Bartlett's Test

The assumption of equal variances across treatment groups may cause serious problems if violated in one-way analysis of variance models. A common test for homogeneity of variances is Bartlett's test. This statistical test checks whether the variances from different groups (or samples) are equal.

Suppose that there are r treatment groups and we want to test

In this context, we assume that we have independently chosen random samples of size ni, i = 1,…, r from each of the r independent populations. Let Xij ∼ N(μi, σ2i) be independently distributed with a normal distribution having mean μi and varianceσi2 for each j = 1,…, ni and each i = 1,…, r. Let

i be the sample mean and S2i the sample variance of the sample taken from the ith group or population. The uniformly most powerful unbiased parametric test of size α for testing for equality of variances among r populations is known as Bartlett's test, and Bartlett's test statistic is given by

where wi = (ni − 1)/(N-r) is known as the weight for the ith group and

is the sum of the individual sample sizes. In the equireplicate case(i.e., n1 = … = nr = n), the weights are equal, and wi =1/r for each i = 1,…, r. The test statistic is the ratio of the weighted geometric mean of the group sample variances to their weighted arithmetic mean. The values of the test statistic are bounded as 0 ≤ll∗≤1 by Jensen's inequality. Large values of 0 ≤ll∗≤1 (i.e., values near 1) indicate agreement with the null hypothesis, whereas small values indicate disagreement with the null. The terminology ll∗ is used to indicate that Bartlett's test is based on M. S. Bartlett's modification of the likelihood ratio test, wherein he replaced the sample sizes ni with their corresponding degrees of freedom, ni − 1. Bartlett did so to make the test unbiased. In the equireplicate case, Bartlett's test and the likelihood ratio test result in the same test statistic and same critical region.

The distribution of l1∗ is complex even when the null hypothesis is true. R. E. Glaser showed that the distribution of l1∗ could be expressed as a product of independently distributed beta random variables. In doing so he renewed much interest in the exact distribution of Bartlett's test. We reject H0 provided l1∗≤bα(n1,…, nr) where Pr(l1∗<bα(n1,…, nr)) = α when H0 is true. The Bartlett critical value bα(n1,…, nr) is indexed by level of significance and the individual sample sizes. The critical values were first tabled in the equireplicate case, and the critical value was simplified to bα(n,…, n = bα(n). Tabulating critical values with unequal sample sizes becomes counterproductive because of possible combinations of groups, sample sizes, and levels of significance.

Example

Consider an experiment in which lead levels are measured at five different sites. The data in Table 1 come from Paul Berthouex and Linfield Brown:

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Table 1 Ten Measurements of Lead Concentration (mG=L) Measured on Waste Water Specimens

Source: Berthouex, P. M., & Brown, L. C. (2002). Statistics for environmental engineers (2nd ed., p. 170). Boca Raton, FL: Lewis.

From these data one can compute the sample variances and weights, which are given

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