Proportions: Inequality, two independent groups (Fisher's exact test)

This procedure calculates power and sample size for tests comparing two independent binomial populations with probabilities π₁ and π₂, respectively. The results of sampling from these two populations can be given in a 2 × 2 contingency table X :

	Standard
	Group 1	Group 2	Total
Success	x1 x2 n1-x1 n2-x2		m
Failure			N-m
Total	n1	n2	N

Here, n₁ and n₂ are the sample sizes, and x₁ and x₂ the observed number of successes in the two populations. N = n₁ + n₂ is the total sample size, and m = x₁ + x₂ the total number of successes.

The null hypothesis states that π₁ = π₂, whereas the alternative hypothesis assumes different probabilities in both populations:

H₀ : π₁ − π₂ = 0
H₁ : π₁ − π₂ ≠ 0.

Effect size index

The effect size is determined by directly specifying the two proportions π₁ and π₂.

Options

This test has no options.

Examples

Will follow later.

Related tests

Proportions: Inequality, two dependent groups (McNemar)

Implementation notes

Exact unconditional power

The procedure computes the exact unconditional power of the (conditional) test. The exact probability of the 2 × 2 contingency table X (see introduction) under H₀, conditional on m = x₁ + x₂, is given by:

Let T be a test statistic, t a possible value of T, and M the set of all tables X with a total number of successes equal to m. We define M_t = {X ∈ M : T ≥ t}, i.e. M_t is the subset of M containing all tables for which the value of the test statistic is equal to or exceeds the value t. The exact null distribution of T is obtained by calculating Pr(T ≥ t|m, H₀) = ∑_X∈Mt Pr(X|m, H₀) for all possible t. The critical value t_α is the smallest value such that Pr(T ≥ t_α |m, H₀) ≤ α. The power is then defined as:

where
For two-sided tests G*Power provides three common test statistics that are asymptotically equivalent.

1. Fisher's exact test:

2. Persons's exact test:

3. Likelihood ratio exact test:
For one-sided tests the test statistic is T = x₂.

Large sample approximation

The large sample approximation is based on a continuity corrected χ2 test with pooled variances. To permit a two-sided test, a z test version is used: The H₀ distribution is the standard normal distribution N(0, 1), and the H₁ distribution given by the normal distribution N(m(k), σ), with

Validation

The results were checked against the values produced by GPower 2.0.