Lecture notes. Written from the material of GTPB Psls20, licensed CC BY 4.0. These are notes, not a transcript: the material has been reorganised and rewritten. This adaptation carries the same licence, and the original is linked above.

16. Kruskal-Wallis Test for Groups¶

What this covers¶

This chapter sets up a practical exercise on comparing more than two groups when the assumptions behind ANOVA cannot be relied on. It assumes the reader already has one-way ANOVA and the idea of a hypothesis test in hand, and is meeting the rank-based alternative — the Kruskal-Wallis test — as something to apply to a real dataset rather than to derive. The source material for this chapter is the exercise brief itself; the worked solution and the test's mechanics live in a linked notebook that was not supplied (see Sources).

The setting: does fertilizer type affect lettuce growth?¶

The exercise is framed around a question from agriculture: growers want a high yield of crops, and one route to more leaf weight is fertilizer. The study compares three fertilizer-related treatments:

biochar
compost
cobc — a combination of biochar and compost

To test whether any of these help, researchers grew lettuce plants in a greenhouse, with pots filled with one of four soil types:

soil only (control)
soil supplemented with biochar (refoak)
soil supplemented with compost (compost)
soil supplemented with both biochar and compost (cobc)

This gives a one-way, four-group design: a single factor (soil treatment) with four levels, and the response is the freshweight of the lettuce plant. The dataset, freshweight_lettuce.txt, records the freshweight of 28 lettuce plants across the four groups.

Why Kruskal-Wallis rather than ANOVA¶

The exercise is explicitly placed under "non-parametric, multigroup analysis." A one-way ANOVA answers the same kind of question — does the group mean differ across more than two groups? — but it leans on assumptions (normally distributed residuals, equal variances) that a small dataset like this one may not support. The Kruskal-Wallis test is the rank-based analogue: it asks whether the four groups' distributions differ, using the ranks of the freshweight values rather than the raw measurements, and so does not require normality. The exercise gives the practical motivation for reaching for it — a four-group comparison on a modest sample — without walking through the test statistic's construction; that derivation is not part of the supplied material.

The research question¶

The concrete question the researchers want answered is whether one or more of the treatments (refoak, compost, cobc) shifts lettuce growth relative to control — i.e., whether soil treatment has any effect at all on freshweight, before asking which treatment is responsible.

Exercises¶

Working from the freshweight_lettuce.txt dataset (28 lettuce plants, one of four soil treatments each: control, refoak, compost, cobc):

State the null and alternative hypotheses for testing whether soil treatment affects lettuce freshweight.
Explain why a rank-based test such as Kruskal-Wallis is a reasonable choice for this design and sample size, rather than a one-way ANOVA.
Carry out the Kruskal-Wallis test on the four groups and state your conclusion about whether the treatments affect growth.

Sources¶

Notes: docs/omics-statistics/gtpb/psls20/tutorialScripts/excercises/09_kruskalWallis/09_kruskalWallis.md (GTPB "Practical Statistics for the Life Sciences" 2020, day 3, non-parametric multigroup exercise).
Referred to but not supplied as input: the worked exercise notebook Kruskal_Wallis_lettuce_plants_half.html and the data file freshweight_lettuce.txt, both linked from the notes but not included in the material given for this chapter.

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