Modeling Repeated Count Data:Some Extensions of the Rasch Poisson Counts Model

We consider data that can be summarized as an N X K table of counts-for example, test data obtained by administering K tests to N subjects. The cell entries y(ij) are assumed to be conditionally independent Poisson-distributed random variables, given the NK Poisson intensity parameters mu(ij). The Rasch Poisson Counts Model (RPCM) postulates that the intensity parameters are products of test difficulty and subject ability parameters. We expand the RPCM by assuming that the subject parameters are random variables having a common gamma distribution with fixed unknown parameters and that the vectors of test difficulty parameters per subject follow a common Dirichlet distribution with fixed unknown parameters. Further we show how additional structures can be imposed on the rest parameters, modeling a within-subjects design. Methods for testing the fit and estimating the parameters of these models are presented and illustrated with the analysis of two empirical data sets.