A Bayesian Version of Rasch’s Multiplicative Poisson Model for the Number of Errors of an Achievement Test

Abstract

Apart from the widely known logistic model for binary scored items, Rasch developed several other models for the analysis of achievement test data. The model considered in this paper is the so-called multiplicative Poisson model for misreadings. This model assumes that the number of reading errors that an individual makes when reading a text is approximately Poisson distributed with an intensity parameter λ, which depends on the ratio of two other parameters, one pertaining to the ability of the individual and one to the difficulty of the words in the text. In this paper a Bayesian procedure is developed for the simultaneous estimation of the ability and difficulty parameters. Similar procedures have been applied to the estimation of the parameters of the logistic model. According to several criteria, the Bayesian estimates are better than comparable maximum likelihood estimates. The method is illustrated by an empirical example. Some evidence on the potential usefulness of the method, based on Monte Carlo studies, is given.