Standardized Likelihood Ratio Test for Homogeneity of Variance Based on Likelihood Ratio Under Normality

Abstract

In this article, standardized likelihood ratio test is proposed for the homogeneity of variances under normality. The proposed method is compared with some of the existing methods via Monte Carlo simulation for various parameter combinations, different group sizes and sample sizes in terms of type I error rate and power of test. According to numeric results, the proposed method performs quite well according to its alternatives. 

Keywords

• Standardized likelihood ratio test,Homogeneity of variances,Type I error

References

1. [1] Bartlett, M.S., “Properties of sufficiency and statistical tests”, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 160(901):268-282, (1937).
2. [2] Bhandary, M., and Dai, H., “An alternative test for the equality of variances for several populations when the underlying distributions are normal”, Communications in Statistics-Simulation and Computation, 38(1):109-117, (2009).
3. [3] Boos, D.D. and Brownie, C., “Comparing variances and other measures of dispersion”, Statistical Science, 19(4):571-578, (2004).
4. [4] Box, G.E. and Andersen, S.L., “Permutation theory in the derivation of robust criteria and the study of departures from assumption”, Journal of the Royal Statistical Society. Series B (Methodological), 17(1):1-34, (1955).
5. [5] Brown, M.B. and Forsythe, A.B., “Robust tests for the equality of variances”, Journal of the American Statistical Association, 69(346):364-367, (1974).
6. [6] Cahoy, D.O., “A bootstrap test for equality of variances”, Computational Statistics & Data Analysis, 54(10):2306-2316, (2010).
7. [7] Chang, C.H., Lin, J.J. and Pal, N. “Testing the equality of several gamma means: a parametric bootstrap method with applications”, Computational Statistics, 26(1), 55-76, (2011).
8. [8] Chang, C.H. and Pal, N., “A revisit to the Behrens–Fisher problem: comparison of five test methods”, Communications in Statistics—Simulation and Computation, 37(6), 1064-1085, (2008).

9. [9] Chang, C.H., Pal, N. and Lin, J.J., “A note on comparing several poisson means”, Communications in Statistics-Simulation and Computation, 39(8):1605-1627, (2010).
10. [10] Chang, C.H., Pal, N. and Lin, J-J., “A Revisit to Test the Equality of Variances of Several Populations”, Communications in Statistics-Simulation and Computation (in press), 2016.
11. [11] Cochran, W.G., “Testing a linear relation among variances”, Biometrics,7(1):17-32, (1951).
12. [12] Conover, W. J., Johnson, M. E. and Johnson, M.M., “A comparative study of tests for homogeneity of variances, with applications to the outer continental shelf bidding data”, Technometrics, 23(4), 351-361, (1981).
13. [13] Davison, A.C. and Hinkley, D.V. Bootstrap methods and their application (Vol. 1). Cambridge university press, New York, (1997).
14. [14] Gökpınar, E. and Gökpınar, F. “A test based on computational approach for equality of means under unequal variance assumption”, Hacettepe Journal of Mathematics and Statistics, 41(4):605-613, (2012).
15. [15] Gökpınar, E., Polat, E., Gokpinar, F. and Gunay S., “A new computational approach for testing equality of ınverse gaussian means under heterogenity”, Hacettepe Journal of Mathematics and Statistics, 42(5):581-590, (2013).
16. [16] Gökpınar, F. and Gökpınar, E., “ A Computational approach for testing of coefficients of variation in k normal population”, Hacettepe Journal of Mathematics and Statistics, 44(5):1197-1213, (2015a).
17. [17] Gokpinar, E. and Gokpinar, F., “Testing equality of variances for several normal populations”, Communications in Statistics-Simulation and Computation (DOI: 10.1080/03610918.2014.955110) (2015b).
18. [18] Gokpinar, F. and Gokpinar, E., "Testing the equality of several log-normal means based on a computational approach." Communications in Statistics-Simulation and Computation 46(3): 1998-2010, (2017).
19. [19] Hall, I.J., “Some comparisons of tests for equality of variances”, Journal of Statistical Computation and Simulation, 1(2):133-194, (1972).
20. [20] Hartley, H.O., “The maximum F-ratio as a short-cut test for heterogeneity of variance”, Biometrika, 37(3/4):308-312, (1950).
21. [21] Jafari, A.A. and Abdollahnezhad, K., “Inferences on the Means of Two Log-Normal Distributions: A Computational Approach Test”, Communications in Statistics-Simulation and Computation, 44(7):1659-1672, (2015).
22. [22] Jafari, A. A. and Kazemi, M. R., “Computational Approach Test for Inference about Several Correlation Coefficients: Equality and Common”. Communications in Statistics-Simulation and Computation, DOI:10.1080/03610918.2015.1030416, (2015).
23. [23] Jen, T. and Gupta, A.K., “On testing homogeneity of variances for Gaussian models”, Journal of Statistical Computation and Simulation, 27(2):155-173, (1987).
24. [24] Keyes, T. K. and Levy, M.S., “Analysis of Levene’s Test under Design Inbalance”, Journal of Educational and Behavioral Statistics, 22(2):227-236, (1997).
25. [25] Krishnamoorthy, K. and Oral, E., "Standardized likelihood ratio test for comparing several log-normal means and confidence interval for the common mean," Statistical methods in medical research, DOI:10.1177/0962280215615160, (2015).
26. [26] Levene, H., “Robust tests for equality of variances”, In Contributions to probability and statistics: Essays in honor of Harold Hotelling, 2:278-292, (1960).
27. [27] Liu, X. and Xu, X. “A new generalized p-value approach for testing the homogeneity of variances”, Statistics & probability letters, 80(19):1486-1491, (2010).
28. [28] Loh, W.Y., “Some Modifications of Levene’s Test of Variance Homogeneity”, Journal of Statistical Computation and Simulation, 28:213-226, (1987).
29. [29] Mutlu, H.T., Gökpınar, F., Gökpınar, E. and Gül, H.H., "A New Computational Approach Test for One-Way ANOVA under Heteroscedasticity" ,Communications in Statistics - Theory and Methods , DOI: 10.1080/03610926.2016.117708, ISSN: 0361-0926, (2016).
30. [30] Nair, V.N. and Pregibon, D., “Analyzing dispersion effects from replicated factorial experiments”, Technometrics, 30(3):247-257, (1988).
31. [31] Neyman, J. and Pearson, E.S., On the problem of k samples. Bull. Ac. Polonaise Sci. Letters, Serie A3: 460-481, (1931).
32. [32] Pal, N., Lim, W.K. and Ling, C.H., “A computational approach to statistical inferences”, Journal of Applied Probability and Statistics, 2(1):13-35, (2007).
33. [33] Taguchi, G., Introduction to quality engineering: designing quality into products and processes, Asian Productivity Organization, Tokyo, (1986).