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1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.5. Quantitative Techniques

1.3.5.10.

Levene Test for Equality of Variances
Purpose:
Test for Homogeneity of Variances 		Levene's test ( Levene 1960) is used to test if k samples have equal variances. Equal variances across samples is called homogeneity of variance. Some statistical tests, for example the analysis of variance, assume that variances are equal across groups or samples. The Levene test can be used to verify that assumption.

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.

Definition The Levene test is defined as:
H0: sigma1 = simga2 = ... = sigmak
Ha: sigmai <> sigmaj    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:
    W = [(N-k)*SUM(i=1 to k)(N(i)*(ZBAR(i.) - ZBAR(..))**2]/ [(k-1)*SUM(i=1 to k)(SUM(j=1 to N(i))(Z(ij) - ZBAR(i.))**2]
where Zij can have one of the following three definitions:
  1. Z(ij) = |Y(ij) - YBAR(i.)|

    where is the mean of the ith subgroup.

  2. Z(ij) = |Y(ij) - YMEDIAN(i.)|

    where YMEDIAN is the median of the ith subgroup.

  3. Z(ij) = |Y(ij) - YTRIMMED(i.)|

    where YTRIMMED is the 10% trimmed mean of the ith subgroup.

ZBAR(i.) are the group means of the Zij and ZBAR(..) 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 Chi-Square(4) (i.e., skewed) distribution. Using the mean provided the best power for symmetric, moderate-tailed, distributions.

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: alpha
Critical Region: The Levene test rejects the hypothesis that the variances are equal if
    W > F(alpha,k-1,N-k)
where F(alpha,k-1,N-k) 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 F(alpha) is the upper critical value from the F distribution and F(1-alpha is the lower critical value. Note that this is the opposite of some texts and software programs. In particular, Dataplot uses the opposite convention.

Sample Output
 		Dataplot generated the following output for Levene's test using the GEAR.DAT data set (by default, Dataplot performs the form of the test based on the median): 
  
              LEVENE F-TEST FOR SHIFT IN VARIATION
                (CASE: TEST BASED ON MEDIANS)
 
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.
Interpretation of Sample Output We are testing the hypothesis that the group variances are equal. The output is divided into three sections.
  1. The first section prints the number of observations (N), the number of groups (k), and the value of the Levene test statistic.

  2. 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-alpha). 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.

  3. 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 alpha = 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 alpha = 0.10 level.
Output from other statistical software may look somewhat different from the above output.
Question Levene's test can be used to answer the following question:
  • Is the assumption of equal variances valid?
Related Techniques Standard Deviation Plot
Box Plot
Bartlett Test
Chi-Square Test
Analysis of Variance
Software The Levene test is available in some general purpose statistical software programs, including Dataplot.
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