Ch10.wilcoxon Rank-Sum Test.pdfVIP

1The Wilcoxon Rank-Sum Test The Wilcoxon rank-sum test is a nonparametric alternative to the two- sample t-test which is based solely on the order in which the observations from the two samples fall. We will use the following as a running example. Example 1 In a genetic inheritance study discussed by Margolin [1988], samples of individuals from several ethnic groups were taken. Blood samples were collected from each individual and several variables measured. For a detailed discussion of the study and a definition of the variable, see Exer- cises 10.1.3 in the text. We shall compare the groups labeled “Native Amer- ican” and “Caucasian” with respect to the variable MSCE (mean sister chro- matid exchange). The data is as follows: Native American: 8.50 9.48 8.65 8.16 8.83 7.76 8.63 Caucasian: 8.27 8.20 8.25 8.14 9.00 8.10 7.20 8.32 7.70 7.5 8.0 8.5 9.0 9.5 Native American Caucasian Figure 1 : Comparing MSCE measurements. Looking at the dot plots for the two groups, several questions come to mind. Firstly, do the data come from Normal distributions? Unfortunately we can’t say much about the distributions as the samples are too small. However there does not seem to be any clear lack of symmetry. Secondly, are the two distributions similar in shape? Again it is hard to say much with such small samples, though the Caucasian data seems to have longer tails. Finally, is there any difference in the centers of location? The plots suggest a difference with Native American values being larger on average. We shall now put this type of problem in a more general context and come back to this example later. Suppose, more generally, that we have samples of observations from each of two populations A and B containing nA and nB observations respectively. We wish to test the hypothesis that the distribution of X-measurements in population A is the same as that in B, which we will write symbolically as H0 : A = B. The departures from H0 that the Wilcoxon test tries to detect are location