B Effect computation

The $$t$$, $$F$$, and $$r$$-values were all transformed into the effect size $$\eta^2$$, which is the explained variance for that test result and ranges between 0 and 1, for comparing observed to expected effect size distributions. For $$r$$-values, this only requires taking the square (i.e., $$r^2$$). $$F$$ and $$t$$-values were converted to effect sizes by $$$\eta^2=\frac{\frac{F\times df_1}{df_2}}{\frac{F\times df_1}{df_2}+1} \tag{B.1}$$$ where $$F=t^2$$ and $$df_1=1$$ for $$t$$-values. Adjusted effect sizes, which correct for positive bias due to sample size, were computed as $$$\eta^2_{adj}=\frac{\frac{F\times df_1}{df_2}-\frac{df_1}{df_2}}{\frac{F\times df_1}{df_2}+1} \tag{B.2}$$$ which shows that when $$F=1$$ the adjusted effect size is zero. For $$r$$-values the adjusted effect sizes were computed as (Ivarsson et al. 2013) $$$\eta^2_{adj}=\eta^2-([1-\eta^2]\times\frac{v}{N-v-1}) \tag{B.3}$$$ where $$v$$ is the number of predictors. It was assumed that reported correlations concern simple bivariate correlations and concern only one predictor (i.e., $$v=1$$). This reduces the previous formula to $$$\eta^2_{adj}=\eta^2-\frac{1-\eta^2}{df} \tag{B.4}$$$

where $$df=N-2$$.

References

Ivarsson, Andreas, Mark B. Andersen, Urban Johnson, and Magnus Lindwall. 2013. “To Adjust or Not Adjust: Nonparametric Effect Sizes, Confidence Intervals, and Real-World Meaning.” Psychology of Sport and Exercise 14 (1). Elsevier BV: 97–102. doi:10.1016/j.psychsport.2012.07.007.