Estimating parameters in the Rasch model in the presence of null categories

A category with a frequency of zero is called a null category. When null categories are present in polytomous responses, then in the Rasch model for such responses, the thresholds that define the categories are inestimable with the commonly used joint maximum likelihood, marginal maximum likelihood, or standard conditional maximum likelihood estimation algorithms. The reason for this situation is that in principle, these estimation algorithms involve frequencies of each category. Andrich and Luo (2003) describe an algorithm in which the thresholds are reparameterized into their principal components and in which the estimate of any threshold is based on a function of the frequencies of all categories of the item rather than the frequency of a particular category. This algorithm works in the presence of null categories. However, in situations where the null categories are at the extremes of a set of categories, the estimates themselves can become too extreme. This paper describes a procedure in which the solution algorithm described by Andrich and Luo is further adapted in the presence of null categories by using their expected frequencies. The procedure is demonstrated with simulated and real data.