A multivariate mixed linear model for meta-analysis

A multivariate mixed-effects approach for meta-analysis is presented. The approach (a) incorporates as outcomes multiple effect sizes per study; (b) allows different studies to have different subsets of effect sizes; and (c) treats each study's effect sizes as random realizations from a population of possible effect sizes. Application is illustrated via reanalysis of data from studies assessing the effects of coaching on verbal and mathematical subtests of the Scholastic Aptitude Test. Covariance components are estimated via restricted maximum likelihood (REML); inferences about regression coefficients and specific study effect sizes are based on their joint conditional distribution given the REML covariance component estimates. The approach can be implemented via now-standard software for unbalanced hierarchical data.