The Problem With the Step Metaphor for Polytomous Models for Ordinal Assessments

P enfield’s (2014) “Instructional Module on Polytomous Item Response Theory Models” begins with a review of dichotomous response models. He refers to these as The Building Blocks of Polytomous IRT Models: The Step Function. The mathematics of these models and their interrelationships with the polytomous models is correct. Unfortunately, the step characterization for dichotomous responses, which he uses to explain the two most commonly used classes of polytomous models for ordered categories, is incompatible with the mathematical structure of these models. These two classes of models are referred to in Penfield’s paper as adjacent category models and cumulative models. At best, taken in the dynamic sense of taking a step, the step metaphor leads to a superficial understanding of the models as mere descriptions of the data; at worst it leads to a misunderstanding of the models and how they can be used to assess if the empirical ordering of the categories is consistent with the intended ordering. The purpose of this note is to explain why the step metaphor is incompatible with both models and to summarize the distinct processes for each. It is also shows, with concrete examples, how one of these models can be applied to better understand assessments in ordered categories.