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A General Family of Limited Information Goodness-of-Fit Statistics for Multinomial Data

Published online by Cambridge University Press:  01 January 2025

Harry Joe  and
Alberto Maydeu-Olivares
Harry Joe
Affiliation:
Department of Statistics, University of British Columbia
Alberto Maydeu-Olivares*
Affiliation:
Faculty of Psychology, University of Barcelona
*
Requests for reprints should be sent to Alberto Maydeu-Olivares, Faculty of Psychology, University of Barcelona, P. Valle de Hebrón, 171, 08035 Barcelona, Spain. E-mail: amaydeu@ub.edu
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Abstract

Maydeu-Olivares and Joe (J. Am. Stat. Assoc. 100:1009–1020, 2005; Psychometrika 71:713–732, 2006) introduced classes of chi-square tests for (sparse) multidimensional multinomial data based on low-order marginal proportions. Our extension provides general conditions under which quadratic forms in linear functions of cell residuals are asymptotically chi-square. The new statistics need not be based on margins, and can be used for one-dimensional multinomials. We also provide theory that explains why limited information statistics have good power, regardless of sparseness. We show how quadratic-form statistics can be constructed that are more powerful than X2 and yet, have approximate chi-square null distribution in finite samples with large models. Examples with models for truncated count data and binary item response data are used to illustrate the theory.

Keywords

categorical data analysiscell-focusingdiscrete dataitem response theoryoverdispersionoverlapping cellsPoisson modelsquadratic form statisticsRasch modelsscore testsparse contingency tableszero-inflation
Type
Original Paper
Information
Psychometrika , Volume 75 , Issue 3 , September 2010 , pp. 393 - 419
Copyright
Copyright © 2010 The Psychometric Society

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Footnotes

This research has been by grant SEJ2006-08204 from the Spanish Ministry of Education, and an NSERC Canada Discovery Grant. We are grateful to the referees and associate editor for comments leading to improvements. Also, we thank Virginia Yue Chen for some early numerical investigations.

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