The Rasch Poisson counts model for incomplete data:An application of the EM algorithm

Rasch's Poisson counts model is a latent trait model for the situation in which K tests are administered to N examinees and the test score is a count [e.g., the repeated occurrence of some event, such as the number of items completed or the number of items answered (in)correctly]. The Rasch Poisson counts model assumes that the test scores are Poisson distributed random variables. In the approach presented here, the Poisson parameter is assumed to be a product of a fixed test difficulty and a gamma-distributed random examinee latent trait parameter. From these assumptions, marginal maximum likelihood estimators can be derived for the test difficulties and the parameters of the prior gamma distribution. For the examinee parameters, there are a number of options. The model can be applied in a situation in which observations result from an incomplete design. When examinees are assigned to different subsets of tests using background information, this information must be taken into account when using marginal maximum likelihood estimation. If the focus is on test calibration and there is no interest in the characteristics of the latent traits in relation to the background information, conditional maximum likelihood methods may be preferred because they are easier to implement and are justified for incomplete data for test parameter estimation.