A Monte Carlo Study of Seven Homogeneity of Variance Tests

Abstract

Problem statement:The decision by SPSS (now PASW) to use the unmodified Levene test to test homogeneity of variance was questioned. It was compared to six other tests. In total, seven homogeneity of variance tests used in Analysis Of Variance (ANOVA) were compared on robustness and power using Monte Carlo studies. The homogeneity of variance tests were (1) Levene, (2) modified Levene, (3) Z-variance, (4) Overall-Woodward Modified Z-variance, (5) O'Brien, (6) Samiuddin Cube Root and (7) F-Max. Approach: Each test was subjected to Monte Carlo analysis through different shaped distributions: (1) normal, (2) platykurtic, (3) leptokurtic, (4) moderate skewed and (5) highly skewed. The Levene Test is the one used in all of the latest versions of SPSS. Results: The results from these studies showed that the Levene Test is neither the best nor worst in terms of robustness and power. However, the modified Levene Test showed very good robustness when compared to the other tests but lower power than other tests. The Samiuddin test is at its best in terms of robustness and power when the distribution is normal. The results of this study showed the strengths and weaknesses of the seven tests. Conclusion/Recommendations: No single test outperformed the others in terms of robustness and power. The authors recommend that kurtosis and skewness indices be presented in statistical computer program packages such as SPSS to guide the data analyst in choosing which test would provide the highest robustness and power.

Comparing the Performance of Approaches for Testing the Homogeneity of Variance Assumption in One-Factor ANOVA Models

“…Overall, five HOV tests maintain adequate Type I error control better than the others across all the conditions, including the Ramsey conditional test, the O’Brien test, the BF test, the Bootstrap BF test, and the Levene test with squared deviations. The robustness of the Ramsey conditional, O’Brien, and BF tests confirmed findings of previous studies (Lee et al, 2010; Ramsey & Ramsey, 2007; Sharma & Kibria, 2013). The differences among the power distributions for these five robust HOV tests are subtle.…”

“…In terms of statistical power, Bartlett, Z -variance, Cochran’s C, F max , and O’Brien achieved high statistical power with the presence of normal or platykurtic distributions, with Cochran’s C having the highest power (Gorbunova & Lemeshko, 2012). Modified Z -variance had lower power than the Z -variance test (Lee et al, 2010). Although with normal or platykurtic distributions Levene had a lower power than those tests, it had higher power when distributions were leptokurtic and skewed.…”

“…Compared with the Bartlett test, the Levene test was less sensitive to departures of normality (Gorbunova & Lemeshko, 2012). But in this case, both the Bartlett and Levene tests were inferior to the O’Brien test that performed very well in terms of Type I error control across all population distributions, except for small sample size coupled with skewed distributions (Lee, Katz, & Restori, 2010).…”

“…F max test was reported to have adequate Type I error control with normal distributions when the group size was relatively small (i.e., 10), while with a larger group size (i.e., 30), Type I error rates were slightly lower than the nominal alpha level. It was extremely conservative with platykurtic distributions, whereas with leptokurtic and skewed distributions Type I error rates were inflated substantially (Lee et al, 2010). The performance of the Z -variance test was very similar to that of the F max test in terms of robustness.…”

Efficiency of Bartlett and Levenes Tests for Testing Homogeneity of Variance Under Varying Number of Replicates and Groups in One – Way ANOVA

“…The test statistic   distribution. This test has been commonly employed by many researchers and has proven to be very powerful ( [19]; [20]; [21]; [22]; [23]; [24]; [25]; [26]).…”