doi:10.1007/s11336-009-9135-y

Paper Title

Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic

Authors

Peter M. Bentler

ALBERT BRUCART SATORRA

Keywords

Moment-Structures
Goodness-Of-Fit Test
Chi-Square Difference Test Statistic
Chi-Square Distribution
Nonnormality
Scaled Difference Test Statistic
Structural Equation Models (SEM)
Scaling Correction
Implicit Function Theorem
Negative Chi-Square Values
Statistical Analysis
Model Evaluation
SEM Software
Asymptotic Equivalence
Multivariate Statistical Analysis
Hand Calculations

Article Type

Research Article

Journal

Psychometrika

Research Impact Tools

DOI

10.1007/s11336-009-9135-y

Issue

Volume : 75 | Issue : 2 | Page No : 243-248

Published On

January, 2010

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Abstract

A scaled difference test statistic 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 is asymptotically equivalent to the scaled difference test statistic 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 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 that avoids negative chi-square values.