A new effect size for meta-analysis of magnitude: lnM

Shinichi Nakagawa , Ayumi Mizuno , Coralie Williams , Malgorzata Lagisz , Rose E O'Dea, Daniel W.A. Noble, Alistair M Senior, Erick Lundgren, Santiago Ortega

Meta-analyses in ecology and evolution often consider the magnitude of differences between groups rather than their direction. Yet, a common practice is to coerce signed effects (e.g., d and response ratio) into magnitudes by taking absolute values. This transformation induces strong upward bias and non-normal (Gaussian) sampling distributions, violating the assumptions of standard meta-analytic models. Here
we introduce lnM, a log-ratio effect size for the magnitude of difference between two groups, defined from standard one-way ANOVA components. Unlike absolute-value approaches, the proposed lnM is asymptotically normal, and can be analysed, using standard multilevel meta-analysis and meta-regression with both categorical and continuous moderators. We combine theory, simulations, and worked examples
to compare lnM with absolute-value approaches. We show when the delta-method and parametric single-fit bootstrap estimators for lnM perform well, and how one may assess publication bias. The lnM effect size provides a direction-free, meta-regression-friendly measure of magnitude that is applicable to both ratio- and interval-scale traits, offering a practical solution for synthesising the magnitude of ecological and evolutionary effects and beyond.