TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Strong familywise error control by Bonferroni

Statement

Let $H_{01}, \dots, H_{0 m}$ be a fixed finite family with $m \geq 1$ in a statistical experiment, and let $p_{i}$ be a valid p-value for $H_{0 i}$ as in Null-valid p-values and familywise and false-discovery error rates. For $α \in [0, 1]$ , reject $H_{0 i}$ exactly when $p_{i} (X) \leq α / m$ . Then, for every $θ$ in the parameter space, $P_{θ} {at least one true null is rejected} = P_{θ} (V_{θ} \geq 1) \leq α .$ The bound holds for every true-null configuration, including the global null and configurations with false nulls, and for arbitrary dependence among the p-values. This is strong FWER control; it is not an FDR guarantee.

Facts & Assumptions

Given: A fixed family of $m \geq 1$ null hypotheses, p-values valid under their respective nulls, and $α \in [0, 1]$ .

[F1]

Null-valid p-values and familywise and false-discovery error rates: $I_{0} (θ) = {i : θ \in H_{0 i}}$ , and FWER is $P_{θ} (V_{θ} \geq 1)$ , where $V_{θ}$ counts rejected true nulls.

[F2]

Valid p-values yield level tests for nonempty disjoint alternatives: for a nonempty null and a p-value valid for it, every threshold $u \in [0, 1]$ satisfies $P_{θ} (p_{i} (X) \leq u) \leq u$ for each parameter in that null.

[F3]

Basic identities for a probability measure: for a finite family of measurable events, the probability of their union is at most the sum of their probabilities, including the empty family.

Proof

technique · direct

Given: A fixed family of valid p-values, $m \geq 1$ , and $0 \leq α \leq 1$ .

1.1givenF1F2

Set $u = α / m \in [0, 1]$ . For each $i \in I_{0} (θ)$ , $θ \in H_{0 i}$ , so $H_{0 i}$ is nonempty; the event $A_{i} : = {x : p_{i} (x) \leq u}$ is measurable because $p_{i}$ is a statistic. By validity and [F2], $P_{θ} (A_{i}) \leq u = α / m$ .

2.1givenF1F3step 1.1algebra∎

The event of at least one false rejection is exactly ${V_{θ} \geq 1} = ⋃_{i \in I_{0} (θ)} A_{i}$ . By [F3] and step 1.1, $P_{θ} (V_{θ} \geq 1) \leq \sum_{i \in I_{0} (θ)} P_{θ} (A_{i}) \leq ∣ I_{0} (θ) ∣ α / m \leq α$ . If $I_{0} (θ) = \emptyset$ , the union and sum are empty and the FWER is $0$ . Since this argument holds for every $θ$ and nowhere uses a joint dependence condition, it proves the strong, arbitrary-dependence claim.

Depends on

Used by

A discrete p-value family under global and mixed nulls Example
Holm step-down controls strong FWER Theorem

Dependency tree · two levels

18 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

Jelle J. Goeman and Aldo Solari, The Sequential Rejection Principle of Familywise Error Control, Section 3, Eqs. (11)–(13) (standard reference, not scraped)
William Fithian, STAT 210A Lecture 26: Multiple Testing, PDF pp. 2–4 (standard reference, not scraped)