Evaluation and Comparison of SEM, ESEM, and BSEM in Estimating Structural Models with Potentially Unknown Cross-loadings

ABSTRACT

Cross-loadings are common in multidimensional instruments; however, they cannot be appropriately addressed in conventional structural equation modeling (SEM) owing to the assumption of zero cross-loadings in standard confirmatory factor analysis (CFA). Although it has been proposed that exploratory structural equation modeling (ESEM) and Bayesian structural equation modeling (BSEM) can address this issue more flexibly, their performance in structural parameter estimation has not been adequately compared. This study uses simulated data to evaluate and compare SEM, ESEM, and BSEM in estimating structural models under different manipulation conditions (i.e., sample size, target loading, cross-loading, and path coefficient). The results demonstrated that the performances of these approaches were similar in the case of zero cross-loadings. SEM performed worse as cross-loadings increased, and the performance of BSEM significantly depended on the accuracy of the priors for cross-loadings. ESEM was inferior to BSEM with correctly specified prior means for cross-loadings in most evaluation measures and exhibits unstable performance in conditions with small target loadings. Recommended strategies for selecting an appropriate modeling approach are discussed based on our findings.

KEYWORDS:

• BSEM
• cross-loadings
• ESEM
• structural parameter estimation

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

For an indicator, for example, $x_1$, loads concurrently on two latent variables (${\xi }_1$ and ${\xi }_2$) $Var\left(x_1\right)=Var\left({\lambda }_T\cdot {\xi }_1+{\lambda }_C\cdot {\xi }_2\right)+Var\left(\epsilon \right)={{\lambda }_T}^{2}Var\left({\xi }_1\right)$ $+{{\lambda }_C}^{2}Var\left({\xi }_2\right)+2{\lambda }_T\cdot {\lambda }_C\cdot \rho \cdot \sqrt{Var\left({\xi }_1\right)Var\left({\xi }_2\right)}+Var\left(\epsilon \right)$. Because the variances of the factors and indicators were equal to 1.0, ${{\lambda }_T}^2+{{\lambda }_C}^2+2{\lambda }_T\cdot {\lambda }_C\cdot \rho =1-Var\left(\epsilon \right)$. Because $Var\left(\epsilon \right)\ge 0$, ${{\lambda }_T}^2+{{\lambda }_C}^2+2{\lambda }_T\cdot {\lambda }_C\cdot \rho \le 1$.

Additional information

Funding

This study was funded by the National Natural Science Foundation of China [31871128] and the MOE (Ministry of Education) Project of Humanities and Social Science of China [18YJA190013].