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Continuous Orthogonal Complement Functions and Distribution-Free Goodness of Fit Tests in Moment Structure Analysis

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  • Robert Jennrich
  • Albert Satorra

Abstract

It is shown that for any full column rank matrix X 0 with more rows than columns there is a neighborhood $\mathcal{N}$ of X 0 and a continuous function f on $\mathcal{N}$ such that f(X) is an orthogonal complement of X for all X in $\mathcal{N}$ . This is used to derive a distribution free goodness of fit test for covariance structure analysis. This test was proposed some time ago and is extensively used. Unfortunately, there is an error in the proof that the proposed test statistic has an asymptotic χ 2 distribution. This is a potentially serious problem, without a proof the test statistic may not, in fact, be asymptoticly χ 2 . The proof, however, is easily fixed using a continuous orthogonal complement function. Similar problems arise in other applications where orthogonal complements are used. These can also be resolved by using continuous orthogonal complement functions. Copyright The Psychometric Society 2013

Suggested Citation

  • Robert Jennrich & Albert Satorra, 2013. "Continuous Orthogonal Complement Functions and Distribution-Free Goodness of Fit Tests in Moment Structure Analysis," Psychometrika, Springer;The Psychometric Society, vol. 78(3), pages 545-552, July.
    • Handle: RePEc:spr:psycho:v:78:y:2013:i:3:p:545-552
    DOI: 10.1007/s11336-013-9320-x
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    References listed on IDEAS

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    1. Albert Satorra & Peter Bentler, 2001. "A scaled difference chi-square test statistic for moment structure analysis," Psychometrika, Springer;The Psychometric Society, vol. 66(4), pages 507-514, December.
    2. Kenneth Bollen & Albert Maydeu-Olivares, 2007. "A Polychoric Instrumental Variable (PIV) Estimator for Structural Equation Models with Categorical Variables," Psychometrika, Springer;The Psychometric Society, vol. 72(3), pages 309-326, September.
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    Cited by:

    1. Albert Satorra & Heinz Neudecker, 2015. "A Theorem on the Rank of a Product of Matrices with Illustration of Its Use in Goodness of Fit Testing," Psychometrika, Springer;The Psychometric Society, vol. 80(4), pages 938-948, December.
    2. Lifang Deng & Ke-Hai Yuan, 2016. "Comparing Latent Means Without Mean Structure Models: A Projection-Based Approach," Psychometrika, Springer;The Psychometric Society, vol. 81(3), pages 802-829, 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. Robert Jennrich & Albert Satorra, 2016. "The Infinitesimal Jackknife and Moment Structure Analysis Using Higher Order Moments," Psychometrika, Springer;The Psychometric Society, vol. 81(1), pages 90-101, March.

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