16%20Permutation%20Test%203_27_08

16%20Permutation%20Test%203_27_08 - Assumption for a...

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1 1 Introduction to Permutation Tests Peng Liu 3/27/2008 2 Assumption for a z-test, t-test or F-test ± When conducting a z-test or a t-test, we are actually assuming that the data (or the random errors) follow a normal distribution. ± Based on this assumption, we know the distribution of the test statistic (T.S.) under the null hypothesis. ± Based on the distribution (z-distribution, t-distribution or F-distribution), we get a p-value for each observed T.S. . ± This can be referred to as “ parametric approaches ”. 3 What if the distributional assumption does not hold? ± If the normal assumption does not hold for the data and the sample size is small, the results of z-test, t- or F-test are not reliable. ± What can we do? Transformation of data to make the data normal Choose some tests that do not make such distributional assumptions – “nonparametric approaches” 4 Permutation test ± Permutation tests (randomization tests) can be used without the normal assumption for the distribution of data. ± Permutation test is a nonparametric approach to establish the null distribution of a test statistic. ± Permutation tests are attractive to microarray study because it makes fewer assumptions.
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2 5 Permutation ± Permutation is the rearrangement of objects or symbols into distinguishable sequences. ± Each unique ordering is called a permutation. ± For example, for A, B, C, and D, each possible ordering of all 4 elements without repetitions is one permutation, such as B, C, D, A. 6 Calculation the number of permutations ± Definition: For a positive number n , n! (read n factorial) is the product of all the positive integers less than or equal to n . That is: n! = n x (n-1) x (n-2) x … x 3 x 2 x 1 e.g.: 4! = 4 x 3 x 2 x 1 = 24 ± The number of permutations with n objects is n!
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16%20Permutation%20Test%203_27_08 - Assumption for a...

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