1.3.5.7. Bartlett's Test

1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.5. Quantitative Techniques

1.3.5.7. Bartlett's Test

Purpose:
Test for Homogeneity of Variances

Bartlett's test (Snedecor and Cochran, 1983) is used to test if k samples have equal variances. Equal variances across samples is called homogeneity of variances. Some statistical tests, for example the analysis of variance, assume that variances are equal across groups or samples. The Bartlett test can be used to verify that assumption.

Bartlett's test is sensitive to departures from normality. That is, if your samples come from non-normal distributions, then Bartlett's test may simply be testing for non-normality. The Levene test is an alternative to the Bartlett test that is less sensitive to departures from normality.

Definition

The Bartlett test is defined as:

H₀:	σ₁² = σ₂² = ... = σ_k²
H_a:	σ_i² ≠ σ_j² for at least one pair (i,j).
Test Statistic:	The Bartlett test statistic is designed to test for equality of variances across groups against the alternative that variances are unequal for at least two groups. \[ T = \frac{(N-k) \ln{s^{2}_{p}} - \sum_{i=1}^{k}(N_{i} - 1)\ln{s^{2}_{i}}}{1 + (1/(3(k-1)))((\sum_{i=1}^{k}{1/(N_{i} - 1))} - 1/(N-k))} \] In the above, *s_i²* is the variance of the ith group, N is the total sample size, N_i is the sample size of the ith group, k is the number of groups, and *s_p²* is the pooled variance. The pooled variance is a weighted average of the group variances and is defined as: \[ s^{2}_{p} = \sum_{i=1}^{k}(N_{i} - 1)s^{2}_{i}/(N-k) \]
Significance Level:	α
Critical Region:	The variances are judged to be unequal if, \( T > \chi^2_{1-\alpha, \, k-1} \) where \( \chi^2_{1-\alpha, \, k-1} \) is the critical value of the chi-square distribution with k - 1 degrees of freedom and a significance level of α.

An alternate definition (Dixon and Massey, 1969) is based on an approximation to the F distribution. This definition is given in the Product and Process Comparisons chapter (chapter 7).

Example	Bartlett's test was performed for the GEAR.DAT data set. The data set contains 10 measurements of gear diameter for ten different batches for a total of 100 measurements. H₀: σ₁² = σ₂² = ... = σ₁₀² H_a: At least one σ_i² is not equal to the others. Test statistic: T = 20.78580 Degrees of freedom: k - 1 = 9 Significance level: α = 0.05 Critical value: Χ²_1-α,k-1 = 16.919 Critical region: Reject H₀ if T > 16.919 We are testing the null hypothesis that the batch variances are all equal. Because the test statistic is larger than the critical value, we reject the null hypotheses at the 0.05 significance level and conclude that at least one batch variance is different from the others.
Question	Bartlett's test can be used to answer the following question: Is the assumption of equal variances valid?
Importance	Bartlett's test is useful whenever the assumption of equal variances is made. In particular, this assumption is made for the frequently used one-way analysis of variance. In this case, Bartlett's or Levene's test should be applied to verify the assumption.
Related Techniques	Standard Deviation Plot Box Plot Levene Test Chi-Square Test Analysis of Variance
Case Study	Heat flow meter data
Software	The Bartlett test is available in many general purpose statistical software programs. Both Dataplot code and R code can be used to generate the analyses in this section. These scripts use the GEAR.DAT data file.