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
- 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 .
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.
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:
- against
Test statistics
If the groups have sample variances and sample sizes , then the test statistic is defined as
with and , the pooled variance.
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 .
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:
- against
Test statistics
If is the number of variables for which the correlation matrix was calculated, then the test statistic is defined as
where is the number of observations and the determinant of .
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 .
Individual evidence
- ^ 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 .
- ↑ SPSS (2007), SPSS 16.0 Algorithms, SPSS Inc., Chicago, Illinois, p. 293.