Sample Size Calculation for Two Co-Primary Continuous Endpoints (Approximate)

Calculates the required sample size for a two-arm superiority trial with two co-primary continuous endpoints using sequential search algorithm.

Usage

ss2Continuous(
  delta1,
  delta2,
  sd1,
  sd2,
  rho,
  r,
  alpha,
  beta,
  known_var = TRUE,
  nMC = 10000
)

Arguments

delta1: Mean difference for the first endpoint
delta2: Mean difference for the second endpoint
sd1: Common standard deviation for the first endpoint
sd2: Common standard deviation for the second endpoint
rho: Common correlation between the two outcomes
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)
known_var: Logical value indicating whether variance is known (TRUE) or unknown (FALSE). If TRUE, power is calculated analytically; if FALSE, Monte Carlo simulation is used
nMC: Number of Monte Carlo simulations when known_var = FALSE (default is 10000)

Value

A data frame with the following columns:

delta1, delta2: Mean differences
sd1, sd2: Standard deviations
rho: Correlation
r: Allocation ratio
alpha: One-sided significance level
beta: Type II error rate
known_var: Variance assumption
nMC: Number of Monte Carlo simulations (NA if known_var = TRUE)
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) for both known and unknown variance cases:

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.

For known variance, the standardized test statistics are: $$Z_k = \frac{\delta_k}{\sigma_k \sqrt{1/n_1 + 1/n_2}}$$ For unknown variance, t-statistics with $\nu = n_1 + n_2 - 2$ degrees of freedom are used, and power is calculated using Monte Carlo simulation following Sozu et al. (2011).

References

Sozu, T., Sugimoto, T., & Hamasaki, T. (2011). Sample size determination in superiority clinical trials with multiple co-primary correlated endpoints. Journal of Biopharmaceutical Statistics, 21(4), 650-668.

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 with known variance
ss2Continuous(
  delta1 = 0.2,
  delta2 = 0.2,
  sd1 = 1,
  sd2 = 1,
  rho = 0.5,
  r = 1,
  alpha = 0.025,
  beta = 0.1,
  known_var = TRUE
)
#> 
#> Sample size calculation for two continuous co-primary endpoints
#> 
#>              n1 = 626
#>              n2 = 626
#>               N = 1252
#>           delta = 0.2, 0.2
#>              sd = 1, 1
#>             rho = 0.5
#>      allocation = 1
#>           alpha = 0.025
#>            beta = 0.1
#>       known_var = TRUE
#> 

# Sample size calculation with unequal allocation
ss2Continuous(
  delta1 = 0.3,
  delta2 = 0.25,
  sd1 = 1,
  sd2 = 1,
  rho = 0.3,
  r = 2,
  alpha = 0.025,
  beta = 0.2,
  known_var = TRUE
)
#> 
#> Sample size calculation for two continuous co-primary endpoints
#> 
#>              n1 = 418
#>              n2 = 209
#>               N = 627
#>           delta = 0.3, 0.25
#>              sd = 1, 1
#>             rho = 0.3
#>      allocation = 2
#>           alpha = 0.025
#>            beta = 0.2
#>       known_var = TRUE
#> 

# \donttest{
# Sample size calculation with unknown variance (uses sequential search)
ss2Continuous(
  delta1 = 0.5,
  delta2 = 0.4,
  sd1 = 1,
  sd2 = 1,
  rho = 0.4,
  r = 1,
  alpha = 0.025,
  beta = 0.1,
  known_var = FALSE,
  nMC = 10000
)
#> 
#> Sample size calculation for two continuous co-primary endpoints
#> 
#>              n1 = 138
#>              n2 = 138
#>               N = 276
#>           delta = 0.5, 0.4
#>              sd = 1, 1
#>             rho = 0.4
#>      allocation = 1
#>           alpha = 0.025
#>            beta = 0.1
#>       known_var = FALSE
#>             nMC = 10000
#> 
# }