An efficiency comparison of test statistics for testing homogeneity of variances for non-normally distributed data

Abstract

This research aimed to study and compare the efficiency of three test statistics for testing homogeneity of variances,
(Levene’s test, Brown-Forsythe’s test, and Figner-Killeen’s test) when data are not normally distributed. The number
of populations considered in this study was three groups and considered only when sample sizes of all groups were
equal. The distributions considered in this study were Gamma distributions ((α, β) = (2, 2), (3, 2), (4, 2), (6, 2), (10,
2))), Weibull distributions ((α, β) = (2, 6.105), (2, 7.478), (2, 8.635), (2, 10.575), (2, 13.652)), Logistic distributions
((µ, s) = (2, 1.559), (2, 1.910), (2, 2.205), (2, 2.701), (2, 3.487)), and Uniform distributions ((a, b) = (0, 9.798),
(0, 12), (0, 13.856), (0, 16.971), (0, 21.909)). The criteria used to compare the efficiency of proposed test statistics were the ability to control the probability of type 1 error, robustness, and power of a test. In this study, the test statistic that could control the probability of type 1 error and had the highest empirical power was concluded to be the best test statistic. The results showed that Levene’s test performed better than Brown-Forsythe’s test and Figner-Killeen’s test in almost all cases when data followed Logistic and Uniform distributions because it could control the probability of type 1 error and had the higher empirical power. In the case of Weibull distribution, Levene’s test still performed better than other tests when the sample size of each group was 30 or more. Finally, Figner-Killeen’s was the best in all cases when data followed the Gamma distribution.