twoCoprimary (version 1.0.0)

ss2Continuous: Sample Size Calculation for Two Co-Primary Continuous Endpoints (Approximate)

Description

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
)

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)

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)

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 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
)

# \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
)
# }