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 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 (Γ^DFU) in robust test statistics. Specifically, using normal theory based parameter estimates with Γ^DFU, 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^DFU. The Satorra–Bentler statistic TSB based on Γ^DFU (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 Γ^DFU and that with Γ^ADF.
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