A Systematic Evaluation and Comparison Between Exploratory Structural Equation Modeling and Bayesian Structural Equation Modeling

Abstract

In this study, we contrast two competing approaches, not previously compared, that balance the rigor of CFA/SEM with the flexibility to fit realistically complex data. Exploratory SEM (ESEM) is claimed to provide an optimal compromise between EFA and CFA/SEM. Alternatively, a family of three Bayesian SEMs (BSEMs) replace fixed-zero estimates with informative, small-variance priors for different subsets of parameters: cross-loadings (CL), residual covariances (RC), or CLs and RCs (CLRC). In Study 1, using three simulation studies, results showed that (1) BSEM-CL performed more closely to ESEM; (2) BSEM-CLRC did not provide more accurate model estimation compared with BSEM-CL; (3) BSEM-RC provided unstable estimation; and (4) different specifications of targeted values in ESEM and informative priors in BSEM have significant impacts on model estimation. The real data analysis (Study 2) showed that the differences in estimation between different models were largely consistent with those in Study1 but somewhat smaller.

Keywords:

• Factor analysis
• Bayesian statistics
• exploratory structural equation modeling
• informative priors

Acknowledgments

The authors would like to acknowledge David Kaplan, Bengt Muthén, and Tihomir Asparouhov for their comments on earlier versions of this paper.

Notes

¹ The rotation function for the Geomin rotation criterion is $f(\mathrm{\Lambda })=\sum _{i=1}^p{\left(\prod _{j=1}^m({\lambda }_{ij}^2+\epsilon )\right)}^{1/m}$

Where $\epsilon$ is a small positive constant added by Browne (2001) to reduce the problem of indeterminacy. Geomin has performed relatively well when numbers of non-zero cross-loadings for each latent variables are greater than 1 in both simulation and empirical examples, when compared with other mechanical rotation criteria (Marsh et al., 2009; McDonald, 2005).

² Once the Markov chain has stabilized, the iterations prior to the stabilization (referred to as the “burn-in” phase) are discarded.

³ In geomin rotation, the constant $\epsilon$ was set to .05 which has been widely used in empirical studies (Marsh et al., 2009; Marsh et al., 2010, 2014). A recent simulation study (Celimli., Myers, & Ahn, 2018) found that the geomin rotation with $\epsilon$ = .05 provided more stable but less accurate factor solutions than the default geomin rotation (where $\epsilon$ = .001 in Mplus) with very small effect size; the accuracy of factor solutions in geomin rotation with $\epsilon$ = .05 increased when the factors are more correlated.

⁴ We also compared BIC between ESEM and BSEM-CL and found that ESEM had consistently smaller BIC than BSEM-CL to a small extent (diff = 53–64 to across different sample sizes). In addition, given that the DIC was developed as the Bayesian counterpart of AIC in frequentist analysis, we compared AIC in ESEM with DIC in BSEM-CL and found that the differences were tiny (diff = 7–12 to across different sample sizes). Even though these fit indices are fairly good approximations for model comparisons between ML and Bayesian estimation, there is no full simulation studies that have confirmed that. Hence, the small differences in these fit indices should be treated as inconclusive (see http://www.statmodel.com/cgi-bin/discus/show.cgi?9/6256 for further discussion in Mplus discussion forum). Also see below for model fit comparison among different BSEM models (Table 5).