Friedman Test — Nonparametric Repeated Measures ANOVA

Friedman Test

The Friedman test is the nonparametric alternative to repeated-measures ANOVA. Used when the same subjects are measured under k conditions and normality cannot be assumed.

import numpy as np
from scipy import stats

np.random.seed(42)

# Scenario: 15 athletes tested under 3 training conditions
n_subjects = 15
# Performance scores (not normally distributed)
condition_a = np.array([8.2, 7.5, 9.1, 6.8, 8.9, 7.2, 9.5, 6.9, 8.3, 7.8, 9.2, 7.1, 8.7, 7.4, 9.0])
condition_b = np.array([7.8, 7.1, 8.6, 6.3, 8.5, 6.9, 9.1, 6.5, 7.9, 7.4, 8.8, 6.7, 8.3, 7.0, 8.6])
condition_c = np.array([9.1, 8.4, 9.8, 7.7, 9.6, 8.2, 10.2, 7.8, 9.4, 8.7, 9.9, 8.0, 9.6, 8.3, 9.7])

# Friedman test
stat, p = stats.friedmanchisquare(condition_a, condition_b, condition_c)
k, n = 3, n_subjects
df = k - 1

print(f"Friedman χ²({df}) = {stat:.4f}, p = {p:.6f}")
print(f"Decision: {'Reject H₀ — conditions differ' if p < 0.05 else 'Fail to reject H₀'}")

# Kendall's W (effect size)
data = np.column_stack([condition_a, condition_b, condition_c])
W = stat / (n * (k - 1))
print(f"Kendall's W = {W:.4f} ({'small' if W < 0.1 else 'medium' if W < 0.3 else 'large'} effect)")

# Post-hoc: pairwise Wilcoxon signed-rank tests (Bonferroni corrected)
print("\nPost-hoc: Pairwise Wilcoxon signed-rank (Bonferroni corrected):")
pairs = [('A vs B', condition_a, condition_b),
         ('A vs C', condition_a, condition_c),
         ('B vs C', condition_b, condition_c)]
alpha_bonf = 0.05 / len(pairs)
for name, g1, g2 in pairs:
    _, p_w = stats.wilcoxon(g1, g2, alternative='two-sided')
    print(f"  {name}: p = {p_w:.6f} → {'*' if p_w < alpha_bonf else 'ns'}")

import matplotlib.pyplot as plt
fig, ax = plt.subplots(figsize=(8, 5))
ax.boxplot([condition_a, condition_b, condition_c],
           labels=['Condition A', 'Condition B', 'Condition C'],
           patch_artist=True)
ax.set_title(f'Performance by Condition (n={n} athletes)
Friedman χ²={stat:.3f}, p={p:.4f}')
ax.set_ylabel('Performance Score')
plt.tight_layout()
plt.savefig('friedman_test.png', dpi=150)
plt.show()

Key Takeaways

1. Nonparametric repeated-measures ANOVA — for within-subjects designs
2. Ranks within each subject (row-wise), then compares across conditions
3. Kendall's W measures agreement across raters/conditions (0 = no agreement, 1 = perfect)
4. Post-hoc: pairwise Wilcoxon with Bonferroni or Nemenyi test
5. Use instead of repeated-measures ANOVA when normality fails