Why are there two forms for the Mann-Whitney U test statistic? I have encountered two forms for calculating the two-sample Mann-Whitney U test statistic, which are:
$$U_1 = R_1 - \frac{n_1(n_1 + 1)}{2}$$
and
$$U_1 = n_1n_2 + \frac{n_1(n_1+1)}{2} - R_1$$
where $n_1$ is the sample size of group 1, $n_2$ is the sample size of group 2, and $R_1$ is the sum of ranks for group 1.
Why are there two forms for the U test statistic? Is this the case where the first equation is actually the Wilcoxon $W$ statistic, which I understand to be functionally equivalent to $U$ (although not numerically equivalent)? I am a biochemist by training, so I apologize for any incorrect statements or assumptions in my question.
 A: There are actually more than two forms of the Mann-Whitney-Wilcoxon test.
Given no ties (which I will assume throughout), the two forms you have there correspond to 
(i) the number of times an observation in sample 1 exceeds an observation from sample 2, and
(ii) the number of times an observation in sample 2 exceeds an observation from sample 1.
We'd do best to distinguish those two definitions. Let's call them $U_{1>2}$ and $U_{2>1}$.
Note that $U_{1>2}+U_{2>1} = n_1 n_2$, the number of pairwise comparisons between sample 1 and sample 2.
$R_1$ is the sum of ranks in sample 1, one of the two common forms most associated with Wilcoxon (mentioned in the original paper) -- sometimes called W or occasionally U or T.
The other form associated with Wilcoxon (in the first tables of the statistic, published shortly after) is $W=R_1- \frac{n_1(n_1 + 1)}{2}$, the sum of ranks in sample 1 minus the smallest possible value for that sum. This form is equivalent to what I called $U_{1>2}$.
(More forms still are possible.)
These forms are all linearly related. As a result they yield equivalent tests (they should reject or fail to reject the null for the same samples under the same conditions).
