Mixture Models for the Analysis of Repeated Count Data

Abstract

SUMMARY Repeated count data showing overdispersion are commonly analysed by using a Poisson model with varying intensity parameter, resulting in a mixed model. A mixed model with a gamma distribution for the Poisson parameter does not adequately fit a data set on 721 children's spelling errors. An alternative approach is a latent class or mixture model in which the distribution of the intensity parameter is a step function. This gives a solution with many classes that is difficult to interpret. A combination of the two models, resulting in a mixture model with two gamma distributions, however, fits the data very well. Moreover, it yields a substantively satisfactory interpretation: two heterogeneous classes of 'good' and 'poor' spelling children can be identified. Therefore, mixture models for the analysis of overdispersed repeated count data are proposed, where the counts have independent Poisson distributions conditional on the Poisson parameter whose distribution is a mixture of gamma distributions. Combining marginal maximum likelihood methods and the EM algorithm leads to straightforward estimations of the models, for which goodness-of-fit tests are also presented.