Wilcoxon rank sum test in R

Question

I have results from the same test applied to two independent samples:

x <- c(17, 12, 13, 16, 9, 19, 21, 12, 18, 17)
y <- c(10, 6, 15, 9, 8, 11, 8, 16, 13, 7, 5, 14)

And I want to compute a Wilcoxon rank sum test.

When I calculate the statistic $T_{W}$ by hand, I get: $$ T_{W}=\sum\text{rank}(X_{i}) = 156.5 $$

When I let R perform a wilcox.test(x, y, correct = F), I get:

W = 101.5

Why is that? Shouldn't the statistic $W^{+}$ only be returned when I perform a signed rank test with paired = T? Or do I misunderstand the rank sum test?

How can I tell R to output $T_{W}$

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As part of the test results, not through something like:

dat <- data.frame(v = c(x, y), s = factor(rep(c("x", "y"), c(10, 12))))
dat$r <- rank(dat$v)
T.W <- sum(dat$r[dat$s == "x"])

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I asked a follow up question about the meaning of the Different ways to calculate the test statistic for the Wilcoxon rank sum test

Answer 1

The Note in the help on the wilcox.test function clearly explains why R's value is smaller than yours:

Note

The literature is not unanimous about the definitions of the Wilcoxon rank sum and Mann-Whitney tests. The two most common definitions correspond to the sum of the ranks of the first sample with the minimum value subtracted or not: R subtracts and S-PLUS does not, giving a value which is larger by m(m+1)/2 for a first sample of size m. (It seems Wilcoxon's original paper used the unadjusted sum of the ranks but subsequent tables subtracted the minimum.)

That is, the definition R uses is $n_1(n_1+1)/2$ smaller than the version you use, where $n_1$ is the number of observations in the first sample.

As for modifying the result, you could assign the output from wilcox.test into a variable, say a, and then manipulate a$statistic - adding the minimum to its value and changing its name. Then when you print a (e.g. by typing a), it will look the way you want.

To see what I am getting at, try this:

a <- wilcox.test(x,y,correct=FALSE)
str(a) 

So for example if you do this:

n1 <- length(x)
a$statistic <- a$statistic + n1*(n1+1)/2
names(a$statistic) <- "T.W"
a

then you get:

        Wilcoxon rank sum test with continuity correction

data:  x and y 
T.W = 156.5, p-value = 0.006768
alternative hypothesis: true location shift is not equal to 0 

It's quite common to refer to the rank sum test (whether shifted by $n_1(n_1+1)/2$ or not) as either $W$ or $w$ or some close variant (e.g. here or here). It also often gets called '$U$' because of Mann & Whitney. There's plenty of precedent for using $W$, so for myself I wouldn't bother with the line that changes the name of the statistic, but if it suits you to do so there's no reason why you shouldn't, either.