Mann-Whitney test statistic?

Question

For a Mann-Whitney test do we use the T value from the smallest sample size as the test statistic or the smallest T value? I presume it is the former since this is how it is with the Wilcoxon rank sum test.

Answer 1

1) There are two common definitions of the Wilcoxon rank-sum statistic, that have been around since the start (each appears in one of the first two papers that relate to the Wilcoxon tests).

One of those is the sum of the ranks in the smallest sample, which sounds like the one you're used to.

The Wilcoxon rank-sum statistic and the Mann-Whitney U statistic are - up to a simple location shift that depends only on the two sample sizes - identical. Since they're actually the same test, they're often referred to as Wilcoxon-Mann-Whitney (or Mann-Whitney-Wilcoxon).

2) It's not clear what you mean by T-value. If you're just referring to a test statistic based on the sum of ranks in one sample or the other, if you do it right it will make no difference which you choose. On the other hand, maybe you're referring to actually calculating a t-statistic on the mean ranks. The Wilcoxon-Mann-Whitney can be performed by computing t-statistic on the ranks, but normally isn't. When you compute such a t-statistic, it's monotonic in the usual W or U statistic, so its permutation distribution yields exactly the same p-values as the normal WMW.

The upshot of all this is, since it's easiest to just do the "sum of the ranks in the smallest group" calculation when working by hand, that's probably what you should do, but either you need to be working with tables that work that way or you need to then convert the statistic to the way your tables work. This is pretty straightforward.