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Basics
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the deviations from the central point of the remaining scores are ranked such that the
smallest deviation has a rank of 1. Tied scores are assigned for a mean rank. The
sums for the ranks of scores with positive and negative deviations from the central
point are then calculated separately. A value S is defined as the smaller of these two
rank sums. S is then compared to a table of all possible distributions of ranks to
calculate p, the statistical probability of attaining S from a population of scores that is
symmetrically distributed around the central point. As the number of used scores, n,
increases, the distribution of all possible ranks S tends towards the z-distribution, so for
an n of greater than 10 this distribution is used to calculate p. This test assumes that
the compared sample sets originate at least from a common distribution.
For details, please refer to F. Wilcoxon, "Individual Comparisons by Ranking Methods",
Biometrics 1, pp 80-83 (1945).
6.4.1.4
Kruskal-Wallis Test
In statistics, the Kruskal-Wallis one-way analysis of variance by ranks is a non-parametric method. Unlike the analogous one-way analysis of variance, the Kruskal-Wallis test
does not assume a normal population. This, like many non-parametric tests, uses the
ranks of the data rather than their raw values to calculate the statistic. Since this test
does not make a distributional assumption, it is not as powerful as the ANOVA test.
The hypotheses for the comparison of two independent groups are:
• H0 (null hypothesis): The samples come from identical populations
• Ha (alternative hypothesis): The samples come from different populations
Notice that the hypothesis makes no assumptions about the distribution of the populations. These hypotheses are also sometimes written as testing the equality of the central tendency of the populations.
The test statistic for the Kruskal-Wallis test is H. This value is compared to a table of
critical values for U based on the sample size of each group. If H exceeds the critical
value for H at some significance level (usually 0.05) it means that there is evidence to
reject the null hypothesis in favor of the alternative hypothesis.
For details, please refer to W. H. Kruskal and W. A. Wallis, "Use of ranks in onecriterion variance analysis", Journal of the American Statistical Association 47 (260): pp
583-621 (1952).
6.4.1.5
Anderson-Darling Test
The Anderson-Darling test (AD test; Stephens, 1974) is used to test if a sample of data
comes from a population with a specific distribution. It is a modification of the Kolmogorov-Smirnov (KS) test and gives more weight to the tails than does the KS test. The
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ClinProTools User Manual, Version 2.2