Exact Distributions of Multiple Comparisons Rank Statistics

Abstract

Computation of exact p values of multiple comparisons rank statistics is often a very time-consuming task. In some cases approximations are available, but they are often unsatisfactory when samples are small or the number of ties is large. Some existing tables of exact critical values contain errors and are limited to small samples without ties and to Wilcoxon scores. Therefore, this article proposes a new algorithm to compute exact p values of generalized Steel statistics for one-way classification. This recursive algorithm builds up a generating function that represents the null distribution of the statistic. Several techniques are used to reduce computing time considerably. An improvement of an existing algorithm is given for computing exact p values, which is valid for a smaller class of statistics than the recursion. However, for this class of statistics, this algorithm outperforms the recursion. How to compute tight bounds on the p value when exact computation is too time-consuming is discussed. Also discussed is when to use which method (exact, bounds, normal approximation, or simulation) for computation of p values or critical values.