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Limited Information Goodness-of-fit Testing in Multidimensional Contingency Tables

Published online by Cambridge University Press:  01 January 2025

Albert Maydeu-Olivares  and
Harry Joe
Albert Maydeu-Olivares*
Affiliation:
University of Barcelona and Instituto De Empresa Business School
Harry Joe
Affiliation:
University of British Columbia
*
Requests for reprints should be sent to Albert Maydeu-Olivares. Faculty of Psychology, University of Barcelona. P. Valle de Hebrn, 171, 08035 Barcelona (Spain). E-mail: amaydeu@ub.edu
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Abstract

We introduce a family of goodness-of-fit statistics for testing composite null hypotheses in multidimensional contingency tables. These statistics are quadratic forms in marginal residuals up to order r. They are asymptotically chi-square under the null hypothesis when parameters are estimated using any asymptotically normal consistent estimator. For a widely used item response model, when r is small and multidimensional tables are sparse, the proposed statistics have accurate empirical Type I errors, unlike Pearson’s X2. For this model in nonsparse situations, the proposed statistics are also more powerful than X2. In addition, the proposed statistics are asymptotically chi-square when applied to subtables, and can be used for a piecewise goodness-of-fit assessment to determine the source of misfit in poorly fitting models.

Keywords

multivariate discrete datacategorical data analysismultivariate multinomial distributioncomposite likelihooditem response theoryLisrel
Type
Original Paper
Information
Psychometrika , Volume 71 , Issue 4 , December 2006 , pp. 713 - 732
Copyright
Copyright © 2007 The Psychometric Society

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Footnotes

This research has been supported by the Department of Universities, Research, and Information Society (DURSI) of the Catalan Government, by grant BSO2003-08507 of the Spanish Ministry of Science and Technology, and an NSERC Canada grant. We are grateful to the referees for comments leading to improvements.

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