Exact Power Calculation for Two Co-Primary Binary Endpoints

Calculates the exact power for a two-arm superiority trial with two co-primary binary endpoints using the bivariate binomial distribution, as described in Homma and Yoshida (2025).

Usage

power2BinaryExact(n1, n2, p11, p12, p21, p22, rho1, rho2, alpha, Test)

Arguments

n1

Sample size for group 1 (test group)

n2

Sample size for group 2 (control group)

p11

True probability of responders in group 1 for the first outcome (0 < p11 < 1)

p12

True probability of responders in group 1 for the second outcome (0 < p12 < 1)

p21

True probability of responders in group 2 for the first outcome (0 < p21 < 1)

p22

True probability of responders in group 2 for the second outcome (0 < p22 < 1)

rho1

Correlation between the two outcomes for group 1

rho2

Correlation between the two outcomes for group 2

alpha

One-sided significance level (typically 0.025 or 0.05)

Test

Statistical testing method. One of:

"Chisq": One-sided Pearson chi-squared test
"Fisher": Fisher exact test
"Fisher-midP": Fisher mid-p test
"Z-pool": Z-pooled exact unconditional test
"Boschloo": Boschloo exact unconditional test

Value

A data frame with the following columns:

n1: Sample size for group 1
n2: Sample size for group 2
p11, p12, p21, p22: Response probabilities
rho1, rho2: Correlations
alpha: One-sided significance level
Test: Testing method used
power1: Power for the first endpoint alone
power2: Power for the second endpoint alone
powerCoprimary: Exact power for both co-primary endpoints

Details

This function calculates exact power using equation (9) in Homma and Yoshida (2025): $$power_A(\theta) = \sum_{(a_{1,1},a_{2,1})\in\mathcal{A}_1} \sum_{(a_{1,2},a_{2,2})\in\mathcal{A}_2} f(a_{1,1}|N_1,p_{1,1}) \times f(a_{2,1}|N_2,p_{2,1}) \times g(a_{1,2}|a_{1,1},N_1,p_{1,1},p_{1,2},\gamma_1) \times g(a_{2,2}|a_{2,1},N_2,p_{2,1},p_{2,2},\gamma_2)$$

where $\mathcal{A}_k$ is the rejection region for endpoint k, and $(Y_{j,1}, Y_{j,2}) \sim BiBin(N_j, p_{j,1}, p_{j,2}, \gamma_j)$ follows the bivariate binomial distribution.

The correlation bounds are automatically checked using corrbound2Binary.

References

Homma, G., & Yoshida, T. (2025). Exact power and sample size in clinical trials with two co-primary binary endpoints. Statistical Methods in Medical Research, 34(1), 1-19.

Examples

# Exact power calculation using Boschloo test
power2BinaryExact(
  n1 = 100,
  n2 = 50,
  p11 = 0.5,
  p12 = 0.4,
  p21 = 0.3,
  p22 = 0.2,
  rho1 = 0.7,
  rho2 = 0.7,
  alpha = 0.025,
  Test = 'Boschloo'
)
#> 
#> Power calculation for two binary co-primary endpoints
#> 
#>              n1 = 100
#>              n2 = 50
#>     p (group 1) = 0.5, 0.4
#>     p (group 2) = 0.3, 0.2
#>             rho = 0.7, 0.7
#>           alpha = 0.025
#>            Test = Boschloo
#>          power1 = 0.651316
#>          power2 = 0.70332
#>  powerCoprimary = 0.563055
#> 

# Exact power with Fisher exact test
power2BinaryExact(
  n1 = 80,
  n2 = 80,
  p11 = 0.6,
  p12 = 0.5,
  p21 = 0.4,
  p22 = 0.3,
  rho1 = 0.5,
  rho2 = 0.5,
  alpha = 0.025,
  Test = 'Fisher'
)
#> 
#> Power calculation for two binary co-primary endpoints
#> 
#>              n1 = 80
#>              n2 = 80
#>     p (group 1) = 0.6, 0.5
#>     p (group 2) = 0.4, 0.3
#>             rho = 0.5, 0.5
#>           alpha = 0.025
#>            Test = Fisher
#>          power1 = 0.658351
#>          power2 = 0.677271
#>  powerCoprimary = 0.517687
#> 

# \donttest{
# Larger sample sizes (computationally intensive)
power2BinaryExact(
  n1 = 200,
  n2 = 100,
  p11 = 0.5,
  p12 = 0.4,
  p21 = 0.3,
  p22 = 0.2,
  rho1 = 0.6,
  rho2 = 0.6,
  alpha = 0.025,
  Test = 'Chisq'
)
#> 
#> Power calculation for two binary co-primary endpoints
#> 
#>              n1 = 200
#>              n2 = 100
#>     p (group 1) = 0.5, 0.4
#>     p (group 2) = 0.3, 0.2
#>             rho = 0.6, 0.6
#>           alpha = 0.025
#>            Test = Chisq
#>          power1 = 0.9219
#>          power2 = 0.949665
#>  powerCoprimary = 0.892527
#> 
# }