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5 permutation - Permutation Test for the Two Sample Problem...

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Permutation Test for the Two Sample Problem we wish to compare results for two groups of experimental units the first group could be some subjects who have been given a treatment, whereas the second group has not in some cases we are unable to assume that the two samples of sizes n 1 and n 2 are from normal populations and/or the populations have the same vari- ance however we may be able to asssume that the groups were obtained by randomly split- ting the subjects n = n 1 + n 2 into two groups with only this assumption, we are able to base the test on the permutation distribu- tion, described below 1
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the hypotheses are H o : no effect of the treatment H a : there is an effect a reasonable test statistic is T = ¯ X 1 - ¯ X 2 which measures the effect of the treat- ment if H o is true the observed differences in the data are due only to variation among the subjects with a different random allocation of sub- jects, a different value for T would be ob- tained 2
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there are exactly n 1 + n 2 n 1 ! = ( n 1 + n 2 )! n 1 ! n 2 ! ways of randomly allocating n 1 of the sub- jects to group 1 and the remaining n 2 to group 2 each of these is equally likely, and each can lead to a different value of the test statistic T the permutation distribution describes the possible values for T for all possible allo- cations of the subjects the P value is the fraction of values for T which are as least as extreme as the observed value T obs
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