Tests of Linear Hypotheses in the ANOVA under Heteroscedasticity
Abstract
It is often interest to undertake a general linear hypothesis testing (GLHT)problem in the one-way ANOVA without assuming the equality of thegroup variances. When the equality of the group variances is valid,it is well known that the GLHT problem can be solved by the classical F-test. The classical F-test, however, may lead to misleading conclusions when the variance homogeneity assumption is seriously violated since it doesnot take the group variance heteroscedasticity into account. To ourknowledge, little work has been done for this heteroscedastic GLHTproblem except for some special cases. In this paper, we propose asimple approximate Hotelling T2 (AHT) test. We show that the AHTtest is invariant under affine-transformations, different choices ofthe coefficient matrix used to define the same hypothesis, anddifferent labeling schemes of the group means. Simulations and realdata applications indicate that the AHT test is comparable with oroutperforms some well-known approximate solutions proposed for the k-sample Behrens-Fisher problem which is a special case of theheteroscedastic GLHT problem.About this article
References
[1] A. A. Aspin, An examination and further development of a formula arising in the problem of comparing two mean values, Biometrika 35 (1948) 88–96.
[2] B. V. Behrens, Ein Beitrag zur Fehlerberechnung bei wenige Beobachtungen, Landwirtschaftliches Jahresbuch 68 (1929) 807–837.
[3] M. B. Brown, A.B. Forsytbe, The small sample behavior of some statistics which test the equality of several means, Technometrics 16 (1974) 129–132.
[4] C.-H. Chang, J.-J. Lin, N. Pal, Testing the equality of several gamma means: a parametric bootstrap method with applications, Comput. Statist. 26 (2011) 55–76.
[5] C.-H. Chang, N. Pal, A Revisit to the Behrens-Fisher Problem: Comparison of Five test methods, Comm.Statist. Theor. & Meth. 37 (2008) 1064–1085.
View more references (27)
[6] C.-H. Chang, N. Pal, W. K. Lim, 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 (2010) 71–95.
[7] Coombs, W. T., J. Algina, D.O. Oltman, Univariate and multivariate omnibus hypothesis tests selected to control Type I error rates when population variances are not necessarily equal, Rev. Educ. Res. 66 (1996) 137–179.
[8] R. A. Fisher, The fiducial argument in statistical inference, Ann. Eugen. 11 (1935) 141–172.
[9] E. B. Foa, B. O. Rothbaum, D. S. Riggs, T. B. Murdock, Treatment of posttraumatic stress disorder in rape victims: a comparison between cognitive-behavioral procedures and counseling, J. Consult. Clin. Psych. 59 (1991) 715–723.
[10] A. K. Gupta, Y. Wang, Some tests with specified size for the Behrens-Fisher problem, Comm. Statist. Theor. Meth. 28 (1999) 511–517.
[11] G.S. James, The comparison of several groups of observations when the ratios of the population variances are unknown, Biometrika 38 (1951) 324–329.
[12] G.S. James, Tests of linear hypotheses in univariate and multivariate analysis when the ratios of the population variances are unknown, Biometrika 41 (1954) 19–43.
[13] S. H. Kim, A. S. Cohen, On the Behrens-Fisher problem: a review, J. Educ. Behavior. Statist. 23 (1998) 356–377.
[14] K. Krishnamoorthy, F. Lu, A parametric bootstrap solution to the MANOVA under heteroscedasticity, J.Statist. Comput. Simul. 80 (2010) 873–887.
[15] K. Krishnamoorthy, F. Lu, T. Mathew, A parametric bootstrap approach for ANOVA with unequal variances:Fixed and random models, Comput. Statist. & Data Anal. 51 (2007) 5731–5742.
[16] K. Krishnamoorthy, Y. Xia, On selecting tests for equality of two normal mean vectors, Multi. Behavior.Res. 41 (2006) 533–548.
[17] K. Krishnamoorthy, J. Yu, Modified Nel and Van der Merwe test for the multivariate Behrens-Fisher problem,Statist. Prob. lett. 66 (2004) 161–169.
[18] R. G. Krutchkoff, One-way fixed effects analysis of variance when the error variances may be unequal, J.Statist. Comput. Simul. 30 (1988) 259–271.24 International Journal of Advanced Statistics and Probability
[19] A. M. Kshirsagar, Multivariate Analysis, Marcel Decker, New York, 1972.
[20] S. Lee, C. H. Ahn, Modified ANOVA for unequal variances, Comm. Statist. Simul. Comput. 32 (2003) 987–1004.
[21] A. Lee, J. Gurland, Size and power of tests for equality of means of two normal populations with unequal variances, J. Amer. Statist. Assoc., 70 (1975) 933–947.
[22] D. G. Nel, C. A. Van der Merwe, A solution to the multivariate Behrens-Fisher problem, Comm. Statist. Theor. Meth. 15 (1986) 3719–3735.
[23] W. R. Rice, S. D. Gaines, One-way analysis of variance with unequal variances, Proc. Natl. Acad. Sci. USA 86 (1989) 8183–8184.
[24] H. Ruben, A simple conservative and robust solution of the BehrensFisher problem, Sankhya Ind. J. Statist. Ser. A 64 (2002) 139–155.
[25] S. Weerahandi, Exact Statistical Methods in Data Analysis, Springer-Verlag, New York, 1995.
[26] B.L. Welch, The significance of the difference between two means when the population variances are unequal,Biometrika 29 (1938) 350–362
[27] B.L. Welch, The generalization of Student’s problem when several different population variances are involved, Biometrika 34 (1947) 28–35.
[28] B.L. Welch, On the Comparison of Several Mean Values: An Alternative Approach, Biometrika 38 (1951) 330–336.
[29] R.R. Wilcox, A new alternative to the ANOVA F and new results on James’s second order method, Brit. J. Math. Statist. Psych., 41 (1988) 109–117.
[30] R.R. Wilcox, Adjusting for unequal variances when comparing means in one-way and two-way fixed effects ANOVA model, J. Educ. Statist. 14 (1989) 269–278.
[31] H. Yanagihara, K.-H. Yuan, Three Approximate Solutions to the Multivariate Behrens-Fisher Problem, Comm. Statist. Simul. Comput. 34 (2005) 975–988.
[32] J.T. Zhang, Tests of linear hypotheses in one-way ANOVA under heteroscedasticity, Manuscript (2011).