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Sample Size (n_A): 25

Calculate Sample Size Needed to Compare 2 Proportions: 2-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 the proportion in group 'A', pAp_A, is non-inferior/superior to the proportion in group 'B', pBp_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, 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:pA−pB≤δH_0: p_A - p_B \le \delta
H1:pA−pB>δH_1: p_A - p_B > \delta
where δ\delta is the superiority or non-inferiority margin and the ratio between the sample sizes of the two groups is
κ=nAnB\kappa = \frac{n_A}{n_B}

Formulas

This calculator uses the following formulas to compute sample size and power, respectively:
nA=κnBandnB=(pA(1−pA)κ+pB(1−pB))(z1−α+z1−βpA−pB−δ)2n_A = \kappa n_B \quad and \quad 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−β=Φ(z−z1−α/2)+Φ(−z−z1−α/2),z=pA−pB−δpA(1−pA)nA+pB(1−pB)nB1-\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: κ=nA/nB\kappa=n_A/n_B is the matching ratio 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

1pA=0.85
2pB=0.65
3delta=-0.10
4kappa=1
5alpha=0.05
6beta=0.20
7(nB=(pA*(1-pA)/kappa+pB*(1-pB))*((qnorm(1-alpha)+qnorm(1-beta))/(pA-pB-delta))^2)
8ceiling(nB) # 25
9z=(pA-pB-delta)/sqrt(pA*(1-pA)/nB/kappa+pB*(1-pB)/nB)
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 90.