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The Unique Correspondence of the Item Response Function and Item Category Response Functions in Polytomously Scored Item Response Models

Published online by Cambridge University Press:  01 January 2025

Hua-Hua Chang  and
John Mazzeo
Hua-Hua Chang*
Affiliation:
Educational Testing Service
John Mazzeo*
Affiliation:
Educational Testing Service
*
Requests for reprints should be sent to Hua-Hua Chang or John Mazzeo, Educational Testing Service, Rosedale Road, Princeton, NJ 08541.
Requests for reprints should be sent to Hua-Hua Chang or John Mazzeo, Educational Testing Service, Rosedale Road, Princeton, NJ 08541.
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Abstract

The item response function (IRF) for a polytomously scored item is defined as a weighted sum of the item category response functions (ICRF, the probability of getting a particular score for a randomly sampled examinee of ability θ). This paper establishes the correspondence between an IRF and a unique set of ICRFs for two of the most commonly used polytomous IRT models (the partial credit models and the graded response model). Specifically, a proof of the following assertion is provided for these models: If two items have the same IRF, then they must have the same number of categories; moreover, they must consist of the 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. Theoretical as well as practical implications of these results are discussed.

Keywords

item response theorypolytomous itempartial credit modelgeneralized partial credit modelgraded response modelinvarianceordered categories
Type
Original Paper
Information
Psychometrika , Volume 59 , Issue 3 , September 1994 , pp. 391 - 404
Copyright
Copyright © 1994 The Psychometric Society

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Footnotes

This research was supported by Educational Testing Service Allocation Projects No. 79409 and No. 79413. The authors wish to thank John Donoghue, Ming-Mei Wang, Rebecca Zwick, and Zhiliang Ying for their useful comments and discussions. The authors also wish to thank three anonymous reviewers for their comments.

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