Power Calculation for Two Co-Primary Mixed Endpoints — power2MixedContinuousBinary • twoCoprimary

Calculates the power for a two-arm superiority trial with two co-primary endpoints where one is continuous and one is binary, as described in Sozu et al. (2012).

Usage

power2MixedContinuousBinary(
  n1,
  n2,
  delta,
  sd,
  p1,
  p2,
  rho,
  alpha,
  Test,
  nMC = 10000
)

Arguments

n1

Sample size for group 1 (test group)

n2

Sample size for group 2 (control group)

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

alpha

One-sided significance level (typically 0.025 or 0.05)

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 (Monte Carlo simulation required)

nMC

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

Value

A data frame with the following columns:

n1: Sample size for group 1
n2: Sample size for group 2
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
alpha: One-sided significance level
Test: Testing method used for binary endpoint
powerCont: Power for the continuous endpoint alone
powerBin: Power for the binary endpoint alone
powerCoprimary: Power for both co-primary endpoints

Details

This function implements the power calculation for mixed endpoints (one continuous and one binary) as described in Sozu et al. (2012). The method assumes that the binary variable is derived from a latent continuous variable via dichotomization at a threshold point.

For Fisher's exact test, Monte Carlo simulation is used because exact calculation is computationally intensive. The continuous endpoint is analyzed using t-test, and the binary endpoint uses Fisher's exact test.

For asymptotic methods (AN, ANc, AS, ASc), analytical formulas are used based on bivariate normal approximation. The correlation between test statistics depends on the biserial correlation rho and the specific testing method.

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.

Examples

# Power calculation using asymptotic normal method
power2MixedContinuousBinary(
  n1 = 100,
  n2 = 100,
  delta = 0.5,
  sd = 1,
  p1 = 0.6,
  p2 = 0.4,
  rho = 0.5,
  alpha = 0.025,
  Test = 'AN'
)
#> 
#> Power calculation for mixed continuous and binary co-primary endpoints
#> 
#>              n1 = 100
#>              n2 = 100
#>           delta = 0.5
#>              sd = 1
#>               p = 0.6, 0.4
#>             rho = 0.5
#>           alpha = 0.025
#>            Test = AN
#>       powerCont = 0.942438
#>        powerBin = 0.812291
#>  powerCoprimary = 0.781111
#> 

# \donttest{
# Power calculation with Fisher's exact test (computationally intensive)
power2MixedContinuousBinary(
  n1 = 50,
  n2 = 50,
  delta = 0.5,
  sd = 1,
  p1 = 0.6,
  p2 = 0.4,
  rho = 0.5,
  alpha = 0.025,
  Test = 'Fisher',
  nMC = 5000
)
#> 
#> Power calculation for mixed continuous and binary co-primary endpoints
#> 
#>              n1 = 50
#>              n2 = 50
#>           delta = 0.5
#>              sd = 1
#>               p = 0.6, 0.4
#>             rho = 0.5
#>           alpha = 0.025
#>            Test = Fisher
#>       powerCont = 0.705414
#>        powerBin = 0.440109
#>  powerCoprimary = 0.364131
#> 
# }