# Covariance structure of effects in multivariate meta-analysis with correlated outcomes and predictors

I'm working on a meta-analysis of studies examining the correlation between SES and mental health. Unfortunately, studies typically have multiple measures of both SES and mental health, which I've been handling through a multilevel, multivariate approach:

1. I am conducting separate analysis for different SES measures (e.g. parental education and family income).
2. I use a three level model to allow for shared variance between studies. (see Van den Noortgate, W., López-López, J. A., Marín-Martínez, F., & Sánchez-Meca, J. (2013). Three-level meta-analysis of dependent effect sizes. Behavior Research Methods, 45, 576–594.) According to that source, this apparently also allows pretty well for correlated otucomes, but just to be sure I am also:
3. Calculating a variance-covariance matrix based on variance components of effects and the correlations between the outcome variables. My approach is similar to that described in the clubSandwich package save that I am providing different estimates of correlation depending on the outcome variables involved, since I have good data about their correlations.

Ok here is the rub: I've been asked to make a combined analysis across SES measures, mostly to allow for better testing of moderators. So now I have a model where there is covariance between effects based on correlation between outcomes AND correlation between SES measures.

If I'm just considering correlated outcomes, the covariance can be calculated as a cross product of the variance estimates with the correlation table between outcomes. How can I additionally consider correlation between the SES variables in this formula?. The outcome metric is Hedge's g.

• title part of your question is not clear. Also, end part of body is confusing. Be brief and explicit. – Subhash C. Davar Jul 11 at 13:11
• Subhash thank you for responding and I apologize if my question was not clear. The brief version of my question is: how can I calculate a variance-covariance matrix for a multivariate meta-analysis, given that I have multiple correlated outcome variables and multiple correlated predictor variables within samples. I have reasonable estimates of the involved correlations. – mrpeverill Jul 11 at 14:21