Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic

Bibliographic Data

ID23313610
AuthorsAlbert Satorra (0000-0001-8974-5241, Universitat Pompeu Fabra, corresponding author), Peter M Bentler (0000-0002-9440-721X, University of California, Los Angeles)
Year2010
Volume75
Issue2
Pages243-248
Publication date2010-06-01
Peer ReviewedYes
Open AccessYes
TypeARTICLE
VenuePsychometrika (JOURNAL)
Journal identifiersISSN: 0033-3123 • E-ISSN: 1860-0980
PublisherCambridge University Press (CUP) (PUBLISHER)
DOI10.1007/s11336-009-9135-y
PMID20640194
PMCIDPMC2905175
OpenAlexW2047317044
LanguageEN
Citations received833
References cited12

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.

Algorithm · Ancillary statistic · Chi-square test · Computation · F-test · PRESS statistic · Scaling · Statistic · Statistical hypothesis testing · Statistics · Test statistic · Advanced Statistical Methods and Models · Advanced Statistical Modeling Techniques · Applied Mathematics · Mathematics · Sensory Analysis and Statistical Methods

Tools

Open DOISci-HubOpen Access