  Finkelstein-Schoenfeld, Buyse, Pocock, and other authors have developed
generalizations of the Mann-Whitney test that allow for pairwise patient
comparisons to include a hierarchy of measurements. Various authors present
either asymptotic or randomized methods for analyzing the wins. We use graph
theory concepts to derive exact means and variances for the number of wins, as
a replacement for approximate values obtained from bootstrap analysis or random
sampling from the permutation distribution. The time complexity of our
algorithm is $O(N^2)$, where $N$ is the total number of patients. In any
situation where the mean and variance of a bootstrap sample are used to draw
conclusions, our methodology will be faster and more accurate than the
randomized bootstrap or permutation test.
