Compare 2 Means 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 mean in group 'A', $\mu_A$, is non-inferior/superior to the mean in group 'B', $\mu_B$. We collect a sample from both groups, and thus will conduct a two-sample test. The idea is that statistically significant differences between the means 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

$H_0:\mu_A-\mu_B\le\delta$
$H_1:\mu_A-\mu_B>\delta$
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: $$ n_A=\kappa n_B \;\text{ and }\; n_B=\left(1+\frac{1}{\kappa}\right) \left(\sigma\frac{z_{1-\alpha}+z_{1-\beta}}{\mu_A-\mu_B-\delta}\right)^2$$
$$1-\beta= \Phi\left(z-z_{1-\alpha}\right)+\Phi\left(-z-z_{1-\alpha}\right) \quad ,\quad z=\frac{\mu_A-\mu_B-\delta}{\sigma\sqrt{\frac{1}{n_A}+\frac{1}{n_B}}}$$ where

$\kappa=n_A/n_B$ is the matching ratio
$\sigma$ is standard deviation
$\Phi$ 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:

muA=5
muB=5
delta=5
kappa=1
sd=10
alpha=0.05
beta=0.20
(nB=(1+1/kappa)*(sd*(qnorm(1-alpha)+qnorm(1-beta))/(muA-muB-delta))^2)
ceiling(nB) # 50
z=(muA-muB-delta)/(sd*sqrt((1+1/kappa)/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 61.