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Unformatted text preview: Mantel Permutation Tests PERMUTATION TESTS Basic Idea: In some experiments a test of treatment effects may be of interest where the null hypothesis is that the different populations are actually from the same population. Or in other tests, the null hypothesis is one of complete randomness. Example 1: ANOVA where H is that the treatment means are all equal. The assumptions that must be true are that each treatment must have the same variance and the same shape. If in fact, the null hypothesis is true, then the observations are not distinguishable by treatment but are instead from the same distribution (one shape, mean and variance) and just happen to be randomly associated with a treatment. Original dataset collected Sample ID Pop 1 Pop 2 1 7 12 2 0 1 3 6 5 4 2 6 5 3 5 6 4 3 7 7 3 8 6 4 9 5 7 Mean 4.44 4.55 Permuted Data Sample ID Pop 1 Pop 2 1 5 7 2 0 7 3 3 5 4 2 4 5 3 12 6 5 1 7 7 6 8 6 4 9 6 3 Mean 4.11 5.44 ALS5932/FOR6934 Fall 2006 1 Mary C. Christman Permutation tests are based on this idea. If H is true then any set of values are just random assignments among treatments. Method Under The Assumptions That The Distributions Are Identical Under H And Sampling Is Random And With Replacement And Treatment Assignment Is Random : 1) Calculate the test statistic for the hypotheses for the original observed arrangement of data. This could be a sample correlation, an Fstat or a MS or some other statistic. Call it κ . 2) Now, randomly rearrange the data among the treatments (shuffle or permute the data according to the experimental design; see below for the case of matrices) and calculate the test statistic for the new arrangement. Call it . * p κ 3) Store the permutation estimate . * p κ 4) Repeat steps 23 many times. Call the total number of times you repeat the permutations P . That is p = 1, 2, …, P. 5) Compare κ to the distribution of the permutation estimates . The p value for the test is * p κ P value p p p ) ( # * κ κ > = − . Example : The most famous use of permutation tests for ecological problems is Mantel’s test of similarity of two symmetric matrices. Mantel’s test was extended to allow more than 2 matrices by Smouse et al. 1986. We’ll look at the simple case (2 matrices). Mantel’s test is a test of the correlation between the elements in one matrix with the elements in the other matrix where the elements within the matrices have been organized in a very specific way (symmetric with zeroes on the diagonal). Original use was to compare two distance matrices and that is still the most common use today. STA 6934 Spring 2007 2 Mary C. Christman Matrix Y Matrix X ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ f e c e d b c b a ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ φ ε χ ε δ β χ β α Question : Are the elementwise pairs, ( a, α ), ( b, β ), ( c, χ ), ( d, δ ), ( e, ε ), ( f, φ ), correlated? Can we use Pearson’s correlation coefficient to test that?...
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This note was uploaded on 01/15/2012 for the course STA 6934 taught by Professor Young during the Fall '08 term at University of Florida.
 Fall '08
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