parameter equal to the square root of the observed deviance of the model divided by its degrees of freedom [ 16].
This deviance can now be tested against chi-square with degrees of freedom equal to the difference in numbers of parameters of the two models, at the desired level of significance (Hogg and Craig, 1994).
