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Modified Distribution-Free Goodness-of-Fit Test Statistic

Published online by Cambridge University Press:  01 January 2025

So Yeon Chun ,
Michael W. Browne  and
Alexander Shapiro
So Yeon Chun*
Affiliation:
Georgetown University
Michael W. Browne
Affiliation:
Ohio State University
Alexander Shapiro
Affiliation:
Georgia Institute of Technology
*
Correspondence should be made to So Yeon Chun, Georgetown University, Washington, DC 20057, USA. Email: soyeon.chun@georgetown.edu
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Abstract

Covariance structure analysis and its structural equation modeling extensions have become one of the most widely used methodologies in social sciences such as psychology, education, and economics. An important issue in such analysis is to assess the goodness of fit of a model under analysis. One of the most popular test statistics used in covariance structure analysis is the asymptotically distribution-free (ADF) test statistic introduced by Browne (Br J Math Stat Psychol 37:62–83, 1984). The ADF statistic can be used to test models without any specific distribution assumption (e.g., multivariate normal distribution) of the observed data. Despite its advantage, it has been shown in various empirical studies that unless sample sizes are extremely large, this ADF statistic could perform very poorly in practice. In this paper, we provide a theoretical explanation for this phenomenon and further propose a modified test statistic that improves the performance in samples of realistic size. The proposed statistic deals with the possible ill-conditioning of the involved large-scale covariance matrices.

Keywords

covariance structuresdistribution-free test statisticasymptoticsChi-square distributionill-conditioned problem
Type
Original Paper
Information
Psychometrika , Volume 83 , Issue 1 , March 2018 , pp. 48 - 66
Copyright
Copyright © 2017 The Psychometric Society

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