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qwilcox

2 messages · Knut M. Wittkowski, Torsten Hothorn

Knut M. Wittkowski
# 
The function 'wilcox.test' in R and S gives (almost) identical results (see 
below). 'qwilcox' however, does not:

 > qwilcox(p,5,5)

p:      0.025   0.975
--------------------
R>       3      22
S>      18      37

I originally wanted to ask a questions, but then I found the answer. Given 
the confusion I run into, I wonder if this experience is worth reporting.

The S-Plus quantiles are almost correct (they are the limits of the region 
of acceptance, rather than the quantiles). The description in the R help file

         Distribution of the Wilcoxon Rank Sum Statistic

suggests that R:qwilcox also gives quantiles for the rank sum (which the 
Wilcoxon rank sum test is based on). In fact, however, it gives quantiles 
for the u-statistic (which the Mann-Whitney test is based upon). While the 
tests are logically equivalent, the particular test statistics

         - sum(Xi>c(X,Y))        rank sum (Wilcoxon)
         - sum(Xi>c(  Y))        u statistic (Mann-Whitney)

are different (apologies for the non-standard notation). Since 
"wilcox.test" relates to the rank sums in both R and S, as does qwilcox in 
S, the name 'qwilcox' in R may be misleading. How about renaming it to 
'qmannwhitney' instead and adding 'qwilcoxon' for a function that 
corresponds to S:qwilcox?

 > x1 <- c(1,2,3,  5,6         )
 > x2 <- c(      4,    7,8,9,10)
 > sum(x1)
[1] 17
 > sum(x2)
[1] 38

R> wilcox.test(x1,x2,alternative="two.sided")
         Wilcoxon rank sum test: p-value = 0.03175

R> wilcox.exact(x1,x2,alternative="two.sided")
         Exact Wilcoxon rank sum test: p-value = 0.03175

S> wilcox.test(x1,x2,alternative="two.sided")
         Exact Wilcoxon rank-sum test: p-value = 0.0317

 > x1 <- c(1,2,  4,5,6         )
 > x2 <- c(    3,      7,8,9,10)
 > sum(x1)
[1] 18
 > sum(x2)
[1] 37

R> wilcox.test(x1,x2,alternative="two.sided")
         Wilcoxon rank sum test: p-value = 0.05556

R> wilcox.exact(x1,x2,alternative="two.sided")
         Exact Wilcoxon rank sum test: p-value = 0.05556

S> wilcox.test(x1,x2,alternative="two.sided")
         Exact Wilcoxon rank sum test: p-value = 0.0556


Knut M. Wittkowski, PhD,DSc
------------------------------------------
The Rockefeller University, GCRC
1230 York Ave #121B, Box 322, NY,NY 10021
+1(212)327-7175, +1(212)327-8450 (Fax)
kmw at rockefeller.edu
http://www.rucares.org/statist/
Torsten Hothorn
#
On Tue, 10 Jun 2003, Knut M. Wittkowski wrote:

            

      
      
      
    

        
That is not true. You did not tell us the whole story:
`wilcox.test' in S-PLUS 2000 reports a statistic of `W = 17' for your
example below whereas R says

R> wilcox.test(x1,x2,alternative="two.sided")

        Wilcoxon rank sum test

data:  x1 and x2
W = 2, p-value = 0.03175
alternative hypothesis: true mu is not equal to 0

and as one can find out easily, `wilcox.test' computes the statistic as

STATISTIC <- sum(r[seq(along = x)]) - n.x * (n.x + 1) / 2

So both R and S are consistent and ?qwilcox will tell you what exactly is
meant by `Wilcoxon rank sum':

     This distribution is obtained as follows.  Let `x' and `y' be two
     random, independent samples of size `m' and `n'. Then the Wilcoxon
     rank sum statistic is the number of all pairs `(x[i], y[j])' for
     which `y[j]' is not greater than `x[i]'.  This statistic takes
     values between `0' and `m * n', and its mean and variance are `m *
     n / 2' and `m * n * (m + n + 1) / 12', respectively.


Best,

Torsten