Fixed-Effects Meta-Analyses as Multiple-Group Structural Equation Models

Abstract

Meta-analysis is the statistical analysis of a collection of analysis results from individual studies, conducted for the purpose of integrating the findings. Structural equation modeling (SEM), on the other hand, is a multivariate technique for testing hypothetical models with latent and observed variables. This article shows that fixed-effects meta-analyses with the following characteristics can be modeled in the SEM framework: (a) using any type of effect size; (b) including categorical and continuous moderators; and (c) including multivariate effect sizes. Empirical examples in LISREL syntax are used to demonstrate the equivalence between the meta-analytic and SEM approaches. Future directions for and extensions to this approach are discussed.

Notes

¹SEM can also be used to model mean structures as in multiple-group analysis (Sörbom, 1974) and latent growth modeling (Willett & Sayer, 1994). Recent developments in SEM focus on analyzing raw data directly. The advantage of this is that missing data can be handled efficiently with full information maximum likelihood estimation (e.g., Arbuckle, 1996; Enders, 2001; Neale, 2000).

²When we construct a model with two variables, there will be one correlation in a correlation structure analysis and one covariance and two variances in a covariance structure analysis. If a correlation structure analysis is used to conduct a meta-analysis, the correlation can be used to represent the effect size. If a covariance structure analysis is used to conduct a meta-analysis, we will create three effect sizes. Although missing effect sizes are allowed (see the discussion in modeling multivariate meta-analysis in SEM), it might complicate the syntax setup.

³The meanings of WLS and GLS are different in meta-analysis and SEM. In the literature of meta-analysis and regression analysis, WLS and GLS are used to handle the error structures with a diagonal and a block-diagonal variance-covariance matrix, respectively. In other words, WLS is a special case of GLS. In the context of SEM, GLS is a special case of WLS. The GLS estimation method is used with a normality assumption on the data, whereas the WLS estimation method can be used for data with arbitrary distribution.

⁴As suggested by an anonymous reviewer, sample size can be used as a potential moderator to explain the variability of effect sizes (see Hox, 2002, for an example).

⁵LISREL prints a warning message showing that “Total sample size is smaller than the number of parameters. Parameter estimates are unreliable.” We can safely ignore it when we are conducting a meta-analysis in SEM.

⁶A more efficient and simple way, which is equivalent to the use of phantom variables, is to create two additional parameters (AP = 2) in LISREL. The additional parameters are used to directly represent the regression coefficients (see the LISREL syntax in the Appendix; M. W. L. Cheung, 2007, 2009).

⁷Missing effect sizes are coded with 0 in the input files in LISREL. The sampling variances and the sampling covariances of the missing effect sizes are coded as 1 and 0, respectively. This setup is the same as that in MASEM (Cheung & Chan, 2005a).

⁸LISREL does not provide overall GFI and SRMR indexes in multiple-group analysis. It only gives GFI and SRMR in each group. The overall GFI and SRMR reported in this article are calculated by GFI = Σ ^k _i=1 and SRMR = where k is the number of studies.