The Kruskal-Wallis test

One-way analysis of variance test is based on an assumption that the underlying population has a normal distribution. If the population distribution is not normal, it is possible to develop a nonparametric alternative to the one-way analysis of variance. This nonparametric test is known as the Kruskal-Wallis test. Like the most of nonparametric the Kruskal-Wallis test is based on the ranks of the sample observations.

Suppose that we have independent random samples of sizes observations, selected from K populations. Let

Denote the total number of observations. The null hypothesis is

To apply Kruskal-Wallis test it is necessary to take following steps:

1. Pool together all sample observations.

2. Rank all of pooled sample observations in ascending order.

3. Denote by the sums of ranks for the K samples

4. Calculate the value of the test statistic

A test of significance level is given by the decision rule

Reject if

where is the number that is exceed with probability by a random variable with degrees of freedom.

Example:

The following table gives the response time (in minutes) of three fire companies in a city for certain randomly selected incidents after a fire was reported.

Company A	Company B	Company C
1.6 0.8 2.7 1.2 3.4 1.9 4.3	1.4 2.6 0.9 3.5 1.2 1.5	0.8 1.3 1.7 0.9 1.1 0.7 2.1

Perform at the 5% significance level the Kruskal-Wallis test to test the null hypothesis that the mean response time for each of these fire companies for all fire incidents are the same.

Solution:

First of all we pool all sample observation together and rank them in ascending order. The following table illustrates this procedure

Company A	Rank	Company B	Rank	Company C	Rank
1.5 0.8 2.7 1.2 3.4 1.9 4.3	2.5 7.5	1.4 2.6 0.9 3.5 1.2 1.6	4.5 7.5	0.8 1.3 1.7 0.9 1.1 0.7 2.1	2.5 4.5
Rank sums

The null hypothesis is

The decision rule is

Reject if

The value of the test statistic is

Since, 3.35 is not greater than 5.99, we fail to reject the null hypothesis. And we accept that the mean response time for each of these fire companies for all fire incidents is the same.

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