Testing the equality of treatment means in one-way ANOVA: Short-tailed symmetric error terms with heterogeneous variances

Abstract

We propose two tests based on fiducial and generalized $p$-value approaches for testing the equality of treatment means in one-way analysis of variance (ANOVA). Modified maximum likelihood (MML) estimators are used in the proposed tests. In contrast to least squares (LS) estimators, MML estimators are highly efficient and robust to plausible deviations from an assumed distribution and to mild data anomalies. In this study, error terms are assumed to have short-tailed symmetric (STS) distributions with heterogeneous variances. The performances of the proposed tests are compared with the fiducial based test using bias-corrected LS estimators via an extensive Monte Carlo simulation study. Finally, two real datasets are analyzed for illustrative purposes.

Keywords

• One-way ANOVA
• short tailed symmetric distribution
• fiducial based test
• generalized F test
• modified maximum likelihood
• Monte Carlo simulation

References

1. [1] Ş. Acıtaş and B. Şenoğlu, Robust factorial ANCOVA with LTS error distributions, Hacet. J. Math. Stat. 47 (2), 347-363, 2018.
2. [2] T. Arslan and B. Şenoğlu, Estimation for the location and the scale parameters of the Jones And Faddy’s Skew t distribution under the doubly Type II censored, Anadolu Univ. J. Sci. Technol. - B - Theor. Sci. 5 (1), 100–110, 2017.
3. [3] N. Celik, B. Şenoğlu and O. Arslan, Estimation and testing in one-way ANOVA when the errors are skew-normal, Rev. Colombiana Estadist. 38 (1), 75-91, 2015.
4. [4] C.H. Chang, N. Pal, W.K. Lim and J.J. Lin, Comparing several population means: a parametric bootstrap method, and its comparison with usual ANOVA F test as well as ANOM, Comput. Statist. 25 (1), 71-95, 2010.
5. [5] R.A. Fisher, Inverse probability, Math. Proc. Cambridge Philos. Soc. 26 (4), 528-535, 1930.
6. [6] R.A. Fisher, The concepts of inverse probability and fiducial probability referring to unknown parameters, Proc. R. Soc. Lond A. 139 (838), 343-348, 1933.
7. [7] R.A. Fisher, The fiducial argument in statistical inference, Ann. Eugen 6 (4), 391-398, 1933.
8. [8] E.B. Foa, B.O. Rothbaum, D.S. Riggs and T.B. Murdock, Treatment of post- traumatic stress disorder in rape victims: a comparison between cognitive behavioral procedures and counselling, J. Consult. Clin. Psychol. 59 (5), 715723, 1991.

9. [9] G. Güven, Ö. Gürer, H. Şamkar and B. Şenoğlu, A fiducial-based approach to the one-way ANOVA in the presence of nonnormality and heterogeneous error variances, J. Stat. Comput. Simul. 89 (9), 1715-1729, 2019.
10. [10] J. Hannig and T.C. Lee Generalized fiducial inference for wavelet regression, Biometrika 96 (4), 847-860, 2009.
11. [11] J. Hartung, G. Knapp and B.K. Sinha, Statistical Meta-Analysis with Applications, John Wiley and Sons, 2008.
12. [12] M. Kendall and A. Stuart, The Advanced Theory of Statistics, 2nd ed., Vol. 2, C, Griffin, London, 1979.
13. [13] K. Krishnamoorthy, F. Lu and T. Mathew A parametric bootstrap approach for ANOVA with unequal variances: Fixed and random models, Comput. Statist. Data Anal. 51 (12), 5731-5742, 2007.
14. [14] K. Krishnamoorthy and E. Oral Standardized likelihood ratio test for comparing several log-normal means and confidence interval for the common mean, Stat. Methods Med. Res. 26 (6), 2919-2937, 2017.
15. [15] K.R. Lee, C.H. Kapadia and D.B. Brock, On estimating the scale parameter of the Rayleigh distribution from doubly censored samples, Stat. Hefte 21 (1), 14-29, 1980.
16. [16] X. Li, A generalized p-value approach for comparing the means of several log-normal populations, Statist. Probab. Lett. 79 (11), 14041408, 2009.
17. [17] X. Li, J. Wang and H. Liang, Comparison of several means: a fiducial based approach, Comput. Statist. Data Anal. 55 (5), 19932002, 2011.
18. [18] Y. Li and A. Xu, Fiducial inference for Birnbaum-Saunders distribution, J. Stat. Comput. Simul. 86 (9), 1673-1685, 2016.
19. [19] C.X. Ma and L. Tian, A parametric bootstrap approach for testing equality of inverse Gaussian means under heterogeneity, Comm. Statist. Simulation Comput. 38 (6), 1153-1160, 2009.
20. [20] F. O’Reilly and R. Rueda, Fiducial inferences for the truncated exponential distribution, Comm. Statist. Theory Methods 36 (12), 2207-2212, 2007.
21. [21] T.P. Ryan, Modern Experimental Design, John Wiley and Sons, 2007.
22. [22] H. Scheffe, The Analysis of Variance, John Wiley and Sons, 1999.
23. [23] B. Şenoğlu, Estimating parameters in one-way analysis of covariance model with short-tailed symmetric error distributions, J. Comput. Appl. Math. 201 (1), 275-283, 2007.
24. [24] B. Şenoğlu and M.L. Tiku, Analysis of variance in experimental design with nonnormal error distributions, Comm. Statist. Theory Methods 30 (7), 1335-1352, 2001.
25. [25] B. Şenoğlu and M.L. Tiku, Linear contrasts in experimental design with non-identical error distributions, Biom J. 44 (3), 359-374, 2002.
26. [26] M.L. Tiku, Estimating the mean and standard deviation from a censored normal sample, Biometrika 54 (1-2), 155–165, 1967.
27. [27] M.L. Tiku and D.C. Vaughan, A family of short-tailed symmetric distributions, Technical Report, 1999.
28. [28] N. Tongmol, W. Srisodaphol and A. Boonyued A Bayesian approach to the one-way ANOVA under unequal variance, Sains Malays. 45 (10), 1565-1572, 2016.
29. [29] D.C. Vaughan, On the Tiku-Suresh method of estimation, Comm. Statist. Theory Methods 21 (2), 451-469, 1992.
30. [30] D.C. Vaughan, The generalized secant hyperbolic distribution and its properties, Comm. Statist. Theory Methods 31 (2), 219-238, 2002.
31. [31] D.C. Vaughan and M.L. Tiku, Estimation and hypothesis testing for a nonnormal bivariate distribution with applications, Math. Comput. Model. 32 (1-2), 53-67, 2000.
32. [32] D.V. Wandler and J. Hannig, A fiducial approach to multiple comparisons, J. Statist. Plann. Inference 142 (4), 878-895, 2012.
33. [33] C.M. Wang, J. Hannig and H.K. Iyer, Fiducial prediction intervals, J. Statist. Plann. Inference 142 (7), 1980-1990, 2012.
34. [34] S. Weerahandi, ANOVA under unequal error variances, Biometrics 51 (2), 589-599, 1995.
35. [35] S. Weerahandi, Generalized Inference in Repeated Measures: exact Methods in MANOVA and Mixed Models, John Wiley and Sons, 2004.
36. [36] B.L. Welch, On the comparison of several mean values: an alternative approach, Biometrika 38 (3-4), 330-336, 1951.
37. [37] G. Zhang, A parametric bootstrap approach for one-way ANOVA under unequal variances with unbalanced data, Comm. Statist. Simulation Comput. 44 (4), 827-832, 2015.