Confirmatory factor analysis with ordinal data : effects of model misspecification and indicator nonnormality on two weighted least squares estimators

Full weighted least squares (full WLS) and robust weighted least squares (robust WLS) are currently the two primary estimation methods designed for structural equation modeling with ordinal observed variables. These methods assume that continuous latent variables were coarsely categorized by the measurement process to yield the observed ordinal variables, and that the model proposed by the researcher pertains to these latent variables rather than to their ordinal manifestations. Previous research has strongly suggested that robust WLS is superior to full WLS when models are correctly specified. Given the realities of applied research, it was critical to examine these methods with misspecified models. This Monte Carlo simulation study examined the performance of full and robust WLS for two-factor, eight-indicator confirmatory factor analytic models that were either correctly specified, overspecified, or misspecified in one of two ways. Seven conditions of five-category indicator distribution shape at four sample sizes were simulated. These design factors were completely crossed for a total of 224 cells. Previously findings of the relative superiority of robust WLS with correctly specified models were replicated, and robust WLS was also found to perform better than full WLS given overspecification or misspecification. Robust WLS parameter estimates were usually more accurate for correct and overspecified models, especially at the smaller sample sizes. In the face of misspecification, full WLS better approximated the correct loading values whereas robust estimates better approximated the correct factor correlation. Robust WLS chi-square values discriminated between correct and misspecified models much better than full WLS values at the two smaller sample sizes. For all four model specifications, robust parameter estimates usually showed lower variability and robust standard errors usually showed lower bias. These findings suggest that robust WLS should likely remain the estimator of choice for applied researchers. Additionally, highly leptokurtic distributions should be avoided when possible. It should also be noted that robust WLS performance was arguably adequate at the sample size of 100 when the indicators were not highly leptokurtic.