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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

Information

Type
Original Paper
Information
Psychometrika , Volume 83 , Issue 1 , March 2018, pp. 48 - 66
Copyright
Copyright © 2017 The Psychometric Society

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References

Amemiya, Y., Anderson, T.W., (1990). Asymptotic chi-square tests for a large class of factor analysis models, Annals of Statistics, 18, 1453–1463.CrossRefGoogle Scholar
Bellman, R.E. (1960) Introduction to matrix analysis New York: McGraw-Hill Book Company.Google Scholar
Bollen, K.A., (1989). Structural equations with latent variables New York: Wiley.CrossRefGoogle Scholar
Boomsma, A., & Hoogland, J. J., (2001). The robustness of lisrel modeling revisited. In Structural equation modeling: Present and future: A Festschrift in Honor of Karl Jöreskog (pp. 139–168). Chicago: Scientific Software International..Google Scholar
Browne, M.W. eds.Hawkins, D.M., (1982). Covariance structures, Topics in applied multivariate analysis Cambridge: Cambridge University Press 72–141.CrossRefGoogle Scholar
Browne, M.W., (1984). Asymptotically distribution-free methods for the analysis of covariance structures, British Journal of Mathematical and Statistical Psychology, 37, 62–83.CrossRefGoogle ScholarPubMed
Browne, M.W., Shapiro, A., (1987). Robustness of normal theory methods in the analysis of linear latent variate models, British Journal of Mathematical and Statistical Psychology, 41, 193–208.CrossRefGoogle Scholar
Browne, M.W., Shapiro, A., (2015). Comments on the asymptotics of a distribution-free goodness of fit test statistic, Psychometrika, 80, 196–199.CrossRefGoogle ScholarPubMed
Byrne, B.M. eds.Hoyle, R.H., (2012). Choosing structural equation modeling computer software: Snapshots of lisrel, eqs, amos, and mplus, Handbook of structural equation modeling, chap. 19, New York: Guilford Press 307–324.Google Scholar
Chun, S.Y., Shapiro, A., (2009). Normal versus noncentral chi-square asymptotics of misspecified models, Multivariate Behavioral Research, 44(6) 803–827.CrossRefGoogle ScholarPubMed
Hoogland, J.J., Boomsma, A., (1998). Robustness studies in covariance structure modeling: An overview and a meta-analysis, Sociological Methods and Research, 26(3) 329–367.CrossRefGoogle Scholar
Hu, L., Bentler, P.M., Kano, Y., (1992). Can test statistics in covariance structure analysis be trusted?, Psychological Bulletin, 112, 351–362.CrossRefGoogle ScholarPubMed
Huang, Y., Bentler, P.M., (2015). Behavior of asymptotically distribution free test statistics in covariance versus correlation structure analysis, Structural Equation Modeling, 22, 489–503.CrossRefGoogle Scholar
Jennrich, R., Satorra, A. The nonsingularity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} in covariance structure analysis of nonnormal data Psychometrika 2013 79, 51–59.Google Scholar
Kano, Y., (2002). Variable selection for structural models, Journal of Statistical Planning and Inference, 108(1–2) 173–187.CrossRefGoogle Scholar
Lee, S-Y, (1990). Multilevel analysis of structural equation models, Biometrika, 77(4) 763–772.CrossRefGoogle Scholar
McManus, D.A., (1991). Who invented local power analysis?, Econometric Theory, 7, 265–268.CrossRefGoogle Scholar
Shapiro, A., (1986). Asymptotic theory of overparametrized structural models, Journal of the American Statistical Association, 81, 142–149.CrossRefGoogle Scholar
Tomarken, A., Waller, N.G., (2005). Structural equation modeling as a data-analytic framework for clinical science: Strengths, limitations, and misconceptions, Annual Review of Clinical Psychology, 1, 31–65.CrossRefGoogle Scholar
Wu, H.a.o., Browne, M.W., (2015). Quantifying adventitious error in a covariance structure as a random effect, Psychometrika, 80(3) 571–6005439333.CrossRefGoogle Scholar
Xu, J., Mackenzie, G., (2012). Modelling covariance structure in bivariate marginal models for longitudinal data, Biometrika, 99(3) 649–662.CrossRefGoogle Scholar
Yuan, K.H., Bentler, P.M., (1998). Normal theory based test statistics in structural equation modelling, British Journal of Mathematical and Statistical Psychology, 51, 289–309.CrossRefGoogle ScholarPubMed
Yuan, K.H., Bentler, P.M., (1999). F tests for mean and covariance structure analysis, Journal of Educational and Behavioral Statistics, 24(3) 225–243.CrossRefGoogle Scholar
Yuan, K.H., Hayashi, K., Bentler, P.M., (2007). Normal theory likelihood ratio statistic for mean and covariance structure analysis under alternative hypotheses, Journal of Multivariate Analysis, 98(6) 1262–1282.CrossRefGoogle Scholar