Modeling dependent effect sizes with three-level meta-analyses: a structural equation modeling approach

Meta-analysis is an indispensable tool used to synthesize research findings in the social, educational, medical, management, and behavioral sciences. Most meta-analytic models assume independence among effect sizes. However, effect sizes can be dependent for various reasons. For example, studies might report multiple effect sizes on the same construct, and effect sizes reported by participants from the same cultural group are likely to be more similar than those reported by other cultural groups. This article reviews the problems and common methods to handle dependent effect sizes. The objective of this article is to demonstrate how 3-level meta-analyses can be used to model dependent effect sizes. The advantages of the structural equation modeling approach over the multilevel approach with regard to conducting a 3-level meta-analysis are discussed. This article also seeks to extend the key concepts of Q statistics, I2, and R2 from 2-level meta-analyses to 3-level meta-analyses. The proposed procedures are implemented using the open source metaSEM package for the R statistical environment. Two real data sets are used to illustrate these procedures. New research directions related to 3-level meta-analyses are discussed.