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Back-transformations in random-effects meta-analysis-Impact and interpretation
Jan-Bernd Igelmann1, Markus Pauly1,2, Wolfgang Viechtbauer3
1Department of Statistics, https://ror.org/01k97gp34TU Dortmund University, Germany.
Research Synthesis Methods
|May 15, 2026
Summary
Standard inverse back-transformations in meta-analysis estimate medians, not means. Integral back-transformations can recover mean effect sizes but require careful application due to potential instability and sensitivity to heterogeneity.
Area of Science:
- Biostatistics
- Quantitative Synthesis
- Statistical Modeling
Background:
- Meta-analyses often transform effect sizes for normality assumptions.
- Back-transformation is used for interpreting results on the original scale.
- Standard inverse back-transformations in random-effects models can yield median, not mean, estimates due to Jensen's inequality.
Purpose of the Study:
- To investigate integral back-transformations for recovering mean effect sizes in meta-analysis.
- To derive general formulations for integral back-transformations and confidence intervals (CIs).
- To provide a software implementation for these methods.
Main Methods:
- Studied integral back-transformations for various effect sizes (correlation coefficients, proportions, odds ratios, risk ratios, Cronbach's alpha).
- Derived general formulations for integral back-transformations and associated CIs.
- Developed a software implementation for practical application.
Main Results:
- Integral back-transformations can recover mean effect size estimates but are sensitive to heterogeneity estimation and can be unstable.
- Asymmetric transformations may lead to inconsistent inference.
- Standard inverse back-transformation is suitable for descriptive purposes and median estimation.
Conclusions:
- The choice of back-transformation method depends on the analyst's objective (mean vs. median estimation).
- Integral back-transformation is recommended only when a mean estimate is explicitly required.
- Use integral back-transformations for CIs cautiously, primarily for estimation, not hypothesis testing.
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