twoCoprimary (version 1.0.0)

power2Continuous: Power Calculation for Two Co-Primary Continuous Endpoints

Description

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
)

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

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)

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
)

# \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
)
# }