THE UNIQUE CORRESPONDENCE OF THE ITEM RESPONSE FUNCTION AND ITEM CATEGORY RESPONSE FUNCTIONS IN POLYTOMOUSLY SCORED ITEM RESPONSE MODELS

The item response function (IRF) for a polytomously scored item is def ined as a weighted sum of the item category response functions (ICRF, the probability of getting a particular score for a randomly sampled e xamined of ability theta). This paper establishes the correspondence b etween an IRF and a unique set of ICRFs for two of the most commonly u sed polytomous IRT models (the partial credit models and the graded re sponse model). Specifically, a proof of the following assertion is pro vided for these models: If two items have the same IRF, then they must have the same number of categories; moreover, they must consist of th e same ICRFs. As a corollary, for the Rasch dichotomous model, if two tests have the same test characteristic function (TCF), then they must have the same number of items. Moreover, for each item in one of the tests, an item in the other test with an identical IRF must exist. The oretical as well as practical implications of these results are discus sed.