Identification of parametric Rasch-type models

Abstract

In modern Item Response Theory, the Rasch model is viewed as a Generalized Linear Mixed Model, where the item parameters correspond to the fixed-effects, whereas the person specific parameters are the random-effects. The statistical model, bearing on the observable variables only, is obtained after integrating out the random-effects. Although it is widely accepted that the parameters of this model are identified, it is hard to find a correct justification. Furthermore, the meaning of the parameters of the Rasch model – as well as of its extensions – is typically based on the fixed-effects specification of the model, that is, when the person specific parameters are also treated as fixed-effects. The contribution of this paper is to provide an explicit proof of the identification of the random-effects Rasch model. The proof is valid for a large class of Rasch-type models. It is also shown that such a proof can be applied to analyze the identification of Explanatory Rasch Models. Finally, the meaning of the parameters of interest with respect to the different data generating process is discussed.