Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic

2009/06/19 by Albert Satorra, Peter M. Bentler · 2,692 citations
Agricultural and Biological Sciences · Computer Science · Mathematics · #Advanced Statistical Methods and Models #Advanced Statistical Modeling Techniques #Algorithm #Ancillary statistic #Applied mathematics #Chi-square test #Computation #F-test #Mathematics #PRESS statistic #Scaling #Sensory Analysis and Statistical Methods #Statistic #Statistical hypothesis testing #Statistics #Test statistic

paper · open access · doi:10.1007/s11336-009-9135-y

published in Psychometrika 75(2), 243-248 (Springer Science+Business Media)

openalex publication_date 2009/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

A scaled difference test statistic \documentclass[12pt]minimal \usepackageamsmath \usepackagewasysym \usepackageamsfonts \usepackageamssymb \usepackageamsbsy \usepackagemathrsfs \usepackageupgreek \setlength\oddsidemargin-69pt \begindocumentT_d\enddocument 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 \usepackageamsmath \usepackagewasysym \usepackageamsfonts \usepackageamssymb \usepackageamsbsy \usepackagemathrsfs \usepackageupgreek \setlength\oddsidemargin-69pt \begindocumentT_d\enddocument is asymptotically equivalent to the scaled difference test statistic \documentclass[12pt]minimal \usepackageamsmath \usepackagewasysym \usepackageamsfonts \usepackageamssymb \usepackageamsbsy \usepackagemathrsfs \usepackageupgreek \setlength\oddsidemargin-69pt \begindocumentT_d\enddocument 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 \usepackageamsmath \usepackagewasysym \usepackageamsfonts \usepackageamssymb \usepackageamsbsy \usepackagemathrsfs \usepackageupgreek \setlength\oddsidemargin-69pt \begindocumentT_d\enddocument 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 \usepackageamsmath \usepackagewasysym \usepackageamsfonts \usepackageamssymb \usepackageamsbsy \usepackagemathrsfs \usepackageupgreek \setlength\oddsidemargin-69pt \begindocumentT_d\enddocument that avoids negative chi-square values.

Citations

• A Scaled Difference Chi-Square Test Statistic for Moment Structure Analysis

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