Definition

The Kruskal-Wallis test is the non-parametric counterpart to the one-way ANOVA F-test of multiple Means. It is used to compare the Medians of continuous Populations that are assumed identical in every way except possibly for the Medians. That is, this test still makes the assumptions of independence within and among Samples and of equal Variances. Unlike the Anova test however, it can be used even if the data are only available in the form of ranks.

Examples

The following three samples are from independent, non-normal populations:

A: 43, 24, 32, 29

B: 31, 37, 20, 12

C: 33, 19, 36, 25

Are the population medians for the three groups the same? To test this, rank the pooled data from all the samples while keeping track of the sample IDs and calculate the sum of the ranks (R_{i}) for each sample:

A: 12, 4, 8, 6

B: 7, 11, 3, 1

C: 9, 2, 10, 5

Sum of the ranks for each sample: R_{A} = 32, R_{B} = 18 and R_{C} = 28.

The test statistic is calculated as follows:

H = [12/N(N+1)]*[SUM(R_{i}^{2}/n_{i}] - 3*(N+1)

where

n_{i} = i^{th} sample size and N = Sum(n_{i}) = size of the pooled sample. Thus,

H = [12/(12*13)]*[(30+22+26)/4] - (3*13) = 2.

H follows a Chi-square distribution with (k-1) degrees of freedom for large samples, where k = the number of samples. The p-value = 0.3679 indicates that there isn't enough evidence to reject the null hypothesis of no difference at the 10% level.

Application

Use this test when sample distributions do not satisfy the assumption of normality.