Kruskal-Wallis test

The Kruskal-Wallis-Test (after William Kruskal and Wilson Allen Wallis ; also H-Test) is a parameter-free statistical test with which an analysis of variance tests whether independent samples (groups or series of measurements) with regard to an ordinally scaled variable of a common population come from. It is similar to a Mann-Whitney U-Test and, like this, is based on rank sums, with the difference that it can be used to compare more than two groups. In the case of dependent samples, the Friedman test can be used instead.

The null hypothesis is that there is no difference between the groups. A so-called H-value is calculated as a test variable for the Kruskal-Wallis test. The H-value is formed as follows: The rank for each of the observations in the union of the samples is determined. This then becomes the sums of ranks for the individual groups and from these the test statistics ${\displaystyle R_{i}}$ ${\displaystyle n}$ ${\displaystyle S_{h}}$

${\displaystyle H={\tfrac {12}{n(n+1)}}\sum _{h}{\tfrac {S_{h}^{2}}{n_{h}}}-3(n+1)}$

or when there are ties

${\displaystyle H={\frac {{\tfrac {12}{n(n+1)}}\sum _{h}{\tfrac {S_{h}^{2}}{n_{h}}}-3(n+1)}{1-{\tfrac {1}{(n^{3}-n)}}\sum t_{r(i)}^{3}-t_{r(i)}}}}$

(with the number of bound observations with rank ) calculated. If the null hypothesis is valid, the test variable is asymptotic, i.e. H. for large sample sizes in all groups, chi-square distributed . The number of degrees of freedom (Df) is calculated according to Df = k-1, where k is the number of classes (groups). The calculated test variable H is compared with a theoretical variable from the chi-square distribution for an error probability selected a priori . If the calculated H value is greater than the H value from the chi-square table, the null hypothesis is rejected, so there is a significant difference between the groups. ${\displaystyle t_{r(i)}}$ ${\displaystyle i}$ ${\displaystyle H}$

If and , then the test statistic is not distributed and tabulated critical values ​​must be used. ${\displaystyle h=3}$ ${\displaystyle n_{h}<6}$ ${\displaystyle H}$ ${\displaystyle \chi ^{2}}$

A similar test to the Kruskal-Wallis test is the Jonckheere-Terpstra test or its generalization, the umbrella test according to Mack and Wolfe. The Scheirer-Ray-Hare test is an extension of the Kruskal-Wallis test to include the application of multi-factor analysis of variance .

Since the H-test only makes a statement about the differences between all samples considered, it makes sense to carry out a post-hoc test that compares the individual samples in pairs. Here, for example, the Bonferroni method is ideal .

Individual evidence

1. ^ WH Kruskal, WA Wallis: Use of ranks in one-criterion variance analysis . In: Journal of the American Statistical Association , 47 (160), 1952, pp. 583-621, doi: 10.1080 / 01621459.1952.10483441 , JSTOR 2280779 .
2. ^ Douglas C. Montgomery: Design and Analysis of Experiments . John Wiley & Sons, Danvers 2005, ISBN 0-471-48735-X , pp. 110-111
3. ^ HB Mack, DA Wolfe: K-sample rank tests for umbrella alternatives. In: Journal of the American Statistical Association , 76 (373), 1981. pp. 175-181, doi: 10.1080 / 01621459.1981.10477625 , JSTOR 2287064
4. ↑ James Scheirer, William S. Ray, Nathan Hare: The Analysis of Ranked Data Derived from Completely Randomized Factorial Designs. In: Biometrics. 32 (2), 1976, pp. 429-434, JSTOR 2529511
5. ^ H Abdi: Bonferroni and Sidak corrections for multiple comparisons . In: NJ Salkind (ed.) (Ed.): Encyclopedia of Measurement and Statistics (PDF), Sage, Thousand Oaks CA 2007.