Estimation and Specification Search of Structural Equation Modeling Using Frequentist and Bayesian Methods

Structural equation modeling (SEM) refers to statistical analyses of the relationships among observed and latent variables based on hypothesized models. In reality, proposed models are rarely perfect, specification search is conducted to correct specification errors between the proposed and population models. Both frequentist and Bayesian methods have strength and limitations in the estimation and specification search of SEM models. Estimation problems that arise from the violation of distributional and/or structural assumptions have not been thoroughly studied. Performance of specification search methods based on different theoretical framework has rarely been compared. Two purposes of this study were: (1) to investigate robust maximum likelihood (RML) and three Bayesian methods for estimating confirmatory factor analysis models under imperfect conditions, and (2) to compare modification index (MI) and Bayesian structural equation modeling (BSEM) in search of cross-loadings in factor analysis models. Two Monte Caro studies were designed for model estimation (Study 1) and specification search (Study 2), respectively. Both studies replicated 2000 datasets for each condition. Design factors included sample size, factor structure, loading size, and item distribution. Study 1 analyzed both correctly specified and misspecified models. Results were evaluated based on model fit, parameter estimates, and standard errors. Study 2 searched for omitted 1, 2, and 4 cross-loadings in data generation models. The evaluation of results focused on the success of specification search and model evaluation. Results showed that the frequentist chi-square test was more powerful than the Bayesian posterior predictive p-value test. Bayesian methods specified with appropriate priors provided accurate parameter estimates similar to RML even under moderate violation of SEM assumptions. Practically however, the selections of Bayesian priors on hypothesized models need to be exceptionally cautious, because they are likely to interact with sample sizes, data distribution, and degree of model misspecification. In specification search, MI generally provided higher model recovery rates than BSEM under the designed conditions. BSEM led to considerable false positive solutions as sample size increased if informative priors were not properly selected. However, MI is not always preferable. The study recommended that practical selections of Bayesian priors may be based on 95% parameter coverage. Future research will investigate the sensitivity of various Bayesian priors in specification search.