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

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 ${\chi }_{df}^2$ 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 (${\widehat{\mathbf{\Gamma }}}_{\mathbf{DF}}^{\mathbf{U}}$) in robust test statistics. Specifically, using normal theory based parameter estimates with ${\widehat{\mathbf{\Gamma }}}_{\mathbf{DF}}^{\mathbf{U}},$ 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 ${\widehat{\mathbf{\Gamma }}}_{\mathbf{DF}}^{\mathbf{U}}.$ The Satorra–Bentler statistic T_SB based on ${\widehat{\mathbf{\Gamma }}}_{\mathbf{DF}}^{\mathbf{U}}$ ($T_{SB}^U$) provided acceptable Type I error rates at $\alpha =.01,$.05, or .1 across all conditions (except a few cases with $\alpha =.01$), regardless of the sample size and the distribution. $T_{SB}^U$ or $T_{\mathit{MVA}2}^U$ typically provided the smallest Anderson-Darling test values, showing the smallest distances between p-values and $\mathit{Uniform}(0,1).$ We use a real data example to compare statistics with ${\widehat{\mathbf{\Gamma }}}_{\mathbf{DF}}^{\mathbf{U}}$ and that with ${\widehat{\mathbf{\Gamma }}}_{\mathbf{ADF}}.$

Keywords:

• Robust test statistics
• distribution free estimator
• goodness-of-fit test statistics
• robust standard error
• non-normal data

Notes

1 In samples, ${\widehat{\mathbf{\Gamma }}}_{\mathbf{N}}$ can be either estimated by sample covariances ${\left\{{\widehat{\mathbf{\Gamma }}}_{\mathbf{N}.\mathbf{S}}\right\}}_{hj,kl}=s_{hk}{s}_{jl}+s_{hl}{s}_{jk}$ or estimated by the model implied covariances ${\left\{{\widehat{\mathbf{\Gamma }}}_{\mathbf{N}.\mathbf{M}}\right\}}_{hj,kl}={\widehat{\sigma }}_{hk}{\widehat{\sigma }}_{jl}+{\widehat{\sigma }}_{hl}{\widehat{\sigma }}_{jk}.$ Du and and Bentler (2021) found that ${\widehat{\mathbf{\Gamma }}}_{\mathbf{N}.\mathbf{M}}$ had a better performance than ${\widehat{\mathbf{\Gamma }}}_{\mathbf{N}.\mathbf{S}}.$ Hence, we focus on ${\widehat{\mathbf{\Gamma }}}_{\mathbf{N}.\mathbf{M}}.$ We do not take derivatives of the weight matrix. We update $\widehat{\mathbf{W}}={\widehat{\mathbf{\Gamma }}}_{\mathbf{N}}^{-\mathbf{1}}$ 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.