Kruskal-Wallis Test

Concept and Example

The idea of the test is to bring all observations of all k samples into a rank order and to assign them an according rank. After this initial procedure, all further calculations are based only on these ranks but not on the original observations anymore. The underlying concept of the test is that these ranks should be equally distributed throughout the k samples, if all observations are from the same population. A simple example is used to demonstrate this.

A researcher made measurements on k = 3 different groups. Overall there are N = 15 observations. Data are arranged according to their group, and an individual rank is assigned to each observation starting with 1 for the smallest observation (see Table 2).

Two things might be noted here. First, in this example, for the sake of simplicity, all groups have the same number of observations; however, this is not a necessary condition. Second, there are several observations with the same value called tie. In this case, all observations sharing the same values are assigned their mean rank. In the current example, two observations resulted in a value of 1280 sharing ranks 4 and 5. Thus, both receive rank 4.5. Furthermore, three samples had a value of 1310 and would have received ranks from 6 to 8. Now they get rank 7 as the mean of 6, 7, and 8.

In the next step, the sum of ranks (R1,R2,R3) for each group is calculated. The overall sum of ranks is N(N + 1)/2. In the example case, this is 15 x 16 / 2 = 120. As a first control, the sum of ranks for all groups should add up to the same value: R1 + R2 + R3 = 59 + 29:5 + 31:5 = 120.

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