Loglinear representations of multivariate Bernoulli Rasch models

In this paper, the extended Rasch model for dichotomously scored items is derived from the general multivariate Bernoulli distribution. The necessary and sufficient conditions for the multivariate Bernoulli distribution to be equal to the extended Rasch model provide a new loglinear representation of the extended Rasch model. Conditions are also given under which the extended Rasch model is equal to the random effects Rasch model, and it is shown under what conditions the extended Rasch model is equal to a random effects Rasch model in which the underlying variable has a normal distribution. In addition, alternative models for the construction of likelihood ratio tests are proposed. One of these alternative models is Haberman's extended interaction model. Furthermore, it is shown how both the SPSS and SAS programs can be used to estimate and test loglinear representations of extended Rasch models.