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

A family of scaling corrections aimed to improve the chi-square approximation of goodness-oft 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 sayM0 implies on a less restricted oneM1. If T0 and T1 denote the goodness-oft test statistics associated toM0 andM1, respectively, then typically the di erence Td = T0 T1 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 modelsM0 and M1. As in the case of the goodness-oft 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 di erence between two SatorraBentler 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-oft test statistics of modelsM0 and M1. A Monte Carlo study is provided to illustrate the performance of the competing statistics.