I am preparing data for a meta-analysis regression of effect sizes, in which I will study the determinants of effect size for a specific variable across several studies. I have some studies in my dataset that have used the logarithm of my predictor variable of interest in their regression models. How should I deal with these studies where this independent variable is log-transformed, so that they are comparable to all the other effect sizes (in the other studies) for that same variable that are not log-transformed?
This situation often arises in systematic reviewing. Unfortunately there is nothing that can be done here apart from contacting the authors of the primary study and asking if they can re-run the analysis. The crucial problem if that the untransformed predictors give the effect as additive whereas if you back-transform the results from the analysis using logged predictors you get a multiplicative effect on the original predictor scale.