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

Published online by Cambridge University Press:  01 January 2025

Robert Jennrich  and
Albert Satorra
Robert Jennrich*
Affiliation:
University of California, Los Angeles
Albert Satorra
Affiliation:
Universitat Pompeu Fabra, Barcelona
*
Requests for reprints should be sent to Robert Jennrich, University of California, Los Angeles, 3400 Purdue Ave., Los Angeles, CA, USA. E-mail: rij@stat.ucla.edu
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Abstract

It is shown that for any full column rank matrix X0 with more rows than columns there is a neighborhood \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mathcal{N}$\end{document} of X0 and a continuous function f on \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mathcal{N}$\end{document} such that f(X) is an orthogonal complement of X for all X in \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mathcal{N}$\end{document}. 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 χ2 distribution. This is a potentially serious problem, without a proof the test statistic may not, in fact, be asymptoticly χ2. 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.

Keywords

χ2 testsorthogonal complementsQR factorizationimplicit function theorem
Type
Original Paper
Information
Psychometrika , Volume 78 , Issue 3 , July 2013 , pp. 545 - 552
Copyright
Copyright © 2013 The Psychometric Society

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