Implied probabilities of polytomous response functions for model-based prediction and comparison

Polytomous response models are typically motivated by assumptions about specific dichotomizations of the responses. We build on this concept by considering all possible dichotomizations. We first enumerate the full range of possible dichotomizations for an outcome in a given number of categories. We then show that many of these dichotomizations lead to “implied probabilities”—the probability associated with a specific dichotomization that can be computed directly from the category response function associated with a given model—that have dramatically different forms when computed for different models. We also illustrate how these differences can be used to evaluate model fit. This consideration of the full range of possible dichotomizations and how the resulting implied probabilities may behave is of both conceptual and applied interest.