Obtaining robust practical fit indices with multiply imputed nonnormal data in structural equation modeling

Evaluating the model fit of a structural equation model under the presence of missing nonnormal data is challenging, especially when multiple imputation (MI) is used. Various robust estimators for nonnormality and strategies to pool model fit indices across imputations could be utilized. Through a comprehensive simulation study, we investigated the performance of various combinations of robust estimators, including Maximum Likelihood with Robust standard errors and test statistic (MLR), Maximum Likelihood with robust standard errors and a Mean-adjusted test statistic (MLM), and Maximum Likelihood with robust standard errors and a Mean- and Variance-adjusted test statistic (MLMV), and pooling strategies in computing two popular practical fit indices: root mean-squared error of approximation (RMSEA) and comparative fit index (CFI). These pooling strategies are extensions of normal-theory-based D₂, D₃, and D₄ pooling methods. Results suggested that MLM and MLMV yielded more accurate Type I error rates than MLR and showed minimal difference on producing practical fit indices. The extensions of D₄ and D₃ (both were comparable) produced better Type I error rates and lower failure rates than that of D₂ strategy. Robust CFI estimates appeared to be less affected than robust RMSEA under MCAR (missing completely at random) and MAR (missing at random) conditions, when the missing data were not drawn from the long tail of skewed distributions; however, interpretations of their performance should be made with caution.