Wilcoxon rank sum test W=0 but p<0.05? [duplicate]

Question

While performing the Wilcoxon rank sum test in R, I sometimes get W values of 0, but significant p values. Since the way R calculates W is the difference between the rank sum of one group minus the expected rank sum, I don't understand how this can be 0 but then still reject the null hypothesis. If one of the groups rank sum is equal to the expected, shouldn't the other be as well and thus there is no difference, i.e. the null hypothesis holds?

Thanks a lot, Ser

PS I'm new here and I don't know how to cross ref to another post, but the way R calculates W is really well explained a post called 'Wilcoxon rank sum test correct vectors order'

Edit: I have read the topic on what the test statistic indicates, however, I am specifically interested in the W=0 case

Answer 1

According to its documentation, the ?wilcox.test function returns a statistic that is:

the number of all pairs (x[i], y[j]) for which y[j] is not greater than x[i], the most common definition of the Mann-Whitney test.

If you get $W=0$, it means that there are no rank of $X$ that is larger than the ranks of $Y$, which is is an extreme value if the null hypothesis is true. If you have a large enough sample size, you will then logically get a small p-value.

If you need more details on how this statistic is computed in R, you can refer to these other threads:

• Is the W statistic output by wilcox.test() in R the same as the U statistic?
• What does the size of the Wilcoxon rank sum test 'W statistic' indicate?
• Wilcoxon rank sum test in R

As pointed out in comments and in other threads, the function documentation also mentions that:

The literature is not unanimous about the definitions of the Wilcoxon rank sum and Mann-Whitney test.

So if you intend to publish your results, no matter if you use R or something else, be explicit about how you computed the test and which software you used, as it is something that may vary between tools and could confuse readers.

Answer 2

There are 3 issues at play here.
The first is that the test actually computes 2 statistics, $R_1$ and $R_2$, namelly the sum of ranks for sample 1, and sample 2. The "custom" is to chose as the statistic reported $Min(R_1,R_2)$; but in fact, one could chose either, as they carry exactly the same information (remember that $R_1+R_2=\frac {(n_1+n_2+1).(n_1+n_2)} {2}$). So if you see $W=0$, that means that "the other W" is quite large. And remember also that, under the null, W will not be equal to 0, but instead to $\frac {(n_1+n_2+1).(n_1+n_2)} {4}$, so $W=0$ is in fact very extreme.
The 2nd is that in your case, all the values of the 2nd sample are larger than all values of the first. That is as "extreme" (against the null) as 2 samples could get. Now if you only had run wilcox.test(b,a), the p-value would have been identical, but W would be larger (36 for your example).
The 3rd, and the major one, is that wilcox.test does not return a sum of ranks. If it did, for your example with 6 values in each sample, the sum of ranks statistic should be in the interval $[21,57]$. It is indeed impossible for a true sum of ranks to be 0 (that would imply that the sample had 0 values?!?). What this R function does instead is subtract the minimum possible value of the statistic (in your example, 21) from the actual, true sum of ranks. It does this, so that what it returns is actually the Mann-Whitney U statistic (remember that $U_1=R_2-\frac {n_2(n_2+1)} {2}$, and vice-versa for $U_2$. But it calls it W (should really call it U), adding to the confusion, as you examplify (the function is called wilcox.test(), but returns the Mann-Whitney statistic?? Yes, they are equivalent, but...)
In fact, if I use other software, I never get $W=0$, for your example, but instead $W=21$, as it should be (e.g. Minitab, this online calculator, or the Real Statistics Excel add-on package, all return $W=21$). And I get W=57 when I test B against A, again as it should be.