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

Published online by Cambridge University Press:  01 January 2025

Albert Satorra  and
Peter M. Bentler
Albert Satorra*
Affiliation:
Universitat Pompeu Fabra, Barcelona
Peter M. Bentler
Affiliation:
University of California, Los Angeles
*
Requests for reprints should be sent to Albert Satorra, Universitat Pompeu Fabra, Ramon Trias Fargas, 25-27, 08005 Barcelona, SPAIN. E-Mail: [email protected]
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Abstract

A family of scaling corrections aimed to improve the chi-square approximation of goodness-of-fit test statistics in small samples, large models, and nonnormal data was proposed in Satorra and Bentler (1994). For structural equations models, Satorra-Bentler's (SB) scaling corrections are available in standard computer software. Often, however, the interest is not on the overall fit of a model, but on a test of the restrictions that a null model say M0 implies on a less restricted one M1. If T0 and T1 denote the goodness-of-fit test statistics associated to M0 andM1, respectively, then typically the difference Td=T0−T1 is used as a chi-square test statistic with degrees of freedom equal to the difference on the number of independent parameters estimated under the models M0 andM1. As in the case of the goodness-of-fit test, it is of interest to scale the statistic Td in order to improve its chi-square approximation in realistic, that is, nonasymptotic and nonormal, applications. In a recent paper, Satorra (2000) shows that the difference between two SB scaled test statistics for overall model fit does not yield the correct SB scaled difference test statistic. Satorra developed an expression that permits scaling the difference test statistic, but his formula has some practical limitations, since it requires heavy computations that are not available in standard computer software. The purpose of the present paper is to provide an easy way to compute the scaled difference chi-square statistic from the scaled goodness-of-fit test statistics of models M0 and M1. A Monte Carlo study is provided to illustrate the performance of the competing statistics.

Keywords

moment-structuresgoodness-of-fit testchi-square difference test statisticchi-square distributionnonnormality
Type
Articles
Information
Psychometrika , Volume 66 , Issue 4 , December 2001 , pp. 507 - 514
Copyright
Copyright © 2001 The Psychometric Society

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Footnotes

This research was supported by the Spanish grants PB96-0300 and BEC2000-0983, and USPHS grants DA00017 and DA01070.

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