Ignoring overdispersion in hierarchical loglinear models: Possible problems and solutions

Poisson data frequently exhibit overdispersion; and, for univariate models, many options exist to circumvent this problem. Nonetheless, in complex scenarios, for example, in longitudinal studies, accounting for overdispersion is a more challenging task. Recently, Molenberghs et.al, presented a model that accounts for overdispersion by combining two sets of random effects. However, introducing a new set of random effects implies additional distributional assumptions for intrinsically unobservable variables, which has not been considered before. Using the combined model as a framework, we explored the impact of ignoring overdispersion in complex longitudinal settings via simulations. Furthermore, we evaluated the effect of misspecifying the random-effects distribution on both the combined model and the classical Poisson hierarchical model. Our results indicate that even though inferences may be affected by ignored overdispersion, the combined model is a promising tool in this scenario.