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Research Articles

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

ORCID Icon & ORCID Icon
Pages 872-887 | Received 13 Jan 2022, Accepted 05 Apr 2022, Published online: 01 Jun 2022
 

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.

Notes

1 In samples, Γ̂N can be either estimated by sample covariances {Γ̂N.S}hj,kl=shksjl+shlsjk or estimated by the model implied covariances {Γ̂N.M}hj,kl=σ̂hkσ̂jl+σ̂hlσ̂jk. Du and and Bentler (Citation2021) found that Γ̂N.M had a better performance than Γ̂N.S. Hence, we focus on Γ̂N.M. We do not take derivatives of the weight matrix. We update Ŵ=Γ̂N1 at the end of each iteration based on the updated parameters and model implied covariances.

2 Under the robust nonnormal conditions, T has the same asymptotic distribution as under the normal condition.

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