Distinctive and Incompatible Properties of Two Common Classes of IRT Models for Graded Responses

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Abstract
Two classes of models for graded responses, the first based on the work of Thurstone and the second based on the work of Rasch, are juxtaposed and shown to satisfy important, but mutually incompatible, criteria and to reflect different response processes. Specifically, in the Thurstone models if adjacent categories are joined to form a new category, either before or after the data are collected, then the probability of a response in the new category is the sum of the probabilities of the responses in the original categories. However, the model does not have the explicit property that if the categories are so joined, then the estimate of the location of the entity or object being measured is invariant before and after the joining. For the Rasch models, if a pair of adjacent categories are joined and then the data are collected, the es-timate of the location of the entity is the same before and after the joining, but the probability of a response in the new category is not the sum of the probabilities of the responses in the original categories. Furthermore, if data satisfy the model and the categories are joined after the data are collected, then they no longer satisfy the same Rasch model with the smaller number of categories. These differences imply that the choice between these two classes of models for graded responses is not simply a matter of preference; they also permit a better understanding of the choice of models for graded response data as a function of the underlying processes they are intended to represent. Index terms: graded responses, joining assumption, polytomous IRT models, Rasch model, Thurstone model.