Kruskal-Wallis Test

Nonparametric Tests

Nonparametric One-Way ANOVA for Multiple Groups

When you need to compare three or more independent groups without assuming normality, the Kruskal-Wallis test extends the Mann-Whitney U to multiple samples. It ranks all observations together and tests whether groups share the same distribution.

Agricultural Research — Compare crop yields across multiple fertilizers with skewed distributions
Education — Assess student performance across teaching methods with ordinal outcomes
Healthcare — Compare recovery times across multiple treatment protocols

When ANOVA's normality assumption breaks, Kruskal-Wallis carries the comparison forward.

The Kruskal-Wallis test is the nonparametric alternative to one-way ANOVA. It tests whether k independent groups have the same distribution (or equivalently, the same median when distributions are identically shaped).

DfKruskal-Wallis Test

A nonparametric test that determines whether k independent groups come from the same distribution, without assuming normality.

Kruskal-Wallis H Statistic

H = \frac{12}{N(N+1)} \sum_{i=1}^k \frac{R_i^2}{n_i} - 3(N+1)

Here,

$H$ =The Kruskal-Wallis test statistic
$N$ =Total number of observations
$k$ =Number of groups
$R_i$ =Sum of ranks in group i
$n_i$ =Sample size of group i


import numpy as np

from scipy import stats

import scikit_posthocs as sp  # pip install scikit-posthocs

import matplotlib.pyplot as plt



np.random.seed(42)



# Test: Does pain relief differ across 3 medication types?

# Data is not normally distributed (skewed)

drug_a = np.random.lognormal(2.0, 0.6, 25)  # pain scores

drug_b = np.random.lognormal(2.3, 0.5, 25)

drug_c = np.random.lognormal(2.6, 0.7, 25)



# Kruskal-Wallis

H, p = stats.kruskal(drug_a, drug_b, drug_c)

df = 3 - 1  # k-1



print(f"Kruskal-Wallis H({df}) = {H:.4f}, p = {p:.4f}")

print(f"Decision: {'Reject H0 — groups differ' if p < 0.05 else 'Fail to reject H0'}")



# Effect size: eta-squared for Kruskal-Wallis

n = len(drug_a) + len(drug_b) + len(drug_c)

eta2 = (H - df + 1) / (n - df)

print(f"Effect size ?²_KW = {eta2:.4f}")



# If significant -> post-hoc pairwise comparisons (Dunn's test)

try:

    import scikit_posthocs as sp

    data_combined = [drug_a, drug_b, drug_c]

    posthoc = sp.posthoc_dunn(data_combined, p_adjust='bonferroni')

    print("\nDunn's Post-hoc Test (Bonferroni corrected):")

    print(posthoc.round(4))

except ImportError:

    # Manual Mann-Whitney pairwise

    pairs = [('A vs B', drug_a, drug_b), ('A vs C', drug_a, drug_c), ('B vs C', drug_b, drug_c)]

    bonf_alpha = 0.05 / 3

    for name, g1, g2 in pairs:

        _, p_mw = stats.mannwhitneyu(g1, g2, alternative='two-sided')

        print(f"{name}: p={p_mw:.4f} -> {'Significant' if p_mw < bonf_alpha else 'Not significant'} (Bonferroni a={bonf_alpha:.4f})")



# Box plots

fig, ax = plt.subplots(figsize=(8, 5))

ax.boxplot([drug_a, drug_b, drug_c], labels=['Drug A', 'Drug B', 'Drug C'], patch_artist=True)

ax.set_title(f'Pain Scores by Drug\nKruskal-Wallis H={H:.3f}, p={p:.4f}')

ax.set_ylabel('Pain Score')

plt.tight_layout()

plt.savefig('kruskal_wallis.png', dpi=150)

plt.show()

Post-Hoc Testing

When the Kruskal-Wallis test is significant, you must use post-hoc tests (like Dunn's test with Bonferroni correction) to determine which specific groups differ.

Key Takeaways

Summary: Kruskal-Wallis Test

Nonparametric alternative to one-way ANOVA — uses ranks
Assumes independence, ordinal+ data, and identically shaped distributions between groups
If significant: use Dunn's test (post-hoc) with Bonferroni or BH correction
More robust than ANOVA for skewed or heavy-tailed distributions
Less powerful than ANOVA when normality truly holds

Kruskal-Wallis Test — Nonparametric One-Way ANOVA