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Essential Independence and Likelihood-Based Ability Estimation for Polytomous Items

Published online by Cambridge University Press:  01 January 2025

Brian W. Junker
Brian W. Junker*
Affiliation:
Department of Statistics, University of Illinois at Urbana-Champaign
*
Requests for reprints should be sent to Brian W. Junker, Department of Statistics, 232 Baker Hall, Carnegie Mellon University, Pittsburgh, PA 15213.
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Abstract

A definition of essential independence is proposed for sequences of polytomous items. For items satisfying the reasonable assumption that the expected amount of credit awarded increases with examinee ability, we develop a theory of essential unidimensionality which closely parallels that of Stout. Essentially unidimensional item sequences can be shown to have a unique (up to change-of-scale) dominant underlying trait, which can be consistently estimated by a monotone transformation of the sum of the item scores. In more general polytomous-response latent trait models (with or without ordered responses), an M-estimator based upon maximum likelihood may be shown to be consistent for θ under essentially unidimensional violations of local independence and a variety of monotonicity/identifiability conditions. A rigorous proof of this fact is given, and the standard error of the estimator is explored. These results suggest that ability estimation methods that rely on the summation form of the log likelihood under local independence should generally be robust under essential independence, but standard errors may vary greatly from what is usually expected, depending on the degree of departure from local independence. An index of departure from local independence is also proposed.

Keywords

item response theory (IRT)polytomous item responsesessential independenceunidimensionalitylatent trait identifiabilitylikelihood-based trait estimationasymptotic standard errorsstructural robustnesslocal dependence

Information

Type
Original Paper
Information
Psychometrika , Volume 56 , Issue 2 , June 1991 , pp. 255 - 278
Copyright
Copyright © 1991 The Psychometric Society

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Footnotes

This work was supported in part by Office of Naval Research Grant N00014-87-K-0277 and National Science Foundation Grant NSF-DMS-88-02556. The author is grateful to William F. Stout for many helpful comments, and to an anonymous reviewer for raising the questions addressed in section 2. A preliminary version of section 6 appeared in the author's Ph.D. thesis.

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