# Modifications of Levene’s and O’Brien’s Tests for Testing the Homogeneity of Variance Based on Median and Trimmed Mean

### Abstract

The propose of this paper is to compare the efficiency of Levene’s test with absolute deviation with the alternative tests consists of Brown-Forsythe test, modified Levene’s test based on trimmed mean, three Levene’s tests with squared deviation and three O’Brien’s tests. In total, nine homogeneity of variance tests used in ANOVA were compared on probability of type I error (also robustness) and power of the test (also power). These tests are analyzed through four different shaped distributions contain normal distribution, Uniform distribution, Logistic distribution, and gamma distribution. The results from these studies showed that Levene’s test with absolute deviation is neither the best nor worst in terms of robustness and power. It is the best choice for logistic distribution. There are better tests that could be used, and some were more preferable depending on the distributional shape of data. For normal distribution, modified Levene’s test with squared deviation based on median is the best choice for equal sample size. For uniform distribution, modified O’Brien’s test based on median is the best choice for small sample size both equal and unequal. For gamma distribution, Levene’s test with absolute deviation based on trimmed mean is the best choice for both small and medium sample sizes whereas modified Levene’s test with squared deviation based on median is the best choice same as Levene’s test with absolute deviation based on trimmed mean and Levene’s test with absolute deviation for large sample size.

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