The estimation of parameters in structural equation modeling (SEM) has been primarily based on the maximum likelihood estimator (MLE) and relies on large sample asymptotic theory. Consequently, the results of the SEM analyses with small samples may not be as satisfactory as expected. In contrast, informative priors typically do not require a large sample, and they may be helpful for improving the quality of estimates in the SEM models with small samples. However, the role of informative priors in the Bayesian SEM has not been thoroughly studied to date. Given the limited body of evidence, specifying effective informative priors remains challenging for applied researchers. Therefore, a study that investigates performances on the parameter estimates of the SEM models with small samples among the MLE, the Bayesian estimator with informative priors, and the Bayesian estimator with non-informative priors is warranted.
Two Monte Carlo studies were designed for this dissertation: one with a confirmatory factor analysis (CFA) model and another with an SEM model. Both studies replicated 1000 datasets for each of the various experimental conditions. Specifically, they included a) sample sizes (30 and 70), b) the number of items per factor (5 and 10), c) mean factor loadings (.30 and .70), and d) estimators (the MLE, the Bayesian estimator with non-informative priors, the Bayesian estimator with correctly specified informative priors, and the Bayesian estimator with incorrectly specified informative priors). Results were evaluated by various criteria, including the convergence rate, relative bias, root mean square error (RMSE), standard error (SE). The study on the CFA model focused on the evaluation of factor loadings, while the study on the SEM model concentrated on the evaluation of path coefficients.
Results demonstrated that the Bayesian estimator with informative priors converged with 100% convergence rates even for the CFA models with small sample sizes, as opposed to the ML estimator that had quite low convergence rates. For SEM models, the Bayesian estimator with informative priors displayed high convergence rates when the sample size was large (N=70 ), while the convergence rates were very low when the sample size was small (N=30 ) and the mean factor loading was small (J=.30). For the other conditions, the differences were not substantially large among estimators. In addition, the Bayesian estimator with correctly specified informative priors outperformed the Bayesian estimator with non-informative priors and the MLE in the recovery of factor loadings, while the Bayesian estimator with incorrectly specified informative priors did not outperform the MLE. Finally, the Bayesian estimator with correctly specified informative priors performed best on the recovery of path coefficients in the SEM models with small sample sizes, compared to the other estimators. Also, it was revealed that the performance between the Bayesian estimator with correctly specified informative priors and the Bayesian estimator with incorrectly specified informative priors was similar in this regard. Thus, it would be practical to use the Bayesian estimator with incorrectly specified informative priors to estimate path coefficients in the SEM models with small samples when researchers specify the prior mean somewhat lower than the mean factor loading, with the belief that the mean factor loading is higher or the mean factor loading is lower but the number of indicators per factor is larger. In contrast, researchers need to choose the location of prior mean in terms of accuracy and precision when the number of indicators per factor is lower and they believe that mean factor loading is smaller. If researchers specify prior mean somewhat higher or lower than the population value, the estimate of path coefficients would be most accurate or inaccurate with the least or largest bias but the largest or smallest SE and RMSE value.
In addition to the Monte Carlo studies, a real dataset with a small sample size was analyzed. The results were interpreted by reflecting the results of the Monte Carlo simulation studies. Finally, study limitations, practical implications, and future research were discussed.
Keywords: Bayesian statistics; Informative prior; Factor analysis; Structural equation modeling; Small samples
Education Policy and Leadership
Professor Akihito Kamata
Assocate Professor Doris Baker
Dr. Yusuf Kara
Professor Paul Yovanoff
Number of Pages
Ma, Hao, "Evaluation Of The Utility Of Informative Priors In Bayesian Structural Equation Modeling With Small Samples" (2020). Department of Education Policy and Leadership Theses and Dissertations. 4.