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Calculate Sample Size Needed to Compare k Proportions: 1-Way ANOVA Pairwise

This calculator is useful for tests concerning whether the proportions in several groups are equal. The statistical model is called an Analysis of Variance, or ANOVA model. This calculator is for the particular situation where we wish to make pairwise comparisons between groups. That is, we test for equality between two groups at a time, and we make several of these comparisons. For example, suppose we want to compare the proportions in three groups called foo, bar, and ack. These groups may represent groups of people that have been exposed to three different medical procedures, marketing schemes, etc. The complete list of pairwise comparisons are foo vs. bar, foo vs. ack, and bar vs. ack. In more general terms, we may have kk groups, meaning there are a total of K≡(k2)=k(k−1)/2K \equiv \binom{k}{2}=k(k-1)/2 possible pairwise comparisons. When we test τ≤K\tau \le K of these pairwise comparisons, we have τ\tau hypotheses of the form:
H0:pA=pBH_0: p_A = p_B
H1:pA≠pBH_1: p_A \ne p_B
where pAp_A and pBp_B represent the proportions in two of the kk groups, groups 'A' and 'B'. We'll compute the required sample size for each of the τ\tau comparisons, and total sample size needed is the largest of these. In the formula below, nn represents the sample size in any one of these τ\tau comparisons; that is, there are n/2n/2 people in the 'A' group, and n/2n/2 people in the 'B' group.

Formulas

This calculator uses the following formulas to compute sample size and power, respectively:
n=(pA(1−pA)+pB(1−pB))(z1−α/(2τ)+z1−βpA−pB)2n = \left(p_A(1-p_A) + p_B(1-pB)\right) \left(\frac{z_{1-\alpha/(2\tau)} + z_{1-\beta}}{p_A - pB}\right)^2
1−β=Φ(z−z1−α/(2τ))+Φ(−z−z1−α/(2τ))1-\beta = \Phi\left(z - z_{1-\alpha/(2\tau)}\right) + \Phi\left(-z - z_{1-\alpha/(2\tau)}\right)
z=pA−pBpA(1−pA)n+pB(1−pB)nz = \frac{p_A - p_B}{\sqrt{\frac{p_A(1-p_A)}{n} + \frac{p_B(1-pB)}{n}}}
where: nn is sample size Φ\Phi is the standard Normal distribution function Φ−1\Phi^{-1} is the standard Normal quantile function α\alpha is Type I error τ\tau is the number of comparisons to be made β\beta is Type II error, meaning 1−β1-\beta is power

R Code

1pA=0.2
2pB=0.4
3tau=2
4alpha=0.05
5beta=0.20
6(n=(pA*(1-pA)+pB*(1-pB))*((qnorm(1-alpha/2/tau)+qnorm(1-beta))/(pA-pB))^2)
7ceiling(n) # 96
8z=(pA-pB)/sqrt(pA*(1-pA)/n+pB*(1-pB)/n)
9(Power=pnorm(z-qnorm(1-alpha/2/tau))+pnorm(-z-qnorm(1-alpha/2/tau)))

References

Chow S, Shao J, Wang H. 2008. Sample Size Calculations in Clinical Research. 2nd Ed. Chapman & Hall/CRC Biostatistics Series. page 100.