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Modified Distribution-Free Goodness-of-Fit Test Statistic

Author

Listed:
  • So Yeon Chun

    (Georgetown University)

  • Michael W. Browne

    (Ohio State University)

  • Alexander Shapiro

    (Georgia Institute of Technology)

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.

Suggested Citation

  • So Yeon Chun & Michael W. Browne & Alexander Shapiro, 2018. "Modified Distribution-Free Goodness-of-Fit Test Statistic," Psychometrika, Springer;The Psychometric Society, vol. 83(1), pages 48-66, March.
    • Handle: RePEc:spr:psycho:v:83:y:2018:i:1:d:10.1007_s11336-017-9574-9
    DOI: 10.1007/s11336-017-9574-9
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    References listed on IDEAS

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    1. Jing Xu & Gilbert Mackenzie, 2012. "Modelling covariance structure in bivariate marginal models for longitudinal data," Biometrika, Biometrika Trust, vol. 99(3), pages 649-662.
    2. Ke-Hai Yuan & Peter M. Bentler, 1999. "F Tests for Mean and Covariance Structure Analysis," Journal of Educational and Behavioral Statistics, , vol. 24(3), pages 225-243, September.
    3. Michael Browne & Alexander Shapiro, 2015. "Comment on the Asymptotics of a Distribution-Free Goodness of Fit Test Statistic," Psychometrika, Springer;The Psychometric Society, vol. 80(1), pages 196-199, March.
    4. Yuan, Ke-Hai & Hayashi, Kentaro & Bentler, Peter M., 2007. "Normal theory likelihood ratio statistic for mean and covariance structure analysis under alternative hypotheses," Journal of Multivariate Analysis, Elsevier, vol. 98(6), pages 1262-1282, July.
    5. Hao Wu & Michael Browne, 2015. "Quantifying Adventitious Error in a Covariance Structure as a Random Effect," Psychometrika, Springer;The Psychometric Society, vol. 80(3), pages 571-600, September.
    6. McManus, Douglas A., 1991. "Who Invented Local Power Analysis?," Econometric Theory, Cambridge University Press, vol. 7(2), pages 265-268, June.
    7. Chun, So Yeon & Alexander, Shapiro, 2009. "Normal versus Noncentral Chi-square Asymptotics of Misspecified Models," MPRA Paper 17310, University Library of Munich, Germany.
    8. Jeffrey J. Hoogland & Anne Boomsma, 1998. "Robustness Studies in Covariance Structure Modeling," Sociological Methods & Research, , vol. 26(3), pages 329-367, February.
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    Cited by:

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    2. Viet Anh Nguyen & Daniel Kuhn & Peyman Mohajerin Esfahani, 2018. "Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator," Papers 1805.07194, arXiv.org.

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