1.3. EDA Techniques
1.3.5. Quantitative Techniques
1.3.5.10. |
Levene Test for Equality of Variances |
Test for Homogeneity of Variances
Levene's test is an alternative to the Bartlett test. The Levene test is less sensitive than the Bartlett test to departures from normality. If you have strong evidence that your data do in fact come from a normal, or nearly normal, distribution, then Bartlett's test has better performance.
| H0: |
|
| Ha: |
for at least one pair
(i,j).
|
| Test Statistic: |
Given a variable Y with sample of size N
divided into k subgroups, where Ni
is the sample size of the ith subgroup, the Levene
test statistic is defined as:
are the group means of the Zij
and  is the overall mean of the
Zij.
The three choices for defining Zij determine the robustness and power of Levene's test. By robustness, we mean the ability of the test to not falsely detect unequal variances when the underlying data are not normally distributed and the variables are in fact equal. By power, we mean the ability of the test to detect unequal variances when the variances are in fact unequal.
Levene's original paper only proposed using the mean.
Brown and Forsythe
(1974)) extended Levene's test to use either the
median or the trimmed mean in addition to the mean.
They performed Monte Carlo studies that indicated that
using the trimmed mean performed best when the underlying
data followed a Cauchy distribution (i.e., heavy-tailed)
and the median performed best when the underlying data
followed a Although the optimal choice depends on the underlying distribution, the definition based on the median is recommended as the choice that provides good robustness against many types of non-normal data while retaining good power. If you have knowledge of the underlying distribution of the data, this may indicate using one of the other choices. |
| Significance Level: |
|
| Critical Region: |
The Levene test rejects the hypothesis that the
variances are equal if
 is the
upper critical value of the
F distribution
with k - 1 and N - k degrees of
freedom at a significance level of
.
In the above formulas for the critical
regions, the Handbook follows the convention that
|
is
the 
is
the 
is the 10% 
(i.e., skewed)
distribution. Using the mean provided the best power
for symmetric, moderate-tailed, distributions.
is the upper critical value from the F distribution
and 
is the lower critical value.
Note that this is the opposite of some
texts and software programs. In particular, Dataplot
uses the opposite convention.
LEVENE F-TEST FOR SHIFT IN VARIATION
(ASSUMPTION: NORMALITY)
1. STATISTICS
NUMBER OF OBSERVATIONS = 100
NUMBER OF GROUPS = 10
LEVENE F TEST STATISTIC = 1.705910
2. FOR LEVENE TEST STATISTIC
0 % POINT = 0.
50 % POINT = 0.9339308
75 % POINT = 1.296365
90 % POINT = 1.702053
95 % POINT = 1.985595
99 % POINT = 2.610880
99.9 % POINT = 3.478882
90.09152 % Point: 1.705910
3. CONCLUSION (AT THE 5% LEVEL):
THERE IS NO SHIFT IN VARIATION.
THUS: HOMOGENEOUS WITH RESPECT TO VARIATION.
- The first section prints the number of observations (N),
the number of groups (k), and the value of the Levene
test statistic.
- The second section prints the upper
critical value of the F
distribution corresponding to various significance levels.
The value in the first column, the confidence level of the
test, is equivalent to 100(1-). We reject the null hypothesis at that
significance level if the value of the Levene F test statistic
printed in section one is greater than the critical value
printed in the last column.
- The third section prints the conclusion for a 95% test.
For a different significance level, the appropriate
conclusion can be drawn from the table printed in section
two. For example, for
= 0.10, we look at the row for 90% confidence and compare
the critical value 1.702 to the Levene test statistic 1.7059.
Since the test statistic is greater than the critical value,
we reject the null hypothesis at the
= 0.10 level.
- Is the assumption of equal variances valid?
Box Plot
Bartlett Test
Chi-Square Test
Analysis of Variance
