Exact Sample Size Calculation for Two Co-Primary Binary Endpoints

Calculates the required sample size for a two-arm superiority trial with two co-primary binary endpoints using exact methods, as described in Homma and Yoshida (2025).

Usage

ss2BinaryExact(p11, p12, p21, p22, rho1, rho2, r, alpha, beta, Test)

Arguments

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

r

Allocation ratio of group 1 to group 2 (group 1:group 2 = r:1, where r > 0)

alpha

One-sided significance level (typically 0.025 or 0.05)

beta

Target type II error rate (typically 0.1 or 0.2)

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:

p11, p12, p21, p22: Response probabilities
rho1, rho2: Correlations
r: Allocation ratio
alpha: One-sided significance level
beta: Type II error rate
Test: Testing method used
n1: Required sample size for group 1
n2: Required sample size for group 2
N: Total sample size (n1 + n2)

Details

This function uses a sequential search algorithm to find the minimum sample size that achieves the target power:

Step 1: Initialize with sample size from approximate method (AN). This provides a good starting point for the exact calculation.

Step 2: Use sequential search algorithm (Homma and Yoshida 2025, Algorithm 1):

Calculate power at initial sample size
If power >= target: decrease n2 until power < target, then add 1 back
If power < target: increase n2 until power >= target

Step 3: Return final sample sizes.

Note: Due to the saw-tooth nature of exact power (power does not increase monotonically with sample size), this sequential search ensures the minimum sample size that achieves the target power.

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

# Quick example with Chi-squared test (faster)
ss2BinaryExact(
  p11 = 0.6,
  p12 = 0.5,
  p21 = 0.4,
  p22 = 0.3,
  rho1 = 0.3,
  rho2 = 0.3,
  r = 1,
  alpha = 0.025,
  beta = 0.2,
  Test = "Chisq"
)
#> 
#> Sample size calculation for two binary co-primary endpoints
#> 
#>              n1 = 123
#>              n2 = 123
#>               N = 246
#>     p (group 1) = 0.6, 0.5
#>     p (group 2) = 0.4, 0.3
#>             rho = 0.3, 0.3
#>      allocation = 1
#>           alpha = 0.025
#>            beta = 0.2
#>            Test = Chisq
#> 

# \donttest{
# More computationally intensive example with Fisher test
ss2BinaryExact(
  p11 = 0.5,
  p12 = 0.4,
  p21 = 0.3,
  p22 = 0.2,
  rho1 = 0.5,
  rho2 = 0.5,
  r = 1,
  alpha = 0.025,
  beta = 0.2,
  Test = "Fisher"
)
#> 
#> Sample size calculation for two binary co-primary endpoints
#> 
#>              n1 = 117
#>              n2 = 117
#>               N = 234
#>     p (group 1) = 0.5, 0.4
#>     p (group 2) = 0.3, 0.2
#>             rho = 0.5, 0.5
#>      allocation = 1
#>           alpha = 0.025
#>            beta = 0.2
#>            Test = Fisher
#> 
# }