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Likelihood Inference on the Underlying Structure of IRT Models

Published online by Cambridge University Press:  01 January 2025

Francesco Bartolucci  and
Antonio Forcina
Francesco Bartolucci*
Affiliation:
Università di Urbino
Antonio Forcina
Affiliation:
Università di Perugia
*
Requests for reprints should be sent to Francesco Bartolucci, Istituto di Scienze Economiche, Unversità du Urbino, Via Saffi, 42, 61029 Urbino, Italy. E-mail: Francesco.Bartolucci@uniurb.it
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Abstract

The assumptions underlying item response theory (IRT) models may be expressed as a set of equality and inequality constraints on the parameters of a latent class model. It is well known that the same assumptions imply that the parameters of the manifest distribution have to satisfy a more complicated set of inequality constraints which, however, are necessary but not sufficient. In this paper, we describe how the theory for likelihood-based inference under equality and inequality constraints may be used to test the underlying assumptions of IRT models. It turns out that the analysis based directly on the latent structure is simpler and more flexible. In particular, we indicate how several interesting extensions of the Rasch model may be obtained by partial relaxation of the basic constraints. An application to a data set provided by Educational Testing Service is used to illustrate the approach.

Keywords

nonparametric mixture modelsorder restricted inferencechi-bar squared distributionmarginal parameterizations
Type
Original Paper
Information
Psychometrika , Volume 70 , Issue 1 , March 2005 , pp. 31 - 43
Copyright
Copyright © 2005 The Psychometric Society

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Footnotes

We thank Dr. Gorman and Dr. Rogers of the Educational Testing Service for providing the data analyzed in Section 4. We also thank three reviewers for comments and suggestions.

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