Converting and Constructing Effect Sizes With the Response Ratio

Data synthesised and published as response ratios in ecology ( $\text{lnRR}$ , or ratio of means, $\mathrm{RoM}$ ) remain isolated from broad secondary analyses because they cannot be converted to other effect size metrics. Here I address this lack of data interoperability by developing a conversion to the widely used Hedges' $d$ (standardised mean difference, $\mathrm{SMD}$ ). This conversion is practical and near exact-as long as assumptions of homogeneity of variances are met, Hedges' $g$ correction is used to adjust for small-sample bias, and only additive and not multiplicative ecological processes are converted. I then generalise this conversion with abstract algebra to develop additional opportunities to reuse effect sizes-first by stating the response ratio as a geometric construction of Pythagorean means, and then $d$ as a proportional compass-and-straightedge construction of the response ratio. Constructability is a new pathway of interoperability for effect sizes, and without collecting new data, allows for the response ratio and $d$ to be repurposed into relative change datatypes such as the arithmetic, harmonic, geometric, quadratic and logarithmic means. Much of what has been synthesised in ecology is only available as response ratios, and I hope these conversions increase their value post-publication and facilitate reuse for bolder, more comprehensive meta-analyses.