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 \[\begin{equation} \eta^2=\frac{\frac{F\times df_1}{df_2}}{\frac{F\times df_1}{df_2}+1} \tag{B.1} \end{equation}\] 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 \[\begin{equation} \eta^2_{adj}=\frac{\frac{F\times df_1}{df_2}-\frac{df_1}{df_2}}{\frac{F\times df_1}{df_2}+1} \tag{B.2} \end{equation}\] 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) \[\begin{equation} \eta^2_{adj}=\eta^2-([1-\eta^2]\times\frac{v}{N-v-1}) \tag{B.3} \end{equation}\] 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 \[\begin{equation} \eta^2_{adj}=\eta^2-\frac{1-\eta^2}{df} \tag{B.4} \end{equation}\]

where \(df=N-2\).


Ivarsson, Andreas, Mark B. Andersen, Urban Johnson, and Magnus Lindwall. 2013. β€œTo 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.