Calculates the required sample size for a two-arm superiority trial with a single binary endpoint using various statistical testing methods.
Arguments
- p1
True probability of responders in group 1 (0 < p1 < 1). Must be greater than
p2, since the design is a superiority trial- p2
True probability of responders in group 2 (0 < p2 < 1)
- 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)
- beta
Target type II error rate (typically 0.1 or 0.2)
- Test
Statistical testing method. One of:
"AN": Asymptotic normal method without continuity correction (default)"ANc": Asymptotic normal method with continuity correction"AS": Arcsine transformation without continuity correction"ASc": Arcsine transformation with continuity correction"Fisher": Fisher's exact test with iterative sample size determination
Value
A data frame with the following columns:
- p1
Probability of responders in group 1
- p2
Probability of responders in group 2
- r
Allocation ratio
- alpha
One-sided significance level
- beta
Type II error rate
- Test
Testing method used
- n1
Required sample size for group 1
- n2
Required sample size for group 2
- N
Total sample size (n1 + n2)
Details
This function implements sample size calculations for single binary endpoint trials using five different methods.
Important: This function is designed for a single binary endpoint.
For co-primary endpoints, use ss2BinaryApprox (for approximate
methods) or ss2BinaryExact (for exact methods).
Notation:
r = n1/n2: allocation ratio (group 1 to group 2)
kappa = 1/r = n2/n1: inverse allocation ratio
p1, p2: response probabilities
theta1 = 1 - p1, theta2 = 1 - p2: non-response probabilities
delta = p1 - p2: treatment effect
All four asymptotic methods return the smallest \(n_2\) whose realized
group sizes reach the target power under power2BinaryApprox
with both endpoints set to the same marginal. The closed forms below give the
starting value of that search. They are not used as the answer, because they
assume that \(n_1\) is exactly \(r n_2\), whereas \(n_1\) is rounded
up to an integer, and because for the two continuity corrected methods the
correction depends on the sample size being solved for.
AN (Asymptotic Normal) Method: Uses the standard normal approximation with pooled variance under H0: $$n_2 = \left\lceil \frac{(1 + \kappa)}{(\pi_1 - \pi_2)^2} \left(z_{1-\alpha} \sqrt{\bar{\pi}(1-\bar{\pi})} + z_{1-\beta} \sqrt{\kappa\pi_1\theta_1 + \pi_2\theta_2}\right)^2 / \kappa \right\rceil$$ where \(\bar{\pi} = (r\pi_1 + \pi_2)/(1 + r)\) is the pooled proportion.
ANc Method: Adds Yates's continuity correction to the AN method.
AS (Arcsine) Method: Uses the variance-stabilizing arcsine transformation: $$n_2 = \left\lceil \frac{(z_{1-\alpha} + z_{1-\beta})^2 (1 + \kappa)}{4(\sin^{-1}\sqrt{\pi_1} - \sin^{-1}\sqrt{\pi_2})^2} \right\rceil$$
ASc Method: Applies continuity correction to the arcsine method. The correction moves the two arms toward each other, and the variance is adjusted for the corrected proportions following Sozu et al. (2010).
Fisher Method: Fisher's exact test does not have a closed-form sample size formula. This method:
Starts with the AN method's sample size as initial value
Incrementally increases n2 by 1
Calculates exact power by summing the binomial probabilities over the rejection region
Stops when power is greater than or equal to 1 - beta
Steps back down while the target is still met, so that a starting value that already reached the target does not hide a smaller one
Note: Due to the saw-tooth nature of exact power (power does not increase monotonically with sample size), a sequential search approach is used. The incremental approach ensures the minimum sample size that achieves the target power.
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
# Balanced design with 1:1 allocation (AN method)
ss1BinaryApprox(p1 = 0.6, p2 = 0.4, r = 1, alpha = 0.025, beta = 0.1, Test = "AN")
#>
#> Sample size calculation for single binary endpoint
#>
#> n1 = 130
#> n2 = 130
#> N = 260
#> p = 0.6, 0.4
#> allocation = 1
#> alpha = 0.025
#> beta = 0.1
#> Test = AN
#>
# Unbalanced design with 2:1 allocation (ANc method)
ss1BinaryApprox(p1 = 0.5, p2 = 0.3, r = 2, alpha = 0.025, beta = 0.2, Test = "ANc")
#>
#> Sample size calculation for single binary endpoint
#>
#> n1 = 156
#> n2 = 78
#> N = 234
#> p = 0.5, 0.3
#> allocation = 2
#> alpha = 0.025
#> beta = 0.2
#> Test = ANc
#>
# Arcsine transformation method
ss1BinaryApprox(p1 = 0.55, p2 = 0.35, r = 1, alpha = 0.025, beta = 0.1, Test = "AS")
#>
#> Sample size calculation for single binary endpoint
#>
#> n1 = 129
#> n2 = 129
#> N = 258
#> p = 0.55, 0.35
#> allocation = 1
#> alpha = 0.025
#> beta = 0.1
#> Test = AS
#>
# Arcsine with continuity correction
ss1BinaryApprox(p1 = 0.65, p2 = 0.45, r = 1, alpha = 0.025, beta = 0.1, Test = "ASc")
#>
#> Sample size calculation for single binary endpoint
#>
#> n1 = 139
#> n2 = 139
#> N = 278
#> p = 0.65, 0.45
#> allocation = 1
#> alpha = 0.025
#> beta = 0.1
#> Test = ASc
#>
# Fisher's exact test
ss1BinaryApprox(p1 = 0.6, p2 = 0.4, r = 2, alpha = 0.025, beta = 0.1, Test = "Fisher")
#>
#> Sample size calculation for single binary endpoint
#>
#> n1 = 206
#> n2 = 103
#> N = 309
#> p = 0.6, 0.4
#> allocation = 2
#> alpha = 0.025
#> beta = 0.1
#> Test = Fisher
#>