Levene’s Test of Variances

Robust test of variance homogeneity across groups

Introduction

Levene’s test checks whether variances are equal across groups. It is used before or alongside ANOVA to verify the homogeneity-of-variance assumption. Compared to Bartlett’s test, it is robust to non-normality.

Prerequisites

One-way ANOVA, variance.

Theory

For \(k\) groups, replace each observation by its absolute deviation from the group mean (Levene original) or median (Brown-Forsythe / median-based Levene), then run a one-way ANOVA on the transformed values:

\[W = \frac{(N - k) \sum_i n_i (\bar{Z}_i - \bar{Z})^2}{(k - 1) \sum_i \sum_j (Z_{ij} - \bar{Z}_i)^2},\]

where \(Z_{ij}\) is the absolute deviation. Under \(H_0\) of equal variances, \(W \sim F_{k-1, N-k}\).

The median-based Levene (Brown-Forsythe) is robust to non-normality and recommended as the default.

Assumptions

• Independent observations.
• Appropriate deviation centre (median for non-normal data).

No normality assumption.

R Implementation

library(car)
set.seed(2026)

df <- data.frame(
  group = factor(rep(c("A", "B", "C"), each = 40)),
  y     = c(rnorm(40, 50, 5),
            rnorm(40, 55, 12),
            rnorm(40, 52, 5))
)

# Levene's test (median-based, default in car)
leveneTest(y ~ group, data = df, center = median)

# Mean-based (original)
leveneTest(y ~ group, data = df, center = mean)

# Bartlett's test (for comparison; assumes normality)
bartlett.test(y ~ group, data = df)

Output & Results

Levene's Test for Homogeneity of Variance (center = median)
      Df  F value  Pr(>F)
group  2   16.98  3.1e-07
      117

Bartlett test of homogeneity of variances
Bartlett's K-squared = 35.2, df = 2, p-value = 2.3e-08

Strong evidence of unequal variances. For ANOVA, switch to Welch’s F-test.

Interpretation

“Levene’s test indicated significant heterogeneity of variance across groups (F(2, 117) = 16.98, p < 0.001); Welch’s ANOVA was used for the mean comparison.”

Practical Tips

• Use median-based Levene by default; it is robust to non-normality.
• A routine pre-test is discouraged: conditionally choosing between Student and Welch ANOVA inflates Type I error. Just default to Welch.
• With very small \(n_i\), Levene has low power; violations may be missed.
• Bartlett assumes normality and is very sensitive to its violation; avoid it for real data.
• For repeated-measures designs, a Levene-style test is not applicable directly; inspect residual plots.