On the use of corrections for overdispersion

In studying fluctuations in the size of a blackgrouse (Tetrao tetrix) popul ation, an autoregressive model using climatic conditions appears to follow the changes quite well. However, the deviance of the model is considerably larger than its number of degrees of freedom. A widely used statistical rul e of thumb holds that overdispersion is present in such situations, but mod el selection based on a direct likelihood approach can produce opposing res ults. Two further examples, of binomial and of Poisson data, have models wi th deviances that are almost twice the degrees of freedom and yet various o verdispersion models do not fit better than the standard model for independ ent data. This can arise because the rule of thumb only considers a point e stimate of dispersion, without regard for any measure of its precision. A r easonable criterion for detecting overdispersion is that the deviance be at least twice the number of degrees of freedom, the familiar Akaike informat ion criterion, but the actual presence of overdispersion should then be che cked by some appropriate modelling procedure.