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Repeated unadjusted tests inflate global-null error

Example

Let $K \geq 2$ be an integer, let $α \in (0, 1)$ , and under a global-null law $P_{0}$ let $P_{1}, \dots, P_{K}$ be mutually independent random variables with $P_{0} (P_{i} \leq u) = u (0 \leq u \leq 1) .$ At each of the $K$ looks, reject the global null if the current value is at most $α$ , without adjusting the threshold. The probability of at least one rejection is $P_{0} ⁣ (⋃_{i = 1}^{K} {P_{i} \leq α}) = 1 - (1 - α)^{K} > α .$ The first rejection time is a stopping time for the filtration $F_{k} = σ (P_{1}, \dots, P_{\min (k, K)})$ for $k \geq 0$ , with $F_{0}$ trivial; thus the filtration stays constant after look $K$ .

Facts & Assumptions

Given: Assume AC for the comparison with the A-page theorem. Under a global null law $P_{0}$ , $K \geq 2$ independent uniform $[0, 1]$ random variables $P_{1}, \dots, P_{K}$ are observed successively. Here uniform means exactly $P_{0} (P_{i} \leq u) = u$ for every $u \in [0, 1]$ .

[F1]

Independent random elements: random variables are independent when the sigma-algebras they generate are independent.

[F2]

Independent random elements have product joint law: the joint law of finitely many independent random elements is the product of their marginal laws.

[F3]

Discrete stopping time: $σ$ is a stopping time when ${σ \leq k} \in F_{k}$ for every $k$ .

[F4]

Discrete Ville inequality and anytime threshold validity: under each null law, a nonnegative test supermartingale with initial expectation at most one has all-time threshold-crossing probability at most the threshold level.

[A1]

AC is required only for applying [F4]; the exact product probability calculation below uses no choice principle (The Axiom of Choice).

Verification

1.1given

For each $i$ , the uniform-law assumption gives $P_{0} (P_{i} > α) = 1 - α$ .

1.2F3givenconstruct

Define $σ = \inf {1 \leq i \leq K : P_{i} \leq α}$ , with $\inf \emptyset = \infty$ . For every $k \geq 0$ , ${σ \leq k} = ⋃_{i = 1}^{\min (k, K)} {P_{i} \leq α} \in F_{k}$ , where the union is empty for $k = 0$ . Thus [F3] makes $σ$ a stopping time, and the rejection event by look $K$ is ${σ \leq K}$ .

2.1F1F2step 1.1

By [F1]–[F2], the probability that every look misses is the product of the marginal probabilities, so $P_{0} (⋂_{i = 1}^{K} {P_{i} > α}) = (1 - α)^{K}$ .

3.1step 2.1algebra

The event of at least one rejection is the complement of the event in step 2.1. Therefore its probability is $1 - (1 - α)^{K}$ .

4.1givenstep 3.1algebra

Since $0 < 1 - α < 1$ and $K \geq 2$ , $(1 - α)^{K} < 1 - α$ , so step 3.1 is strictly greater than $α$ .

5.1F4A1step 1.2step 4.1∎

[F4] controls arbitrary threshold stopping only when its nonnegative test-supermartingale hypotheses hold. The calculation above concerns repeated fixed-look uniform p-values and does not assume such a process, so the excess rejection probability is not a violation of Ville's inequality.

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