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Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic

Published online by Cambridge University Press:  01 January 2025

Albert Satorra  and
Peter M. Bentler
Albert Satorra*
Affiliation:
Universitat Pompeu Fabra
Peter M. Bentler
Affiliation:
University of California
*
Requests for reprints should be sent to Albert Satorra, Department of Economics and Business, Universitat Pompeu Fabra, Ramon Trias Fargas 25-27, Barcelona, Spain. E-mail: albert.satorra@upf.edu
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Abstract

A scaled difference test statistic \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\tilde{T}{}_{d}$\end{document} that can be computed from standard software of structural equation models (SEM) by hand calculations was proposed in Satorra and Bentler (Psychometrika 66:507–514, 2001). The statistic \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\tilde{T}_{d}$\end{document} is asymptotically equivalent to the scaled difference test statistic \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\bar{T}_{d}$\end{document} introduced in Satorra (Innovations in Multivariate Statistical Analysis: A Festschrift for Heinz Neudecker, pp. 233–247, 2000), which requires more involved computations beyond standard output of SEM software. The test statistic \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\tilde{T}_{d}$\end{document} has been widely used in practice, but in some applications it is negative due to negativity of its associated scaling correction. Using the implicit function theorem, this note develops an improved scaling correction leading to a new scaled difference statistic \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\bar{T}_{d}$\end{document} that avoids negative chi-square values.

Keywords

moment-structuresgoodness-of-fit testchi-square difference test statisticchi-square distributionnonnormality
Type
Theory and Methods
Information
Psychometrika , Volume 75 , Issue 2 , June 2010 , pp. 243 - 248
Copyright
Copyright © 2010 The Psychometric Society

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Footnotes

Research supported by grants SEJ2006-13537 and PR2007-0221 from the Spanish Ministry of Science and Technology and by USPHS grants DA00017 and DA01070.

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