40-Year Old Unbiased Distribution Free Estimator Reliably Improves SEM Statistics for Nonnormal Data. Issue 6 (2nd November 2022) - Du, H.; Bentler, P. M.
Abstract: In structural equation modeling, researchers conduct goodness-of-fit tests to evaluate whether the specified model fits the data well. With nonnormal data, the standard goodness-of-fit test statistic T does not follow a chi-square distribution. Comparing T to χ d f 2 can fail to control Type I error rates and lead to misleading model selection conclusions. To better evaluate model fit, researchers have proposed various robust test statistics, but none of them consistently control Type I error rates under all examined conditions. To improve model fit statistics for nonnormal data, we propose to use an unbiased distribution free weight matrix estimator ( Γ ^ DF U ) in robust test statistics. Specifically, using normal theory based parameter estimates with Γ ^ DF U, we calculate various robust test statistics and robust standard errors. We conducted a simulation study to compare 63 existing robust statistic combinations with the 4 proposed robust statistics with Γ ^ DF U . The Satorra–Bentler statistic TSB based on Γ ^ DF U ( T S B U ) provided acceptable Type I error rates at α = .01, .05, or .1 across all conditions (except a few cases with α = .01 ), regardless of the sample size and the distribution. T S B U or T MVA 2 U typically provided the smallest Anderson-Darling test values, showing the smallest distances between p -values and Uniform ( 0, 1 ) . We use a real data example to compare statistics with Γ ^ DF U and that with Γ ^ ADF .