An Approximate Hotelling T^{2}-Test for Heteroscedastic One-Way MANOVA

Author(s)
Jin-Ting Zhang

ABSTRACT

In this paper, we consider the general linear hypothesis testing (GLHT) problem in heteroscedastic one-way MANOVA. The well-known Wald-type test statistic is used. Its null distribution is approximated by a Hotelling T^{2} distribution with one parameter estimated from the data, resulting in the so-called approximate Hotelling T^{2} (AHT) test. The AHT test is shown to be invariant under affine transformation, different choices of the contrast matrix specifying the same hypothesis, and different labeling schemes of the mean vectors. The AHT test can be simply conducted using the usual F-distribution. Simulation studies and real data applications show that the AHT test substantially outperforms the test of [1] and is comparable to the parametric bootstrap (PB) test of [2] for the multivariate k-sample Behrens-Fisher problem which is a special case of the GLHT problem in heteroscedastic one-way MANOVA.

In this paper, we consider the general linear hypothesis testing (GLHT) problem in heteroscedastic one-way MANOVA. The well-known Wald-type test statistic is used. Its null distribution is approximated by a Hotelling T

Cite this paper

J. Zhang, "An Approximate Hotelling T^{2}-Test for Heteroscedastic One-Way MANOVA," *Open Journal of Statistics*, Vol. 2 No. 1, 2012, pp. 1-11. doi: 10.4236/ojs.2012.21001.

J. Zhang, "An Approximate Hotelling T

References

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[2] K. Krishnamoorthy and F. Lu, “A Parametric Bootstrap Solution to the MANOVA under Heteroscedasticity,” Journal of Statistical Computation and Simulation, Vol. 80, No. 8, 2010, pp. 873-887. doi:10.1080/00949650902822564

[3] T. W. Anderson, “An Introduction to Multivariate Statistical Analysis,” Wiley, New York, 2003.

[4] K. Krishnamoorthy and Y. Xia, “On Selecting Tests for Equality of Two Normal Mean Vectors,” Multivariate Behavioral Research, Vol. 41, No. 4, 2006, pp. 533-548. doi:10.1207/s15327906mbr4104_5

[5] K. Krishnamoorthy and J. Yu, “Modified Nel and van der Merwe Test for the Multivariate Behrens-Fisher Problem,” Statistics and Probability Letters, Vol. 66, No. 2, 2004, pp. 161-169. doi:10.1016/j.spl.2003.10.012

[6] G. S. James, “Tests of Linear Hypotheses in Univariate and Multivariate Analysis When the Ratios of the Population Variances Are Unknown,” Biometrika, Vol. 41, No. 1-2, 1954, pp. 19-43.

[7] Y. Yao, “An Approximate Degrees of Freedom Solution to the Multivariate Behrens-Fisher Problem,” Biometrika, Vol. 52, 1965, pp. 139-147.

[8] D. G. Nel and C. A. van der Merwe, “A Solution to the Multivariate Behrens-Fisher Problem,” Communication Statistics: Theory and Methods, Vol. 15, No. 12, 1986, pp. 3719-3735. doi:10.1080/03610928608829342

[9] S. Kim, “A Practical Solution to the Multivariate Behrens- Fisher Problem,” Biometrika, Vol. 79, No. 1, 1992, pp. 171-176. doi:10.1093/biomet/79.1.171

[10] H. Yanagihara and K. H. Yuan, “Three Approximate Solutions to the Multivariate Behrens-Fisher Problem,” Communication Statistics: Simulation and Computation, Vol. 34, No. 4, 2005, pp. 975-988. doi:10.1080/03610910500308396

[11] A. Belloni and G. Didier, “On the Behrens-Fisher Problem: A Globally Convergent Algorithm and a Finite- Sample Study of the Wald, LR and LM Tests,” Annals of Statistics, Vol. 36, No. 5, 2008, pp. 2377-2408. doi:10.1214/07-AOS528

[12] W. F. Christensen and A. C. Rencher, “A Comparison of Type I Error Rates and Power Levels for Seven Solutions to the Multivariate Behrens-Fisher Problem,” Communication Statistics: Theory and Methods, Vol. 26, 1997, pp. 1251-1273.

[13] B. L. Welch, “On the Comparison of Several Mean Values: An Alternative Approach,” Biometrika, Vol. 38, 1951, pp. 330-336.

[14] J. Gamage, T. Mathew and S. Weerahandi, “Generalized p-Values and Generalized Confidence Regions for the Multivariate Behrens-Fisher Problem and MANOVA,” Journal of Multivariate Analysis, Vol. 88, No. 1, 2004, pp. 177-189. doi:10.1016/S0047-259X(03)00065-4

[15] K. L. Tang and J. Algina, “Performing of Four Multivariate Tests under Variance-Covariance Heteroscedasticity,” Multivariate Behavioral Research, Vol. 28, No. 4, 1993, pp. 391-405. doi:10.1207/s15327906mbr2804_1

[16] K. Krishnamoorthy, F. Lu and T. Mathew, “A Parametric Bootstrap Approach for ANOVA with Unequal Variances: Fixed and Random Models,” Computational Statistics and Data Analysis, Vol. 51, No. 12, 2007, pp. 5731-5742. doi:10.1016/j.csda.2006.09.039

[17] J. T. Zhang, “Tests of Linear Hypotheses in the ANOVA under Heteroscedasticity,” Manuscript, 2012.

[18] J. T. Zhang, “An Approximate Degrees of Freedom Test for Heteroscedastic Two-Way ANOVA,” Journal of Statistical Planning and Inference, Vol. 142, 2012, pp. 336-346.

[19] F. E. Satterthwaite, “An Approximate Distribution of Estimate of Variance Components,” Biometrics Bulletin, Vol. 2, No. 6, 1946, pp. 110-114. doi:10.2307/3002019

[20] J. T. Zhang, “Approximate and Asymptotic Distribution of χ2-Type Mixtures with Application,” Journal of American Statistical Association, Vol. 100, No. 469, 2005, pp. 273-285. doi:10.1198/016214504000000575

[21] A. M. Kshirsagar, “Multivariate Analysis,” Marcel Decker, New York, 1972.

[1] S. Johansen, “The Welch-James Approximation to the Distribution of the Residual Sum of Squares in a Weighted linear Regression,” Biometrika, Vol. 67, No. 1, 1980, pp. 85-95. doi:10.1093/biomet/67.1.85

[2] K. Krishnamoorthy and F. Lu, “A Parametric Bootstrap Solution to the MANOVA under Heteroscedasticity,” Journal of Statistical Computation and Simulation, Vol. 80, No. 8, 2010, pp. 873-887. doi:10.1080/00949650902822564

[3] T. W. Anderson, “An Introduction to Multivariate Statistical Analysis,” Wiley, New York, 2003.

[4] K. Krishnamoorthy and Y. Xia, “On Selecting Tests for Equality of Two Normal Mean Vectors,” Multivariate Behavioral Research, Vol. 41, No. 4, 2006, pp. 533-548. doi:10.1207/s15327906mbr4104_5

[5] K. Krishnamoorthy and J. Yu, “Modified Nel and van der Merwe Test for the Multivariate Behrens-Fisher Problem,” Statistics and Probability Letters, Vol. 66, No. 2, 2004, pp. 161-169. doi:10.1016/j.spl.2003.10.012

[6] G. S. James, “Tests of Linear Hypotheses in Univariate and Multivariate Analysis When the Ratios of the Population Variances Are Unknown,” Biometrika, Vol. 41, No. 1-2, 1954, pp. 19-43.

[7] Y. Yao, “An Approximate Degrees of Freedom Solution to the Multivariate Behrens-Fisher Problem,” Biometrika, Vol. 52, 1965, pp. 139-147.

[8] D. G. Nel and C. A. van der Merwe, “A Solution to the Multivariate Behrens-Fisher Problem,” Communication Statistics: Theory and Methods, Vol. 15, No. 12, 1986, pp. 3719-3735. doi:10.1080/03610928608829342

[9] S. Kim, “A Practical Solution to the Multivariate Behrens- Fisher Problem,” Biometrika, Vol. 79, No. 1, 1992, pp. 171-176. doi:10.1093/biomet/79.1.171

[10] H. Yanagihara and K. H. Yuan, “Three Approximate Solutions to the Multivariate Behrens-Fisher Problem,” Communication Statistics: Simulation and Computation, Vol. 34, No. 4, 2005, pp. 975-988. doi:10.1080/03610910500308396

[11] A. Belloni and G. Didier, “On the Behrens-Fisher Problem: A Globally Convergent Algorithm and a Finite- Sample Study of the Wald, LR and LM Tests,” Annals of Statistics, Vol. 36, No. 5, 2008, pp. 2377-2408. doi:10.1214/07-AOS528

[12] W. F. Christensen and A. C. Rencher, “A Comparison of Type I Error Rates and Power Levels for Seven Solutions to the Multivariate Behrens-Fisher Problem,” Communication Statistics: Theory and Methods, Vol. 26, 1997, pp. 1251-1273.

[13] B. L. Welch, “On the Comparison of Several Mean Values: An Alternative Approach,” Biometrika, Vol. 38, 1951, pp. 330-336.

[14] J. Gamage, T. Mathew and S. Weerahandi, “Generalized p-Values and Generalized Confidence Regions for the Multivariate Behrens-Fisher Problem and MANOVA,” Journal of Multivariate Analysis, Vol. 88, No. 1, 2004, pp. 177-189. doi:10.1016/S0047-259X(03)00065-4

[15] K. L. Tang and J. Algina, “Performing of Four Multivariate Tests under Variance-Covariance Heteroscedasticity,” Multivariate Behavioral Research, Vol. 28, No. 4, 1993, pp. 391-405. doi:10.1207/s15327906mbr2804_1

[16] K. Krishnamoorthy, F. Lu and T. Mathew, “A Parametric Bootstrap Approach for ANOVA with Unequal Variances: Fixed and Random Models,” Computational Statistics and Data Analysis, Vol. 51, No. 12, 2007, pp. 5731-5742. doi:10.1016/j.csda.2006.09.039

[17] J. T. Zhang, “Tests of Linear Hypotheses in the ANOVA under Heteroscedasticity,” Manuscript, 2012.

[18] J. T. Zhang, “An Approximate Degrees of Freedom Test for Heteroscedastic Two-Way ANOVA,” Journal of Statistical Planning and Inference, Vol. 142, 2012, pp. 336-346.

[19] F. E. Satterthwaite, “An Approximate Distribution of Estimate of Variance Components,” Biometrics Bulletin, Vol. 2, No. 6, 1946, pp. 110-114. doi:10.2307/3002019

[20] J. T. Zhang, “Approximate and Asymptotic Distribution of χ2-Type Mixtures with Application,” Journal of American Statistical Association, Vol. 100, No. 469, 2005, pp. 273-285. doi:10.1198/016214504000000575

[21] A. M. Kshirsagar, “Multivariate Analysis,” Marcel Decker, New York, 1972.