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Valid p-values, Bonferroni, simultaneous coverage — Examples

1 · Prerequisites

2 · Summary

One finite model uses discrete superuniform p-values and a mixed configuration to compare FWER with FDR; another constructs confidence intervals with identical membership events to show that Bonferroni simultaneous coverage remains valid under dependence.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

Simultaneous coverage for dependent interval sets

Example

Take the location experiment with $Θ = R$ and sample space $(X, A) = (R, B (R))$ : under parameter $θ$ , $X$ equals $θ + k / 10$ with probability $1 / 10$ for each $k = 1, \dots, 10$ . For both coordinates set $ψ_{i} (θ) = θ$ and report the interval $C_{i} (x) = [x - 9 / 10, x]$ , $i = 1, 2$ . Each interval has marginal coverage $9 / 10$ . The two coverage events are identical, so their simultaneous coverage is also $9 / 10$ . With $α_{1} = α_{2} = 1 / 10$ and $α = 1 / 5$ , Bonferroni guarantees at least $4 / 5$ ; the bound is conservative here because of perfect dependence.

Facts & Assumptions

Given: The location experiment and two confidence-set procedures in the Example, together with the confidence-set definition and simultaneous-coverage theorem below.

[F1]

Statistical experiments: an experiment is a measurable sample space, parameter set, and family of probability measures on that sample space.

[F2]

Confidence sets and coverage: a confidence-set procedure has a measurable membership section for each candidate parameter, and its coverage is the probability that it contains the true parameter.

[F3]

Bonferroni simultaneous coverage: if $m$ marginal confidence sets have coverage at least $1 - α_{i}$ and $\sum_{i} α_{i} \leq α$ , their joint coverage is at least $1 - α$ without an independence assumption.

[A1]

The sample space is $(X, A) = (R, B (R))$ . For each $θ \in R$ , $P_{θ}$ places mass $1 / 10$ at each of the ten distinct points $θ + k / 10$ , $1 \leq k \leq 10$ ; $ψ_{1} (θ) = ψ_{2} (θ) = θ$ and $C_{1} (x) = C_{2} (x) = [x - 9 / 10, x]$ .

Verification

technique · direct finite calculation

Given: The family of ten-point laws, targets, and interval procedures in the Example.

1.1givenA1F1F2algebra

For each candidate $η \in R$ , the membership section of either interval procedure is ${x : η \in [x - 9 / 10, x]} = [η, η + 9 / 10]$ , a Borel set, so [F2] applies. Under $P_{θ}$ , the true value $θ$ is included exactly for $k = 1, \dots, 9$ ; hence each marginal coverage is $9 / 10$ .

2.1givenA1F3step 1.1algebra∎

The two interval procedures are equal for every observation, so their membership events coincide and joint coverage is $9 / 10$ . Since $α_{1} + α_{2} = 1 / 5$ , [F3] gives the general lower bound $1 - 1 / 5 = 4 / 5$ , which this example satisfies. The equality of the two events shows that independence is not assumed.

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

A discrete p-value family under global and mixed nulls

Example

Let $Q = {1 / 4, 1 / 2, 3 / 4, 1}$ , $Q^{*} = Q \cup {1 / 20}$ , and use the finite sample space $X = Q^{*} \times Q^{*}$ with its full power-set sigma-algebra. The parameter set is $Θ = {0, 1}$ . Under $θ = 0$ , give each point $(u, u)$ , $u \in Q$ , probability $1 / 4$ . Under $θ = 1$ , give each point $(1 / 20, u)$ , $u \in Q$ , probability $1 / 4$ . Define $p_{1} (s, t) = s$ , $p_{2} (s, t) = t$ , with null hypotheses $H_{01} = {0}$ and $H_{02} = {0, 1}$ . For $α = 1 / 2$ , Bonferroni rejects when $p_{i} \leq α / 2 = 1 / 4$ . Under the global null $θ = 0$ , both p-values are the same discrete uniform variable: FWER and FDR are both $1 / 4$ . Under the mixed configuration $θ = 1$ , $p_{1} = 1 / 20$ is always rejected but is not a true null; $p_{2}$ is a valid discrete p-value. There FWER is $1 / 4$ and FDR is $1 / 8$ . The true-null p-values are perfectly dependent under $θ = 0$ .

Facts & Assumptions

Given: The two-parameter finite experiment, p-value statistics and null hypotheses in the example, the Bonferroni rule at $α = 1 / 2$ , and the definitions and theorem below.

[F1]

Statistical experiments: an experiment is a measurable sample space, parameter set, and family of probability measures on that sample space.

[F2]

Null-valid p-values and familywise and false-discovery error rates: $p_{i}$ is valid for $H_{0 i}$ when $P_{θ} (p_{i} \leq u) \leq u$ for all $θ \in H_{0 i}$ and $u \in [0, 1]$ ; FWER is $P_{θ} (V_{θ} \geq 1)$ and FDR is $E_{θ} [V_{θ} / (R \lor 1)]$ .

[F3]

Strong familywise error control by Bonferroni: for any fixed family of $m \geq 1$ valid p-values, rejecting each at $p_{i} \leq α / m$ controls FWER by $α$ for every true-null configuration and arbitrary dependence.

[A1]

$Q = {1 / 4, 1 / 2, 3 / 4, 1}$ , $Q^{*} = Q \cup {1 / 20}$ , $X = Q^{*} \times Q^{*}$ with $A = P (X)$ , $H_{01} = {0}$ , $H_{02} = {0, 1}$ , $p_{1} (s, t) = s$ , $p_{2} (s, t) = t$ , and $P_{0}, P_{1}$ are the four-point uniform laws specified in the Example.

Verification

technique · direct finite calculation

Given: The finite laws and p-value family in the Example.

1.1givenA1F1F2algebra

Each $P_{θ}$ assigns mass $1 / 4$ to four distinct points, so its masses sum to $1$ . Under either parameter where $H_{01}$ is true, namely $θ = 0$ , and under both parameters where $H_{02}$ is true, the corresponding null p-value has the law of $U$ uniform on $Q$ . For every $u \in [0, 1]$ , $P (U \leq u) = ⌊ 4 u ⌋ / 4 \leq u$ ; hence both p-values are valid for their stated nulls by [F2].

1.2givenA1F2algebra

At $θ = 0$ , $I_{0} (0) = {1, 2}$ and $p_{1} = p_{2} = U$ . A true null is rejected exactly when $U = 1 / 4$ , which has probability $1 / 4$ . On that event $V_{0} = R = 2$ and $V_{0} / (R \lor 1) = 1$ ; otherwise both counts are zero. Thus FWER $_{0} =$ FDR $_{0} = 1 / 4$ .

2.1givenA1F2F3step 1.2algebra∎

At $θ = 1$ , $I_{0} (1) = {2}$ , $p_{1} = 1 / 20 < 1 / 4$ always, and $p_{2} = U$ . A true null is rejected exactly when $U = 1 / 4$ , so FWER $_{1} = 1 / 4$ . On that event $(V_{1}, R) = (1, 2)$ and otherwise $(V_{1}, R) = (0, 1)$ , giving FDR $_{1} = (1 / 4) (1 / 2) = 1 / 8$ . The theorem [F3] also gives the strong FWER bound $1 / 4 \leq 1 / 2$ at both parameter values, without an independence hypothesis.