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Calculate Sample Size Needed to Test 1 Mean: 1-Sample Non-Inferiority or Superiority

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, δ\delta, is positive or negative. In this setting, we wish to test whether a mean, μ\mu, is non-inferior/superior to a reference value, μ0\mu_0. The idea is that statistically significant differences between the mean and the reference value may not be of interest unless the difference is greater than a threshold, δ\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
H0:μ−μ0≤δH_0: \mu - \mu_0 \le \delta
H1:μ−μ0>δH_1: \mu - \mu_0 > \delta
and δ\delta is the superiority or non-inferiority margin.

Formulas

This calculator uses the following formulas to compute sample size and power, respectively:
n=(σz1−α+z1−βμ−μ0−δ)2n=\left(\sigma\frac{z_{1-\alpha}+z_{1-\beta}}{\mu-\mu_0-\delta}\right)^2
1−β=Φ(z−z1−α)+Φ(−z−z1−α),z=μ−μ0−δσ/n1-\beta= \Phi\left(z-z_{1-\alpha}\right)+\Phi\left(-z-z_{1-\alpha}\right) \quad ,\quad z=\frac{\mu-\mu_0-\delta}{\sigma/\sqrt{n}}
where: nn is sample size σ\sigma is standard deviation Φ\Phi is the standard Normal distribution function Φ−1\Phi^{-1} is the standard Normal quantile function α\alpha is Type I error β\beta is Type II error, meaning 1−β1-\beta is power δ\delta is the testing margin

R Code

1mu=2
2mu0=1.5
3delta=-0.5
4sd=1
5alpha=0.05
6beta=0.20
7(n=(sd*(qnorm(1-alpha)+qnorm(1-beta))/(mu-mu0-delta))^2)
8ceiling(n)# 7
9z=(mu-mu0-delta)/sd*sqrt(n)
10(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 52.