Bartlett's test

In statistics, Bartlett's test (see Snedecor and Cochran, 1989) is used to test if k samples are from populations with equal variances. Equal variances across populations 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. Levene's test and the Brown–Forsythe test are alternatives to the Bartlett test that are less sensitive to departures from normality.^[1]

The test is named after Maurice Stevenson Bartlett.

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 $n_{i}$ and sample variances $S_{i}^{2}$ then Bartlett's test statistic is

\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 $N=\sum _{{i=1}}^{k}n_{i}$ 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 $\chi _{{k-1}}^{2}$ distribution. Thus the null hypothesis is rejected if $\chi ^{2}>\chi _{{k-1,\alpha }}^{2}$ (where $\chi _{{k-1,\alpha }}^{2}$ is the upper tail critical value for the $\chi _{{k-1}}^{2}$ distribution).

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

Notes

The test statistics may be written in some sources with logarithms of base 10 as:^[2]

\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)}}

References

↑ NIST/SEMATECH e-Handbook of Statistical Methods. Available online, URL: http://www.itl.nist.gov/div898/handbook/eda/section3/eda357.htm. Retrieved December 31, 2013.
↑ F., Gunst, Richard; L., Hess, James (2003-01-01). Statistical design and analysis of experiments : with applications to engineering and science. Wiley. p. 98. ISBN 0471372161. OCLC 856653529.

Bartlett, M. S. (1937). "Properties of sufficiency and statistical tests". Proceedings of the Royal Statistical Society, Series A 160, 268–282 JSTOR 96803
Snedecor, George W. and Cochran, William G. (1989), Statistical Methods, Eighth Edition, Iowa State University Press. ISBN 978-0-8138-1561-9

External links

NIST page on Bartlett's test

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[1] NIST/SEMATECH e-Handbook of Statistical Methods. Available online, URL: http://www.itl.nist.gov/div898/handbook/eda/section3/eda357.htm. Retrieved December 31, 2013.

[2] F., Gunst, Richard; L., Hess, James (2003-01-01). Statistical design and analysis of experiments : with applications to engineering and science. Wiley. p. 98. ISBN 0471372161. OCLC 856653529.