Compare 2 Proportions 2-Sample Non-Inferiority or Superiority | Power and Sample Size Calculators

This calculator is useful for the types of tests known as non-inferiority and superiority tests. Whether the null hypothesis represents 'non-inferiority' or 'superiority' depends on the context and whether the non-inferiority/superiority margin, $\delta$, is positive or negative. In this setting, we wish to test whether the proportion in group 'A', $p_A$, is non-inferior/superior to the proportion in group 'B', $p_B$. We collect a sample from both groups, and thus will conduct a two-sample test. The idea is that statistically significant differences between the proportions may not be of interest unless the difference is greater than a threshold, $\delta$. This is particularly popular in clinical studies, where the margin is chosen based on clinical judgement and subject-domain knowledge. The hypotheses to test are

$H_0:p_A-p_B\le\delta$
$H_1:p_A-p_B>\delta$
where $\delta$ is the superiority or non-inferiority margin and the ratio between the sample sizes of the two groups is
$$\kappa=\frac{n_A}{n_B}$$

Formulas

This calculator uses the following formulas to compute sample size and power, respectively: $$ n_A=\kappa n_B \;\text{ and }\; n_B=\left(\frac{p_A(1-p_A)}{\kappa}+p_B(1-p_B)\right) \left(\frac{z_{1-\alpha}+z_{1-\beta}}{p_A-p_B-\delta}\right)^2$$
$$1-\beta= \Phi\left(z-z_{1-\alpha/2}\right)+\Phi\left(-z-z_{1-\alpha/2}\right) \quad ,\quad z=\frac{p_A-p_B-\delta}{\sqrt{\frac{p_A(1-p_A)}{n_A}+\frac{p_B(1-p_B)}{n_B}}}$$ where

$\kappa=n_A/n_B$ is the matching ratio
$\Phi$ is the standard Normal distribution function
$\Phi^{-1}$ is the standard Normal quantile function
$\alpha$ is Type I error
$\beta$ is Type II error, meaning $1-\beta$ is power
$\delta$ is the testing margin

R Code

R code to implement these functions:

pA=0.85
pB=0.65
delta=-0.10
kappa=1
alpha=0.05
beta=0.20
(nB=(pA*(1-pA)/kappa+pB*(1-pB))*((qnorm(1-alpha)+qnorm(1-beta))/(pA-pB-delta))^2)
ceiling(nB) # 25
z=(pA-pB-delta)/sqrt(pA*(1-pA)/nB/kappa+pB*(1-pB)/nB)
(Power=pnorm(z-qnorm(1-alpha))+pnorm(-z-qnorm(1-alpha)))

References

Chow S, Shao J, Wang H. 2008. Sample Size Calculations in Clinical Research. 2nd Ed. Chapman & Hall/CRC Biostatistics Series. page 90.