Unmixing Rasch scales: How to score an educational test

One of the important questions in the practice of educational testing is how a particular test should be scored. In this paper we consider what an appropriate simple scoring rule should be for the Dutch as a second language test consisting of listening and reading items. As in many other applications, here the Rasch model which allows to score the test with a simple sumscore is too restrictive to adequately represent the data. In this study we propose an exploratory algorithm which clusters the items into subscales each fitting a Rasch model and thus provides a scoring rule based on observed data. The scoring rule produces either a weighted sumscore based on equal weights within each subscale or a set of sumscores (one for each of the subscales). An MCMC algorithm which enables to determine the number of Rasch scales constituting the test and to unmix these scales is introduced and evaluated in simulations. Using the results of unmixing, we conclude that the Dutch language test can be scored with a weighted sumscore with three different weights.