[Paper Review] The FDR-Linking Theorem

Weijie Su|arXiv (Cornell University)|Dec 21, 2018

Statistical Methods in Clinical Trials48 references4 citations

TL;DR

This paper introduces the FDR-linking theorem, a novel framework that establishes a non-asymptotic FDR control bound for the Benjamini-Hochberg procedure under arbitrary dependence by showing that FDR depends primarily on the joint distribution of null p-values. The key result is a tight upper bound on FDR that depends only on null dependence, revealing that non-null p-values have a bounded influence, and enabling new dependence conditions and FDR consistency analysis.

ABSTRACT

This paper introduces the exttt{FDR-linking} theorem, a novel technique for understanding extit{non-asymptotic} FDR control of the Benjamini--Hochberg (BH) procedure under arbitrary dependence of the $p$-values. This theorem offers a principled and flexible approach to linking all $p$-values and the null $p$-values from the FDR control perspective, suggesting a profound implication that, to a large extent, the FDR of the BH procedure relies mostly on the null $p$-values. To illustrate the use of this theorem, we propose a new type of dependence only concerning the null $p$-values, which, while strictly extit{relaxing} the state-of-the-art PRDS dependence (Benjamini and Yekutieli, 2001), ensures the FDR of the BH procedure below a level that is independent of the number of hypotheses. This level is, furthermore, shown to be optimal under this new dependence structure. Next, we present a concept referred to as extit{FDR consistency} that is weaker but more amenable than FDR control, and the exttt{FDR-linking} theorem shows that FDR consistency is completely determined by the joint distribution of the null $p$-values, thereby reducing the analysis of this new concept to the global null case. Finally, this theorem is used to obtain a sharp FDR bound under arbitrary dependence, which improves the $\log$-correction FDR bound (Benjamini and Yekutieli, 2001) in certain regimes.

Motivation & Objective

• To address the long-standing gap in non-asymptotic FDR control of the BH procedure under arbitrary dependence of p-values.
• To identify a minimal, flexible dependence condition that still ensures FDR control independent of the number of hypotheses.
• To introduce and analyze FDR consistency, a weaker but more tractable alternative to FDR control.
• To demonstrate that FDR control is fundamentally governed by the null p-value dependence, simplifying analysis.

Proposed method

• Derives the FDR-linking theorem, which bounds the FDR of the BH procedure using only the null p-value joint distribution.
• Establishes that the FDR upper bound is a function of the null FDR at level x, denoted FDR₀(x), integrated over x ∈ [π₀α, 1].
• Uses the connection between FDR₀(x) and the Simes method’s type I error to link FDR control to a well-known multiple testing procedure.
• Introduces a new dependence condition—PRDN (positive regression dependence within nulls)—which is strictly weaker than PRDS and still ensures FDR control.
• Defines FDR consistency as a concept that depends solely on the null p-value distribution, reducing analysis to the global null case.
• Applies the theorem to derive a sharp FDR bound under arbitrary dependence, improving upon the log-correction bound in certain regimes.

Experimental results

Research questions

• RQ1 Can non-asymptotic FDR control of the BH procedure be established under arbitrary dependence without restrictive assumptions on the full joint distribution?
• RQ2 Is there a dependence condition that is strictly weaker than PRDS but still ensures FDR control independent of the number of hypotheses?
• RQ3 Does FDR consistency, a weaker form of FDR control, depend only on the null p-value distribution?
• RQ4 Can the FDR-linking theorem be used to derive tighter FDR bounds than existing methods like the log-correction bound?
• RQ5 Is the FDR-linking bound tight under general null dependence structures?

Key findings

• The FDR of the BH procedure is bounded by π₀α + π₀α ∫_{π₀α}¹ FDR₀(x)/x² dx, a bound that depends only on the joint distribution of null p-values.
• The FDR-linking theorem holds unconditionally under arbitrary dependence, without requiring independence or specific dependence structures.
• Under the PRDN condition, the FDR of the BH procedure is bounded by α, and this bound is optimal under this dependence structure.
• FDR consistency is completely determined by the null p-value distribution, making it analyzable via the global null case.
• The FDR-linking theorem improves upon the log-correction FDR bound of Benjamini and Yekutieli (2001) in certain regimes, particularly when the null p-values exhibit weak dependence.
• The theorem reveals that the BH procedure is robust to adversarial dependence between null and non-null p-values, as the FDR bound remains controlled by null dependence alone.