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

Hua‐Hua Chang · John Mazzeo

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Psychometric Methodologies and Testing Psychological Testing and Assessment Educational Assessment and Pedagogy

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.

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References

1982A Rasch Model for Partial Credit Scoring 1978A Rating Formulation for Ordered Response Categories 1968ESTIMATION OF LATENT ABILITY USING A RESPONSE PATTERN OF GRADED SCORES1 1969Estimation of Latent Ability Using a Response Pattern of Graded Scores 1982Rating scale analysis 1992A Generalized Partial Credit Model: Application of an EM Algorithm 1972Estimating Item Parameters and Latent Ability when Responses are Scored in Two or More Nominal Categories 1993A Model-Based Standardization Approach that Separates True Bias/DIF from Group Ability Differences and Detects Test Bias/DTF as well as Item Bias/DIF

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