Power Calculation for Two Co-Primary Continuous Endpoints

Calculates the power for a two-arm superiority trial with two co-primary continuous endpoints, as described in Sozu et al. (2011).

Usage

power2Continuous(
  n1,
  n2,
  delta1,
  delta2,
  sd1,
  sd2,
  rho,
  alpha,
  known_var = TRUE,
  nMC = 10000
)

Arguments

n1: Sample size for group 1 (test group)
n2: Sample size for group 2 (control group)
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
alpha: One-sided significance level (typically 0.025 or 0.05)
known_var: Logical value indicating whether variance is known (TRUE) or unknown (FALSE). If TRUE, power is calculated analytically; otherwise, Monte Carlo simulation is used for unknown variance
nMC: Number of Monte Carlo simulations when known_var = FALSE (default is 10000)

Value

A data frame with the following columns:

n1: Sample size for group 1
n2: Sample size for group 2
delta1: Mean difference for endpoint 1
delta2: Mean difference for endpoint 2
sd1: Standard deviation for endpoint 1
sd2: Standard deviation for endpoint 2
rho: Correlation between endpoints
alpha: One-sided significance level
known_var: Variance assumption
nMC: Number of Monte Carlo simulations (NA if known_var = TRUE)
power1: Power for the first endpoint alone
power2: Power for the second endpoint alone
powerCoprimary: Power for both co-primary endpoints

Details

For known variance, the power is calculated using the bivariate normal distribution as described in Sozu et al. (2011). The test statistics are: $$Z_k = \frac{\delta_k}{\sigma_k \sqrt{1/n_1 + 1/n_2}}$$ for k = 1, 2. The co-primary power is: $$1 - \beta = \Phi_2\left(-z_{1-\alpha} + Z_1, -z_{1-\alpha} + Z_2 \mid \rho\right)$$ where $\Phi_2$ is the cumulative distribution function of the bivariate standard normal distribution.

For unknown variance, Monte Carlo simulation is used with Wishart-distributed variance-covariance matrices to account for variance estimation uncertainty, following equation (6) in Sozu et al. (2011): $$\text{Power} = E_W\left[\Phi_2(-c_1^*\sqrt{w_{11}}, -c_2^*\sqrt{w_{22}} | \rho)\right]$$ where $c_k^* = t_{\alpha,\nu}\sqrt{\frac{1}{\nu}} - \frac{Z_k}{\sqrt{w_{kk}}}$ and $W$ follows a Wishart distribution with $\nu = n_1 + n_2 - 2$ degrees of freedom.

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.

Examples

# Example parameters for comparison across methods
n1_ex <- 100
n2_ex <- 100
delta1_ex <- 0.5
delta2_ex <- 0.5
sd1_ex <- 1
sd2_ex <- 1
rho_ex <- 0.3
alpha_ex <- 0.025

# Power calculation with known variance
power2Continuous(
  n1 = n1_ex,
  n2 = n2_ex,
  delta1 = delta1_ex,
  delta2 = delta2_ex,
  sd1 = sd1_ex,
  sd2 = sd2_ex,
  rho = rho_ex,
  alpha = alpha_ex,
  known_var = TRUE
)
#> 
#> Power calculation for two continuous co-primary endpoints
#> 
#>              n1 = 100
#>              n2 = 100
#>           delta = 0.5, 0.5
#>              sd = 1, 1
#>             rho = 0.3
#>           alpha = 0.025
#>       known_var = TRUE
#>          power1 = 0.942438
#>          power2 = 0.942438
#>  powerCoprimary = 0.893807
#> 

# \donttest{
# Power calculation with unknown variance (Monte Carlo)
power2Continuous(
  n1 = n1_ex,
  n2 = n2_ex,
  delta1 = delta1_ex,
  delta2 = delta2_ex,
  sd1 = sd1_ex,
  sd2 = sd2_ex,
  rho = rho_ex,
  alpha = alpha_ex,
  known_var = FALSE,
  nMC = 10000
)
#> 
#> Power calculation for two continuous co-primary endpoints
#> 
#>              n1 = 100
#>              n2 = 100
#>           delta = 0.5, 0.5
#>              sd = 1, 1
#>             rho = 0.3
#>           alpha = 0.025
#>       known_var = FALSE
#>             nMC = 10000
#>          power1 = 0.940427
#>          power2 = 0.940427
#>  powerCoprimary = 0.890195
#> 
# }