# How can one get consistent (i.e. direct+indirect=total) effects in a Meta-Analytic SEM model with latent variables?

I have a mediation model with latent variables using datasets from a few studies. In particular, one of the latent variables is the outcome, while the others are tested as possible mediators of a given treatment. By performing a SEM model in each study and then pooling estimates together in RE models, I noticed that the total effect needs not be the sum of single effects. I believe this is due to the fact that each effect (including the total one) has a different weighting pattern, and the covariance structure among coefficient estimates of the same regression (I didn't force covariances between mediators to be zero) is neglected. I am aware of the existence of MASEM models (see, for example, here: Whether to apply structural equation modelling separately to each of a set of heterogeneous correlation matrices in a meta-analysis context?), that estimate all effects simultaneously. However, as far as I know, such models are based on (covariances or correlations between) observed (not latent) variables. (see, for example, here: https://openmx.ssri.psu.edu/node/4214). My question is: is there an acceptable way to force the sum of the direct and indirect effects to equal the total effect in a meta-analytic framework with latent variables?

## 1 Answer

I've found out what I believe is the theoretical answer to my question. See also here: How can I perform bivariate random-effects meta-regression if the implied between-study covariance matrix is not positive definite? What I need to do is a Bivariate random-effects meta-analysis (pdf). Multivariate results do not have to be consistent with univariate results, since they make use of further information (i.e., the one about the correlation between effects), thus typically leading to more precise estimates (see, for example, here:(pdf). Obviously, results are still internally consistent, i.e. direct effect + indirect effect = total effect, since the parameters of interest are 2, and the third is just a linear combination (simply a sum, or difference, in this case) of them. In practice, if I meta-analyse any pair of these 3 variables. This means that the total effect implied by the sum of direct and indirect effect (and its variance implied by the between-study covariance matrix), the one calculated by considering the pair direct/total and the one from the pair indirect/total need to coincide.