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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.

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