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Fisher's exact test from two Bernoulli samples

Example

Take two independent Bernoulli samples of sizes $n_{1}, n_{2} \geq 1$ and test the one-sided alternative that the first success probability exceeds the second. Under the null, all $N = n_{1} + n_{2}$ trials have a common success probability $p \in [0, 1]$ . Let $K$ be the number of successes in the first sample and $S$ the total number in both samples. For every integer margin $s \in {0, \dots, N}$ , define the supplied conditional-law kernel $Q$ by $Q (s, {k}) = \frac{(\binom{n_{1}}{k}) (\binom{n_{2}}{s - k})}{(\binom{N}{s})}, \max (0, s - n_{2}) \leq k \leq \min (n_{1}, s),$ and set $Q (s, {k}) = 0$ outside the displayed range. For each $s$ , these weights sum to one because they count all $(\binom{N}{s})$ ways to choose the $s$ success positions according to how many lie in the first sample. At margins of positive probability this is the conditional law and it does not depend on $p$ ; at endpoint nulls $p = 0, 1$ , the formula also supplies a version at the other, probability-zero margins. Large values of $K$ support the one-sided alternative that the first sample has a larger success probability. For $n_{1} = n_{2} = 2$ and $s = 2$ , the conditional masses at $k = 0, 1, 2$ are $1 / 6, 4 / 6, 1 / 6$ ; the inclusive upper-tail p-values are $1, 5 / 6, 1 / 6$ .

Facts & Assumptions

Given: Positive integers $n_{1}, n_{2}$ , the null product Bernoulli model with common parameter $p$ , the statistic $K$ conditioned on $S$ , the supplied kernel $Q$ defined for every margin in the Example, and an auxiliary $U \sim Unif [0, 1]$ independent of all trials.

[F1]

A product of finite probability spaces assigns a sequence the product of its coordinate weights. The finite product of finite probability spaces

[F2]

Product weights normalize and the coordinate trials are independent. Product weights normalize, and coordinate events are mutually independent, Pairwise and mutual independence of finite-valued random variables

[F3]

For an event of positive probability, conditional probability is the ratio of the joint event probability to the conditioning-event probability. Conditional probability $P (A ∣ B)$ for $P (B) > 0$

[F4]

The nonrandomized conditional upper-tail value uses the inclusive tail; the randomized value uses the strict tail plus an independent uniform fraction of the tie mass. Randomized conditional upper-tail p-values

[F5]

For each fixed null value $p$ , under the supplied kernel the randomized upper-tail value is conditionally uniform and the inclusive value is conditionally superuniform. Randomized conditional p-values are uniform

[F6]

A test has unconditional size equal to the supremum of its null rejection expectations; conditional level at most $α$ implies unconditional level at most $α$ . Test size and power, Conditional level implies unconditional level

Verification

1.1F1F2givenalgebra

Write $Ω = {0, 1}^{N}$ and give each coordinate the two-point Bernoulli $(p)$ law. By [F1], a sequence with $s$ ones has weight $p^{s} (1 - p)^{N - s}$ ; the product weights sum to $(p + (1 - p))^{N} = 1$ , and [F2] gives the independent-trial model. For $0 < p < 1$ , every sequence with $S = s$ has the same positive weight.

2.1F1F3step 1.1givenconstructalgebra

There are $(\binom{N}{s})$ sequences with $s$ successes. Exactly $(\binom{n_{1}}{k}) (\binom{n_{2}}{s - k})$ of them have $K = k$ : choose $k$ of the first $n_{1}$ positions and $s - k$ of the other $n_{2}$ . These counts partition all size- $s$ subsets, so the displayed kernel weights sum to one for every $s$ , and when $P_{p} (S = s) > 0$ , dividing by the total count using [F3] gives the conditional mass; the feasible range is precisely where both binomial coefficients are nonzero. At $p = 0$ only $S = 0$ occurs and $K = 0$ ; at $p = 1$ only $S = N$ occurs and $K = n_{1}$ , so the supplied kernel has the required conditional law on the realized margin. At the other, probability-zero margins the same formula is simply the chosen kernel extension.

3.1F3F4step 2.1givenalgebra

For $n_{1} = n_{2} = 2$ and $s = 2$ , the feasible values are $k = 0, 1, 2$ , with counts $1, 4, 1$ among the six two-success sequences. Their probabilities are therefore $1 / 6, 4 / 6, 1 / 6$ . Summing the masses at or above each observed value gives inclusive p-values $1, 5 / 6, 1 / 6$ . The strict-tail-plus-tie formulas are respectively $5 / 6 + U / 6$ , $1 / 6 + 2 U / 3$ , and $U / 6$ ; their intervals $[5 / 6, 1]$ , $[1 / 6, 5 / 6]$ , and $[0, 1 / 6]$ have lengths matching the three conditional masses.

4.1F3F5F6step 2.1step 3.1given∎

For each margin $s$ with $P_{p} (S = s) > 0$ , [F5] makes the nonrandomized p-value conditionally superuniform and the randomized value conditionally uniform under the null. Rejecting when the nonrandomized value is at most $α$ is therefore a conditional level- $α$ test, so [F6] gives unconditional size at most $α$ . The test function of the randomized rule is $φ_{α} (x) = P_{U} (p_{U} (x, U) \leq α)$ ; it is measurable because the sample space is finite. By finite conditioning and independence of $U$ , $E_{p} [φ_{α} (X) ∣ S = s] = P_{p} (p_{U} \leq α ∣ S = s) = α$ , so the randomized rule has conditional and unconditional size exactly $α$ . At $p = 0$ and $p = 1$ the only observed margin has a degenerate conditional law; its inclusive value is $1$ and its randomized value is $U$ , so the same validity conclusions hold.

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Sources

Berkeley Stat 210A Course Reader, Testing with Nuisance Parameters, Conditional testing section (standard reference, not scraped)
George Deligiannidis, Foundations of Statistical Inference, Chapter 13, PDF pp. 77–78 (standard reference, not scraped)