In this article we focus on three partial effect sizes for the correlation ( family of effects: the standardized slope , the partial correlation , and the semi-partial correlation . These partial effect sizes are useful for meta-analyses in two common situations: when primary studies reporting regression models do not report bivariate correlations, and when it is of specific interest to partial out the effects of other variables. We clarify the use of these three indices in the context of meta-analysis and describe how the indices can be estimated and analyzed. We provide examples of syntheses of these partial effect sizes using a published social work meta-analysis. Finally, we share practical recommendations for meta-analysts wanting to use such indices.
Key words: meta-analysis, research synthesis, partial effect sizes, regression analysisMeta-analysis is the statistical analysis of empirical results of a series of studies (Glass, 1977). Quantitative procedures in meta-analysis typically begin by transforming study results to a common metric, which often represents results as effect sizes. An effect size is a scale-free index that assesses the magnitude and direction of some relationship between variables. Almost two decades ago researchers argued that results from regression models can be treated as effect-size indices (e.g., Becker & Schram, 1994;Cooper & Hedges, 1994;Greenwald, Hedges, & Laine, 1996;Valentine, DuBois, & Cooper, 2004). More recently, different approaches have been proposed to synthesize regression models (Aloe & Becker, 2012;Becker & Wu, 2007;Peterson & Brown, 2005;Wu & Becker, 2013). These approaches can be organized into three methodological groups. The first group focuses on methods that combine full regression models (e.g., Wu & Becker, 2013), the second group focuses on attempting to approximate the bivariate correlation when it is not directly reported (e.g., Peterson & Brown, 2005), and the third group focuses on methods that combine partial effect sizes that have been extracted from regression models (e.g., Aloe & Becker, 2012;Keef & Roberts, 2004).This article focuses on the estimation and use of the partial effect sizes for the family of effects, which can be computed when multiple predictor variables are included in a primary study. A partial effect size is an index that describes the magnitude of an effect after controlling for the influence of other variables in a model. Part of our rationale for focusing on partial effect sizes is that they are quantities that researchers typically find familiar. Furthermore, partial effect sizes are often directly reported or easily estimated from results found in primary studies. Finally, typical meta-analysis techniques can be used once partial effects are extracted or computed from primary studies.We begin by briefly introducing the meta-analysis of correlational data. Next we introduce required notation for effects and meta-analytic models, including three partial effect sizes (partial correlation, semi-partial correlation, and standardiz...