Academic journal article North American Journal of Psychology

A Monte Carlo Study Comparing the Levene Test to Other Homogeneity of Variance Tests

Academic journal article North American Journal of Psychology

A Monte Carlo Study Comparing the Levene Test to Other Homogeneity of Variance Tests

Article excerpt

The impetus for the current study was from a student inquiry into the Levene Test that is used in the very popular statistical package SPSS (Statistical Package for the Social Sciences) to test for homogeneity of variance prior to conducting tests of the equality of means. The student had read in David Howell's book Statistical Methods for Psychology, 5th Edition (Howell, 2002) that the Levene test may not be the best one available. The student wanted to know what research, if any, led to SPSS's decision to use this particular test for homogeneity of variance that is used in the independent samples t-test and one-way ANOVA. A subsequent literature search produced some research on the Levene Test (Brown & Forsyth, 1974; Levene, 1960; Tomarken & Serlin, 1986). A further search found other tests of homogeneity of variance in studies by Overall and Woodward (1974, 1976); O'Brien, (1981), and Levy (1975) that may have been better choices than the Levene Test.

Brown and Forsyth (1974) compared the original Levene Test to two modifications of it. The original Levene Test used sample means. The modified Levene tests introduced by Brown and Forsyth (1974) used the median and the trimmed mean. They demonstrated through Monte Carlo studies that the median and the trimmed means outperformed the original test when the homogeneity of variance assumption was violated.

Of particular interest is the modified Z-variance test presented by Overall and Woodward (1976). Overall and Woodward had compared the robustness and power of this modification against four other homogeneity-of-variance tests: (1) Z-variance unmodified, (2) Wilson-Hilferty (3) Bartlett and (4) Box. Using a series of Monte Carlo studies, Overall and Woodward (1976) demonstrated the superiority of the modified Z-variance test over the other four tests. Unfortunately, the Overall-Woodward modification of the Z-variance test is not well known. This modification appears only in a technical report that may no longer be available or easily accessible. A Google search for this modification did not produce any results.

The O'Brien Test is also mentioned by Howell (2003) but little research could be found on it. From O'Brien's article (O'Brien, 1981) this test appears promising. With all of these more complicated formulas developed to measure homogeneity of variance, there is a simple one that will also be used in this study. The Fmax test developed by Hartley (1950) is very simple, involving no more than computing the ratio of the greatest subgroup variance and the smallest subgroup variance.

In this study, all five homogeneity of variance tests were compared using a Monte Carlo approach. Of particular interest are the Overall-Woodward modification of the Z-variance test and the O'Brien Test. A major goal of the present study is to evaluate just how good the original Levene test is when compared to these other alternatives. "Goodness" of each test is determined by examining the robustness and power for each test. Should SPSS and other statistical packages consider using other tests along with the Levene?

The Levene Test

In 1960, Levene proposed an alternative method to the Bartlett Test (Bartlett, 1937) for testing the assumption of homogeneity of variance for the independent sample t-test and ANOVA designs. The Bartlett test works well for data that are normally or approximate normally distributed. The Bartlett test does not fare well for data that follow a leptokurtic or skewed distribution (Overall & Woodward, 1974). According to Levene (1960), the test he proposed was less sensitive to departures from normality. This says that the Levene Test had fewer Type 1 errors than the Bartlett Test for distributions that were aberrant from normality.

The Levene Test is defined as the following:

[H.sub.0]: [[sigma].sup.2.sub.1] = [[sigma].sup.2.sub.2] = [[sigma].sup.2.sub.3] = ... = [[sigma].sup.2.sub. …

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