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,
  n_grid = 100
)

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

n_grid

Number of grid points used to maximize the null tail probability over the nuisance parameter in the two exact unconditional tests, that is "Z-pool" and "Boschloo" (default is 100). The other three tests read their p-values off a distribution and ignore this argument. A finer grid locates the maximum more accurately at a proportionally higher computational cost, and the default reproduces the results of earlier versions of the package.

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(11), 2183-2201.

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
#> 
# }