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

作者：Albert Satorra, Peter M. Bentler · 发表于：Psychometrika · 年份：2001 · DOI：10.1007/bf02296192 · 被引用次数：5029 · 研究领域：Statistical Methods and Bayesian Inference、Psychometric Methodologies and Testing、Advanced Statistical Methods and Models

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 M 0 implies on a less restricted one M 1 . If T 0 and T 1 denote the goodness-of-fit test statistics associated to M 0 and M 1 , respectively, then typically the difference T d = T 0 − T 1 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 M 0 and M 1 . As in the case of the goodness-of-fit test, it is of interest to scale the statistic T d 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 ...