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BH controls FDR under PRDS

Statement

Let $E = ((X, A), Θ, (P_{θ})_{θ \in Θ})$ be a statistical experiment, let $m \geq 1$ , and fix a finite family of null hypotheses $H_{01}, \dots, H_{0 m}$ with p-value statistics $p_{1}, \dots, p_{m}$ valid for their respective nulls. For each $θ$ , write

$I_{0} (θ) : = {i : θ \in H_{0 i}}, m_{0} (θ) : = ∣ I_{0} (θ) ∣ .$

Assume the Axiom of Choice as required for the regular conditional-law interface in Positive regression dependence on true-null coordinates (PRDS). Suppose that, under every $P_{θ}$ , the p-value vector $(p_{1} (X), \dots, p_{m} (X))$ is PRDS on $I_{0} (θ)$ . Then for every $q \in (0, 1)$ , the ordinary BH procedure in Benjamini–Hochberg step-up procedure satisfies

${FDR}_{θ} \leq \frac{m_{0} (θ)}{m} q \leq q .$

The PRDS assumption is imposed only on the true-null coordinates; no independence assumption is made.

Facts & Assumptions

Given: A finite family of valid p-values, $q \in (0, 1)$ , and under each $P_{θ}$ the PRDS condition on the true-null set $I_{0} (θ)$ . The PRDS regular conditional-law convention uses the Axiom of Choice as recorded in Positive regression dependence on true-null coordinates (PRDS).

[F1]

Null-valid p-values and familywise and false-discovery error rates: $I_{0} (θ)$ is the true-null index set, $V_{θ}$ counts rejected true nulls, and ${FDR}_{θ} = E_{θ} [V_{θ} / (R \lor 1)]$ .

[F2]

Benjamini–Hochberg step-up procedure: BH takes the largest $R$ with $p_{(R)} \leq R q / m$ and rejects exactly ranks $1, \dots, R$ .

[F3]

Positive regression dependence on true-null coordinates (PRDS): for each $i \in I_{0} (θ)$ , every increasing Borel event has a conditional-probability version nondecreasing in $p_{i}$ on its support.

[F4]

A PRDS bound for each BH true-null contribution: under validity and PRDS on $I_{0} (θ)$ , each true-null contribution satisfies $E_{θ} [I_{i} / (R \lor 1)] \leq q / m$ .

[F5]

Linearity, monotonicity, and the modulus bound for expectation: expectation is linear on finite sums of integrable random variables.

[F6]

Expectation of a nonnegative or integrable random variable: expectation is integration with respect to the specified law.

[A1]

Every p-value is valid under its own null, so each $i \in I_{0} (θ)$ meets the superuniformity hypothesis of [F4].

[A2]

Under every $P_{θ}$ , the vector of all p-values is PRDS on the full set $I_{0} (θ)$ , not merely pairwise positively correlated.

[A3]

The standard-Borel regular conditional-law existence interface assumes the Axiom of Choice (The Axiom of Choice) as stated in Positive regression dependence on true-null coordinates (PRDS).

Proof

technique · sum the true-null contributions

Given: The experiment, the fixed family and validity assumptions in the Statement, $q \in (0, 1)$ , and the PRDS condition for every true-null set.

1.1F1F2F5F6givenconstruct

Fix $θ \in Θ$ . Let $I_{i}$ be the indicator that the BH rule [F2] rejects hypothesis $i$ . By [F1], $V_{θ} = \sum_{i \in I_{0} (θ)} I_{i}$ and ${FDR}_{θ} = E_{θ} [V_{θ} / (R \lor 1)]$ . The ratio is bounded by one, so it is integrable by [F6]; finite linearity [F5] gives ${FDR}_{θ} = \sum_{i \in I_{0} (θ)} E_{θ} [I_{i} / (R \lor 1)]$ .

2.1F1F2F3F4A1A2A3step 1.1algebra∎

If $I_{0} (θ) = \emptyset$ , step 1.1 is an empty sum and FDR is zero. Otherwise, for each $i \in I_{0} (θ)$ , [A1] gives validity and [A2] gives PRDS on a set containing $i$ under the AC-based conditional-law interface [A3], so [F4] bounds its contribution by $q / m$ . Summing step 1.1 gives ${FDR}_{θ} \leq ∣ I_{0} (θ) ∣ q / m = (m_{0} (θ) / m) q \leq q$ . Only the designated true-null coordinates are needed; independence is not assumed.

Remarks

The Axiom of Choice enters through the standard-Borel regular conditional-law interface that gives the stated PRDS condition its kernel formulation. The finite sum of null contributions uses no additional choice. Pairwise covariance information alone does not supply the hypothesis; the companion examples include a valid-p-value witness.

Depends on

Used by

BY and BH thresholds for independent valid p-values Example

Dependency tree · two levels

39 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Yoav Benjamini and Daniel Yekutieli, The Control of the False Discovery Rate in Multiple Testing under Dependency, The Annals of Statistics 29(4) (2001), Theorem 1.2 and Section 4, PDF pp. 4 and 13–16 (standard reference, not scraped)
Yoav Benjamini, Discovering the False Discovery Rate, full 75-slide lecture, slides 33–35 (standard reference, not scraped)