Knockoffs are a powerful tool for controlled variable selection with false discovery rate (FDR) control. However, while they are frequently used in high-dimensional regressions, they lack power in low-dimensional and sparse signal settings. One of the main reasons is that knockoffs require a minimum number of selections, depending on the nominal FDR level. In this paper, we leverage e-values to allow the nominal level to be switched after looking at the data and applying the knockoff procedure. In this way, we can increase the nominal level in cases where the original knockoff procedure does not make any selections to potentially make discoveries. Also, in cases where the original knockoff procedure makes discoveries, we can often decrease the nominal level to increase the precision. These improvements come without any costs, meaning the results of our post-hoc knockoff procedure are always more informative than the results of the original knockoff procedure. Furthermore, we apply our technique to recently proposed derandomized knockoff procedures.

翻译：Knockoffs是一种用于控制错误发现率（FDR）的变量选择方法。然而，尽管该方法常被用于高维回归分析，但在低维和稀疏信号场景下其检验功效不足。主要原因之一是Knockoffs方法要求最小选择数量，该数量取决于名义FDR水平。本文利用e值技术，允许在观察数据并应用Knockoffs程序后调整名义水平。通过这种方式，当原始Knockoffs程序未选择任何变量时，我们可以提高名义水平以可能获得发现；而在原始程序已有发现的情况下，我们通常可以降低名义水平以提高精确度。这些改进无需任何代价，意味着我们的事后Knockoffs程序结果始终比原始程序结果更具信息量。此外，我们将该技术应用于近期提出的去随机化Knockoffs方法。