Extended Poisson process modelling and analysis of grouped binary data

Abstract

A simple extension of the Poisson process results in binomially distributed counts of events in a time interval. A further extension generalises this to probability distributions under‐ or over‐dispersed relative to the binomial distribution. Substantial levels of under‐dispersion are possible with this modelling, but only modest levels of over‐dispersion – up to Poisson‐like variation. Although simple analytical expressions for the moments of these probability distributions are not available, approximate expressions for the mean and variance are derived, and used to re‐parameterise the models. The modelling is applied in the analysis of two published data sets, one showing under‐dispersion and the other over‐dispersion. More appropriate assessment of the precision of estimated parameters and reliable model checking diagnostics follow from this more general modelling of these data sets.