Wilcoxon Rank-Sum Test

1
Wilcoxon Rank-Sum Test

• If X1, X2, Xn is a sample of size n from a
  population, then the rank of Xi , R(Xi), is given
  by
• R(Xi) number of Xjs Xi for each i.
  Compute midranks if there are tied values (i.e.,
  average the ranks - later). The Wilcoxon
  statistic is based on the sum of the ranks of the
  data in one of the two groups
• Here's the process
• pool the m observations from group1 with the n
  observations from group2 (total of mn) and order
  them all from smallest to largest.
• assign ranks (or midranks) to the ordered data
  smallestrank1, next smallestrank 2, etc.
• Let W sum of the ranks of the observations from
  group1 (or group2)
• determine the p-value associated with your value
  of W and decide whether to reject the null
  hypothesis of no difference in the two population
  distributions
• if there are no ties, the p-value can be computed
  by looking at the actual distribution of all the
  permutations of the mn ranks or
• by looking at many samples of permutations of
  ranks or
• by looking in Table A3 of the Appendix of our
  book
• Let's go over Example 2.4.2 on page 38hypothesis
  is that there is no difference in distributions
  of dry weights of herbicide treated and untreated
  plants.

2

• The alternative is that the untreated plants will
  have a larger distribution of weights, so we'll
  use an upper-tailed test. Table A3 has critical
  values (both upper and lower) for a.05, .025 and
  .01 when m,n range from 4 to 10 more extensive
  tables are available in the library and R and SAS
  will both give exact (and approximate) p-values
  for the Rank-Sum test.
• Look at the R code for doing the Wilcoxon test in
  the R3 document
• Let's look at the SAS solution to this
• dm output 'clear' dm log 'clear'
• options ls80
• data ex2_4_2
• input group weight _at__at_
• datalines
• u .55 u .67 u .63 u .79 u .81 u .85 u .68
• t .65 t .59 t .44 t .60 t .47 t .58 t .66 t .52 t
  .51
• proc rank dataex2_4_2 outrankwts var weight
  ranks rwt run
• proc sort datarankwts by group run
• proc print datarankwts by group sumby group
  run
• proc npar1way wilcoxon exact
• class group var weight run quit