Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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Tests of equality of several variances with the likelihood ratio principle

  • Received : 2018.01.10
  • Accepted : 2018.05.08
  • Published : 2018.07.31
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Abstract

In this study, we propose tests for equality of several variances with the normality assumption. First of all, we propose the likelihood ratio test by applying the permutation principle. Then by using the p-values for the pairwise tests between variances and combination functions, we propose combination tests. We apply the permutation principle to obtain the overall p-values. Also we review the well- known test statistics for the completion of our discussion and modify a statistic with the p-values. Then we illustrate proposed tests by numerical and simulated data and compare their efficiency with the reviewed ones through a simulation study by obtaining empirical p-values. Finally, we discuss some interesting features related to the resampling methods and tests for equality among several variances.

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References

  1. Bartlett MS (1937). Properties of sufficiency and statistical tests. In Proceedings of the Royal Statistical Society, Series A, 160, 268-282.
  2. Boos DD and Zhang J. (2000). Monte Carlo evaluation of resampling-based hypothesis tests, Journal of American Statistical Association, 95, 486-492. https://doi.org/10.1080/01621459.2000.10474226
  3. Brown MB and Forsythe AB (1974). Robust tests for the equality of variances, Journal of the American Statistical Association, 69, 364-367. https://doi.org/10.1080/01621459.1974.10482955
  4. Chang CH, Pal N, and Lin JJ (2017). A revisit to test the equality of variances of several populations, Communications in Statistics-Simulation and Computation, 46, 6360-6384. https://doi.org/10.1080/03610918.2016.1202277
  5. Gokpinar E and Gokpinar F (2017). Testing equality of variances for several normal populations, Communications in Statistics-Simulation and Computation, 46, 38-52. https://doi.org/10.1080/03610918.2014.955110
  6. Good P (2000). Permutation Tests, A Practical Guide to Resampling Methods for Testing Hypotheses, Springer, New York.
  7. Hand HA and Nagaraja HN (2003). Order Statistics (3rd ed), Wiley, New York.
  8. Hartley HO (1950). The Use of Range in Analysis of Variance, Biometrika, 37, 271-280. https://doi.org/10.1093/biomet/37.3-4.271
  9. Jayalath KP, Ng HKT, Manage AB, and Riggs KE (2017). Improved tests for homogeneity of variances, Communications in Statistics-Theory and Methods, 46, 7423-7446.
  10. Levene H (1960). Robust tests for equality of variances, In Ingram Olkin; Harold Hotelling; et al. Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling, Stanford University Press. pp. 278-292.
  11. Lowe B (1935). Data from the Iowa Agricultural Experiment Station.
  12. O'Brien RG (1979). A general ANOVA method for robust tests of additive models for variance, Journal of American Statistical Association, 74, 877-880. https://doi.org/10.1080/01621459.1979.10481047
  13. O'Brien RG (1981). A simple test for variance effects in experimental designs, Psychological Bulletin, 89, 570-574. https://doi.org/10.1037/0033-2909.89.3.570
  14. Oden NL (1991). Allocation of effort in Monte Carlo simulation for power of permutation tests, Journal of American Statistical Association, 86, 1074-1076. https://doi.org/10.1080/01621459.1991.10475153
  15. Park HI (2015). Simultaneous test for the mean and variance with an application to the statistical process control, Journal of Statistical Theory and Practice, 9, 868-881. https://doi.org/10.1080/15598608.2015.1034387
  16. Park HI (2017). On tests detecting difference in means and variances simultaneously under normality, Communications in Statistics-Theory and Methods, 46, 10025-10035. https://doi.org/10.1080/03610926.2016.1228964
  17. Park HI (2018). A note on the test for the covariance matrix under normality, Communications for Statistical Applications and Methods, 25, 71-78. https://doi.org/10.29220/CSAM.2018.25.1.071
  18. Pearson ES and Hartley HO (1970). Biometrika Tables for Statisticians, Vol 1.
  19. Pesarin F (2001). Multivariate Permutation Tests with Applications in Biostatistics, Wiley, West Sussex, England.
  20. Piepho HP (1996). A Monte Carlo test for variance homogeneity in linear models, Biometrical Journal, 38, 461-473 https://doi.org/10.1002/bimj.4710380411
  21. Snedecor GW and Cochran WG (1989). Statistical Methods, Eighth Edition, Iowa State University Press. Westfall
  22. PH and Young SS (1993). Resamplimg-Based Multiple Testing, Examples and Methods for p-Value Adjustment, Wiley, New York.