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BY controls FDR under arbitrary dependence

Statement

Let $E = ((X, A), Θ, (P_{θ})_{θ \in Θ})$ be a statistical experiment (Statistical experiments), let $m \geq 1$ , and fix a finite family of null hypotheses $H_{01}, \dots, H_{0 m}$ . For each $i$ , let $p_{i}$ be a p-value statistic valid for $H_{0 i}$ in the sense of Null-valid p-values and familywise and false-discovery error rates. For $θ \in Θ$ , put

$I_{0} (θ) : = {i \in {1, \dots, m} : θ \in H_{0 i}}, m_{0} (θ) : = ∣ I_{0} (θ) ∣ .$

For every $q \in (0, 1)$ , the BY procedure in Benjamini–Yekutieli step-up procedure satisfies

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

No dependence condition on the joint law of the p-values is required.

Facts & Assumptions

Given: The experiment and fixed finite family in the Statement, $q \in (0, 1)$ , and p-value validity under each corresponding null law.

[F1]

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

[F2]

Benjamini–Yekutieli step-up procedure: BY uses the critical values $c_{k} = k q / (m H_{m})$ and rejects exactly the first $R$ ranks, where $R$ is the largest passing rank.

[F3]

Leave-one-coordinate rejection partition for a step-up rule: for any nondecreasing step-up critical values, ${I_{i} = 1, R = k} = {p_{i} \leq c_{k}} \cap {p_{- i} \in C_{i, k}}$ , and the sets $C_{i, k}$ partition the remaining-coordinate space.

[F4]

Valid p-values yield level tests for nonempty disjoint alternatives: for a valid p-value and a nonempty null hypothesis, $P_{θ} (p_{i} \leq u) \leq u$ for every parameter in that null and every $u \in [0, 1]$ . Here $i \in I_{0} (θ)$ means $θ \in H_{0 i}$ , so the null $H_{0 i}$ is nonempty and the supplier applies.

[F5]

The expectation of an indicator is the probability of the event: $E_{θ} [1_{A}] = P_{θ} (A)$ for every event $A$ .

[F6]

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

[F7]

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

[A1]

Each p-value is superuniform under every parameter in its corresponding null hypothesis.

Proof

technique · finite rejection-count decomposition and summation by parts

Given: A fixed family of $m \geq 1$ null hypotheses, p-values valid for their respective nulls, and $q \in (0, 1)$ .

1.1F1F5F6F7givenconstruct

Fix $θ \in Θ$ and let $I_{i}$ indicate that BY rejects hypothesis $i$ . Pointwise, $V_{θ} / (R \lor 1) = \sum_{i \in I_{0} (θ)} \sum_{k = 1}^{m} k^{- 1} 1 {I_{i} = 1, R = k}$ : if $R = 0$ both sides are zero, and if $R > 0$ only $k = R$ contributes, with weight $1 / R$ for each rejected true null. This finite sum is bounded and integrable by [F7]; [F5] and [F6] therefore give ${FDR}_{θ} = \sum_{i \in I_{0} (θ)} \sum_{k = 1}^{m} k^{- 1} P_{θ} (I_{i} = 1, R = k)$ .

1.2F2F3F4givenalgebra

Fix $i \in I_{0} (θ)$ and put $t = q / (m H_{m})$ and $J_{j} = ((j - 1) t, j t]$ for $j = 1, \dots, m$ . Since $θ \in H_{0 i}$ , that null is nonempty and [F4] at $u = 0$ gives $P_{θ} (p_{i} = 0) = 0$ . Since BY has $c_{k} = k t$ , [F3] splits each event ${I_{i} = 1, R = k}$ into the disjoint events ${p_{i} \in J_{j}, p_{- i} \in C_{i, k}}$ for $j \leq k$ , apart from that null zero event. Using $1 / k \leq 1 / j$ for $k \geq j$ and that the $C_{i, k}$ partition the remaining coordinates, $\sum_{k = 1}^{m} k^{- 1} P_{θ} (I_{i} = 1, R = k) = \sum_{j = 1}^{m} \sum_{k = j}^{m} k^{- 1} P_{θ} (p_{i} \in J_{j}, p_{- i} \in C_{i, k}) \leq \sum_{j = 1}^{m} j^{- 1} P_{θ} (p_{i} \in J_{j})$ .

2.1F2F4step 1.2algebra

Set $F_{j} = P_{θ} (p_{i} \leq j t)$ for $j = 0, \dots, m$ . Then $F_{0} = P_{θ} (p_{i} = 0) = 0$ , $F_{j} \leq j t$ by [F4], and $P_{θ} (p_{i} \in J_{j}) = F_{j} - F_{j - 1}$ . Finite summation by parts gives $\sum_{j = 1}^{m} (F_{j} - F_{j - 1}) / j = F_{m} / m + \sum_{j = 1}^{m - 1} F_{j} (1 / j - 1 / (j + 1)) \leq t + t \sum_{j = 1}^{m - 1} 1 / (j + 1) = t H_{m} = q / m$ . Here $m t = q / H_{m} \leq q < 1$ , and $H_{m} = \sum_{j = 1}^{m} 1 / j$ .

3.1F1step 1.1step 1.2step 2.1algebra∎

Combining steps 1.2 and 2.1 bounds each true-null contribution in step 1.1 by $q / m$ . Summing over the $m_{0} (θ)$ true nulls gives ${FDR}_{θ} \leq (m_{0} (θ) / m) q \leq q$ ; with no true nulls the outer sum is empty and FDR is zero. No dependence condition is used, and the CDF summation-by-parts argument covers discrete superuniform p-values.

Remarks

For general superuniform p-values, a single interval $((j - 1) t, j t]$ can have probability greater than its width $t$ . The proof therefore bounds the weighted sum of interval probabilities by summation by parts of the superuniform CDF; it does not assume each interval has mass at most $t$ .

Depends on

Used by

BY and BH thresholds for independent valid p-values Example

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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 1.3 and Section 5, PDF pp. 5 and 18–19 (standard reference, not scraped)
Alexandru Katsevich, STAT 9610 Lecture Notes, Section 6.3.2, Proposition 6.6, printed p. 138 / PDF p. 142 (standard reference, not scraped)