Kruskal-Wallis Test
The Kruskal-Wallis test is a non-parametric test that generalizes the Mann-Whitney U test. It tests the null hypothesis, which states that ‘k’ samples come from the same population with identical medians. In the Kruskal-Wallis test, the null hypothesis is S1 = S2 = … = Sk. Obviously, the alternative hypothesis would be that Si is not equal to Sj. This means that at least one pair of groups or samples has different pairs.
To apply the Kruskal-Wallis test, arrange the data in a two-way format, with each column representing a successive sample. In the computation, each of the ‘N’ observations replaces with ranks. All values from the ‘k’ samples combine and rank in a single series.
The smallest in the Kruskal-Wallis test is replaced by the rank 1. We replace the next smallest with rank 2 and the largest with rank ‘N,’ where ‘N’ represents the total number of observations in the ‘k’ samples. Next, we sum the ranks in each sample or column.
From the sum of the ranks, the researcher computes the average rank for each sample or group. If the samples are from an identical population then the average rank should be about the same. On the other hand, if the samples are from populations with different medians, then the average rank will differ.
The Kruskal-Wallis test compares the differences against average ranks to determine if samples likely come from the same population.
If the ‘k’ samples come from the same population, we can table the sampling distribution and probability.
In the Kruskal-Wallis test, with more than three groups and five or more observations, the distribution approximates chi-square. This approximation improves as the number of groups and observations in each group increase.
There are certain assumptions in the Kruskal-Wallis test.
- We assume that the observations in the data set are independent of each other.
- We assume that the population distribution does not need to be normal, and the variances do not need to be equal.
- We assume that the observations come from the population through random sampling.
The sample sizes in the Kruskal-Wallis test should be as equal as possible, but some differences are allowed. It also has one limitation. If the researcher does not find a significant difference in his data while conducting it, then he cannot say that the samples are the same.
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