Levene's Test for Homogeneity of Variances

Levene's Test

Levene's test checks whether k groups have equal population variances (homoscedasticity). Unlike the F-test, it doesn't assume normality.

$W = \frac{(N-k)}{(k-1)} \cdot \frac{\sum_{i=1}^k n_i (\bar{Z}_{i.} - \bar{Z}_{..})^2}{\sum_{i=1}^k \sum_{j=1}^{n_i} (Z_{ij} - \bar{Z}_{i.})^2}$

where $Z_{ij} = |Y_{ij} - \bar{Y}_{i.}|$ (absolute deviations from group means).

import numpy as np
from scipy import stats
import matplotlib.pyplot as plt

np.random.seed(42)

# Compare 3 groups for ANOVA — check variance assumption first
group_a = np.random.normal(50, 5, 30)
group_b = np.random.normal(55, 5, 28)
group_c = np.random.normal(48, 8, 32)  # larger variance

# Levene's test
stat, p = stats.levene(group_a, group_b, group_c)
print(f"Levene's Test for 3 Groups:")
print(f"W = {stat:.4f}, p = {p:.4f}")
print(f"Groups have {'equal' if p > 0.05 else 'UNEQUAL'} variances (α=0.05)")

# Visualize variance differences
fig, ax = plt.subplots(figsize=(8, 5))
ax.boxplot([group_a, group_b, group_c], labels=['Group A', 'Group B', 'Group C'],
           patch_artist=True)
ax.set_title(f"Group Variance Comparison
Levene's: W={stat:.3f}, p={p:.4f}")
ax.set_ylabel('Values')
for i, (g, label) in enumerate(zip([group_a, group_b, group_c], ['A','B','C']), 1):
    ax.text(i, g.max()+0.5, f'SD={g.std(ddof=1):.2f}', ha='center', fontsize=9)
plt.tight_layout()
plt.savefig('levenes_test.png', dpi=150)
plt.show()

# What to do if variances are unequal in ANOVA
print("
If Levene's is significant (p < 0.05):")
print("→ Use Welch's ANOVA (one-way): stats.f_oneway is not appropriate")
print("→ Or use Kruskal-Wallis nonparametric test")

Key Takeaways

1. Levene's test uses absolute deviations, making it robust to non-normality
2. Brown-Forsythe uses median deviations — even more robust
3. p > 0.05 → fail to reject equal variances → proceed with standard ANOVA
4. p < 0.05 → use Welch's ANOVA or nonparametric alternative
5. ANOVA is fairly robust to moderate variance inequality when group sizes are equal