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,
  n_grid = 100
)

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

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:

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(11), 2183-2201.

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
#> 
# }