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ANOVA - KRUSKAL-WALLIS
Kruskal–Wallis is defined as the one-way analysis of variance by ranks which is a non-parametric statistics that is used for testing whether samples originate from the same distribution. It is used for comparison of more than two samples which are independent, or that are not related and the parametric equivalent of the Kruskal-Wallis test is the one-way analysis of variance. When the Kruskal-Wallis test leads to significant results, then at least one of the samples is diverse from the other samples and the test does not identify where the differences occur or how many differences actually occur. It is an expansion of the Mann–Whitney U test to 3 or more groups where the Mann-Whitney would help analyze the specific sample pairs for significant differences.
Since this is not a parametric method, the Kruskal–Wallis test does not assume a normal distribution unlike the analogous one-way analysis of variance and the test does assume an identically shaped and scaled distribution for each group, except for any difference in median.
Kruskal–Wall is is also used when the examined groups are of unequal size with different number of participants. It rank all data from all groups together and so it rank the data from 1 to N ignoring group membership and assign any tied values the average of the ranks they would have received had they not been tied and the test statistic is given by:
Where is the number of observations in group , is the rank (among all observations) of observation from group , is the total number of observations across all groups and
is the average of all the.
It is a correction for ties if using the short-cut formula which is described can be made by dividing K where G is the number of groupings of different tied ranks.
If the statistic is not significant then there is no evidence of differences between the samples and if the test is significant then a difference exists between at least two of the samples. Therefore, a researcher use sample that is in contrast between individual sample pairs, or post hoc tests. This is to determine which of the sample pairs are significantly different and when performing multiple sample contrasts the Type I error rate tends to become exaggerated.
