Bias and Efficiency in Structural Equation Modeling: Maximum Likelihood Versus Robust Methods.

In the structural equation modeling literature, the normal-distribution-based maximum likelihood (ML) method is most widely used, partly because the resulting estimator is claimed to be asymptotically unbiased and most efficient. However, this may not hold when data deviate from normal distribution. Outlying cases or nonnormally distributed data, in practice, can make the ML estimator (MLE) biased and inefficient. In addition to ML, robust methods have also been developed, which are designed to minimize the effects of outlying cases. But the properties of robust estimates and their standard errors (SEs) have never been systematically studied. This article studies two robust methods and compares them against the ML method with respect to bias and efficiency using a confirmatory factor model. Simulation results show that robust methods lead to results comparable with ML when data are normally distributed. When data have heavy tails or outlying cases, robust methods lead to less biased and more efficient estimators than MLEs. A formula to obtain consistent SEs for one of the robust methods is also developed. The formula-based SEs for both robust estimators match the empirical SEs very well with medium-size samples. A sample of the Cross Racial Identity Scale with a 6-factor model is used for illustration. Results also confirm conclusions of the simulation study.