Fig. 1

Illustration of the 2dFDR procedure using simulated datasets. A The decision boundaries for 1dFDR-A (blue line) and 2dFDR (red line) at 5% FDR for a highly confounded scenario (ρ ≈ 0.8). 1dFDR-A relies on adjusted statistic |Z^A| only (one dimension), while 2dFDR is based on both the adjusted and unadjusted statistic |Z^A| and |Z^U| (two dimensions). |Z^U| is used to exclude a large number of irrelevant features (red vertical line). After that, a less stringent cutoff of |Z^A| (red horizontal line) can be used to achieve a higher power while maintaining the same FDR. The aim of 2dFDR is thus to find the best cutoffs on the two dimensions to maximize the power. B, C The power (true positive rate) difference between 1dFDR-A and 2dFDR increases with the correlation between the variable of interest and the confounder. When the correlation is low (“+,” ρ ≈ 0.2), |Z^A| and |Z^U| are highly correlated, and |Z^U| provides little extra information. The power of 2dFDR is thus similar to that of 1dFDR-A. When the correlation is higher (“++,” “+++,” ρ ≈ 0.6, 0.8), the signals (brown) and noises (blue) are more difficult to separate on |Z^A|. By using |Z^U|, 2dFDR excludes a large number of noises without losing many signals. The signal density on |Z^A| is thus enriched, leading to a significant power gain