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A PRDS bound for each BH true-null contribution

Statement

Let $P = (P_{1}, \dots, P_{m})$ be p-value random variables in a probability space, let $m \geq 1$ , $q \in (0, 1)$ , and fix an index $i$ in a designated true-null set $I_{0}$ . Suppose $P_{i}$ is valid under this law and the law of $P$ is PRDS on $I_{0}$ as defined in Positive regression dependence on true-null coordinates (PRDS). Apply BH at level $q$ , and let $I_{i}$ indicate that $i$ is rejected and $R$ be the total number rejected. Then

$E ⁣ [\frac{I_{i}}{R \lor 1}] \leq \frac{q}{m} .$

Facts & Assumptions

Given: A finite vector of p-values, a fixed true-null set containing $i$ , $q \in (0, 1)$ , validity of $P_{i}$ , and PRDS on that set. The PRDS conditional-law existence convention carries the Axiom of Choice as stated in Positive regression dependence on true-null coordinates (PRDS).

[F1]

Positive regression dependence on true-null coordinates (PRDS): for each $i \in I_{0}$ and each increasing Borel $D \subseteq [0, 1]^{m}$ , there is a regular conditional-law version $K_{i} (x, D)$ that is nondecreasing on the support of $P_{i}$ .

[F2]

Leave-one-coordinate rejection partition for a step-up rule: for $c_{k} = k q / m$ , the events ${I_{i} = 1, R = k}$ are ${P_{i} \leq c_{k}} \cap {P_{- i} \in C_{i, k}}$ ; the sets $C_{i, k}$ partition the remaining-coordinate space, and $D_{i, k} : = ⋃_{j \leq k} C_{i, j}$ are increasing Borel sets with $D_{i, m}$ the whole space.

[F3]

Null-valid p-values and familywise and false-discovery error rates: validity means $P (P_{i} \leq u) \leq u$ for every $u \in [0, 1]$ under the corresponding true-null law.

[F4]

Conditional law given a random element: if $K_{i}$ is a conditional law of $P$ given $P_{i}$ , then for every Borel event $D$ , $P (P_{i} \in A, P \in D) = \int_{A} K_{i} (x, D) P_{P_{i}} (d x)$ for Borel $A \subseteq [0, 1]$ .

[F5]

The expectation of an indicator is the probability of the event: an event indicator has expectation equal to its probability.

[F6]

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

[F7]

Expectation of a nonnegative or integrable random variable: expectation of a bounded measurable random variable is its integral under the law.

[F8]

The usual topology of $R$ has interval neighborhoods, relatively open subsets of $[0, 1]$ are traces of open subsets of $R$ , the rationals are dense and countable, products of two countable sets are countable, and measures are countably subadditive (The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded, Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace, The rationals embed densely in the reals, $Q$ is countably infinite, A product of two at most countable sets is at most countable, Finite and countable subadditivity of measures).

[F9]

Every nonempty at-most-countable set has a surjection from $N$ (A nonempty set is at most countable iff it is a surjective image of $N$ ).

[A1]

The conditional-law version in [F1] has the monotonicity required by PRDS on the support of the marginal law of $P_{i}$ .

[A2]

The standard-Borel conditional-law existence interface is used under the Axiom of Choice (The Axiom of Choice) as recorded in Positive regression dependence on true-null coordinates (PRDS).

Proof

technique · decompose by the rejection count and use PRDS to fold the conditional event probabilities

Given: $m, q, P, I_{0}, i$ as in the Statement.

1.1F2F5F6F7givenconstruct

Put $c_{k} = k q / m$ , $A_{k} : = {P_{i} \leq c_{k}}$ , and $B_{i, k} : = {P_{- i} \in C_{i, k}}$ . If $P (A_{k}) > 0$ , set $a_{k} : = P (A_{k} \cap B_{i, k}) / P (A_{k})$ ; otherwise set $a_{k} = 0$ . By [F2], $I_{i} / (R \lor 1) = \sum_{k = 1}^{m} k^{- 1} 1_{A_{k} \cap B_{i, k}}$ . The terms are bounded and the sum is finite, so [F5]–[F7] give $E [I_{i} / (R \lor 1)] = \sum_{k = 1}^{m} (P (A_{k}) / k) a_{k}$ ; a zero-probability $A_{k}$ contributes zero by the definition of $a_{k}$ .

1.2F1A1A2F2given

For each $k$ , lift $D_{i, k}$ to ${\tilde{D}}_{i, k} : = {p \in [0, 1]^{m} : p_{- i} \in D_{i, k}}$ . The coordinate-deletion map from $[0, 1]^{m}$ to the remaining-coordinate product is measurable: the preimage of each measurable rectangle is a finite intersection of coordinate cylinders, and such rectangles generate the product sigma-algebra. Hence ${\tilde{D}}_{i, k}$ is Borel, and it is coordinatewise increasing. Let $g_{k} (x) : = K_{i} (x, {\tilde{D}}_{i, k})$ , nondecreasing on the support by [F1] and [A1]; the kernel exists on this standard-Borel interface under [A2].

1.3F1F8F9givenconstruct

Write $S_{i} = supp (μ_{i})$ . The family $U : = {(a, b) \cap [0, 1] : a, b \in Q, a < b}$ is a basis for the relative topology on $[0, 1]$ . If $U$ is relative open and $x \in U$ , write $U = O \cap [0, 1]$ with $O$ open in $R$ by [F8]; choose $ε > 0$ with $(x - ε, x + ε) \subseteq O$ using [F8], then use rational density from [F8] to choose $a, b \in Q$ with $x - ε < a < x < b < x + ε$ . Thus $x \in (a, b) \cap [0, 1] \subseteq U$ . Since $Q \times Q$ is nonempty and at most countable by [F8], [F9] gives an enumeration $((a_{n}, b_{n}))_{n \in N}$ of it, whose interval images enumerate $U$ ; hence the basis is countable. Let $E_{n} = (a_{n}, b_{n}) \cap [0, 1]$ if $a_{n} < b_{n}$ and $μ_{i} ((a_{n}, b_{n}) \cap [0, 1]) = 0$ , and $E_{n} = \emptyset$ otherwise. By the definition of support in Positive regression dependence on true-null coordinates (PRDS), $[0, 1] ∖ S_{i} = ⋃_{n \in N} E_{n} .$ Indeed, a point outside the support has a relative open null neighborhood and hence a basis neighborhood contained in it, which occurs in the enumeration; conversely, membership in any such null basis neighborhood excludes support membership. The right-hand side is a countable union of null Borel sets, so [F8] gives $μ_{i} ([0, 1] ∖ S_{i}) = 0$ . Thus $S_{i}$ is Borel and $μ_{i} (S_{i}) = 1$ .

2.1F1A1F4F2step 1.3givenalgebra

By [F4], whenever $P (A_{k}) > 0$ , $\frac{P (A_{k} \cap {P_{- i} \in D_{i, k}})}{P (A_{k})} = \frac{\int_{[0, c_{k}]} g_{k} (x) μ_{i} (d x)}{μ_{i} ([0, c_{k}])} .$ For fixed $k < m$ , this average is at most the corresponding average over $[0, c_{k + 1}]$ . By step 1.3, both averages may be restricted to $S_{i}$ without changing them. Every supported $x \leq c_{k}$ is below every supported $y \in (c_{k}, c_{k + 1}]$ , so $g_{k} (x) \leq g_{k} (y)$ by [F1] and [A1]. The larger-cutoff average is a mixture of the first average and the average over the added interval; the latter is at least the first whenever it has positive mass, and if it has zero mass the two averages agree.

2.2F3step 1.1algebra

By [F3], $P (A_{k}) \leq c_{k} = k q / m$ for every $k$ . Substituting into step 1.1 yields $E [I_{i} / (R \lor 1)] \leq (q / m) \sum_{k = 1}^{m} a_{k}$ .

3.1F2F4step 1.1step 2.1givenalgebra

Let $k_{0}$ be the first index with $P (A_{k_{0}}) > 0$ , if one exists; then $a_{j} = 0$ for $j < k_{0}$ , and $C_{i, k_{0}} \subseteq D_{i, k_{0}}$ gives $a_{k_{0}} \leq P (P_{- i} \in D_{i, k_{0}} ∣ A_{k_{0}}) \leq 1$ by step 2.1. For $k > k_{0}$ , suppose $\sum_{j < k} a_{j} \leq P (P_{- i} \in D_{i, k - 1} ∣ A_{k - 1})$ . Since $P (A_{k - 1}) > 0$ and $A_{k - 1} \subseteq A_{k}$ , step 2.1 applied to the nondecreasing $g_{k - 1}$ makes this at most $P (P_{- i} \in D_{i, k - 1} ∣ A_{k})$ . Also $P (A_{k}) > 0$ , so [F4] gives $a_{k} = P (P_{- i} \in C_{i, k} ∣ A_{k})$ . The disjoint union $D_{i, k} = D_{i, k - 1} \dot{\cup} C_{i, k}$ from [F2] then yields $\sum_{j \leq k} a_{j} \leq P (P_{- i} \in D_{i, k} ∣ A_{k})$ , completing the induction.

4.1F2step 2.2step 3.1algebra∎

If $k_{0}$ exists, step 3.1 at $k = m$ and $D_{i, m}$ equal to the whole space give $\sum_{k = 1}^{m} a_{k} \leq 1$ . If no such $k_{0}$ exists, all $P (A_{k}) = 0$ and every $a_{k} = 0$ , so the same bound holds. Substituting into step 2.2 proves the contribution inequality.

Remarks

The algebraic contribution bound uses only the PRDS monotonicity for the finite family of sets ${\tilde{D}}_{i, k}$ . The inherited Axiom of Choice is used to formulate the standard-Borel conditional-law interface; no choice is used in the finite event decomposition or induction.

Depends on

Used by

BH controls FDR under PRDS Theorem

Dependency tree · two levels

91 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), Section 4, PDF pp. 13–16 (standard reference, not scraped)
Yoav Benjamini, Discovering the False Discovery Rate, full 75-slide lecture, slides 31–35 (standard reference, not scraped)