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Polytomous IRT Models and Monotone Likelihood Ratio of the Total Score

Published online by Cambridge University Press:  01 January 2025

Bas T. Hemker ,
Klaas Sijtsma ,
Ivo W. Molenaar  and
Brian W. Junker
Bas T. Hemker*
Affiliation:
Utrecht University
Klaas Sijtsma
Affiliation:
Utrecht University
Ivo W. Molenaar
Affiliation:
University of Groningen
Brian W. Junker
Affiliation:
Carnegie Mellon University
*
Requests for reprints should be sent to Bas T. Hemker, National Institute for Educational Measurement (Cito), PO Box 1034, 6801 MG Arnhem, THE NETHERLANDS.
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Abstract

In a broad class of item response theory (IRT) models for dichotomous items the unweighted total score has monotone likelihood ratio (MLR) in the latent trait θ. In this study, it is shown that for polytomous items MLR holds for the partial credit model and a trivial generalization of this model. MLR does not necessarily hold if the slopes of the item step response functions vary over items, item steps, or both. MLR holds neither for Samejima's graded response model, nor for nonparametric versions of these three polytomous models. These results are surprising in the context of Grayson's and Huynh's results on MLR for nonparametric dichotomous IRT models, and suggest that establishing stochastic ordering properties for nonparametric polytomous IRT models will be much harder.

Keywords

IRT modelsmonotone likelihood rationonparametric IRT modelsparametric IRT modelspolytomous items
Type
Original Paper
Information
Psychometrika , Volume 61 , Issue 4 , December 1996 , pp. 679 - 693
Copyright
Copyright © 1996 The Psychometric Society

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Footnotes

Hemker's research was supported by the Netherlands Research Council, Grant 575-67-034. Junker's research was supported in part by the National Institutes of Health, Grant CA54852, and by the National Science Foundation, Grant DMS-94.04438.

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