Relative Performance of MLR, WLSMV, and Bayes Estimators: An Investigation Using Confirmatory Factor Analysis with Ordinal Data

This study investigated the performance of four robust estimators— robust Maximum Likelihood for continuous data (MLR-CON), robust Maximum Likelihood for categorical data (MLR-CAT), robust Weighted Least Squares (WSLMV), and Bayes estimation—in the context of Confirmatory Factor Analysis (CFA) with ordinal data. The objective was to conduct a comprehensive assessment of their efficacy across varied data conditions. A simulation study was designed to comprise conditions that may be encountered with empirical research. The design included 90 cells with two levels of model specification (correct, misspecified), three sample sizes (100, 200, 500), five item distributions (symmetric, moderately asymmetric, moderately asymmetric-alternating, extremely asymmetric, extremely asymmetric-alternating), and three item categories (2, 4, 5). Outcomes included convergence rates, relative bias in parameter estimates (e.g., factor loading, factor correlations, standard errors), model rejection rates, and fit indices. The findings indicated that categorical estimators such as MLR-CAT, WLSMV, and Bayes were able to yield unbiased factor loadings and factor correlations for ordinal data under correct model specification and adequate convergence, with MLR-CAT providing the most accurate parameter estimates. MLR-CON tended to produce accurate estimates only for ordinal data with at least five categories and a symmetric distribution. In cases of slight model misspecification where two insignificant secondary paths were omitted, a similar pattern is observed in the recovery of factor loadings by the estimators, although all estimators tended to overestimate factor correlations. Model misspecification had little impact on the estimation of standard errors for the parameter estimates. The standard errors yielded by MLR-CON, MLR-CAT and Bayes were largely unbiased with the only exception being asymmetric dichotomous data at a sample size of 100. However, WLSMV only produced satisfactory results for dichotomous data and showed large, negative bias in all other conditions. Regarding model fit indices, Bayes-based Posterior Predictive P-value (PPP) exhibited no type I error for the correctly specified model and considerable tolerance for minor model misspecification. WLSMV-based χ^2 and alternative fit indices (CFI, TLI, RMSEA) generally recommended retaining a correct model across all sample sizes and demonstrated moderate tolerance for minor model misspecification in small samples. MLR-CON χ^2 and alternative fit indices are considered acceptable only for normally distributed data with a large sample size (N > 200). Based on the study, the categorical estimators Bayes, MLR-CAT, and WLSMV are generally recommended for ordinal CFA in small to medium sample sizes. If the standard errors of the parameter estimates are a concern, Bayes and MLR-CAT are preferred over WLSMV. MLR-CAT is particularly suitable for obtaining the most accurate parameter estimates when applicable, while Bayes can effectively handle convergence difficulties that other estimators might encounter. MLR-CON is only recommended for symmetric data with at least five response categories, but its results are less accurate than those yielded by the categorical estimators.