Hierarchical Covariance Estimation Approach to Meta-Analytic Structural Equation Modeling

Abstract

We present a fully Bayesian approach to meta-analytic SEM based on hierarchical modeling of sample covariance matrices. The approach allows for flexible models that would not be identified under a traditional maximum likelihood approach. The approach allows for the inclusion of moderators, produces a global fit index, and permits the investigation of local misspecification. Simulation-based calibration studies show that the Bayesian computation procedure produces valid inferences for commonplace meta-analytic SEM applications. We demonstrate the approach with diverse data analysis examples and provide accompanying R code to support adoption and additional study of the approach. Finally, we lay out proposals that have the potential to extend the approach to accommodate the wide variety of analyses and data conditions that comprise meta-analytic SEM applications.

Keywords:

• Bayesian estimation
• hierarchical covariance
• meta-analytic SEM

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Data Availability Statement

All code for simulation studies, data analyses, and Stan scripts are available at https://osf.io/rstzk/.

Notes

1 Wu and Browne (2015) refer to this distribution as the second type of matrix variate beta distribution citing chapter 5 of Gupta and Nagar (1999). However, Gupta and Nagar (1999) in definition 5.2.4 include the term generalized to describe this distribution.

2 We thank an anonymous reviewer for suggesting that the degrees of freedom be dependent on study characteristics.

3 Based on the empirical rule, 95% of values will fall within $\approx 2$ standard deviations of the mean. The claim above follows from the prior under which the SRCs are assumed to have a 0-mean and ${\tau }_{\psi }^{\prime }$ standard deviation.

4 The TSSEM literature does not provide precise cut-offs for SEM fit indices, though cut-offs in Hu and Bentler (1999) were cited by Cheung and Chan (2005).

5 For reference, the machine is a Ryzen 9 3900 3.1 GHz 12-core processor with 32 GB of RAM.

6 This step differentiates SBC from traditional simulations where the population parameter is usually (but not always) kept fixed across replications (within a design condition).

7 OSMASEM as implemented in the metaSEM package took 17 min, no moderators in the model.