In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, Bartlett's test, named after
Maurice Stevenson Bartlett, is used to test
homoscedasticity
In statistics, a sequence (or a vector) of random variables is homoscedastic () if all its random variables have the same finite variance. This is also known as homogeneity of variance. The complementary notion is called heteroscedasticity. The s ...
, that is, if multiple samples are from populations with equal
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
s. Some statistical tests, such as the
analysis of variance
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statisticia ...
, 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 In statistics, Levene's test is an inferential statistic used to assess the equality of variances for a variable calculated for two or more groups. Some common statistical procedures assume that variances of the populations from which different sam ...
and the
Brown–Forsythe test
The Brown–Forsythe test is a statistical test for the equality of group variances based on performing an Analysis of Variance (ANOVA) on a transformation of the response variable. When a one-way ANOVA is performed, samples are assumed to have bee ...
are alternatives to the Bartlett test that are less sensitive to departures from normality.
Specification
Bartlett's test is used to test the null hypothesis, ''H''
0 that all ''k'' population variances are equal against the alternative that at least two are different.
If there are ''k'' samples with sizes
and
sample variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
s
then Bartlett's test statistic is
:
where
and
is the pooled estimate for the variance.
The test statistic has approximately a
distribution. Thus, the null hypothesis is rejected if
(where
is the upper tail critical value for the
distribution).
Bartlett's test is a modification of the corresponding
likelihood ratio test
In statistics, the likelihood-ratio test assesses the goodness of fit of two competing statistical models based on the ratio of their likelihoods, specifically one found by maximization over the entire parameter space and another found after ...
designed to make the approximation to the
distribution better (Bartlett, 1937).
Notes
The test statistics may be written in some sources with logarithms of base 10 as:
:
See also
*
Box's M test
Box's ''M'' test is a multivariate statistical test used to check the equality of multiple variance-covariance matrices. The test is commonly used to test the assumption of homogeneity of variances and covariances in MANOVA and linear discrimina ...
*
Levene's test In statistics, Levene's test is an inferential statistic used to assess the equality of variances for a variable calculated for two or more groups. Some common statistical procedures assume that variances of the populations from which different sam ...
*
Kaiser–Meyer–Olkin test
The Kaiser–Meyer–Olkin (KMO) test is a statistical measure to determine how suited data is for factor analysis. The test measures sampling adequacy for each variable in the model and the complete model. The statistic is a measure of the proport ...
References
External links
NIST page on Bartlett's test
{{statistics
Analysis of variance
Statistical tests