Kruskall-Wallis test

4.6. Kruskall-Wallis test#

When the sample size is small, or when the data are not normally distributed, it is preferable to use the Kruskal-Wallis test which is the non-parametric equivalent of ANOVA. The equation for the K-W test is:

H = [\frac{12}{N (N + 1)} \sum_{i = 1}^{K} \frac{R_{i}^{2}}{n_{i}}] - 3 (N + 1)

Where the test statistic, $H$ , has a chi-squared distribution.

In this equation, $K$ references the number of groups, $n_{i}$ indicates the number of observations per group, and $R_{i}$ is the sum of the ranks of observations in the group.

https://raw.githubusercontent.com/jillxoreilly/StatsCourseBook/main/images/regression4_bee.jpg

Longhand calculation example: A research paper published in the Journal of Insect Conservation in 2019 examined the attractiveness of different wasteland sites to bees. The table below shows a record of the richness of bee species in 10 wasteland sites, classified by the former land use into extractive industry, suburban/ residential and chemical industry.

Bee Species Richness:

	Ob1	Ob2	Ob3	Ob4
Chemical industry	33	27	40
Extractive industry	95	117	105	71
Suburban/residential	55	73	67

Test whether species richness differs by type of site using the Kruskal-Wallis test.

State your hypotheses.

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$H_{0}$ : the group rank means are equal, i.e., there is no association between site type and bee species richness.

$H 1$ : at least two of the rank means are unequal, i.e., there is an association between site type and bee species richness.

What are the rank sums for each group?

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	n	Rank Sum
Chemical industry	3	6
Extractive industry	4	33
Suburban/residential	3	16

Practice your ability to work with equations. Plug the values in and calculate H.

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H = [\frac{12}{N (N + 1)} \sum_{i = 1}^{K} \frac{R_{i}^{2}}{n_{i}}] - 3 (N + 1)

H = [[\frac{12}{10 (11)}] \times [\frac{6^{2}}{3} + \frac{33^{2}}{4} + \frac{16^{2}}{3}]] - [3 \times 11]

H = 7.318

With 2 df, the critical value of $H$ = 5.99 (Look up here again).

As 7.318 > 5.99, we reject the null hypothesis and conclude that there is a difference between groups.