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Stochastic Ordering Using the Latent Trait and the Sum Score in Polytomous IRT Models

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), P.O. Box 1034, 6801 MG Arnhem, THE NETHERLANDS.
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Abstract

In a restricted class of item response theory (IRT) models for polytomous items the unweighted total score has monotone likelihood ratio (MLR) in the latent trait ϑ. MLR implies two stochastic ordering (SO) properties, denoted SOM and SOL, which are both weaker than MLR, but very useful for measurement with IRT models. Therefore, these SO properties are investigated for a broader class of IRT models for which the MLR property does not hold.

In this study, first a taxonomy is given for nonparametric and parametric models for polytomous items based on the hierarchical relationship between the models. Next, it is investigated which models have the MLR property and which have the SO properties. It is shown that all models in the taxonomy possess the SOM property. However, counterexamples illustrate that many models do not, in general, possess the even more useful SOL property.

Keywords

monotone likelihood rationonparametric IRT modelsparametric IRT modelspolytomous IRT modelsstochastic ordering
Type
Original Paper
Information
Psychometrika , Volume 62 , Issue 3 , September 1997 , pp. 331 - 347
Copyright
Copyright © 1997 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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