Bartlett test

As Bartlett's test (also: Bartlett's test ), two different statistical tests indicated:

• the Bartlett test for equality of variances in samples and ${\displaystyle k}$
• the Bartlett's test of sphericity for performing factor analysis .

Both tests are based on a likelihood ratio test and assume a normal distribution .

Bartlett's test for equality of variances

This test checks whether samples come from populations with equal variances. A series of statistical tests, e.g. For example, analysis of variance assumes that the variance of the groups in the population is the same. The Bartlett test is used to check this requirement. It was developed by Maurice Bartlett in 1937 . ${\displaystyle k}$ ${\displaystyle k}$

requirement

The Bartlett test assumes a normal distribution in the groups and is sensitive to the violation of this requirement. Alternatives are then the Levene test or Brown-Forsythe test , which are less sensitive to a violation of this requirement. ${\displaystyle k}$

Hypotheses

The Bartlett test tests the null hypothesis that all group variances are equal against the alternative hypothesis that at least two group variances are not equal:

${\displaystyle H_{0}:\sigma _{1}^{2}=\dots =\sigma _{k}^{2}}$ against ${\displaystyle H_{1}:\exists i,j\quad {\text{mit}}\quad \sigma _{i}^{2}\neq \sigma _{j}^{2}}$

Test statistics

If the groups have sample variances and sample sizes , then the test statistic is defined as ${\displaystyle k}$ ${\displaystyle S_{i}^{2}}$ ${\displaystyle n_{i}}$

${\displaystyle X^{2}={\frac {(N-k)\ln(S_{p}^{2})-\sum _{i=1}^{k}(n_{i}-1)\ln(S_{i}^{2})}{1+{\frac {1}{3(k-1)}}\left(\sum _{i=1}^{k}({\frac {1}{n_{i}-1}})-{\frac {1}{N-k}}\right)}}}$

with and , the pooled variance. ${\displaystyle N=\sum _{i=1}^{k}n_{i}}$ ${\displaystyle S_{p}^{2}={\frac {1}{N-k}}\sum _{i}(n_{i}-1)S_{i}^{2}}$

The test statistic is -approximatively -distributed with degrees of freedom . I.e. the null hypothesis is rejected if the realization of the test statistic is greater than . The Bartlett test is a modification of a corresponding likelihood ratio test . ${\displaystyle X^{2}}$ ${\displaystyle \chi _{k-1}^{2}}$ ${\displaystyle k-1}$ ${\displaystyle \chi _{k-1,\alpha }^{2}}$

Bartlett's test for sphericity

As part of the factor analysis, he checks whether the correlation matrix of the observed variables in the population is the same as the identity matrix . If this null hypothesis cannot be rejected, the factor analysis should not be carried out.

requirement

The test assumes a multivariate normal distribution of the data and reacts sensitively to the violation of this requirement.

Hypotheses

The test tests the null hypothesis that the correlation matrix is equal to the identity matrix against the alternative hypothesis that the two are not equal: ${\displaystyle R}$ ${\displaystyle E}$

${\displaystyle H_{0}:R=E\,}$ against ${\displaystyle H_{1}:R\neq E}$

Test statistics

If is the number of variables for which the correlation matrix was calculated, then the test statistic is defined as ${\displaystyle p}$ ${\displaystyle R}$

${\displaystyle X^{2}=-\left(n-1-{\frac {2p+5}{6}}\right)\log(|R|)}$

where is the number of observations and the determinant of . ${\displaystyle n}$ ${\displaystyle |R|}$ ${\displaystyle R}$

The test statistic is approximately -distributed with degrees of freedom. I.e. the null hypothesis is rejected if the realization of the test statistic is greater than . ${\displaystyle X^{2}}$ ${\displaystyle \chi _{p(p-1)/2}^{2}}$ ${\displaystyle p(p-1)/2}$ ${\displaystyle \chi _{p(p-1)/2,\alpha }^{2}}$

Individual evidence

1. ^ Maurice Bartlett: Properties of sufficiency and statistical tests . In: Proceedings of the Royal Statistical Society Series A . tape 160 , 1937, pp. 268-282 , doi : 10.1098 / rspa.1937.0109 , JSTOR : 96803 .
2. ↑ SPSS (2007), SPSS 16.0 Algorithms, SPSS Inc., Chicago, Illinois, p. 293.

Web links

• NIST page on Bartlett's test