Sample Size Calculation for Two Co-Primary Binary Endpoints (Asymptotic Approximation)

Calculates the required sample size for a two-arm superiority trial with two co-primary binary endpoints using asymptotic normal approximation or arcsine transformation, as described in Sozu et al. (2010).

Usage

ss2BinaryApprox(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:

"AN": Asymptotic normal method without continuity correction
"ANc": Asymptotic normal method with continuity correction
"AS": Arcsine method without continuity correction
"ASc": Arcsine method with continuity correction

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 (Homma and Yoshida 2025, Algorithm 1) to find the minimum sample size:

Step 1: Initialize with sample sizes from single endpoint formulas.

Step 2: Use sequential search:

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.

The asymptotic normal (AN) and arcsine (AS) methods use normal approximation with or without continuity correction. For small sample sizes or extreme probabilities, consider using exact methods via ss2BinaryExact.

References

Sozu, T., Sugimoto, T., & Hamasaki, T. (2010). Sample size determination in clinical trials with multiple co-primary binary endpoints. Statistics in Medicine, 29(21), 2169-2179.

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

# Sample size calculation using asymptotic normal method
ss2BinaryApprox(
  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 = 'AN'
)
#> 
#> Sample size calculation for two binary co-primary endpoints
#> 
#>              n1 = 109
#>              n2 = 109
#>               N = 218
#>     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 = AN
#> 

# With continuity correction
ss2BinaryApprox(
  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 = 'ANc'
)
#> 
#> Sample size calculation for two binary co-primary endpoints
#> 
#>              n1 = 119
#>              n2 = 119
#>               N = 238
#>     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 = ANc
#> 

# Using arcsine transformation
ss2BinaryApprox(
  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 = 'AS'
)
#> 
#> Sample size calculation for two binary co-primary endpoints
#> 
#>              n1 = 109
#>              n2 = 109
#>               N = 218
#>     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 = AS
#>