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

Universitat Pompeu Fabra, Department of Economics Working Paper No. 412

14 Pages Posted: 5 Jan 2000

See all articles by Albert Satorra

Albert Satorra

Universitat Pompeu Fabra

Peter M. Bentler

University of California, Los Angeles (UCLA) - Department of Psychology

Date Written: August 3, 1999

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 and M1, 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 and M1. 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, i.e., nonasymptotic and nonnormal, applications. In a recent paper, Satorra (1999) shows that the difference between two Satorra-Bentler 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.

JEL Classification: C12, C13, C14, C15, C21, C23

Suggested Citation

Satorra, Albert and Bentler, Peter M., A Scaled Difference Chi-Square Test Statistic for Moment Structure Analysis (August 3, 1999). Universitat Pompeu Fabra, Department of Economics Working Paper No. 412, Available at SSRN: https://ssrn.com/abstract=199064 or http://dx.doi.org/10.2139/ssrn.199064

Albert Satorra (Contact Author)

Universitat Pompeu Fabra ( email )

Ramon Trias Fargas 25-27
Barcelona, 08005
Spain
(34-93) 542 17 58 (Phone)
(34-93) 542 17 46 (Fax)

Peter M. Bentler

University of California, Los Angeles (UCLA) - Department of Psychology ( email )

1285 Franz Hall
Box 951563
Los Angeles, CA 90095-1563
United States

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