Bartlett's test (of homogeneity of variances)

Homogeneity of variances

These k groups of observations were generated by k independent normal distributions with respective variances σ²₁, σ²₂, ..., σ²_k.

Are these k variances equal ?

The means (horizontal positions) are irrelevant.

This question is important because some major techniques such as ANOVA explicitely assume that the considered groups of observations were generated by normal distributions with identical variances. It is therefore appropriate to check is this assumption is plausible in regard of the samples, and therefore to conduct first a test of homogeneity of variances such as Bartlett's test.

A test of homogeneity of variances pits :

    * The null hypothesis H₀ : σ²₁ = σ²₂ = ... = σ²_k ( = σ²),

against

    * The alternative hypothesis H₁ : "At least one variance is different from the others.".

1) Note the formal similarity with the hypothesis of ANOVA, with "mean" being now replaced by "variance".
2) When k = 2, the problem is succesfully addressed by the F-test.

Bartlett's statistic

Denote n₁, n₂, ..., n_k the respective sizes of the samples.

Bartlett's test uses the following quantity as a test statistic :

with :

    * ν_i = n_i - 1,

    * s²_i is the unbiased estimate of σ²_i,

    * And s² is the "pooled variance" estimate of the common variance σ² :

It is the barycenter of the sample variances ponderated by their respective sample sizes minus 1.

-----

The functional part of the statistic is the numerator A, which does not look too bad.

The denominator B contains no random quantity, and is just a "correction factor" that helps the distribution of Bartlett's statistic cling closer to its theoretical asymptotic form (see below). Its exact role is to make the expectation of the Q test statistic equal to the mean of this asymptotic distribution.

It is immediately seen that B tends to 1 as the samples sizes grow without limit.

Distribution of Bartlett's statistic

We'll show that the numerator A has the form -2log(Λ), where Λ is the statistic of the Likelihood Ratio Test (LRT) built for the purpose of testing the above hypothesis. So it appears that Bartlett's test is the improved LRT for testing the homogeneity of variances of independent normal distributions.

Consequently, Q is asymptotically Chi-square distributed. We'll show that the number of degrees of freedom is k - 1. As the exact distribution of Bartlett's statistic is not known, it is customary to use its asymptotic distribution as an approximation of this exact but unknown distribution.

Weaknesses of Bartlett's test

Small samples

Because Bartlett's statistic is only approximately Chi-square distributed, Bartlett's test performs poorly for small samples.

Normality assumption

Bartlett's test is very sensitive to departures of the distributions from strict normality.

___________________________________________________________

In this Tutorial, we calculate the standard statistic Λ of the Likelihood Ratio Test designed for testing the homogeneity of variances of several independent normal distributions. We then show that the numerator A of Bartlett's statistic is just a small modification of -2log(Λ).

No justification will be given for the "correction factor" B of Bartlett's statistic, whose derivation is difficult.

BARTLETT'S TEST OF

HOMOGENEITY OF VARIANCES

 _____________________________________________________

Related readings :