As of 2008, the National Cancer Institute describes balanced permutations in their page on statistical tests at. Since then, they have been applied numerous times in the literature on statistical analysis of microarrays. This is the earliest mention we have found. These balanced permutations are mentioned in remark E on page 1159 of Efron et al. Balanced permutations require n to be even, but nearly balanced permutations are possible for odd n. 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. S ample reuse methods such as permutation testing and the bootstrap are invaluable tools in high-throughput genomic settings, such as microarray analyses.
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