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Calculate Sample Size Needed to Compare 2 Proportions: 2-Sample, 2-Sided Equality

This calculator is useful for tests concerning whether the proportions in two groups are different. Suppose the two groups are 'A' and 'B', and we collect a sample from both groups -- i.e. we have two samples. We perform a two-sample test to determine whether the proportion in group A, pAp_A, is different from the proportion in group B, pBp_B. The hypotheses are
H0:pA−pB=0H_0:p_A-p_B=0
H1:pA−pB≠0H_1:p_A-pB\neq0
. where 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=κnB   and   nB=(pA(1−pA)κ+pB(1−pB))(z1−α/2+z1−βpA−pB)2 n_A=\kappa n_B \;\text{ and }\; n_B=\left(\frac{p_A(1-p_A)}{\kappa}+pB(1-pB)\right) \left(\frac{z_{1-\alpha/2}+z_{1-\beta}}{p_A-p_B}\right)^2
1−β=Φ(z−z1−α/2)+Φ(−z−z1−α/2),z=pA−pBpA(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}{\sqrt{\frac{p_A(1-p_A)}{n_A}+\frac{pB(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

R Code

1pA=0.65
2pB=0.85
3kappa=1
4alpha=0.05
5beta=0.20
6(nB=(pA*(1-pA)/kappa+pB*(1-pB))*((qnorm(1-alpha/2)+qnorm(1-beta))/(pA-pB))^2)
7ceiling(nB) # 70
8z=(pA-pB)/sqrt(pA*(1-pA)/nB/kappa+pB*(1-pB)/nB)
9(Power=pnorm(z-qnorm(1-alpha/2))+pnorm(-z-qnorm(1-alpha/2)))
10

References

Chow S, Shao J, Wang H. 2008. Sample Size Calculations in Clinical Research. 2nd Ed. Chapman & Hall/CRC Biostatistics Series. page 89.