Accuracy of conversion formula for effect sizes: A Monte Carlo simulation

In meta-analysis, effect sizes often need to be converted into a common metric. For this purpose conversion formulas have been constructed; some are exact, others are approximations whose accuracy has not yet been systematically tested. We performed Monte Carlo simulations where samples with pre-specified population correlations between the x and y-variables were drawn from a normally distributed population. A number of commonly used effect size measures and statistics were calculated from each sample. Using several available conversion formula these statistics were converted into Pearson r and Cohen's d and compared to r and d calculated directly from the original data. Converted values were systematically lower than the directly calculated values. While conversions to d were quite accurate, some of the conversions to r resulted in large biases. These systematic errors can in most cases be adjusted for by simply multiplying the converted values with a corresponding correction factor.