A Comparison of Penalized Maximum Likelihood Estimation and Markov Chain Monte Carlo Techniques for Estimating Confirmatory Factor Analysis Models with Small Sample Sizes

With small to modest sample sizes and complex models, maximum likelihood (ML) estimation of confirmatory factor analysis (CFA) models can show serious estimation problems such as nonconvergence or parameter estimates that are outside the admissible parameter space. In the present article, we discuss two Bayesian estimation methods for stabilizing parameter estimates of a CFA: Penalized maximum likelihood (PML) estimation and Markov Chain Monte Carlo (MCMC) methods. We clarify that these use different Bayesian point estimates from the joint posterior distribution—the mode (PML) of the joint posterior distribution, and the mean (EAP) or mode (MAP) of the marginal posterior distribution—and discuss under which conditions the two methods produce different results. In a simulation study, we show that the MCMC method clearly outperforms PML and that these performance gains can be explained by the fact that MCMC uses the EAP as a point estimate. We also argue that it is often advantageous to choose a parameterization in which the main parameters of interest are bounded and suggest the four-parameter beta distribution as a prior distribution for loadings and correlations. Using simulated data, we show that selecting weakly informative four-parameter beta priors can further stabilize parameter estimates, even in cases when the priors were mildly misspecified. Finally, we derive recommendations and propose directions for further research.