Levene and Brown-Forsythe Tests for Equal Variance

Why Test for Equal Variances?

In one-way ANOVA, the equal-variance assumption means that the population response variances are the same across the groups being compared. Residual plots, as discussed in the earlier material on checking equal variances with residual plots, can reveal patterns such as a changing spread. Levene’s and Brown-Forsythe tests provide another check: they turn each observation into a measure of its distance from a group center, then compare the average distances across groups.

These tests are useful as diagnostic evidence, not as automatic pass-or-fail rules. A large p-value does not establish that the population variances are equal, and a small p-value does not tell us how serious the variance differences are for a particular analysis. The study design and the data’s distributional features still matter.

How the Tests Are Calculated

Write \(Y_{ij}\) for observation \(j\) in group \(i\), where group \(i\) has \(n_i\) observations. Choose a center \(C_i\) for each group, and calculate a transformed value \(Z_{ij}=|Y_{ij}-C_i|\). For Levene’s test, \(C_i=\bar{x}_i\), the sample mean in group \(i\). For Brown-Forsythe’s test, \(C_i\) is the sample median in group \(i\). The test is a one-way ANOVA of the \(Z_{ij}\) values.

Let \(N=\sum_{i=1}^{k}n_i\), let \(\bar{Z}_i\) be the mean of the transformed values in group \(i\), and let \(\bar{Z}\) be their mean across all groups. The between-group and within-group sums of squares for these transformed values are

Divide each sum of squares by its degrees of freedom to obtain \(MS_{\text{between}}=SS_{\text{between}}/(k-1)\) and \(MS_{\text{within}}=SS_{\text{within}}/(N-k)\). The test statistic is their ratio. Its reference distribution is an approximate \(F\) distribution with \(k-1\) and \(N-k\) degrees of freedom.

A large \(F\) indicates that the group means of the absolute deviations are separated relative to the variation in those deviations within groups. The p-value is calculated using the upper tail of the reference \(F\) distribution. At significance level \(\alpha\), reject \(H_0\) if the p-value is at most \(\alpha\); otherwise, do not reject it.

Assumptions and a Justification

Before interpreting the approximate \(F\) test, consider its conditions. The observations should be independent within and across groups, as supported by how the data were collected. The response should be quantitative, and the group-specific transformed values should be suitable for the approximate ANOVA reference distribution. The approximation is most dependable when the group data are not dominated by severe skewness or extreme observations; small samples can make the test particularly sensitive to such features. No test of equal variances can establish independence, which must be assessed from the study design.

Why can comparing absolute deviations provide information about spread? If a group’s response is normally distributed with mean \(\mu_i\) and standard deviation \(\sigma_i\), then its expected absolute distance from its population mean is proportional to \(\sigma_i\):

To see this, write \(Y=\mu_i+\sigma_i U\), where \(U\sim N(0,1)\). Then \(|Y-\mu_i|=\sigma_i|U|\), so \(E(|Y-\mu_i|)=\sigma_i E(|U|)\). Integrating the standard normal density gives

which yields the result. Thus, when group distributions have a common normal shape, different standard deviations correspond to different expected absolute deviations. In practice the tests use sample centers, not population centers, and use an approximate \(F\) reference distribution; the argument explains the connection to spread, not a guarantee that every distribution meets the test’s conditions.

Why the Choice of Center Matters

Levene’s mean-centered deviations can be affected by an extreme observation, because that observation also pulls the group mean. The median is less affected by an extreme value, so Brown-Forsythe’s version is often preferred when outliers or skewness are concerns. This greater resistance does not make it immune to unusual data or guarantee a correct conclusion for every distribution.

Interpretation and Limitations

The formal null concerns equal population variances. However, the calculations compare means of absolute deviations. For distributions with different shapes, equal variances need not imply equal expected absolute deviations, and different variances need not translate into a simple difference in those means. The tests are therefore best understood as assumption checks whose behavior depends partly on distribution shape, sample size, and outliers. Brown-Forsythe’s median centering improves resistance to some departures from normality, but does not remove these limitations.

A non-significant result is not evidence that the assumption has been proved. The test may have limited ability to detect variance differences, particularly with small samples. Conversely, a significant result does not by itself settle how an ANOVA analysis should proceed. Consider the result alongside residual plots, the study context, and the size of the observed spreads. These tests assess variance patterns; they do not test normality, independence, or whether group means are equal.

Common Mistakes

• Using the response values instead of absolute deviations. The test’s ANOVA is applied to \(Z_{ij}=|Y_{ij}-C_i|\), not directly to the original responses.
• Reversing the centers. Levene’s test uses group means; Brown-Forsythe’s version uses group medians.
• Calling a large p-value proof of equality. A large p-value means the data do not provide strong evidence against the null; it does not establish that the variances are identical.
• Treating the test as a complete assumption check. Neither procedure establishes independence or checks every feature of the ANOVA model.