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Calculate Sample Size Needed to Test 1 Proportion: 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 proportion, pp, is non-inferior/superior to a reference value, p0p_0. The idea is that statistically significant differences between the proportion 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:p−p0≤δH_0:p-p_0\le\delta
H1:p−p0>δH_1:p-p_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=p(1-p)\left(\frac{z_{1-\alpha}+z_{1-\beta}}{p-p_0-\delta}\right)^2
1−β=Φ(z−z1−α)+Φ(−z−z1−α),z=p−p0−δp(1−p)n1-\beta= \Phi\left(z-z_{1-\alpha}\right)+\Phi\left(-z-z_{1-\alpha}\right) \quad ,\quad z=\frac{p-p_0-\delta}{\sqrt{\frac{p(1-p)}{n}}}
where where
n
issamplesize is sample size
p_0
isthecomparisonvalue is the comparison value
\Phi
isthestandardNormaldistributionfunction is the standard Normal distribution function
\Phi^{-1}
isthestandardNormalquantilefunction is the standard Normal quantile function
\alpha
isTypeIerror is Type I error
\betaisTypeIIerror,meaning is Type II error, meaning 1-\beta
ispower is power
\delta$ is the testing margin

R Code

1p=0.5
2p0=0.3
3delta=-0.1
4alpha=0.05
5beta=0.20(n=p*(1-p)*((qnorm(1-alpha)+qnorm(1-beta))/(p-p0-delta))^2)
6ceiling(n) # 18
7z=(p-p0-delta)/sqrt(p*(1-p)/n)(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 86.