Model selection with overdispersed data

A common problem with survey data is the potential of overdispersion. Overdispersion occurs when the data display more variability than is predicted by the variance--mean relationship for the assumed sampling model. When there is overdispersion, all model selection criteria would generally be expected to perform poorly. If overdispersion is ignored, a model with too many parameters is likely to be selected and can thus lead to overinterpretation of these parameters. In this paper we describe a simple modification of model selection criteria, such as the likelihood ratio statistic, the Akaike information criterion and the Bayesian information criterion, that allows for overdispersion. The proposed adjustment is based on Efron's double-exponential family formulation and can be incorporated into the generalized linear model framework straightforwardly. The proposed adjustment is illustrated using survey data on the relationship between social class and support for political parties in successive British general elections between 1964 and 1992.