Bartlett's test

In statistics, Bartlett's test (Snedecor and Cochran, 1983) is used to test if k samples are from populations with equal variances. Equal variances across samples is called homoscedasticity or 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 the samples come from non-normal distributions, then Bartlett's test may simply be testing for non-normality. The Levene test and Brown–Forsythe test are alternatives to the Bartlett test that are less sensitive to departures from normality.

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 size $n_i$ and sample variance $S_i^2$ then Bartlett's test statistic is

$X^2 = \frac{(N-k)\ln(S_p^2) - \sum_{i=1}^k(n_i - 1)\ln(S_i^2)}{1 %2B \frac{1}{3(k-1)}\left(\sum_{i=1}^k(\frac{1}{n_i-1}) - \frac{1}{N-k}\right)}$

where $N = \sum_{i=1}^k n_i$ and $S_p^2 = \frac{1}{N-k} \sum_i (n_i-1)S_i^2$ is the pooled estimate for the variance.

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

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

References

Bartlett, M. S. (1937). Properties of sufficiency and statistical tests. Proceedings of the Royal Statistical Society Series A 160, 268–282.
Snedecor, George W. and Cochran, William G. (1989), Statistical Methods, Eighth Edition, Iowa State University Press.

External links

NIST page on Bartlett's test

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Bartlett's test

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