Scaling Corrections for Statistics in Covariance Structure Analysis

Research supported by the Spanish DGES grant PB96-0300, and USPHS grants DA00017 and DA01070.

Abstract

A family of scaling corrections aimed to improve the chi-square approximation of goodness-of-t 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 t of a model, but on a test of the restrictions that a null model say M 0 implies on a less restricted one M 1 .I f T 0 and T 1 denote the goodness-of-t test statistics associated to M 0 and M 1 , respectively, then typically the di erence T d = T 0 , T 1 is used as a chi-square test statistic with degrees of freedom equal to the di erence on the number of independent parameters estimated under the models M 0 and M 1 .As in the case of the goodness-of-t test, it is of interest to scale the statistic T d 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 di erence between two Satorra-Bentler scaled test statistics for overall model t does not yield the correct SB scaled di erence test statistic.Satorra developed an expression that permits scaling the di erence 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 di erence chi-square statistic from the scaled goodness-of-t test statistics of models M 0 and M 1 .A Monte Carlo study is provided to illustrate the performance of the competing statistics.