16.6. 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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$$ H = \left[ \frac{12}{N(N+1)} \sum_{i=1}^K \frac{R^2_i}{n_i} \right] - 3(N+1) $$
$$ H = \left[ \left[ \frac{12}{10(11)} \right] \times \left[ \frac{6^2}{3} + \frac{33^2}{4} + \frac{16^2}{3} \right] \right] - \left[ 3 \times 11\right] $$
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.