Kruskal Wallis test in R, Kruskal Wallis test is one of the frequently used methods in nonparametric statistics for analyzing data in one-way classification.

It is equivalent to a one-way analysis of variance in parametric methods.

When we test the identicalness of the k population from which the independent samples have been drawn. There is no restriction of sample sizes.

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## Assumptions

Mainly Kruskal Wallis test is based on the following assumptions.

- The observations are independent within and between samples.
- The variable under study is continuous
- The populations are identical in respect to the median.

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## Hypothesis

Ho: All the populations are identical

H1: At least one pair of the population do not have the same median.

The test statistic is approximately distributed as chi-square with (k-1) degrees of freedom. , subject to the condition n should be large or at least n should not be less than 5.

## Kruskal Wallis test in R

### Load Package

library(tidyverse) library(ggpubr) library(rstatix)

### Getting Data

set.seed(345) PlantGrowth %>% sample_n_by(group, size = 1)

Output:-

weight group

1 5.18 ctrl

2 4.41 trt1

3 5.26 trt2

Ordering the group is really important when you are doing Duncan’s multiple comparison tests.

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PlantGrowth <- PlantGrowth %>% reorder_levels(group, order = c("ctrl", "trt1", "trt2"))

### Summary

PlantGrowth %>% group_by(group) %>% get_summary_stats(weight, type = "common")

Output:-

group variable n min max median iqr mean sd se ci 1 ctrl weight 10 4.17 6.11 5.155 0.743 5.032 0.583 0.184 0.417 2 trt1 weight 10 3.59 6.03 4.550 0.662 4.661 0.794 0.251 0.568 3 trt2 weight 10 4.92 6.31 5.435 0.467 5.526 0.443 0.140 0.317

### Visualization

ggboxplot(PlantGrowth, x = "group", y = "weight", fill="group")

Based on the box plot, it evident that some difference exist between treatment 1 and treatment 2.

### Kruskal Wallis Test

res.kruskal <- PlantGrowth %>% kruskal_test(weight ~ group) res.kruskal

Output:-

.y. n statistic df p method 1 weight 30 7.988229 2 0.0184 Kruskal-Wallis

Based on the p-value significant difference was observed between the group pairs.

### Effect size

The effect size values normally interpreted as 0.01- < 0.06 (small effect), 0.06 – < 0.14 (moderate effect) and >= 0.14 (large effect).

PlantGrowth %>% kruskal_effsize(weight ~ group)

.y. n effsize method magnitude 1 weight 30 0.2217862 eta2[H] large

If effect size is large easily we can identify the significant differences based on small number of sample sizee.

### Pairwise comparisons

Based on the Kruskal Wallis test we identified a significant difference, but we don’t which pair is significantly different. A pairwise comparison will help us to identify the significant pair.

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res1<- PlantGrowth %>% dunn_test(weight ~ group, p.adjust.method = "bonferroni") res1

Output:-

.y. group1 group2 n1 n2 statistic p p.adj p.adj.signif 1 weight ctrl trt1 10 10 -1.117725 0.26368427 0.79105280 ns 2 weight ctrl trt2 10 10 1.689290 0.09116394 0.27349183 ns 3 weight trt1 trt2 10 10 2.807015 0.00500029 0.01500087 *

Based on the pairwise comparison significant difference was observed between Treatment and Traetment2.

res2 <- PlantGrowth %>%

wilcox_test(weight ~ group, p.adjust.method = "bonferroni")

res2

.y. group1 group2 n1 n2 statistic p p.adj p.adj.signif 1 weight ctrl trt1 10 10 67.5 0.199 0.597 ns 2 weight ctrl trt2 10 10 25.0 0.063 0.189 ns 3 weight trt1 trt2 10 10 16.0 0.009 0.027 *

Based on Wilcoxon test also significant difference was observed between treatment 1 and treatment 2.

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### Visualization with p-values

res1 <- res1 %>% add_xy_position(x = "group") ggboxplot(PlantGrowth, x = "group", y = "weight") + stat_pvalue_manual(res1, hide.ns = TRUE) + labs( subtitle = get_test_label(res.kruskal, detailed = TRUE), caption = get_pwc_label(res1))

## Conclusion

Kruskal-Wallis test is an alternative to the one-way ANOVA when there are more than two groups to compare.

When ANOVA assumptions are not met It’s recommended.