# 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.

### References

Bradley JV. Robustness?. Br J Math Stat Psychol. 1978; 31: 144-152.

Brown MB, Forsythe AB. Robust tests for equality of variances. J Am Stat Assoc. 1974; 69: 364-367.

Carroll RJ, Schneider H. A note on Levene’s tests for equality of variances. Stat Probab Lett. 1985; 3: 191-194.

Cochran WG. The distribution of the largest of a set of estimated variances as a fraction of their total. Ann Hum Genet. 1941; 11: 47-52.

Craparo RM. Significance level. In: Salkind NJ, editor. Encyclopedia of Measurement and Statistics 3. Thousand Oaks CA: SAGE Publications; 2007, 889-891.

Game PA, Winkler H, Probert DA. Robust tests for homogeneity of variance. Educ Psychol Meas. 1972; 32: 887-910.

Gastwirth JL, Gel YR, Miao W. The impact of Levene’s test of equality of variances on statistical theory and practice. Stat Sci. 2010; 24: 343-360.

Gogoi P, Gogoi B. Multi-sample scale tests for comparing scale parameters with equal and unequal location differences. Int J Adv Res. 2015; 2: 26-31.

Gorbunova AA, Lemeshko BY. Application of parametric homogeneity of variances tests under violation of classical assumption. Proceedings of 2nd Stochastic Modeling Techniques and Data Analysis International Conference; 2012 June 5-8, Chania Crete Greece. 2012.

Hanif HM, Norazan MR, Rasmani KA, Nor AMG. Enhancing hotelling T2 control chart performance with decile and trimmed means. ICIOMS2013: Proceedings of the International Conference on Information, Operations Management and Statistics; 2013 Sep 1-3; Malaysia. Australia: IEOM Research Solutions Pty Ltd; 2013, 1-7.

Lee HB, Katz GS, Restori AF. A monte carlo study of seven homogeneity of variance tests. J Math Stat. 2010; 6: 359-366.

Levene H. Robust tests for equality of variances. In: Olkin I, editor. Contributions to Probability and Statistics. Palo Alto. CA: Stanford: University Press; 1960, 278-292.

Lu MH, Yiğit S, Mollaoğulla A, Genç S, Mendeş M. Influence of using alternative means on type-I error rate in the comparison of independent groups. JAPS. 2014; 24: 344-349.

Mendes M, Pala A. Evaluation of four tests when normality and homogeneity of variance assumptions are violated. J Appl Sci. 2004; 4: 38-42.

O’Brien RG. A simple test for variance effects in experimental designs. Psychol Bull. 1981; 89: 570-574.

Oladejo NK, Adetunde IA. Assessing computer knowledge in senior high school: A case study of the upper east region in Ghana. J Math Stat. 2009; 5: 287-297.

Othman AR, Yin TS, Keselman H, Wilcox RR, Algina J. Robust Modifications of the Levene and O’Brien Tests for spread. Journal of Modern applied Statistical Methods. 2012; 11: 54-68.

Overall JE, Woodward JA. A simple test for homogeneity of variance in complex factorial design. Psychometrika. 1974; 39: 311-318.

Parra FI. Testing homogeneity of variances with unequal sample sizes. Comput Stat. 2012;28: 1269-1297.

Reiczigel J. Confidence intervals for the binomial parameter: some new considerations. Statist Med. 2003; 22: 611-621.

Sharma D, Kibria BMG. On some test statistics for testing homogeneity of variances: A comparison study. J Stat Comput Simulat. 2013; 83:1944-1963.

Tormarkan AJ, Serlin RC. Comparison of ANOVA alternatives under variance heterogeneity and specific noncentrality structures. Psychol Bull. 1986; 99: 90-99.

Tukey JW. Some elementary problems of importance to small sample practice. Hum Biol. 1948; 20: 205-214.

Wang Y, Rodríguez PDG, Chen YH, et al. Comparing the Performance of approaches for testing the homogeneity of variance assumption in one-factor ANOVA models. Educ Psychol Meas. 2016; 77: 305-329.