1.3.EDA Techniques
1.3.5.Quantitative Techniques
1.3.5.10. | Levene Test for Equality of Variances |
Test for hom*ogeneity of Variances
Levene's test is an alternative to the Bartlett test. The Levene test is less sensitive than the Bartlett test to departures from normality. If you have strong evidence that your data do in fact come from a normal, or nearly normal, distribution, then Bartlett's test has better performance.
| H0: | \( \sigma_{1}^{2} = \sigma_{2}^{2} = \ldots = \sigma_{k}^{2} \) |
| Ha: | \( \sigma_{i}^{2} \ne \sigma_{j}^{2} \) for at least one pair (i,j). |
| Test Statistic: | Given a variable Y with sample of size N divided into k subgroups, where Ni is the sample size of the ith subgroup, the Levene test statistic is defined as:
The three choices for defining Zij determine the robustness and power of Levene's test. By robustness, we mean the ability of the test to not falsely detect unequal variances when the underlying data are not normally distributed and the variables are in fact equal. By power, we mean the ability of the test to detect unequal variances when the variances are in fact unequal. Levene's original paper only proposed using the mean. Brown and Forsythe (1974)) extended Levene's test to use either the median or the trimmed mean in addition to the mean. They performed Monte Carlo studies that indicated that using the trimmed mean performed best when the underlying data followed a Cauchy distribution (i.e., heavy-tailed) and the median performed best when the underlying data followed a \(\chi^{2}_{4}\) (i.e., skewed) distribution. Using the mean provided the best power for symmetric, moderate-tailed, distributions. Although the optimal choice depends on the underlying distribution, the definition based on the median is recommended as the choice that provides good robustness against many types of non-normal data while retaining good power. If you have knowledge of the underlying distribution of the data, this may indicate using one of the other choices. |
| Significance Level: | α |
| Critical Region: | The Levene test rejects the hypothesis that the variances are equal if
In the above formulas for the critical regions, the Handbook follows the convention that Fα is the upper critical value from the F distribution and F1-α is the lower critical value. Note that this is the opposite of some texts and software programs. |
H0: σ12 = ... = σ102Ha: σ12 ≠ ... ≠ σ102We are testing the hypothesis that the group variances areequal. We fail to reject the null hypothesis at the 0.05 significance level since the value of the Levene test statistic is less than the critical value. We conclude that there is insufficient evidence to claim that the variancesare not equal.Test statistic: W = 1.705910Degrees of freedom: k-1 = 10-1 = 9 N-k = 100-10 = 90Significance level: α = 0.05Critical value (upper tail): Fα,k-1,N-k = 1.9855 Critical region: Reject H0 if F > 1.9855
- Is the assumption of equal variances valid?
Box Plot
Bartlett Test
Chi-Square Test
Analysis of Variance
