Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-01-08T15:49:26.107Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.

References

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 516–573.CrossRefGoogle Scholar
Bock, R. D. (1972). Estimating item parameters and latent ability when the responses are scored in two or more nominal categories. Psychometrika, 37, 29–51.CrossRefGoogle Scholar
Chang, H., Mazzeo, J., & Roussos, L. (1993, April). Extension of Shealy-Stout's DIF procedures to polytomously scored items. Paper presented at the 1993 NCME meeting, Atlanta, GA.Google Scholar
Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.CrossRefGoogle Scholar
Muraki, E. (1992). A generalized partial credit model: application of an EM algorithm. Applied Psychological Measurement, 16, 159–176.CrossRefGoogle Scholar
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph No. 17, 34 (4, Pt. 2).Google Scholar
Samejima, F. (1972). A general model for free-response data. Psychometrika Monograph No. 18, 37 (1, Pt. 2).Google Scholar
Shealy, R., Stout, W. (1993). A model-based standardization approach that separates true bias/DIF from group ability differences and detects test bias/DTF as well as item bias/DIF. Psychometrika, 58, 159–194.CrossRefGoogle Scholar
Thissen, D., Steinberg, L. (1986). A taxonomy of item response models. Psychometrika, 51, 567–577.CrossRefGoogle Scholar
Wilson, M., Masters, G. N. (1993). The partial credit model and null categories. Psychometrika, 58, 87–99.CrossRefGoogle Scholar
Wright, B., Masters, G. N. (1982). Rating scale analysis, Chicago: MESA Press.Google Scholar