Speaker
Description
Meta-analyses often involve transforming bounded effect size measures, such as correlation coefficients or odds ratios, onto a real-valued scale prior to estimation. The results are then back-transformed to the original scale for interpretation purposes. However, in the standard random effects model for meta-analysis, simply applying the inverse transformation function generally does not yield an estimate of the mean but of the median effect size, a phenomenon known as transformation bias. This issue is frequently overlooked in practice, leading to an incorrect definition of the estimate or, equivalently, systematic bias in the mean effect size estimates. Integral back-transformations provide a more accurate approach.
We give an overview of different types of back-transformations and a general formulation of the integral back-transformation. In addition, methods for deriving corresponding back-transformed confidence intervals (CIs) are presented. Approaches are compared, aiming for CIs for the mean, median, and mode effect size.
We analyze differences between these back-transformation approaches in a simulation study and visualize the transformation bias analytically and with example data sets. Furthermore, we address an inconsistency that arises when non-symmetrical transformation functions are applied and the CI for the back-transformed mean is interpreted as a hypothesis test. Software implementations of the integral back-transformations in the R package 'metafor' are presented for various effect sizes including correlation coefficients, proportions, odds and risk ratios, and Cronbach's alpha.
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