twoCoprimary (version 1.0.0)

ss1BinaryApprox: Sample Size Calculation for a Single Binary Endpoint

Description

Calculates the required sample size for a two-arm superiority trial with a single binary endpoint using various statistical testing methods.

Usage

ss1BinaryApprox(p1, p2, r, alpha, beta, Test = "AN")

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)

Arguments

p1

True probability of responders in group 1 (0 < p1 < 1)

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

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

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 continuity correction to the AN method. Uses iterative calculation because the correction term depends on sample size. Converges when the difference between successive iterations is less than or equal to 1.

AS (Arcsine) Method: Uses the variance-stabilizing arcsine transformation: $$n_2 = \left\lceil \frac{(z_{1-\alpha} + z_{1-\beta})^2}{4(\sin^{-1}\sqrt{\pi_1} - \sin^{-1}\sqrt{\pi_2})^2} \times \frac{1 + \kappa}{\kappa} \right\rceil$$

ASc Method: Applies continuity correction to the arcsine method. Uses iterative procedure with convergence criterion.

Fisher Method: Fisher's exact test does not have a closed-form sample size formula. This method:

1. Starts with the AN method's sample size as initial value

2. Incrementally increases n2 by 1

3. Calculates exact power using hypergeometric distribution

4. Stops when power is greater than or equal to 1 - beta

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")

# 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")

# Arcsine transformation method
ss1BinaryApprox(p1 = 0.55, p2 = 0.35, r = 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")

# Fisher's exact test
ss1BinaryApprox(p1 = 0.6, p2 = 0.4, r = 2, alpha = 0.025, beta = 0.1, Test = "Fisher")