• Introduction
• Data
• Kruskal-Wallis test
  • Aim and hypotheses
  • Assumptions
  • In R
  • Interpretations
• Post-hoc tests
  • Dunn test
  • Pairwise Wilcoxon test
• Combination of statistical results and plot
• Summary
• References

Introduction

In a previous article, we showed how to do an ANOVA in R to compare three or more groups.

Remember that, as for many statistical tests, the one-way ANOVA requires that some assumptions are satisfied in order to be able to use and interpret the results. In particular, the ANOVA requires that residuals follow approximately a normal distribution.¹

Luckily, if the normality assumption is not satisfied, there is the nonparametric version of the ANOVA: the Kruskal-Wallis test.

In the rest of the article, we show how to perform the Kruskal-Wallis test in R and how to interpret its results. We will also briefly show how to do post-hoc tests and how to present all necessary statistical results directly on a plot.

Data

Data for the present article is based on the penguins dataset (an alternative to the well-known iris dataset), accessible via the {palmerpenguins} package:

# install.packages("palmerpenguins")
library(palmerpenguins)

The original dataset contains data for 344 penguins of 3 different species (Adelie, Chinstrap and Gentoo). It contains 8 variables, but we focus only on the flipper length and the species for this article, so we keep only those 2 variables:

library(tidyverse)

dat <- penguins %>%
  select(species, flipper_length_mm)

(If you are unfamiliar with the pipe operator (%>%), you can also select variables with penguins[, c("species", "flipper_length_mm")]. Learn more ways to select variables in the article about data manipulation.)

It is always a good practice to do some descriptive statistics for the entire sample and by group before doing the test, so we have a broad overview of the data at hand.

# entire sample
summary(dat)
##       species    flipper_length_mm
##  Adelie   :152   Min.   :172.0    
##  Chinstrap: 68   1st Qu.:190.0    
##  Gentoo   :124   Median :197.0    
##                  Mean   :200.9    
##                  3rd Qu.:213.0    
##                  Max.   :231.0    
##                  NA's   :2
# by group
library(doBy)
summaryBy(flipper_length_mm ~ species,
  data = dat,
  FUN = median,
  na.rm = TRUE
)
## # A tibble: 3 × 2
##   species   flipper_length_mm.median
##   <fct>                        <dbl>
## 1 Adelie                         190
## 2 Chinstrap                      196
## 3 Gentoo                         216
# boxplot by species
ggplot(dat) +
  aes(x = species, y = flipper_length_mm, fill = species) +
  geom_boxplot() +
  theme(legend.position = "none")

Based on the boxplots and the summary statistics, we already see that, in our sample, penguins from the Adelie species seem to have the smallest flippers, while those from the Gentoo species seem to have the biggest flippers. However, only a sound statistical test will tell us whether we can infer this conclusion to our population.

Kruskal-Wallis test

kruskal.test(flipper_length_mm ~ species,
  data = dat
)
## 
##  Kruskal-Wallis rank sum test
## 
## data:  flipper_length_mm by species
## Kruskal-Wallis chi-squared = 244.89, df = 2, p-value < 2.2e-16

Post-hoc tests

We have just showed that at least one species is different from the others in terms of flippers length. Nonetheless, here comes the limitations of the Kruskal-Wallis test: it does not say which group(s) is(are) different from the others.

To know this, we need to use other types of test, referred as post-hoc tests (in Latin, “after this,” so after obtaining statistically significant Kruskal-Wallis results) or multiple pairwise-comparison tests. For the interested reader, a more detailed explanation of post-hoc tests can be found here.

The most common post-hoc tests after a significant Kruskal-Wallis test are:

• Dunn test
• Pairwise Wilcoxont test
• Conover test
• Nemenyi test

The Dunn test and pairwise Wilcoxon test being the most common ones among the four, here is how to do them in R.⁴

library(FSA)

dunnTest(flipper_length_mm ~ species,
  data = dat,
  method = "holm"
)
##           Comparison          Z      P.unadj        P.adj
## 1 Adelie - Chinstrap  -3.629336 2.841509e-04 2.841509e-04
## 2    Adelie - Gentoo -15.476612 4.990733e-54 1.497220e-53
## 3 Chinstrap - Gentoo  -8.931938 4.186100e-19 8.372200e-19

pairwise.wilcox.test(dat$flipper_length_mm, dat$species,
  p.adjust.method = "holm"
)
## 
##  Pairwise comparisons using Wilcoxon rank sum test with continuity correction 
## 
## data:  dat$flipper_length_mm and dat$species 
## 
##           Adelie Chinstrap
## Chinstrap 3e-08  -        
## Gentoo    <2e-16 <2e-16   
## 
## P value adjustment method: holm

Combination of statistical results and plot

A very good alternative for performing a Kruskal-Wallis and the post-hoc tests in R is with the ggbetweenstats() function from the {ggstatsplot} package:

library(ggstatsplot)

ggbetweenstats(
  data = dat,
  x = species,
  y = flipper_length_mm,
  type = "nonparametric", # ANOVA or Kruskal-Wallis
  plot.type = "box",
  pairwise.comparisons = TRUE,
  pairwise.display = "significant",
  centrality.plotting = FALSE,
  bf.message = FALSE
)

This method has the advantage that all necessary statistical results are displayed directly on the plot.

The results of the Kruskal-Wallis test are shown in the subtitle above the plot (the p-value is after p =). Moreover, the results of the post-hoc test are displayed between each group via accolades, and the boxplots allow to visualize the distribution for each species.

Summary

In this post, we reviewed the aim and hypotheses of the Kruskal-Wallis test and its underlying assumptions. We then showed how to do the test in R and how to interpret the results.

We also showed the most common post-hoc tests after a significant Kruskal-Wallis test—the Dunn and pairwise Wilcoxon tests.

Last but not least, we presented a concise way to present both the data by group and all the statistical results on the same plot.

Thanks for reading.

As always, if you have a question or a suggestion related to the topic covered in this article, please add it as a comment so other readers can benefit from the discussion.

References

1. This is the case for small sample sizes. For large sample sizes, normality is not required (this is a common misconception!). By the central limit theorem, sample means of large samples are often well-approximated by a normal distribution even if the data are not normally distributed (Stevens 2013). See more details in this note.↩︎

2. Note that in theory, Kruskal-Wallis test can also be used for only two groups. However, in practice we use the Mann-Whitney test for two groups and Kruskal-Wallis for three or more groups.↩︎

3. See more information about the difference in this article.↩︎

4. Note that there are other p-value adjustment methods. See ?dunnTest and ?pairwise.wilcox.test for more options.↩︎