LEVENE TEST

Name:

LEVENE TEST Type:

Analysis Command Purpose:

Description:

The assumption of homogeneous variances arises in other contexts in addition to analysis of variance. Levene's test can be applied in these cases as well.

The Levene test is an alternative to the Bartlett test. Although it is more commonly used, the Bartlett test is known to be sensitive to departures from normality. The Levene test is less sensitive to non-normality than the Bartlett test.

The Levene test is defined as:

H₀: \( \sigma_{1} = \sigma_{2} = \ldots = \sigma_{k} \)

H_a: \( \sigma_{i} \neq \sigma_{j} \) for at least one pair (i,j).

Test Statistic: Given a variable Y with sample of size N divided into k sub-groups, where N_i is the sample size of the i-th sub-group, the Levene test statistic is defined as:
\( W = \frac{(N-k)} {(k-1)} \frac{\sum_{i=1}^{k}N_{i}(\bar{Z}_{i.}- \bar{Z}_{..})^{2} } {\sum_{i=1}^{k}\sum_{j=1}^{N}(Z_{ij}-\bar{Z}_{i.})^{2} } \)
where Z_ij can have one of the following three definitions:

\( Z_{ij} = |Y_{ij} - \bar{Y}_{i.}| \)
where \( \bar{Y}_{i.} \) is the mean of the i-th subgroup.

\( Z_{ij} = |Y_{ij} - \tilde{Y}_{i.}| \)
where \( Z_{ij} = |Y_{ij} - \tilde{Y}_{i.}| \) is the median of the i-th subgroup.

\( Z_{ij} = |Y_{ij} - \bar{Y}_{i.}'| \)
where \( \bar{Y}_{i.}' \) is the 10% trimmed mean of the i-th subgroup.
\( \bar{Z}_{i.} \) are the group means of the Z_ij and \( \bar{Z}_{..} \) is the overall mean of the Z_ij.
The three choices for defining Z_ij determine the robustness and power of Levene's test. By robustness, we mean the ability of the test to not falsely detect non-homogeneous groups when the underlying data is not normally distributed and the groups are in fact homogeneous. By power, we mean the ability of the test to detect non-homogeneous groups when the groups are in fact non-homogenous.
The definition based on the median is recommended as the choice that provides good robustness against many types of non-normal data but retains good power.

Significance Level: \( \alpha \) (typically 0.05).

Critical Region: The Levene test rejects the hypothesis that the variances are homogeneous if

\( W > F_{(1 - \alpha,k-1,N-k)} \)

where \( F_{(1 - \alpha,k-1,N-1)} \) is the upper critical value of the F distribution with k - 1 and N - 1 degrees of freedom at a significance level of \( \alpha \).

Syntax 1:

This syntax computes the median based Levene test.

Syntax 2:

This syntax computes the median based Levene test.

Syntax 3:

This syntax computes the mean based Levene test.

Syntax 4:

This syntax computes the trimmed mean based Levene test. It trims the lowest 10% and the highest 10% of the data.

Examples:

Note:

The various values printed by the LEVENE TEST command are saved as parameters that can be used later by the analyst. Enter the command STATUS PARAMETERS after the LEVENE TEST command to see a list of the saved parameters. Note:

The HOMOGENEITY PLOT is a graphical technique for testing for unequal variances. Default:

The default is to to compute the Levene test based on group medians. Synonyms:

None Related Commands:

BARTLETT TEST	= Compute Bartlett's test.
HOMOSCEDASTICITY PLOT	= Plot group standard deviations against group means.
CONFIDENCE LIMITS	= Compute the confidence limits for the mean of a sample.
F TEST	= Performs a two-sample F test.
T TEST	= Performs a two-sample t test.
CHI-SQUARE TEST	= Performs a one sample chi-square test that the standard deviation is equal to a given value.
STANDARD DEVIATION	= Computes the standard deviation of a variable.

Reference:

Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling

Applications:

Analysis of Variance, Regression Implementation Date:

1998/5 Program:

Dataplot generated the following output:

 
       **************************
       **      LEVENE TEST Y X **
       **************************
            Levene F-Test for Shift in Variation
               (Case: Test Based on Medians)
 
Response Variable: Y
Group-ID Variable: X
 
H0: Homogeneous Variances
Ha: Variances Are Not Homogeneous
 
Summary Statistics:
Total Number of Observations:                        45
Number of Groups:                                     3
 
Levene Test Statistic Value:                   10.88618
CDF of Test Statistic:                          0.99984
P-Value:                                        0.00016
 
 
Percent Points of the Reference Distribution
-----------------------------------
  Percent Point               Value
-----------------------------------
            0.0    =          0.000
           50.0    =          0.705
           75.0    =          1.433
           90.0    =          2.434
           95.0    =          3.220
           97.5    =          5.149
           99.0    =          5.149
           99.9    =          8.179
 
Conclusions (Upper 1-Tailed Test)
----------------------------------------------
  Alpha    CDF   Critical Value     Conclusion
----------------------------------------------
    10%    90%            2.434      Reject H0
     5%    95%            3.220      Reject H0
   2.5%  97.5%            5.149      Reject H0
     1%    99%            5.149      Reject H0