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.

33. Kruskal-Wallis and Pairwise Wilcoxon Tests¶

What this covers¶

This chapter works through a single lab exercise: re-analysing a dataset on lettuce growth, previously analysed with a one-way ANOVA, using the Kruskal-Wallis test, followed by a post-hoc pairwise comparison. It assumes you already know what the Kruskal-Wallis test is and why it is the rank-based counterpart to one-way ANOVA, what the Wilcoxon rank-sum test and its "probabilistic index" interpretation are, and how to correct a family of p-values for multiple testing (e.g. with the Bonferroni method) — none of that machinery is derived here, only applied. It also assumes the earlier ANOVA analysis of the same dataset, referred to in the source material as "chapter 7".

The lettuce experiment¶

A greenhouse experiment grew lettuce plants in four soil conditions:

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

Seven plants were grown per condition, 28 in total, and the response recorded is each plant's fresh weight in grams. The dataset (freshweight_lettuce.txt) has one row per plant and is read directly from the course's repository. The original question — does any treatment change lettuce growth, and if so, which one — was addressed in an earlier tutorial with a one-way ANOVA followed by a Tukey post-hoc test.

Why redo it without ANOVA¶

A one-way ANOVA rests on its residuals being approximately normal with equal variance across the four groups. With only seven values per group there is very little basis for judging whether either assumption actually holds: a departure from normality in a sample of seven is nearly impossible to tell apart from ordinary sampling variation. The Kruskal-Wallis test sidesteps this — it works on the ranks of the freshweights rather than the raw values, and is the standard alternative to one-way ANOVA when the ANOVA assumptions cannot be established.

Structure of the analysis¶

The exercise follows the same shape as the earlier ANOVA one:

import the data and inspect it (glimpse),
tidy it — in particular, make sure treatment is stored as a factor with the right four levels,
explore it — count the observations per treatment and plot freshweight by treatment as a boxplot,
state the hypotheses the Kruskal-Wallis test is testing here,
run the test and interpret the result.

Post-hoc: which treatments differ¶

A significant Kruskal-Wallis result only says that the four treatments are not all identical; it does not say which ones differ. The follow-up here is a pairwise Wilcoxon rank-sum test, run once for each of the six pairs of treatments. Because six tests are being read together rather than one, the resulting p-values need to be corrected for multiple testing (Bonferroni, via p.adjust()) before they are interpreted.

One way to get there, and the way the tutorial's own code builds it, is to enumerate the pairs directly:

treatments <- levels(lettuce$treatment)
freshweight <- lettuce$freshweight

pvalues <- combn(treatments, 2, function(x){
  test <- wilcox_test(freshweight ~ treatment,
                       subset(lettuce, treatment %in% x),
                       distribution = 'exact')
  pvalue(test)
})

pvalues_bonf <- p.adjust(pvalues, method = 'bonferroni')
names(pvalues_bonf) <- combn(levels(lettuce$treatment), 2, paste, collapse = "_VS_")
pvalues_bonf

combn(treatments, 2, ...) walks through all \(\binom{4}{2} = 6\) unordered pairs of treatments, runs an exact Wilcoxon test on each pair's subset of the data, and collects the six p-values before they are Bonferroni-adjusted and labelled by which two treatments they compare.

As the analysis does not assume a location shift between two treatments, the tutorial has us read the result as a probabilistic index instead, rather than as a difference in location.

Exercises¶

These are the tasks the tutorial sets, in order; none is solved here.

Data preparation. Load tidyverse, read freshweight_lettuce.txt into lettuce, and check its structure with glimpse(). Convert the treatment column to a factor. Count how many observations there are per treatment, and produce a boxplot of freshweight by treatment. What does the boxplot suggest about the four treatments?
Kruskal-Wallis test. State a correct null and alternative hypothesis for the Kruskal-Wallis test on this dataset. Run the test (e.g. with kruskal_test()) and interpret the result.
Post-hoc hypotheses. State a correct null and alternative hypothesis for the pairwise Wilcoxon post-hoc analysis.
Post-hoc test. Run the pairwise Wilcoxon rank-sum test across the four treatments (pairwise.wilcox.test(), or the pair-by-pair construction shown above), correcting the resulting p-values for multiple testing. What do you observe?
Effect size. From the analysis above, extract the point estimates of the probabilistic index for each pair of treatments, and interpret them.
Conclusion. Write a conclusion that answers the researchers' original question: does biochar, compost, or their combination affect lettuce growth, and if so, how?

Sources¶

Both sections are drawn from the converted tutorial script Kruskal_Wallis_lettuce_plants_half.Rmd (GTPB PSLS20, licensed CC BY 4.0):
Background, dataset description and the Kruskal-Wallis exercise: 01-introduction.md
Post-hoc pairwise Wilcoxon exercise: 02-post-hoc-analysis.md
No slides or lecture transcript were supplied for this chapter.
The source refers to an earlier ANOVA analysis of the same dataset ("chapter 7", ANOVA_lettuce_plants_half.rmd), which was not among the supplied inputs.

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