Power of FDR Control Methods: The Impact of Ranking Algorithm, Tampered Design, and Symmetric Statistic
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As the power of FDR control methods for high-dimensional variable selections has been mostly evaluated empirically, we focus here on theoretical power analyses of two recent such methods, the knockoff filter and the Gaussian mirror. We adopt the Rare/Weak signal model, popular in multiple testing and variable selection literature, and characterize the rate of convergence of the number of false positives and the number of false negatives of FDR control methods for particular classes of designs. Our analyses lead to several noteworthy discoveries. First, the choice of the symmetric statistic in FDR control methods crucially affects the power. Second, with a proper symmetric statistic, the operation of adding "noise" to achieve FDR control yields almost no loss of power compared with its prototype, at least for some special classes of designs. Third, the knockoff filter and Gaussian mirror have comparable power for orthogonal designs, but they behave differently for non-orthogonal designs. We study the block-wise diagonal designs and show that the knockoff filter has a higher power when the regression coefficient vector is extremely sparse, and the Gaussian mirror has a higher power when the coefficient vector is moderately sparse.
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