Distribution-Free Multiple Comparison

The analysis of rank data, what is generally called nonparametric statistical analysis, is an exact parallel of the more traditional (and familiar) parametric methods. There are methods for the single comparison case (just as Student s t-test is used) and for the multiple comparison case (just as analysis of variance is used) with appropriate post hoc tests for exact identification of the significance with a set of groups. Four tests are presented for evaluating statistical significance in rank data the Wilcoxon Rank Sum Test, distribution-free multiple comparisons, Mann-Whitney U Test, and the Kruskall-Wallis nonparametric analysis of variance. For each of these tests, tables of distribution values for the evaluations of results can be found in any of a number of reference volumes (Gad, 1998). 

The distribution-free multiple comparison test should be used to compare three or more groups of nonparametric data. These groups are then analyzed two at a time for any significant differences (Hollander and Wolfe, 1973, pp. 124-129). The test can be used for data similar to those compared by the rank-sum test. We often employ this test for reproduction and mutagenicity studies (such as comparing survival rates of offspring of rats fed various amounts of test materials in the diet). 

When we find a significant difference, we do not know which groups are different. It is not correct to then perform a Mann-Whitney U Test on all possible combinations rather, a multiple comparison method must be used, such as the distribution-free multiple comparisons. 

Pairwise comparisons among 80 adult E. rufescens using a distribution-free multiple comparison test based upon the Kruskal-Wallis ranked sums test (Hollander Wolfe, 1973) indicate that the active surface area of paired males is significantly larger than that of single males and those of both paired and single females (P < 0.05 for each comparison)... 

Multiple studies have reported a population of shorter grafted polymer chains with broader distribution in comparison with free/solution polymer chains [88, 89, 113]. Gorman et al. conducted SI-ATRP from a silicon wafer, porous silicon, and anodically etched aluminum oxide. The results were compared with those of ATRP conducted in solution under similar conditions. The grafted polymer chains from a flat substrate were shown to be shorter than the free chains formed in the parallel solution polymerization. The polymers obtained from concave substrates have an even lower molecular weight and broader distribution. 

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