note04 - Chapter 2 Two-Sample Methods 3 Wilcoxon Rank-Sum...

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Unformatted text preview: Chapter 2: Two-Sample Methods 3 Wilcoxon Rank-Sum Test • Assumptions: – X 1 ,...,X m ∼ iid F 1 and Y 1 ,...,Y n ∼ iid F 2 ( F 1 and F 2 are continuous cumulative distribution) – X 1 ,...,X m and Y 1 ,...,Y n are independent • No tie in observations – Rank of Y j : R ( Y j ) = – Combine S = m + n obs ⇒ sort in ascending orders Q = ( Q 1 ,...,Q m ) : the rank vector of X ’s among all S obs R = ( R 1 ,...,R n ) : the rank vector of Y ’s among all S obs – Note that R ∗ = ( Q 1 ,...,Q m ,R 1 ,...,R n ) is the rank vector of all S obs and R ∗ is a permutations of the integer 1 ,...,S – Test Stat W obs = n summationdisplay j =1 R j – N = ( m + n m ) possible two-sample data set For all permutations of the ranks, ( k = 1 ,...,N ) W k = n summationdisplay j =1 R ( k ) j where R ( j ) is the rank vector of Y ’s at k th permutation – For H a : F 1 ( x ) ≤ F 2 ( x ), – A test based on the difference of mean ranks reach the same conclusion as a test based on the sum of rank of one group 1 – Suppose X ≥ Y ∗ if W j = ∑ m i =1 Q ( j ) i , ∗ if W j = ∑ n i =1 R ( j ) i , • Ties in observations – Adjusted Ranks : the average rank to the tie observations – W ties : the Wilcoxon rank-sum statisti adjusted for ties – if the number of ties is samll, the approximate critical values may be obtained from the distribution of the rank-sum statistic without ties...
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note04 - Chapter 2 Two-Sample Methods 3 Wilcoxon Rank-Sum...

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