Levene's Test

The Modified Levene Test Supplement to Section 6.4 Brian Habing – University of South Carolina Last Updated: July 16, 2004 While the F -tests -tests for regression and ANOVA are fairly robust, the standard F  test  test for two variances is incredibly non-robust non-robust to lack lack of normality. normality. The F -max -max test for testing equality of several variances (described on pg. 238) iiss similarly non-robust. non-robust. Two major reasons for this can be seen by simply examining the formula for the variance: n

∑ ( y i −  y )

2

s =

2

i =1

n −1

A single outlier outlier will cause the mean to change greatly and the squaring squaring amplifies amplifies the effect. An alternative would be to use some procedure that replaced the mean with median, and squaring with absolute values. The difficulty in doing doing this directly is that the calculus that underlies underlies the distributional distributional theory we commonly use becomes very difficult for either medians or absolute values. The Modified Levene test (called the Brown and Forsythe test by SAS) begins by considering what it means for different populations populations to have equal standard deviations. deviations. If populations populations have the same standard deviation, deviation, then the average deviation from the the center of each population should be be the same. In  |yi-median( y  y)| should be equal for  particular, the average of the  y for each population. population. The test is is constructed constructed  by calculating this this “absolute deviation deviation from the sample sample median” for each observation, observation, and then using using ANOVA to test that the means of this quantity are the same for all of the populations. populations. This is worked out  below for the example example discussed on page page 238 (data on page page 227).

Sample from Population… 1 2 3 4

Observation

Sample Median

|Observation –Median|

934

1041

1028

935

47.5

59.5

46.5

46.5

880

963

924

946

55

28

11

11

987

951

976

840

23.5

12.5

12.5

123.5

992

1143

1140

1191

149.5

1.5

1.5

49.5

981.5 935 963.5 1066

DATA deviations; INPUT sample $ deviations @@; CARDS; 1 47.5 1 59.5 1 46.5 2 55 2 28 2 11 3 23.5 3 12.5 3 12.5 4 149.5 4 1.5 4 1.5 ;

1 2 3 4

46.5 11 123.5 49.5

PROC GLM DATA=deviations; CLASS sample; MODEL deviations = sample; RUN; The GLM Procedure Dependent Variable: deviations Sum of Source

DF

Squares

Mean Square

F Value

Pr > F

Model Error

3 12

1538.18750 24740.75000

512.72917 2061.72917

0.25

0.8607

Corrected Total

15

26278.93750

The test of H0: σ12=σ22=σ32=σ42 versus the alternate that at least one is different gives an F-statistic of 0.25 for 3 and 12 degrees of freedom, resulting in a p-value of 0.8607. We would therefore fail to reject the null hypothesis that the variances were equal. (Note that these values are different than those shown on page 239, where the means were used instead of the medians). The commands to carry this out from the original data using SAS are given below. The HOVTEST stands for “homogeneity of variances test” and the BF stands for “Brown and Forsythe”. DATA original; INPUT sample $ values @@; CARDS; 1 934 1 1041 1 2 880 2 963 2 3 987 3 951 3 4 992 4 1143 4 ;

1028 924 976 1140

1 2 3 4

935 946 840 1191

PROC GLM DATA=original; CLASS sample; MODEL values = sample; MEANS sample / HOVTEST=BF; RUN; The GLM Procedure Brown and Forsythe's Test for Homogeneity of values Variance ANOVA of Absolute Deviations from Group Medians

Source sample Error

DF

Sum of Squares

Mean Square

F Value

Pr > F

0.25

0.8607

3

1538.2

512.7

12

24740.8

2061.7

2