One-stage and two-stage adaptive step-up procedures for false discovery rate control under independence and dependence

⚠️ Curated Limitations & Negative Results

• For the one-stage procedure BR-1S-alpha with thresholds Delta(i) = min(alpha, alpha (1 - alpha) i / (m - i + 1)), the first threshold Delta(1) = alpha (1 - alpha) / m is slightly smaller than the standard linear step-up threshold alpha / m by a factor of (1 - alpha), and BR-1S-alpha only strictly exceeds linear step-up when the rejection fraction satisfies i / m > alpha + 1/m; by Lemma 2, in sparse or low-signal regimes where the Gaussian signal strength satisfies mu < mu* = Phi_bar^{-1}(alpha^2) - Phi_bar^{-1}(alpha (1 - pi_0 alpha) / (1 - pi_0)), BR-1S-alpha and two-stage procedures using lambda = alpha provide negligible power gain over non-adaptive Benjamini-Hochberg (Section 3.1 and Section 5.1).
• While Theorem 3.2 proves finite-sample FDR <= alpha for plug-in procedures G_1..G_4 under independent p-values (where G(p) must be coordinate-wise non-increasing so that G(p_{0,h}) can be bounded via inverse-binomial expectations), positive regression dependency (PRDS) reverses the required monotonicity direction in Lemma 3 to coordinate-wise non-decreasing G(p); consequently, standard one-stage and two-stage adaptive procedures (Storey-lambda, BKY06-lambda, BR-1S-lambda, BR-2S-lambda) do not have a theoretical PRDS guarantee for arbitrary correlation matrices and can exceed nominal FDR when lambda = 1/2 under strong positive correlation (Section 3.3, Proposition 1, and Section 4).
• The provably valid two-stage adaptive procedures under PRDS and arbitrary dependence (Theorems 4.1 and 4.2) must split the error budget across two stages on the same p-value set (alpha = alpha_0 + alpha_1 in Theorem 4.1 or alpha = alpha_1 + kappa alpha_0 in Theorem 4.2), which makes them conservative when pi_0 is moderate or large: for instance, Theorem 4.2 with kappa = 2, alpha_0 = alpha/4, alpha_1 = alpha/2 only produces second-stage thresholds larger than non-adaptive step-up when the first-stage rejection fraction exceeds F_2^{-1}(2) = 0.625 (requiring pi_0 < 0.375; Section 4).
• All 989 lines of Blanchard & Roquain (2009, Journal of Machine Learning Research 10:2837-2871, arXiv:0707.0536v3) were read. Constructed numerical checks in research/discovery/one-and-two-stage-adaptive-step-up-fdr-under-dependence-boundaries.py verify (1) Section 3.1 Theorem 3.1 exact finite-sample FDR bound for BR-1S-alpha (max FDR = 0.04649744 <= 0.05 across m_0 in {1..10}) and Lemma 2 Gaussian signal threshold mu* = 1.511105 (asymptotic LSU rejection proportion 0.008 < 0.05 at mu = 1.2 vs 0.184 > 0.05 at mu = 2.0), (2) Section 3.2 Theorem 3.2 & Corollary 1 inverse-binomial bound max_q pi_0 E[G_1(p_{0,h})] = 1.0 and strict plug-in hierarchy G_3 = 1.142857 < G_4 = 1.333333 < G_1 = 1.6 yielding second-stage rejections LSU = 1 < BKY06 = 2 < BR-2S = 3 < Storey = 4, (3) Section 3.3 Proposition 1 comonotonic maximal-dependence FDR formulas at m = 20, alpha = 0.05 (Storey-1/2 FDR = 0.5 = 10 alpha, Quant-1/2 FDR = 0.083333, vs exact FDR = 0.05 for BR-1S-alpha, BR-2S-alpha, BKY06-alpha, and Storey-alpha when lambda = alpha in [0.045455, 0.5]), and (4) Section 4 Theorems 4.1 & 4.2 Markov factor F_2(0.75) = 3.414214 with crossover proportion F_2^{-1}(2) = 0.625 and rejections increasing from 15 (LSU) to 18 (Theorems 4.1 & 4.2).