Power Calculation for Two Co-Primary Binary Endpoints (Approximate)

Calculates the power for a two-arm superiority trial with two co-primary binary endpoints using various asymptotic normal approximation methods, as described in Sozu et al. (2010).

Usage

power2BinaryApprox(n1, n2, p11, p12, p21, p22, rho1, rho2, alpha, Test)

Arguments

n1

Sample size for group 1 (test group)

n2

Sample size for group 2 (control group)

p11

True probability of responders in group 1 for the first outcome (0 < p11 < 1)

p12

True probability of responders in group 1 for the second outcome (0 < p12 < 1)

p21

True probability of responders in group 2 for the first outcome (0 < p21 < 1)

p22

True probability of responders in group 2 for the second outcome (0 < p22 < 1)

rho1

Correlation between the two outcomes for group 1

rho2

Correlation between the two outcomes for group 2

alpha

One-sided significance level (typically 0.025 or 0.05)

Test

Statistical testing method. 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

Value

A data frame with the following columns:

n1: Sample size for group 1
n2: Sample size for group 2
p11, p12, p21, p22: Response probabilities
rho1, rho2: Correlations
alpha: One-sided significance level
Test: Testing method used
power1: Power for the first endpoint alone
power2: Power for the second endpoint alone
powerCoprimary: Power for both co-primary endpoints

Details

This function implements four approximate power calculation methods:

Asymptotic Normal (AN): Uses the standard normal approximation without continuity correction (equations 3-4 in Sozu et al. 2010).

Asymptotic Normal with Continuity Correction (ANc): Includes Yates's continuity correction (equation 5 in Sozu et al. 2010).

Arcsine (AS): Uses arcsine transformation without continuity correction (equation 6 in Sozu et al. 2010).

Arcsine with Continuity Correction (ASc): Arcsine method with continuity correction (equation 7 in Sozu et al. 2010).

The correlation between test statistics for the two endpoints depends on the method:

For AN and ANc methods: $$\rho_{nml} = \frac{\sum_{j=1}^{2} \rho_j \sqrt{\nu_{j1}\nu_{j2}}/n_j} {se_1 \times se_2}$$ where $\nu_{jk} = p_{jk}(1-p_{jk})$.

For AS method: $$\rho_{arc} = \frac{n_2 \rho_1 + n_1 \rho_2}{n_1 + n_2}$$ This is the weighted average of the correlations from both groups.

For ASc method: $$\rho_{arc,c} = \frac{1}{se_1 \times se_2} \left(\frac{\rho_1 \sqrt{\nu_{11}\nu_{12}}}{4n_1\sqrt{\nu_{11,c}\nu_{12,c}}} + \frac{\rho_2 \sqrt{\nu_{21}\nu_{22}}}{4n_2\sqrt{\nu_{21,c}\nu_{22,c}}}\right)$$ where $\nu_{jk,c} = (p_{jk} + c_j)(1 - p_{jk} - c_j)$, $c_1 = -1/(2n_1)$, and $c_2 = 1/(2n_2)$.

The correlation bounds are automatically checked using corrbound2Binary.

References

Sozu, T., Sugimoto, T., & Hamasaki, T. (2010). Sample size determination in clinical trials with multiple co-primary binary endpoints. Statistics in Medicine, 29(21), 2169-2179.

Examples

# Power calculation using asymptotic normal method
power2BinaryApprox(
  n1 = 200,
  n2 = 100,
  p11 = 0.5,
  p12 = 0.4,
  p21 = 0.3,
  p22 = 0.2,
  rho1 = 0.7,
  rho2 = 0.7,
  alpha = 0.025,
  Test = 'AN'
)
#> 
#> Power calculation for two binary co-primary endpoints
#> 
#>              n1 = 200
#>              n2 = 100
#>     p (group 1) = 0.5, 0.4
#>     p (group 2) = 0.3, 0.2
#>             rho = 0.7, 0.7
#>           alpha = 0.025
#>            Test = AN
#>          power1 = 0.91929
#>          power2 = 0.949617
#>  powerCoprimary = 0.894946
#> 

# Power calculation with continuity correction
power2BinaryApprox(
  n1 = 200,
  n2 = 100,
  p11 = 0.5,
  p12 = 0.4,
  p21 = 0.3,
  p22 = 0.2,
  rho1 = 0.7,
  rho2 = 0.7,
  alpha = 0.025,
  Test = 'ANc'
)
#> 
#> Power calculation for two binary co-primary endpoints
#> 
#>              n1 = 200
#>              n2 = 100
#>     p (group 1) = 0.5, 0.4
#>     p (group 2) = 0.3, 0.2
#>             rho = 0.7, 0.7
#>           alpha = 0.025
#>            Test = ANc
#>          power1 = 0.898088
#>          power2 = 0.933117
#>  powerCoprimary = 0.867311
#> 

# Power calculation using arcsine method
power2BinaryApprox(
  n1 = 150,
  n2 = 150,
  p11 = 0.6,
  p12 = 0.5,
  p21 = 0.4,
  p22 = 0.3,
  rho1 = 0.5,
  rho2 = 0.5,
  alpha = 0.025,
  Test = 'AS'
)
#> 
#> Power calculation for two binary co-primary endpoints
#> 
#>              n1 = 150
#>              n2 = 150
#>     p (group 1) = 0.6, 0.5
#>     p (group 2) = 0.4, 0.3
#>             rho = 0.5, 0.5
#>           alpha = 0.025
#>            Test = AS
#>          power1 = 0.936701
#>          power2 = 0.945629
#>  powerCoprimary = 0.897574
#>