Sums of Random Partitions of Ranks

Abstract

Suppose that the integers $1, 2, \cdots, N$ are randomly distributed among $k$ distinguishable classes with equal probability and without restrictions. It is natural to denote the class sums by $s_1, s_2, \cdots, s_k$ and largest of these by $S$. A generating function is obtained for the upper half of the range of $S$, namely $\frac{1}{4}N(N - 1) \leqq S \leqq \frac{1}{2} N(N - 1)$. For $k \leqq 6$, this is shown to provide the usual percentage points for $N$ up to and beyond 10. Tables of 5% and 1% points are provided for $k = 2, 3, 6$ and $N = 1(1)10$. For $k = 2$, the distribution is that of Wilcoxon's paired sample test [3]. This suggests the application of $k = 6$ to the six possible orders of three responses. This is a possible procedure but the peculiarities of its power are such that its use is not recommended. However, when three treatments with a natural order are examined in randomized blocks, a significance procedure can be based on the same distribution which is specifically sensitive to average responses in either exactly the same or exactly the opposite order as the treatments. 5% and 1% levels are given for $N = 1(1)10$. The procedure may be promising. The basic distribution used here is inappropriate for situations, such as analysis of variance of ranks, where the number of ranks in each class is restricted, as by being the same in all classes.