Converting and Constructing Effect Sizes With the Response Ratio

ABSTRACT Data synthesised and published as response ratios in ecology (, or ratio of means, ) 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' (standardised mean difference, ). This conversion is practical and near exact—as long as assumptions of homogeneity of variances are met, Hedges' 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 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 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.