Conditional Maximum Likelihood Estimation in Polytomous Rasch Models Using SAS

IRT models are widely used but often rely on distributional assumptions about the latent variable. For a simple class of IRT models, the Rasch models, conditional inference is feasible. This enables consistent estimation of item parameters without reference to the distribution of the latent variable in the population. Traditionally, specialized software has been needed for this, but conditional maximum likelihood estimation can be done using standard software for fitting generalized linear models. This paper describes an SAS macro %rasch_cml that fits polytomous Rasch models. The macro estimates item parameters using conditional maximum likelihood (CML) estimation and person locations using maximum likelihood estimator (MLE) and Warm's weighted likelihood estimation (WLE). Graphical presentations are included: plots of item characteristic curves (ICCs), and a graphical goodness-of-fit-test is also produced. 1. Introduction Item response theory (IRT) models were developed to describe probabilistic relationships between correct responses on a set of test items and continuous latent traits [1]. In addition to educational and psychological testing, IRT models have been also used in other areas of research, for example, in health status measurement and evaluation of Patient-Reported Outcomes (PROs) like physical functioning and psychological well-being wich are typical in applications of IRT models. Traditional applications in education often use dichotomous (correct/incorrect) item scoring, but polytomous items are common in other applications. Formally, IRT models deal with the situation where several questions (called items) are used for ordering of a group of subjects with respect to a unidimensional latent variable. Before the ordering of subjects can be done in a meaningful way, a number of requirements must be met.(i)Items should measure only one latent variable. (ii)Items should increase with the underlying latent variable. (iii)Items should be sufficiently different to avoid redundance. (iv)Items should function in the same way in any subpopulation. These requirements are standard in educational tests where (i) items should deal with only one subject (e.g., not being a mixture of math and language items), (ii) the probability of a correct answer should increase with ability, (iii) items should not ask the same thing twice, and (iv) the difficulty of an item should depend only on the ability of the student, for example, an item should not have features that makes it easier for boys than for girls at the same level of ability. Let denote the latent