Two simple sufficient conditions for FDR control

By Gilles Blanchard, Étienne Roquain. 2008

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We show that the control of the false discovery rate (FDR) for a multiple testing procedure is implied by two coupled simple sufficient con- ditions. The first one, which we call "self-consistency condition", concerns the algorithm itself, and the second, called "dependency control condition" is related to the dependency assumptions on the p-value family. Many stan- dard multiple testing procedures are self-consistent (e.g. step-up, step-down or step-up-down procedures), and we prove that the dependency control condition can be fulfilled when choosing correspondingly appropriate rejec- tion functions, in three classical types of dependency: independence, pos- itive dependency (PRDS) and unspecified dependency. As a consequence, we recover earlier results through simple and unifying proofs while extend- ing their scope to several regards: weighted FDR, p-value reweighting, new family of step-up procedures under unspecified p-value dependency and adaptive step-up procedures. We give additional examples of other possi- ble applications. This framework also allows for defining and studying FDR control for multiple testing procedures over a continuous, uncountable space of hypotheses. AMS 2000 subject classifications: Primary 62J15; secondary 62G10.

Subfields: Hardware and Architecture, Software, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/087f9482b1ad71e449c7ad758a3b235626168cc5

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Citations in the corpus

Covariate powered cross‐weighted multiple testing (2017, methodology): “Theorem 4 may appear as a consequence of Theorem 4.2. of Blanchard and Roquain (2008), who extended the results of Benjamini and Yekutieli (2001) and proved that the wBY procedure (Definition 3) controls the FDR for any choice of weights and any p- value distribution.”
Optimal weighting for false discovery rate control (2008, methodology): “Note that this result covers the earlier result of Genovese et al. (2006); Blanchard and Roquain (2008), by taking W i ( u ) = w i constant in u .”
Adaptive FDR control under independence and dependence (2007, methodology): “This result was extended by Blanchard and Fleuret (2007) and Blanchard and Roquain (2008) (see also a related result of Sarkar, 2008a): it can be shown that self-consistent procedures (not necessarily non-increasing) based on a particular class of shape functions have controlled FDR: Theorem 2.7.”
p-Values for High-Dimensional Regression (2008, methodology): “Benjamini and Yekutieli (2001) showed that this procedure is conservative under a wider range of dependencies between p-values (see Blanchard and Roquain 2008 for related work).”
Exact calculations for false discovery proportion with application to least favorable configurations (2010, methodology): “Next, by definition of a step-up procedure, we have SU(t) = {i | pi ≤ tk̂} with k̂ = |SU(t)| (this expression is related to the “self-consistency” condition introduced in [4]).”
Multiple hypothesis testing in genomics (2014, methodology): “Alternatives to the Benjamini & Yekutieli method have been formulated by Sarkar [25] and by Blanchard and Roquain [60].”
False discovery rate control with e‐values (2020, methodology): “Following Blanchard and Roquain (2008), who defined self-consistent ptesting procedures, an e-testing procedure G is said to be self-consistent at level α ∈ (0, 1) if, denoting by RG the number of rejections, every rejected e-value ek satisfies ek ≥ K αRG .”
A unified treatment of multiple testing with prior knowledge using the p-filter (2017, methodology): “[16] when only prior weights are used, (d) the penalty-weighted BH procedure of Benjamini and Hochberg [4] when only penalty weights are used, (e) the doubly weighted BH procedure of Blanchard and Roquain [10] when both sets of weights and reshaping are used and (f) the StoreyBH procedure of Storey ”