Levene's Test

Levene's test is a robust statistical procedure for assessing the equality of variances across two or more groups. Unlike the classic F-test, Levene's test is less sensitive to departures from normality because it works on transformed absolute deviations from a group center (mean or median). In the context of YouTube analytics, for example, you might test whether the variability of watch time differs across content categories or upload days, ensuring your downstream models respect the assumption of equal variance where required.

Levene's Test Statistic (W)

$W$$=$$($$($$N - k$$)$$\div$$k - 1$$)$$\times$$Σ n_i (Z_{i·} - Z_{··})^2$$\div$$Σ Σ (Z_{ij} - Z_{i·})^2$

Definition of Z

$Z_{ij}$$=$$|$$Y_{ij} - ̅Y_i$$|$

Assumptions

  • Independent observations within and across groups
  • Groups are sampled from populations with similar shape (but not necessarily normal)
  • The response variable is measured on at least an interval scale

Hypothesis

  • H₀: σ₁² = σ₂² = … = σ_k² (All group variances are equal)
  • Hₐ: At least one group variance differs

Steps

  1. Choose a measure of central tendency for each group (mean for classic Levene; median for Brown–Forsythe variant).
  2. Compute transformed values: Zij = |Yij – centeri|.
  3. Calculate the mean of Zij within each group and the overall mean across groups.
  4. Form the ANOVA F-ratio on the transformed Z-values: the ratio of between-group to within-group sums of squares (this is W).
  5. Compare W to the critical F-value (with k – 1 and N – k degrees of freedom) or use its p-value.

Interpretation

If the computed W-statistic is significant (p-value < α), reject H₀ and conclude that at least one group has a different variance. This signals potential heteroscedasticity and may motivate variance-stabilizing transformations, weighted regression, or robust modeling strategies.

Variants & Practical Tips

  • Brown–Forsythe Test: Uses the median instead of the mean for the center; even more robust to non-normality and outliers.
  • Sample Size Sensitivity: Very large samples can flag trivial differences in variance; always consider effect size and practical significance.
  • Post-hoc Checks: If H₀ is rejected, explore which groups differ using pairwise variance comparisons or graphical diagnostics (e.g., residual plots).