Surf Wiki
Save to docs
general/analysis-of-variance

From Surf Wiki (app.surf) — the open knowledge base

Bartlett's test

Statistical test used to test homoscedasticity

Summary

Statistical test used to test homoscedasticity

In statistics, Bartlett's test, named after Maurice Stevenson Bartlett, is used to test homoscedasticity (or "homogeneity of variance"), that is, if multiple samples are from populations with equal variances. Some statistical tests, such as the analysis of variance, assume that variances are equal across groups or samples, which can be checked 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 unbiased in the Neyman–Pearson sense and consistent, but 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.

Some statistical methods, assume that variances are equal across groups or samples. The Bartlett test can be used to verify that assumption. For example, in ANOVA, one can first verify that different populations have the same variance using Bartlett's test, then check that they have the same mean. However, George Box argued that using Bartlett's test to check whether it is appropriate to apply ANOVA would be like "putting a rowing boat to sea to find out whether conditions are sufficiently calm for an ocean liner to leave a port", because when different populations are of roughly equal sample size, the F-test is robust to both non-normality and different variances.

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.

For each of i\in 1:k, take n_i samples from that population. Let its sample variances be S_i^2. 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^2_{k-1} distribution. Thus, the null hypothesis is rejected if \chi^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 .

Notes

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

:\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

References

  1. Bartlett, M. S.. (1937). "Properties of sufficiency and statistical tests". Proceedings of the Royal Statistical Society, Series A.
  2. (see [[George W. Snedecor. Snedecor, George W.]] and [[William Gemmell Cochran. Cochran, William G.]] (1989), ''Statistical Methods'', Eighth Edition, Iowa State University Press. {{ISBN. 978-0-8138-1561-9
  3. Pitman, E. J. G.. (1939). "Tests of Hypotheses Concerning Location and Scale Parameters". Biometrika.
  4. Brown, George W.. (1939). "On the Power of the L1 Test for Equality of Several Variances". The Annals of Mathematical Statistics.
  5. Box, G. E. P.. (1953). "Non-Normality and Tests on Variances". Biometrika.
  6. ''NIST/SEMATECH e-Handbook of Statistical Methods''. Available online, URL: http://www.itl.nist.gov/div898/handbook/eda/section3/eda357.htm {{Webarchive. link. (4 May 2020 . Retrieved 31 December 2013.)
  7. (1 January 2003). "Statistical design and analysis of experiments : with applications to engineering and science". Wiley.
Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

analysis-of-variance statistical-tests
Want to explore this topic further?

Ask Mako anything about Bartlett's test — get instant answers, deeper analysis, and related topics.

Research with Mako

Free with your Surf account

Content sourced from Wikipedia, available under CC BY-SA 4.0.

This content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.

Report