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Lecture03

# 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 ...
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
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
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
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
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
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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