Optimal weighting for false discovery rate control

Abstract

How to weigh the Benjamini-Hochberg procedure? In the context of multiple hypothesis testing, we propose a new step-wise procedure that controls the false discovery rate (FDR) and we prove it to be more powerful than any weighted Benjamini-Hochberg procedure. Both finite-sample and asymptotic results are presented. Moreover, we illustrate good performance of our procedure in simulations and a genomics application. This work is particularly useful in the case of heterogeneous $p$-value distributions

Power-enhanced multiple decision functions controlling family-wise error and false discovery rates

“…Crucially, p -value based procedures also do not exploit the power characteristics of the individual tests, contrary to Neyman and Pearson’s [27] adage that such considerations are germane in constructing optimal tests. Such p -value based procedures are fine in exchangeable settings where power characteristics of the individual tests are identical, but not in situations where genes or subclasses of genes have different structures; see [11, 13, 29]. …”

“…The use of weighted p -values to improve type II performance have also been explored in [16, 21, 29, 30, 46]. Other approaches for optimal procedures are those in [42, 43] which employ a Neyman–Pearson approach and [45] where oracle and adaptive compound rules were obtained.…”

“…We focus on the most fundamental setting where the null and alternative hypotheses for each gene are both simple. This is also the setting in [29]. This admits, as starting point, the Neyman–Pearson most powerful (MP) test for each pair of hypotheses.…”

Classes of multiple decision functions strongly controlling FWER and FDR

“…This idea was implemented in a more restricted setting in [4] when each of the pairs of hypotheses contained simple null and simple alternative hypotheses. We point out that previous works have usually focussed in developing a particular MDF and then verifying that it controls the FWER or the FDR, such as, for example, in [3] (more comprehensively, see [27]), with notable exceptions being the papers [6,8,7,28,9,18]. It is our hope that by providing a class of MDFs where each member strongly controls the FWER, given by $${ℭ}^{†}=false\{{\delta }^{†}false(q;\mathrm{bold\Delta },\mathrm{boldA}false):\mathrm{bold\Delta }\in 𝔇,\mathrm{boldA}\in 𝔖false\};$$ or a class of MDFs where each member controls the FDR, given by $${ℭ}^{*}=false\{{\delta }^{*}false(q;\mathrm{bold\Delta },\mathrm{boldA}false):\mathrm{bold\Delta }\in 𝔇,\mathrm{boldA}\in 𝔖,\text{with}false(\mathrm{bold\Delta },\mathrm{boldA}false)\text{satisfying}false(\mathrm{normalC}false)false\},$$ then we acquire the possibility of selecting from these classes MDFs that possess other desirable properties with respect to some suitable Type II error rate, such as the MDR.…”

“…The paper [8] provides two sufficient conditions for FDR control, while [9] concerns aspects of optimality. The papers [10,11] present sequential rejection procedures which control FWER.…”