The Wilcoxon, Ties, and the Computer

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Abstract
In testing for equality of distribution in two unpaired samples with ordered categorical data or data with many ties, the two sample Wilcoxon average rank test is attractive. Unfortunately, the null distribution of the statistic depends upon the pattern of ties and is generally difficult to compute. In an attempt to improve existing approximations, moments through order four are derived. It is regretted that in small to moderate sample sizes, no consistent improvement over the simple normal approximation was found. The erratic nature of both the values of the statistic and the jumps of the distribution appears to rule out any good, smooth approximation in small samples. For calculation of the exact distribution, an algorithm is given to generate restricted compositions efficiently on a computer. The algorithm provides a useful means of obtaining exact probabilities in many cases.

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