Bartlett's test

In statistics, Bartlett's test, named after Maurice Stevenson Bartlett,[1] is used to test homoscedasticity, that is, if multiple samples are from populations with equal variances.[2] Some statistical tests, such as the analysis of variance, assume that variances are equal across groups or samples, which can be verified with Bartlett's test.

In a Bartlett test, we construct the null and alternative hypothesis. For this purpose several test procedures have been devised. The test procedure due to M.S.E (Mean Square Error/Estimator) Bartlett test is represented here. This test procedure is based on the statistic whose sampling distribution is approximately a Chi-Square distribution with (k − 1) degrees of freedom, where k is the number of random samples, which may vary in size and are each drawn from independent normal distributions. Bartlett's test is sensitive to departures from normality. That is, if the samples come from non-normal distributions, then Bartlett's test may simply be testing for non-normality. Levene's test and the Brown–Forsythe test are alternatives to the Bartlett test that are less sensitive to departures from normality.[3]

Specification

Bartlett's test is used to test the null hypothesis, H0 that all k population variances are equal against the alternative that at least two are different.

If there are k samples with sizes ${\displaystyle n_{i}}$ and sample variances ${\displaystyle S_{i}^{2}}$ then Bartlett's test statistic is

${\displaystyle \chi ^{2}={\frac {(N-k)\ln(S_{p}^{2})-\sum _{i=1}^{k}(n_{i}-1)\ln(S_{i}^{2})}{1+{\frac {1}{3(k-1)}}\left(\sum _{i=1}^{k}({\frac {1}{n_{i}-1}})-{\frac {1}{N-k}}\right)}}}$

where ${\displaystyle N=\sum _{i=1}^{k}n_{i}}$ and ${\displaystyle S_{p}^{2}={\frac {1}{N-k}}\sum _{i}(n_{i}-1)S_{i}^{2}}$ is the pooled estimate for the variance.

The test statistic has approximately a ${\displaystyle \chi _{k-1}^{2}}$ distribution. Thus, the null hypothesis is rejected if ${\displaystyle \chi ^{2}>\chi _{k-1,\alpha }^{2}}$ (where ${\displaystyle \chi _{k-1,\alpha }^{2}}$ is the upper tail critical value for the ${\displaystyle \chi _{k-1}^{2}}$ distribution).

Bartlett's test is a modification of the corresponding likelihood ratio test designed to make the approximation to the ${\displaystyle \chi _{k-1}^{2}}$ distribution better (Bartlett, 1937).

Notes

The test statistics may be written in some sources with logarithms of base 10 as:[4]

${\displaystyle \chi ^{2}=2.3026{\frac {(N-k)\log _{10}(S_{p}^{2})-\sum _{i=1}^{k}(n_{i}-1)\log _{10}(S_{i}^{2})}{1+{\frac {1}{3(k-1)}}\left(\sum _{i=1}^{k}({\frac {1}{n_{i}-1}})-{\frac {1}{N-k}}\right)}}}$