Repeated count data showing overdispersion are commonly analysed by us
ing a Poisson model with varying intensity parameter. resulting in a m
ixed model. A mixed model with a gamma distribution for the Poisson pa
rameter 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 t
wo 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. The
refore, mixture models for the analysis of overdispersed repeated coun
t data are proposed, where the counts have independent Poisson distrib
utions conditional on the Poisson parameter whose distribution is a mi
xture of gamma distributions. Combining marginal maximum likelihood me
thods and the EM algorithm leads to straightforward estimations of the
models, for which goodness-of-fit tests are also presented.