Abstract
Overdispersion often occurs when fitting a binomial, multinomial or Poisson model to count data. In the Bayesian setting, failure to allow for overdispersion leads to the posteriors for the parameters being too narrow. A simple and natural approach is to incorporate parameter heterogeneity in the model, e.g. by adding a random effect to the linear predictor. However, overdispersion can also be caused by a lack of independence, which may not be straightforward to model explicitly. In addition, there may still be some residual overdispersion after allowing for heterogeneity or lack of independence. In many settings where overdispersion is present, it is reasonable to assume that the variance of the response variable is proportional to that assumed by the model. When this is the case, we propose estimating the amount of overdispersion, and discuss the link between this estimate and the use of a posterior predictive p-value to check lack-of-fit. We also provide a residual plot that can be used to check the assumption of proportionality. We show how to use the estimate of overdispersion to make a simple adjustment to the posterior distribution for each parameter, analogous to the use of quasi-likelihood in the frequentist setting. We use two examples, regression modelling of count data and estimation of survival from a mark-recapture study, to illustrate the calculation of the estimate of overdispersion, and the resulting adjustment to the posteriors. We perform simulation studies based on the examples to assess the frequentist coverage properties of the adjusted posteriors. In both simulation studies, the adjusted posteriors lead to credible intervals that have approximately the correct coverage for a range of overdispersion scenarios. Our approach provides a new, simple and robust tool for Bayesian modelling of overdispersed data, when it is reasonable to assume that the variance of the response variable is proportional to that assumed by the model.