Back-transformations in random-effects meta-analysis—Impact and interpretation

In meta-analyses, effect size measures with bounded or non-normal sampling distributions are commonly analyzed on a transformed scale to justify normality assumptions. While point estimates and confidence intervals (CIs) are routinely back-transformed to the original scale for interpretation, this practice is nontrivial in random-effects models. In particular, standard inverse back-transformations yield estimates of the median rather than the mean effect size due to Jensen's inequality. Integral back-transformations provide a principled solution for recovering the mean on the original scale, but their use entails practical issues. We study integral back-transformations for several effect size measures, including correlation coefficients, proportions, odds and risk ratios, and Cronbach's alpha. We derive general formulations for integral back-transformations and corresponding CIs that are applicable across different transformation functions and provide a software implementation. Although required to obtain correct mean estimates, these approaches must be used with caution, as they are sensitive to heterogeneity estimation and can be unstable for unbounded transformations. Certain asymmetric transformations can also lead to inconsistent inference results. We illustrate these issues using analytical considerations and re-analyses of several meta-analyses. Importantly, we stress that the choice of back-transformation depends on the analyst's goals. The standard inverse back-transformation remains well suited for descriptive purposes and is often preferable in practice, provided that it is correctly interpreted as a back-transformed median effect size. We recommend using the integral back-transformation only when a mean estimate is explicitly required, and restricting its use for CIs to cases where they are demanded, but not for hypothesis testing.