Levene test
In statistics , the Levene test is a significance test that checks for equality of variances ( homoscedasticity ) of two or more populations (groups). The Brown – Forsythe test is derived from the Levene test. It comes from Howard Levene .
Similar to the Bartlett test, the Levene test tests the null hypothesis that all group variances are equal. The alternative hypothesis is that at least one group pair has unequal variances ( heteroscedasticity ):
Null hypothesis : | |
Alternative hypothesis : | for at least one group pair with |
If the p-value of the test is below a predetermined level, then the differences in the variance of the samples are over random (significant) and the null hypothesis of equality of variance can be rejected.
example
The graphic above shows the distribution of net income by gender and month of birth. The output of car::leveneTest
in R :
- The Levene test by gender gives a p-value less than and is therefore highly significant:
Levene’s Test for Homogeneity of Variance Df F value Pr(>F) group 1 106.09 < 2.2e-16 *** 2404 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
With such a p-value it can be assumed that the variances in the population are different. The null hypothesis of equal variances is accordingly rejected.
- The Levene test by month of birth gives a p-value of and is not significant at a given significance level of 5%:
Levene’s Test for Homogeneity of Variance Df F value Pr(>F) group 11 1.6621 0.076. 2384 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Test statistics
Are ( and ) the sample variables and
with number of groups (samples), the number of observations in the group, and the sample mean of the group . Then is the test statistic
approximately -distributed with the number of all observations:
- ,
the sample mean over all groups and the sample mean over group .
The test statistic is identical to the test statistic of the simple analysis of variance (test for equality of group means). By transforming from to the group mean values are
robust estimators of the group variances. The assumption of normal distribution for the analysis of variance does not apply, but they often have a right skewed distribution for which the analysis of variance can be used.
Brown – Forsythe test
In the Brown – Forsythe test, the group median is used instead of the group mean when calculating . In order to obtain a good test strength, the location parameter must be selected depending on the underlying distribution. Brown and Forsythe showed in simulation studies that the mean is a good choice if the distribution is symmetrical and has "normal" distribution ends ( excess 0), e.g. B. is similar to a normal distribution. The median should be used when the distributions are severely skewed and the trimmed mean when the distribution has severe distribution ends (excess <0).
Individual evidence
- ^ Howard Levene: Robust tests for equality of variances . In: Ingram Olkin, Harold Hotelling et al. (Ed.): Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling . Stanford University Press, 1960, pp. 278-292. .
- ^ Jürgen Janssen, Wilfried Laatz: Statistical data analysis with SPSS for Windows . 8th edition. Springer Verlag, 2007, p. 246 .
- ^ Maxwell J. Roberts, Riccardo Russo: Student's Guide to Analysis of Variance . Routledge Chapman & Hall, 1999, ISBN 978-0-415-16565-5 , pp. 71 .
- ^ Morton B. Brown, Alan B. Forsythe: Robust tests for equality of variances . In: Journal of the American Statistical Association . tape 69 , 1974, pp. 364-367 , doi : 10.1080 / 01621459.1974.10482955 .
literature
- Biostatistics: An Introduction for Life Scientists. (2008). Munich: Pearson studies. Pp. 150-154.