Measurement Equivalence, Symmetry, Effect Sizes, and Meta-Analysis

In meta-analysis, different measures are often used in different studies, so the meta-analyst must combine effect sizes (ESs) based on different measures with unique and arbitrary metrics. It has been argued that standardization in ESs places them on a common metric, which makes them comparable. However, recent theoretical development and research has challenged this assertion. In this article, especially relevant for social work meta-analysts, I draw from symmetry and mathematical group theory to create a conceptual and theoretical basis for the measurement conditions needed for the invariance and comparability of ESs based on different measures. This is the first time that mathematical group theory arguments have been used in the social work literature on measurement. Symmetry is defined and illustrated, as is mathematical group theory—a theory of symmetry. I demonstrate that measurement equivalence manifests as symmetries between true-score distributions based on different measures, which leads to invariant true-score ESs and measurement-independent meta-analytic results. I also show that the equal reliabilities of observed scores leads to approximate symmetries between observed-score distributions and invariant ESs and approximately symmetric meta-analytic outcomes. I argue that measurement equivalence is necessary for meta-analyses. Implications of these results for social work research synthesis are considered.