Saturday, 12 December 2009

Levene Tests (Levene 1, Levene 2, Levene 3 and Levene 4)

Levene (1960) demonstrated that Bartlett’s test performs poorly when the homogeneity assumption is violated. A number of alternative tests were introduced as possible alternatives to Bartlett’s test. Four variants of Levene’s test (Levene 1, Levene 2, Levene 3 and Levene 4) are investigated in this paper. The Levene’s tests are known to be homogeneity-of-variance tests that exhibit improved result (over many of the other tests) when conditions of non-normality exist. The Levene tests involve computing the absolute difference between the value of that case and its cell mean (Note: some of the variants of Levene use the median in place of the mean). It then performs a one-way analysis of variance on the differences of those values.

Each of the four Levene tests used is defined completely (with derivations) in Chapter 2. The Levene 1 test (Lev-1) necessitates substituting each observation in a group by the absolute deviation from the group mean. The substitute observations, clip_image002are then treated as raw observations in the standard ANOVA test. The Levene 1 test with the new values of clip_image004is thus given by:

clip_image006 (F 1.8)

The assumptions that must be met for the Levene tests to hold true include:

1. The samples from the populations under consideration are independent.

2. The populations under consideration are approximately normally distributed.

The first assumption is validated by ensuring that the samples used have been selected independently of one another. The second assumption can be checked using a set of side-by-side boxplots or Q-Q plots. These plots are used by treatment to assess normality. Several of the other tests of normality can also be used. It should be noted that at least one of the samples must have 3 or more observations or else the Levene’s statistic will be undefined as the denominator will equal zero in the event that a sample has only 1 or 2 observations.

The notation used in the Levene test is defined as follows:

The testing for the equality of means of K populations given samples clip_image008given for (F 1.4) and (F 1.5).

  • K = number of sample sets used,
  • xij = sample observation j from sample set i (j = 1, 2,…, ni & i = 1, 2,…, K),
  • ni = number of observations from treatment i (at least one ni must be 3 or more),
  • clip_image010= total number of data in all samples (overall size of combined samples),
  • clip_image012 = mean of sample data from treatment i,
  • clip_image014 This is the absolute deviation of observation j from treatment ith mean,
  • clip_image016is the average of the ni absolute deviations from treatment i,
  • clip_image018is the average value of all n absolute deviations.

The test procedure for Levene’s test[1] is:

Step 1: Validate that the assumptions (above) hold true.

Step 2: State the null and alternative hypothesis. The Null hypothesis, clip_image020is tested against the alternative, clip_image022.

Step 3: Select a significance level (in this paper a value of clip_image024has been used)

Step 4: Select the critical value and rejection region to be used.

Classical Approach

p-Value Approach

Critical Value

clip_image026

Not Valid

Rejection Region

clip_image028

clip_image030

Step 5: Compute the Levene’s statistic. In this example, the Levene-1 test is given.

clip_image006[1] (F 1.9)

Step 6: In the case where the value of the test statistic, clip_image032, lies within the rejection region or in the event that the clip_image030[1], we reject the null hypothesis,clip_image034. Otherwise, if these conditions are not met, we fail to reject the null hypothesis,clip_image034[1].

Step 7: The final step is to state the conclusion:

Rejectclip_image034[2]: At the α level of significance, there is enough evidence to conclude clip_image036.

Fail to Reject clip_image034[3]: At the α level of significance, there is not enough evidence clip_image036[1].

Levene (1960) proposed four separate tests. Levene’s Tests 2 to 4 are defined below with the Levene 1 test in the section above. It has been demonstrated (Brown & Forsythe, 1974) that the Levene 1 test is robust when distribution is asymmetric and values of clip_image038are used.

Levene 2

The Levene 2 test involves the application of the ANOVA procedure to a set of transformed observations. These observations are derived from the taking the following clip_image040value:

clip_image042 (F 1.10)

Using the Levene 2 test, the null hypothesis (F 1.1) will be rejected were the Levene 2 statistic exceeds the clip_image044percentile of an F-distribution with (K-1) and (n-K) degrees of freedom. In all Monte Carlo simulations conducted, the Levene 2 test showed low testing power.

Levene 3

The Levene 3 test is based on another set of transformed variables. In this case, the clip_image040[1]value is given by:

clip_image046 (F 1.11)

As with the Levene 2 test, an ANOVA procedure is applied to the transformed variables. Using the Levene 3 test, the null hypothesis (F 1.1) will be rejected were the Levene 3 statistic exceeds the clip_image044[1]percentile of an F-distribution with (K-1) and (n-K) degrees of freedom.

Levene 4

The Levene 4 test is based on yet another set of transformed variables. In this case, the clip_image040[2]value is given by:

clip_image048 (F 1.12)

Again, the Levene 4 test has an ANOVA procedure applied to the transformed variables. Using the Levene 4 test, the null hypothesis (F 1.1) will be rejected were the Levene 4 statistic exceeds the clip_image044[2]percentile of an F-distribution with (K-1) and (n-K) degrees of freedom.


[1] N.B for each of these steps.

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