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Power of Fisher's Exact Test (Noncentral Hypergeometric)

Source: R/power_fisher_exact.R
power_fisher_exact.Rd

Computes the power of Fisher's exact test (Fisher, 1934) for the 2 \(\times\) 2 table under Fisher's noncentral hypergeometric distribution, where the alternative is parameterized by the true odds ratio \(\psi\). The power is the probability that the (conditional) exact test rejects \(H_0: \psi = 1\) when in fact \(\psi = \psi_1 \ne 1\).

Usage

power_fisher_exact(
  n_1,
  n_2,
  p_1,
  p_2,
  alpha_level = 0.05,
  alternative = c("two_sided", "less", "greater")
)

Arguments

n_1, n_2

Group sample sizes for the two columns of the 2 \(\times\) 2 table (e.g., treatment vs control).

p_1, p_2

Success probabilities in the two groups under the alternative. The odds ratio under the alternative is \(\psi = (p_1 / (1 - p_1)) / (p_2 / (1 - p_2))\).

alpha_level

Two-sided significance level. Default 0.05.

alternative

One of "two_sided" (default; the base-R spelling "two.sided" is accepted as an alias), "less", or "greater".

Value

A data.frame with rows for the power, the alternative-odds-ratio \(\psi_1\), the alternative success probabilities \(p_1\) and \(p_2\), the expected column-1 total \(E[S]\) across both groups, and the row / column totals.

Details

Setup. Fisher's exact test conditions on the marginal totals of the 2 \(\times\) 2 table:

SuccessFailureTotal
Group 1\(X_1\)\(n_1 - X_1\)\(n_1\)
Group 2\(S - X_1\)\((n_1 + n_2) - n_1 - S + X_1\)\(n_2\)
Total\(S\)\(n_1 + n_2 - S\)\(n_1 + n_2\)

Under \(H_0: \psi = 1\), \(X_1\) given the marginals follows the central hypergeometric. Under the alternative \(\psi_1\), \(X_1\) follows Fisher's noncentral hypergeometric with odds ratio parameter \(\psi_1\) (Fisher, 1935; Fog, 2008), the conditional distribution of one binomial count given the total of two independent binomials. (Wallenius' noncentral hypergeometric, which arises from sequential biased urn sampling, is a different distribution and is not the relevant one here.)

Power calculation. For each possible value of the column-1 total \(S = 0, 1, \ldots, n_1 + n_2\):

  1. Determine the rejection region under \(H_0: \psi = 1\) using the central hypergeometric.

  2. Compute \(\Pr(X_1 \in \mathrm{reject} \mid \psi = \psi_1, S)\) under the noncentral hypergeometric.

  3. Weight by \(\Pr(S \mid \psi_1)\), the marginal probability of total column-1 successes under the alternative.

The power is the resulting weighted sum.

References

Fisher, R. A. (1934). Statistical methods for research workers (5th ed.). Oliver & Boyd.

Fisher, R. A. (1935). The logic of inductive inference. Journal of the Royal Statistical Society, 98(1), 39–82.

Fog, A. (2008). Sampling methods for Wallenius' and Fisher's noncentral hypergeometric distributions. Communications in Statistics – Simulation and Computation, 37(2), 241–257. doi:10.1080/03610910701790236

Good, P. I. (2000). Permutation tests: A practical guide to resampling methods for testing hypotheses (2nd ed.). Springer.

O'Brien, R. G. (1998). A tour of UnifyPow: A SAS module/macro for sample size analysis. Proceedings of the 23rd SAS Users Group International Conference, 1346–1355.

See also

fisher.test

Other sample size for power: ss_aipe_mixed_effects(), ss_power_R2(), ss_power_R2_sensitivity(), ss_power_c(), ss_power_c_ancova(), ss_power_composite_ancova(), ss_power_composite_ancova_2group(), ss_power_composite_anova(), ss_power_composite_factorial_ancova(), ss_power_composite_factorial_ancova_het(), ss_power_composite_factorial_anova(), ss_power_composite_sem(), ss_power_contrast(), ss_power_equivalence_c(), ss_power_factorial_ancova(), ss_power_factorial_anova(), ss_power_indirect_effect(), ss_power_mixed_effects(), ss_power_one_way_anova(), ss_power_pcm(), ss_power_r(), ss_power_rc(), ss_power_reg_coef(), ss_power_reg_coef_sensitivity(), ss_power_rm_anova(), ss_power_sc(), ss_power_sem(), ss_power_smd(), ss_power_split_plot_anova()

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. Power for n_1 = n_2 = 30 when the two population proportions are
#    0.6 and 0.3. The value comes from enumerating the conditional
#    reference set the test itself uses, not from a normal
#    approximation, so it is the power of the test as conducted.
power_fisher_exact(n_1 = 30, n_2 = 30, p_1 = 0.6, p_2 = 0.3)
#>  term           value
#>  power          0.561
#>  odds_ratio_alt 3.5  
#>  p_1            0.6  
#>  p_2            0.3  
#>  expected_s     27   
#>  n_1            30   
#>  n_2            30   
#>  alpha_level    0.05 

# 2. A difference of 0.10 between proportions is much harder to detect:
#    100 per group is not close to enough. Sample size requirements grow
#    quickly as the difference between the two proportions shrinks.
power_fisher_exact(n_1 = 100, n_2 = 100, p_1 = 0.45, p_2 = 0.35)
#>  term           value
#>  power          0.258
#>  odds_ratio_alt 1.52 
#>  p_1            0.45 
#>  p_2            0.35 
#>  expected_s     80   
#>  n_1            100  
#>  n_2            100  
#>  alpha_level    0.05