Performance Test Levene’s Tooltips
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Brown and Forsythe (Brown and Forsythe 1974) suggested a modification to Levene’s test statistic for the equality of variances. Simply, Levene proposed a one-way ANOVA of deviations of group observations from the group center. Where Levene used absolute differences from the mean, \(z_{ij} = |y_{ij} - \bar{y}_{i .}|\), Brown and Forsythe recommend using differences from the group median, and found that this measure is more robust when group errors are not normally distributed (i.e. long-tailed). This method is also more robust when data are skewed (Carroll and Schneider 1985, Nordstokke and Zumbo (2007)). Conover, Johnson and Johnson (Conover, Johnson, and Johnson 1981), using extensive simulations, compared the Brown and Forsythe statistic with other test of homogeneity of variance tests and found this method to be the best of the non-ranked based tests.
Since Levene initially proposed a family of tests, including \(\sqrt{z_{ij}}\) and \(\log{z_{ij}}\), Brown and Forsythe’s test is commonly referred to as Levene (Med) (Kuehl 2000, Schabenberger and Pierce (2001)), or more simply state the modification to Levene’s test (Milliken and Johnson 1992). Thus, the SAS option for Levene (Med) test is hovtest=bf.
To illustrate, we use data from (Milliken and Johnson 1992, Table 2.1) as an example:
Table2.1 <- data.frame(
trt=as.factor(c(1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4)),
reps=as.factor(c(1,2,3,4,5,6,7,8,1,2,3,4,5,6,7,8,1,2,3,4,5,6,7,8,1,2,3,4,5,6,7,8)),
Score = c(1,8,9,9,4,0,1,NA,12,10,13,13,12,10,NA,NA,12,4,11,7,8,10,12,5,13,14,14,17,11,14,13,14)
)Compute the median for each treatment group and analyze the absolute values of the differences as a response.
z.med <- tapply(Table2.1$Score,list(Table2.1$trt),median, na.rm=TRUE)
Table2.1$z.med <- z.med[Table2.1$trt]
summary(aov(abs(z.med-Score) ~ trt, data=subset(Table2.1,!is.na(Table2.1$Score))))## Df Sum Sq Mean Sq F value Pr(>F)
## trt 3 31.38 10.459 5.494 0.00486 **
## Residuals 25 47.59 1.904
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1We can also compute Levene’s orginal by
z.mean <- tapply(Table2.1$Score,list(Table2.1$trt),mean, na.rm=TRUE)
Table2.1$z.mean <- z.mean[Table2.1$trt]
summary(aov(abs(z.mean-Score) ~ trt, data=Table2.1))## Df Sum Sq Mean Sq F value Pr(>F)
## trt 3 30.60 10.201 6.966 0.00145 **
## Residuals 25 36.61 1.464
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 3 observations deleted due to missingnessWe can compare this with the leveneTest function in R library car
library(car)## Warning: package 'car' was built under R version 3.3.2leveneTest(Score ~ trt,data=Table2.1) #default is center=median## Levene's Test for Homogeneity of Variance (center = median)
## Df F value Pr(>F)
## group 3 5.4943 0.004861 **
## 25
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1leveneTest(Score ~ trt,data=Table2.1,center=mean) #original Levene## Levene's Test for Homogeneity of Variance (center = mean)
## Df F value Pr(>F)
## group 3 6.9664 0.001454 **
## 25
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1and using SAS:
data Milliken21;
input trt reps Score;
datalines;
1 1 1
1 2 8
1 3 9
1 4 9
1 5 4
1 6 0
1 7 1
2 1 12
2 2 10
2 3 13
2 4 13
2 5 12
2 6 10
3 1 12
3 2 4
3 3 11
3 4 7
3 5 8
3 6 10
3 7 12
3 8 5
4 1 13
4 2 14
4 3 14
4 4 17
4 5 11
4 6 14
4 7 13
4 8 14
;
/* Test for homogeneity of variance */
proc glm data=Milliken21;
class trt;
model Score = trt;
means trt / hovtest=bf; /*Levene (Med)*/
means trt / hovtest=levene; /*Levene Mean*/
run;which produces the results
The SAS System 10:25 Thursday, August 31, 2017 1
The GLM Procedure
Class Level Information
Class Levels Values
trt 4 1 2 3 4
Number of Observations Read 29
Number of Observations Used 29
The GLM Procedure
Dependent Variable: Score
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 3 347.7842775 115.9280925 14.91 <.0001
Error 25 194.4226190 7.7769048
Corrected Total 28 542.2068966
R-Square Coeff Var Root MSE Score Mean
0.641424 28.78028 2.788710 9.689655
Source DF Type I SS Mean Square F Value Pr > F
trt 3 347.7842775 115.9280925 14.91 <.0001
Source DF Type III SS Mean Square F Value Pr > F
trt 3 347.7842775 115.9280925 14.91 <.0001
The GLM Procedure
Brown and Forsythe's Test for Homogeneity of Score Variance
ANOVA of Absolute Deviations from Group Medians
Sum of Mean
Source DF Squares Square F Value Pr > F
trt 3 31.3762 10.4587 5.49 0.0049
Error 25 47.5893 1.9036
The GLM Procedure
Level of ------------Score------------
trt N Mean Std Dev
1 7 4.5714286 4.03555625
2 6 11.6666667 1.36626010
3 8 8.6250000 3.11390889
4 8 13.7500000 1.66904592
The GLM Procedure
Levene's Test for Homogeneity of Score Variance
ANOVA of Squared Deviations from Group Means
Sum of Mean
Source DF Squares Square F Value Pr > F
trt 3 698.5 232.8 7.36 0.0011
Error 25 791.2 31.6468
The GLM Procedure
Level of ------------Score------------
trt N Mean Std Dev
1 7 4.5714286 4.03555625
2 6 11.6666667 1.36626010
3 8 8.6250000 3.11390889
4 8 13.7500000 1.66904592
The GLM Procedure
Class Level Information
Class Levels Values
trt 4 1 2 3 4
Number of Observations Read 29
Number of Observations Used 29
The GLM Procedure
Dependent Variable: Score
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 3 347.7842775 115.9280925 14.91 <.0001
Error 25 194.4226190 7.7769048
Corrected Total 28 542.2068966
R-Square Coeff Var Root MSE Score Mean
0.641424 28.78028 2.788710 9.689655
Source DF Type I SS Mean Square F Value Pr > F
trt 3 347.7842775 115.9280925 14.91 <.0001
Source DF Type III SS Mean Square F Value Pr > F
trt 3 347.7842775 115.9280925 14.91 <.0001
The GLM Procedure
Brown and Forsythe's Test for Homogeneity of Score Variance
ANOVA of Absolute Deviations from Group Medians
Sum of Mean
Source DF Squares Square F Value Pr > F
trt 3 31.3762 10.4587 5.49 0.0049
Error 25 47.5893 1.9036
The GLM Procedure
Level of ------------Score------------
trt N Mean Std Dev
1 7 4.5714286 4.03555625
2 6 11.6666667 1.36626010
3 8 8.6250000 3.11390889
4 8 13.7500000 1.66904592
The SAS System 10:25 Thursday, August 31, 2017 11
The GLM Procedure
Levene's Test for Homogeneity of Score Variance
ANOVA of Squared Deviations from Group Means
Sum of Mean
Source DF Squares Square F Value Pr > F
trt 3 698.5 232.8 7.36 0.0011
Error 25 791.2 31.6468
The GLM Procedure
Level of ------------Score------------
trt N Mean Std Dev
1 7 4.5714286 4.03555625
2 6 11.6666667 1.36626010
3 8 8.6250000 3.11390889
4 8 13.7500000 1.66904592Brown, Morton B, and Alan B Forsythe. 1974. “Robust Tests for the Equality of Variances.” Journal of the American Statistical Association 69 (June): 364–67.
Carroll, Raymond J, and Helmut Schneider. 1985. “A Note on Levene’s Test for Equality of Variances.” Statistics Probability Letters 3: 191–94.
Conover, W J, Mark E Johnson, and Myrle M Johnson. 1981. “A Comparative Study of Tests for Homogeneity of Variances, with Applications to the Outer Continental Shelf Bidding Data.” Technometrics 23 (November): 351–61.
Kuehl, R O. 2000. Design of experiments: statistical principles of research design and analysis. 2nd ed. Brooks/Cole.
Milliken, George A., and Dallas E. Johnson. 1992. Analysis of Messy Data. 1st ed. Vol. Designed Experiments. Chapman; Hall/CRC.
Nordstokke, David W, and Bruno D Zumbo. 2007. “A Cautionary Tale About Levenes Tests for Equal Variances.” Journal of Educational Research and Policy Studies 7.
Schabenberger, Oliver, and Francis J. Pierce. 2001. Contemporary Statistical Models for the Plant and Soil Sciences [Hardcover]. CRC Press.