On Multi-Sided Permutation Tests

In fields such as clinical trials, the environment, epidemiology, genetics, pharmacology, social experiments, etc., situations in which the effect of a treatment can be positive on some individuals and negative on the rest may sometimes occur. In a two-sample design, this situation can be expressed by a response model where a random effect Δ in the alternative is such that , and . Such a situation is essentially different from the traditional two-sided test, in which the alternative is assumed to be active only on one of two directions. We consider alternatives in which two sub-alternatives (Δ < 0) and (Δ > 0) can be jointly active. In order to deal with such an atypical situation, we suggest to firstly apply two goodness-of-fit tests, one for the positive deviations from the empirical distribution function of control sample and the other for the negative, and then proceed with their nonparametric combination within a permutation framework. A simulation study inspects on properties of such a solution.