Sample size determination in two-sided distribution-free treatment versus control multiple comparisons

The problem of power and sample size determination for distribution-free multiple comparison tests of K treatments versus a control group is addressed. We define the power as the probability of correctly rejecting one specified or all K hypotheses, corresponding to the per-pair and all-pairs power, respectively. The power formulas are derived for both joint ranking and pairwise ranking mechanism for general multiple comparison problems, followed by explicit form of these formulas when the single-step, step-down, or step-up adjustments are applied. The proposed power and sample size calculation methods apply to scenarios both when the underlying distributions are known and when they are unknown but a pilot study is available. Numerical methods via quasi-Monte Carlo integration and Monte Carlo integration are assessed. Our simulation studies show the accuracy of the power and sample size calculation formulas. We recommend the Monte Carlo integration as the calculation algorithm. An example from a mouse peritoneal cavity study is used to demonstrate the application of the methods.