When Unequal Variances Matter in ANOVA — and When They Don’t

 

🔴 When the violation of homoscedasticity affects ANOVA

     1. Positive kurtosis (above 2): the F test loses power. That is, it tends not to reject the null hypothesis, even when it is false.
     2. Skewed distributions: in this case, variance tends to increase with the mean, which can seriously bias ANOVA results.

🟢 When the violation does not seriously compromise the analysis

     1. Equal sample sizes across groups: if the groups have the same number of observations (r₁ = r₂ = ... = rₖ), small differences in variances are usually acceptable — unless one is highly discrepant.
     2. Large samples: with more than 10 observations per group, the F test tends to remain robust to mild heteroscedasticity.

How to test the homogeneity of variances?

 The goal is to test the null hypothesis:
                      H₀: s²₁ = s²₂ = s²₃ = ... = s²ₖ (i = 1, 2, ..., k)

against the alternative that at least one variance is different.

Among the available tests, we highlight:
     · Levene’s test ✅
     · Bartlett’s test ⚠️
     · Cochran’s test and Hartley’s test (less common)

⚠️ Beware of Bartlett’s test:
It may mask differences in platykurtic distributions and indicate spurious differences in leptokurtic ones.

Understanding Levene’s Test

Levene’s test evaluates whether the groups have similar dispersions. The logic is simple: if the groups have homogeneous variances, the residuals (or their transformations) should not differ significantly.

✔️ Traditional procedure (with squared residuals)

     1. Calculate the residuals.
     2. Square these residuals.
     3. Perform a new one-way ANOVA using the squared residuals as the variable.

❗ If the F value is not significant, homoscedasticity is assumed.

    EXAMPLE

                                                                   Table 1 – Raw data by group

Table 2 – Squared residuals

Table 3 – Levene’s test result (SAS output)

✔️ Practical alternative (using absolute residuals)

      Another version of Levene’s test — more common in software like SPSS — uses the absolute                values of residuals instead of squares.

      ❗ The procedure is the same: perform an ANOVA using the absolute values.

📊 Table 4 – Levene’s test result (SPSS output)

📌 Note: The F values may differ slightly, but the conclusion is the same — here, there is no evidence of heteroscedasticity.

✔️ Alternative versions:
It is also possible to compute residuals based on the trimmed mean or the median, which may make the test more robust to outliers.

When homoscedasticity fails: what to do?

If the hypothesis of equal variances is rejected, the classic ANOVA may be inappropriate. But there are alternatives:

1. Data transformations
They can stabilize variances and make the data more normally distributed:
· Logarithmic (for positive, skewed data)
· Square root (ideal for counts)
· Arcsine square root (for proportions)
· Standardization (z-scores)

2. Nonparametric tests
· Kruskal-Wallis: replaces ANOVA when assumptions are not met.

3. Other solutions
· Remove outliers, if justified
· Increase sample size
· Redesign the experiment

💡 In summary:

The homogeneity of variances is a key assumption of ANOVA, but its violation is not always fatal. Understanding your data, choosing the right test, and interpreting the results carefully are essential attitudes for any good researcher.

References:

1. Dean, A. & Voss, D. (1999). *Design and Analysis of Experiments*. Springer.
2. Scheffé, H. (1959). *The Analysis of Variance*. Wiley.
3. Zaiontz, C. Levene’s test. Retrieved from http://www.real-statistics.com/one-way-analysis-of-variance-anova/homogeneity-variances/levenes-test/

4. And, why not?