How to use Stata's sem command with nonnormal data: A new nonnormality correction for the RMSEA and incremental fit indices, CFI and TLI

Traditional fit measures like RMSEA, TLI, or CFI are based on noncentral chi-square distribution assuming the multinormal distribution of the observed indicators (Jöreskog 1970). If this assumption has violated programs like Stata, EQS or LISREL calculate the fit indices using the Sattora-Bentler correction. It rescales the likelihood ratio chi2 test statistics of the baseline and the hypothesized model (Satorra & Bentler 1994, Newitt & Hancock 2000). Brosseu-Liard et al. (2012, 2014) and Savalei (2018) showed two results in their simulation studies with nonnormal data: First, they demonstrated that the ad hoc nonnormality corrections of the fit indices provided by the SEM software made the fit worse, better, or unchanged compared with their uncorrected counterparts. Second, the authors proposed new robust versions of RMSEA, CFI, and TLI that performed very well in their simulation studies. They systematically varied the sample size, the extent of misspecification, and nonnormality. Therefore, the same rules of thumb or criteria that are used for normal distributed data can be applied to assess the fit of the structural equation model. My robust_gof.ado ado-file stimates the robust RMSEA, CFI, and TLI fit measures using the corrections proposed by Brosseu-Liard et al. and Savalei. It also estimates a 90% confidence interval for the root mean squared error of approximation. robust_gof.ado can be executed after the sem command with the vce(sbentler) option and estat gof, stats(all) as a postestimation command by simply typing robust_gof. It returns the estimated fit indices and scalars as r containers. I will present a survey example of islamophobia analysis in Germany to demonstrate the usefulness of robust_gof.ado.