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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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- Spring '08