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:

\[ H = \left[ \frac{12}{N(N+1)} \sum_{i=1}^K \frac{R^2_i}{n_i} \right] - 3(N+1) \]

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.

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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\(\mathcal{H_0}\): the group rank means are equal, i.e., there is no association between site type and bee species richness.

\(\mathcal{H1}\): 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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\[ H = \left[ \frac{12}{N(N+1)} \sum_{i=1}^K \frac{R^2_i}{n_i} \right] - 3(N+1) \]

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

\[ H = 7.318 \]

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.