Levene test

Distribution of net income in Germany 2008 ( ALLBUS ) by gender and month of birth of the respondent.

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 :	${\ displaystyle H_ {0} \ colon \ sigma _ {1} ^ {2} = \ sigma _ {2} ^ {2} = \ ldots = \ sigma _ {k} ^ {2}}$
Alternative hypothesis :	${\ displaystyle H_ {1} \ colon \ sigma _ {i} ^ {2} \ neq \ sigma _ {j} ^ {2}}$ for at least one group pair with ${\ displaystyle i, j}$ ${\ displaystyle i \ neq j}$

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::leveneTestin R :

The Levene test by gender gives a p-value less than and is therefore highly significant: ${\ displaystyle 2 {,} 2 \ times 10 ^ {- 16}}$

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%: ${\ displaystyle 0 {,} 076}$

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 ${\ displaystyle X_ {ji}}$ ${\ displaystyle j = 1, \ dots, k}$ ${\ displaystyle i = 1, \ dots, n_ {j}}$

{\ displaystyle Y_ {ji} = | X_ {ji} - {\ bar {X}} _ {j} |}

with number of groups (samples), the number of observations in the group, and the sample mean of the group . Then is the test statistic ${\ displaystyle k}$ ${\ displaystyle n_ {j}}$ ${\ displaystyle j}$ ${\ displaystyle {\ bar {X}} _ {j}}$ ${\ displaystyle j}$

{\ displaystyle L = {\ frac {{\ frac {1} {k-1}} \ sum _ {j = 1} ^ {k} n_ {j} ({\ bar {Y}} _ {j} - {\ bar {Y}}) ^ {2}} {{\ frac {1} {nk}} \ sum _ {j = 1} ^ {k} \ sum _ {i = 1} ^ {n_ {j} } (Y_ {ji} - {\ bar {Y}} _ {j}) ^ {2}}}}

approximately -distributed with the number of all observations: ${\ displaystyle F (k-1, nk)}$ ${\ displaystyle n}$

{\ displaystyle n = \ sum _ {j = 1} ^ {k} n_ {j}}

,

${\ displaystyle {\ bar {Y}}}$ the sample mean over all groups and the sample mean over group . ${\ displaystyle {\ bar {Y}} _ {j}}$ ${\ displaystyle j}$

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 ${\ displaystyle Y_ {ji}}$ ${\ displaystyle k}$ ${\ displaystyle X_ {ji}}$ ${\ displaystyle Y_ {ji}}$

{\ displaystyle {\ bar {Y}} _ {j} = {\ frac {1} {n_ {j}}} \ sum _ {i = 1} ^ {n_ {j}} Y_ {ji} = {\ frac {1} {n_ {j}}} \ sum _ {i = 1} ^ {n_ {j}} | X_ {ji} - {\ bar {X}} _ {j} |}

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. ${\ displaystyle Y_ {ji}}$

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). ${\ displaystyle Y_ {ji}}$ ${\ displaystyle \ approx}$

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.

[Levene1960-1] 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. .

[Janssen2007-2] Jürgen Janssen, Wilfried Laatz: Statistical data analysis with SPSS for Windows . 8th edition. Springer Verlag, 2007, p. 246 .

[3] Maxwell J. Roberts, Riccardo Russo: Student's Guide to Analysis of Variance . Routledge Chapman & Hall, 1999, ISBN 978-0-415-16565-5 , pp. 71 .

[4] 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 .