Fig. 2

Performance on simulated datasets across varying density and strength of the true signals. Average false discovery proportions and true positive rates are compared at 5% FDR using simulated datasets. 1dFDR-U and 1dFDR-A represent the one-dimensional FDR control procedures based on the unadjusted model and confounder-adjusted model, respectively. 1dFDA-H is a heuristic adaptive procedure that uses the adjusted or unadjusted model depending on whether the confounder effect is significant (nominal p value < 0.05). The performance is evaluated at varying signal strength (left: weak, right: strong), signal density (top: low, bottom: high), and the correlation between the variable of interest and the confounder (inside the panel, “+,” “++,” and “+++” represent a low, medium, and high correlation (ρ ≈ 0.2, 0.6, 0.8), respectively). The density of the confounding signals is 10%, and the strength is moderate. 2dFDR and 1dFDR-A control the FDR at the target level (dashed line), while 1dFDR-U and 1dFDR-H fail to control the FDR properly when the confounding is not weak (A). 2dFDR becomes substantially more powerful than 1dFDR-A as the correlation between the variable of interest and the confounder increases (B). The difference is more pronounced when the signals are weak and sparse, as indicated by the percent increase shown on top of the bars