The Derivation of Polytomous Rasch Models

Abstract

This chapter, analogous to Chapter 2, derives polytomous Rasch models from certain sets of assumptions. First, it is shown that the multidimensional polytomous Rasch model follows from the assumption that there exists a vector-valued minimal sufficient statistic T for the vector-valued person parameter θ, where T is independent of the item parameters β_i; the sufficient statistic T is seen to be equivalent to the person marginal vector. Second, it is shown that, within a framework for measuring change between two time points, the partial credit model and the rating scale model follow from the assumption that conditional inference on (or specifically objective assessment of) change is feasible. The essential difference between the partial credit model and the rating scale model is that the former allows for a change of the response distribution in addition to a shift of the person parameter, whereas the rating scale model characterizes change exclusively in terms of the person parameter θ. A family of power series models is derived from the demand for conditional inference which accomodates the partial credit model, the rating scale model, the multiplicative Poisson model, and the dichotomous Rasch model.