[R] qwilcox

[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
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