40-Year Old Unbiased Distribution Free Estimator Reliably Improves SEM Statistics for Nonnormal Data

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 χdf2 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 (ΓDFU) in robust test statistics. Specifically, using normal theory based parameter estimates with ΓDFU, 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 ΓDFU. The Satorra–Bentler statistic TSB based on ΓDFU (TSBU) 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. TSBU or TMVA2U 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 ΓDFU and that with ΓADF.