![]() The number of balanced permutations is which can be much smaller than the total number P of permutations. In a balanced permutation, we make sure that after relabeling, the new treatment group has exactly n/2 members that came from the original treatment group and n/2 from the original control group. Recently, a special form of permutation analysis, called balanced permutation, has been employed. Holds for all D, under the null hypothesis that X i and Y i are all independent and identically distributed. If is the actual difference and is the difference for any reassignment of labels, chosen without looking at the X and Y values, then One reason why permutation tests work and are intuitively reasonable is their symmetry. The smallest available p value is 1/ P because the actual treatment allocation is always included in the reference set. More generally, if the observed effect beats (is larger than) exactly b of these values we can claim p = 1 − b/P. ![]() If the actual treatment effect is larger than that from all of the other permutations, then we may claim a p-value of 1/ P. There are ways to redo the assignment of treatment versus control labels, and they each give a value for the treatment effect. We might measure the treatment effect via where and are averages of the treatment and control observations, respectively. Suppose for example, that there are n observations in the treatment group and also n observations in the control group. Permutation tests, described in more detail below, work by permuting the treatment labels of the data and comparing the resulting values of a test statistic to the original one. ![]() They are used to test hypotheses and compute p-values without making strong parametric assumptions about the data, and they adapt readily to complicated test statistics.
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