Sample Size Calculation for Two Co-Primary Endpoints: One Continuous and One Binary

Determines the sample size for a two-arm superiority trial with two co-primary endpoints where one is continuous and one is binary, to achieve a specified power at a given significance level.

Usage

ss2MixedContinuousBinary(
  delta,
  sd,
  p1,
  p2,
  rho,
  r,
  alpha,
  beta,
  Test,
  nMC = 10000
)

Arguments

delta

Mean difference for the continuous endpoint (group 1 - group 2)

sd

Common standard deviation for the continuous endpoint

p1

Probability of response in group 1 for the binary endpoint (0 < p1 < 1)

p2

Probability of response in group 2 for the binary endpoint (0 < p2 < 1)

rho

Biserial correlation between the latent continuous variable underlying the binary endpoint and the observed continuous endpoint

r

Allocation ratio n1/n2 where n1 is sample size for group 1

alpha

One-sided significance level (typically 0.025 or 0.05)

beta

Type II error rate (typically 0.1 or 0.2). Power = 1 - beta

Test

Statistical testing method for the binary endpoint. 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
"Fisher": Fisher's exact test (uses sequential search)

nMC

Number of Monte Carlo replications when Test = "Fisher" (default: 10000)

Value

A data frame with the following columns:

delta: Mean difference for continuous endpoint
sd: Standard deviation for continuous endpoint
p1: Response probability in group 1 for binary endpoint
p2: Response probability in group 2 for binary endpoint
rho: Biserial correlation
r: Allocation ratio
alpha: One-sided significance level
beta: Type II error rate
Test: Testing method used for binary endpoint
nMC: Number of Monte Carlo replications (NA if Test != "Fisher")
n1: Required sample size for group 1
n2: Required sample size for group 2
N: Total sample size (n1 + n2)

Details

This function implements the sample size calculation for mixed continuous-binary co-primary endpoints following the methodology in Sozu et al. (2012).

The sequential search algorithm (Homma and Yoshida 2025, Algorithm 1) is used for all testing methods:

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.

Biserial Correlation: The biserial correlation rho represents the correlation between the latent continuous variable underlying the binary endpoint and the observed continuous endpoint. This is not the same as the point-biserial correlation observed in the data.

References

Sozu, T., Sugimoto, T., & Hamasaki, T. (2012). Sample size determination in clinical trials with multiple co-primary endpoints including mixed continuous and binary variables. Biometrical Journal, 54(5), 716-729.

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
ss2MixedContinuousBinary(
  delta = 0.5,
  sd = 1,
  p1 = 0.6,
  p2 = 0.4,
  rho = 0.5,
  r = 1,
  alpha = 0.025,
  beta = 0.1,
  Test = 'AN'
)
#> 
#> Sample size calculation for mixed continuous and binary co-primary endpoints
#> 
#>              n1 = 135
#>              n2 = 135
#>               N = 270
#>           delta = 0.5
#>              sd = 1
#>               p = 0.6, 0.4
#>             rho = 0.5
#>      allocation = 1
#>           alpha = 0.025
#>            beta = 0.1
#>            Test = AN
#> 

# With continuity correction
ss2MixedContinuousBinary(
  delta = 0.5,
  sd = 1,
  p1 = 0.6,
  p2 = 0.4,
  rho = 0.5,
  r = 1,
  alpha = 0.025,
  beta = 0.1,
  Test = 'ANc'
)
#> 
#> Sample size calculation for mixed continuous and binary co-primary endpoints
#> 
#>              n1 = 143
#>              n2 = 143
#>               N = 286
#>           delta = 0.5
#>              sd = 1
#>               p = 0.6, 0.4
#>             rho = 0.5
#>      allocation = 1
#>           alpha = 0.025
#>            beta = 0.1
#>            Test = ANc
#> 

# \donttest{
# Fisher's exact test (computationally intensive)
ss2MixedContinuousBinary(
  delta = 0.5,
  sd = 1,
  p1 = 0.6,
  p2 = 0.4,
  rho = 0.5,
  r = 1,
  alpha = 0.025,
  beta = 0.1,
  Test = 'Fisher',
  nMC = 5000
)
#> 
#> Sample size calculation for mixed continuous and binary co-primary endpoints
#> 
#>              n1 = 143
#>              n2 = 143
#>               N = 286
#>           delta = 0.5
#>              sd = 1
#>               p = 0.6, 0.4
#>             rho = 0.5
#>      allocation = 1
#>           alpha = 0.025
#>            beta = 0.1
#>            Test = Fisher
#>             nMC = 5000
#> 
# }