Bartlett's Test

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A Bartlett's Test is a statistical testing of the null hypothesis that states all given samples are from populations with equal variances.



References

2016

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 [math]\displaystyle{ n_i }[/math] and sample variances [math]\displaystyle{ S_i^2 }[/math] then Bartlett's test statistic is

[math]\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)} }[/math]
where [math]\displaystyle{ N = \sum_{i=1}^k n_i }[/math] and [math]\displaystyle{ S_p^2 = \frac{1}{N-k} \sum_i (n_i-1)S_i^2 }[/math] is the pooled estimate for the variance.
The test statistic has approximately a [math]\displaystyle{ \chi^2_{k-1} }[/math] distribution. Thus the null hypothesis is rejected if [math]\displaystyle{ \chi^2 \gt \chi^2_{k-1,\alpha} }[/math] (where [math]\displaystyle{ \chi^2_{k-1,\alpha} }[/math] is the upper tail critical value for the [math]\displaystyle{ \chi^2_{k-1} }[/math] distribution).
Bartlett's test is a modification of the corresponding likelihood ratio test designed to make the approximation to the [math]\displaystyle{ \chi^2_{k-1} }[/math] distribution better (Bartlett, 1937).