notion of a most stringent test which minimizes the maximum power loss over a class of alternatives.) The newer idea addresses a limitation of FisherGLYPH<146>s method when used in the presence of interference. FisherGLYPH<146>s method yields a valid test of the null hypothesis of no e⁄ect. If a treatment e⁄ect has a simple form, say an additive constant e⁄ect or shift, then it is possible to invert FisherGLYPH<146>s test of no e⁄ect to yield a conGLYPH<133>dence statement for the magnitude of this constant e⁄ect (e.g., Lehmann 1975); however, by its nature, interference precludes such a simple form for an e⁄ect. The newer idea is to invert the randomization test of no e⁄ect to yield a conGLYPH<133>dence interval for an attributable e⁄ect in the presence of interference that contrasts the results seen with an active treatment to the results that would have been seen in an experiment of identical design but with no active treatment, a so-called GLYPH<147>uniformity trialGLYPH<148> common in the early years of randomized agricultural experimentation (Rosenbaum 2007a). This newer idea is applicable with distribution-free statistics whose distribution in the uniformity trial is known without conducting the uniformity trial. The classes of distribution-free statistics and of rank statistics overlap substantially but are not the same, and it is the distribution-free property that is needed here. The third, gradually evolving idea made a GLYPH<133>rst appearance in a paper by Lehmann (1953) concerned with the power of rank tests. After showing that WilcoxonGLYPH<146>s test was the locally most powerful rank test for a constant, additive e⁄ect in the presence of logistic errors, Lehmann went on to show that it was also locally most powerful against a very speciGLYPH<133>c mixture alternative in which only a fraction of subjects respond to treatment. Conover and Salsburg (1988) generalized the mixture alternative and derived the form of the corresponding locally most powerful test; this was no longer WilcoxonGLYPH<146>s test, but rather a test that gave greater emphasis to larger responses. Although they substantially increase power when some trials fail to elicit the intended e⁄ect, the ranks used by Conover 5