Another look at the Lady Tasting Tea and permutation-based randomization tests
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Fisher's famous Lady Tasting Tea experiment is often referred to as the first permutation test or as an example of such a test. Permutation tests are special cases of the general group invariance test. Recently it has been emphasized that the set of permutations used within a permutation test should have a group structure, in the algebraic sense. If not, the test can be very anti-conservative. In this paper, however, we note that in the Lady Tasting Tea experiment, the type I error rate is controlled even if the set of permutations used does not correspond to a group. We explain the difference between permutation-based tests that fundamentally rely on a group structure, and permutation-based tests that do not. The latter are tests based on randomization of treatments. When using such tests, it can be useful to consider a randomization scheme that does correspond to a group. In particular, we can use randomization schemes where the number of possible treatment patterns is larger than in standard permutation-based randomization tests. This leads to exact p-values of improved resolution, providing increased power for very small significance levels. We discuss applications in clinical trials and elsewhere.
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