Empirically Corrected Rescaled Statistics for SEM with Small N and Large p

Survey data often contain many variables. Structural equation modeling (SEM) is commonly used in analyzing such data. With typical nonnormally distributed data in practice, a rescaled statistic T_rml proposed by Satorra and Bentler was recommended in the literature of SEM. However, T_rml has been shown to be problematic when the sample size N is small and/or the number of variables p is large. There does not exist a reliable test statistic for SEM with small N or large p, especially with nonnormally distributed data. Following the principle of Bartlett correction, this article develops empirical corrections to T_rml so that the mean of the empirically corrected statistics approximately equals the degrees of freedom of the nominal chi-square distribution. Results show that empirically corrected statistics control type I errors reasonably well even when N is smaller than 2p, where T_rml may reject the correct model 100% even for normally distributed data. The application of the empirically corrected statistics is illustrated via a real data example.