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Calculate Sample Size Needed to Compare 2 Proportions: 2-Sample Equivalence

This calculator is useful when we wish to test whether the proportions in two groups are equivalent, without concern of which group's proportion is larger. Suppose we collect a sample from a group 'A' and a group 'B'; that is we collect two samples, and will conduct a two-sample test. For example, we may wish to test whether a new product is equivalent to an existing, industry standard product. Here, the 'burden of proof', so to speak, falls on the new product; that is, equivalence is actually represented by the alternative, rather than the null hypothesis.
H0:∣pA−pB∣≥δH_0:|p_A-p_B| \ge \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−β/2∣pA−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/2}}{|p_A-p_B|-\delta}\right)^2
1−β=2[Φ(z−z1−α)+Φ(−z−z1−α)]−1,z=∣pA−pB∣−δpA(1−pA)nA+pB(1−pB)nB1-\beta= 2\left[\Phi\left(z-z_{1-\alpha}\right)+\Phi\left(-z-z_{1-\alpha}\right)\right]-1 \quad ,\quad z=\frac{|p_A-p_B|-\delta}{\sqrt{\frac{p_A(1-p_A)}{n_A}+\frac{p_B(1-pB)}{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=0.65
2pB=0.85
3delta=0.05
4kappa=1
5alpha=0.05
6beta=0.20
7(nB=(pA*(1-pA)/kappa+pB*(1-pB))*((qnorm(1-alpha)+qnorm(1-beta/2))/(abs(pA-pB)-delta))^2)
8ceiling(nB) # 136
9z=(abs(pA-pB)-delta)/sqrt(pA*(1-pA)/nB/kappa+pB*(1-pB)/nB)
10(Power=2*(pnorm(z-qnorm(1-alpha))+pnorm(-z-qnorm(1-alpha)))-1)

References

Chow S, Shao J, Wang H. 2008. Sample Size Calculations in Clinical Research. 2nd Ed. Chapman & Hall/CRC Biostatistics Series. page 91.