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When you have $k$ treatment levels or $k$ comparison groups, there are a total of $K=k(k-1)/2$ possible pairwise comparisons that could be made. Suppose you wish to make $\tau\le K$ pairwise equality-comparisons between two group means. That is, we have $\tau$ hypotheses of the form

where $\mu_A$ and $\mu_B$ represent the means of two of the $k$ groups, groups 'A' and 'B'.

Power

This sample size calculator uses the following formula: $$n=\left(\sigma\frac{\Phi^{-1}(\alpha/(2\tau))+\Phi^{-1}(\beta)}{\mu_1-\mu_0}\right)^2$$ where

- $n$ is sample size
- $\sigma$ is standard deviation
- $\Phi^{-1}$ is the standard Normal quantile function
- $\alpha$ is Type I error
- $\tau$ is the number of comparisons to be made
- $\beta$ is Type II error, meaning $1-\beta$ is power

R code to implement this function:

mu0=1.5 mu1=2 sd=1 tau=1 alpha=0.05 beta=0.20 (n=ceiling((sd*(qnorm(alpha/2/tau)+qnorm(beta))/(mu1-mu0))^2)) # 32

Reference: Chow, page 71