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Lecture03 - Applied Nonparametrics STA 4502/5507 Yiyuan She...

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Applied Nonparametrics STA 4502/5507 Yiyuan She Department of Statistics, Florida State University Fall 2009
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Possible Course Flow Estimating success probabilities Single location: estimates, tests, intervals Two locations: testing, estimating di erences between locations Scale comparisons Multiple locations and factors Independence Nonparametric regression Other topics ...
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Location Compare two population centers (paired replicates data) Perhaps before and after of one population One population, where is center? Will use signs and ranks of data
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Paired Replicates Assumptions Data comes in pairs ( X i ; Y i ), i = 1 ; 2 ; : : : ; n Focus on Z i = Y i X i Z i are (mutually) independent Z i come from continuous populations (not necessarily the same) Z i symmetric about : F i ( + t ) + F i ( t ) = 1 for all t ; i : treatment e ect
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Wilcoxon Test Distribution-free ( X ; Y ) Some assumptions (pairs, symmetry, independence) Null: H 0 : = 0 No di erence before and after
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Wilcoxon Test Set i = 8 < : 1 ; Z i > 0 ; 0 ; Z i < 0 : Get ranks R i of j Z i j Test statistic is T + = n X i =1 R i i
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Example Patient i X i Y i 1 1.83 0.878 2 0.50 0.647 3 1.62 0.598 4 2.48 2.05 5 1.68 1.06 6 1.88 1.29 7 1.55 1.06 8 3.06 3.14 9 1.30 1.29
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Example i Z i j Z i j R i i R i i 1 -.952 .952 8 0 0 2 .147 .147 3 1 3 3 -1.022 1.022 9 0 0 4 -.430 .430 4 0 0 5 -.620 .620 7 0 0 6 -.590 .590 6 0 0 7 -.490 .490 5 0 0 8 .080 080 2 1 2 9 -.010 .010 1 0 0
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Wilcoxon Test Suppose > 0 Should be more positive Z i s Should be large (higher ranks) T + will be bigger Reject = 0 when T + big
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Wilcoxon Statistic Under H o , 0 T + n ( n +1) 2 E ( T + ) = n ( n +1) 4 var ( T + ) = n ( n +1)(2 n +1) 24 Symmetry: T + is symmetric about ET + T + d = P n 1 ib i , where b i i : i : d : Bernoulli(1 = 2) Distribution available; see Table A.4
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Wilcoxon Test One-tail alternative H 1 : > 0 Reject H 0 if T + t t : Table A.4 in NSM or wilcox.test in R
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Wilcoxon Test One-tail alternative H 1 : < 0 Reject H 0 if T + n ( n +1) 2 t Symmetry when H 0 true Center of T + is n ( n +1) 4 Two-tail alternative H 1 : 6 = 0 Reject H 0 if T + n ( n +1) 2 t = 2 or T + t = 2
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Wilcoxon Test Large sample approximation E ( T + ) = n ( n +1) 4 var ( T + ) = n ( n +1)(2 n +1) 24 T = T + E ( T + ) p var ( T + ) _ N (0 ; 1) CLT: Is T + the sum of i.i.d. random variables? T + = P i R i : identically distributed, but not independent T + d = P ib i : independent, but not identically distributed (minor) Use z ; z = 2 for di erent alternatives
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Wilcoxon Test Continuous ) No ties, strictly increasing ranks, no Z i = 0 In practice, will get Z i = 0 (rounding, for example) Toss these, set n to be number of non-zero Z i Works if number of zeros not too large Alternative methods exist
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