Abstract
In this paper, we present results for testing main, simple and interaction effects in heteroscedastic two factor MANOVA models. In particular, we suggest modifications to the MANOVA sum of squares and cross product matrices to account for heteroscedasticity. Based on these modified matrices, we define some multivariate test statistics and derive their asymptotic distributions under non-normality for the null as well as non-null cases. Derivation of these results relies on the perturbation method and limit theorems for independently distributed random matrices. Based on the asymptotic distributions, we devise small sample approximations for the quantiles of the null distributions. The numerical accuracy of the large sample as well as small sample approximations are favorable. A real data set from a Smoking Cessation Trial is analyzed to illustrate the application of the methods.
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References
Akritas M.A., Arnold S.F. (2000) Asymptotics for analysis of variance when the number of levels is large. Journal of the American Statistical Association 95: 212–226
Anderson T.W. (2003) An introduction to multivariate analysis. Wiley, New York
Azzalini A., Capitanio A. (1999) Statistical applications of the multivariate skew normal distribution. Journal of the Royal Statistical Society B 61(3): 579–602
Bathke A.C. (2002) ANOVA for large number of treatments. Mathematical Methods of Statistics 11: 118–132
Bathke A.C. (2004) The ANOVA F test can still be used in some balanced designs with unequal variances and nonnormal data. Journal of Statistical Planning and Inference 126: 413–422
Bathke A.C., Harrar S.W. (2008) Nonparametric methods in multivariate factorial designs for large number of factor levels. Journal of Statistical Planning Inference 138: 588–610
Boos D.D., Brownie C. (1995) ANOVA and rank tests when number of treatments is large. Statistics and Probability Letters 20: 183–191
Brunner E., Dette H., Munk A. (1997) Box-type approximations in nonparametric factorial designs. Journal of the American Statistical Association 92: 1494–1502
Dempster A.P. (1958) A high dimensional two sample significance test. Annals of Mathematical Statistics 29: 995–1010
Dempster A.P. (1960) A significance test for the separation of two highly multivariate small samples. Biometrics 16: 41–50
Fujikoshi Y. (1975) Asymptotic formulas for the non-null distributions of three statistics for multivariate linear hypothesis. Annals of the Institute of Statistical Mathematics 27: 99–108
Fujikoshi Y. (2002) Asymptotic expansions for the distribution of multivariate basic statistics and one-way MANOVA tests under nonnormality. Journal of Statistical Planning and Inference 108: 263–282
Fujikoshi Y., Ulyanov V., Shimizu R. (2010) Multivariate statistics: High-dimensional and large-sample approximations. Wiley, New York
Gupta A.K., Harrar S.W., Fujikoshi Y. (2006) Asymptotics for testing hypothesis in some multivariate variance components model under non-normality. Journal of Multivariate Analysis 97: 148–178
Gupta A.K., Harrar S.W., Fujikoshi Y. (2008) MANOVA for large hypothesis degrees of freedom under non-normality. Test 17(1): 120–137
Gupta A.K., Nagar D.K. (2000) Matrix variate distributions. Chapman & Hall/CRC Press, Boca Raton
Harrar S.W., Bathke A.C. (2008) Nonparametric methods for unbalanced multivariate data and many factor levels. Journal of Multivariate Analysis 99: 1635–1664
Harrar S.W., Gupta A.K. (2007) Asymptotic expansion for the null distribution of the F-statistic in one-way ANOVA under non-normality. Annals of the Institute of Statistical Mathematics 59(3): 531–556
Hatsukami D.K., Hughes J.R., Pickens R.W., Svikis D. (1984) Tobacco withdrawal symptoms: An experimental analysis. Psychopharmacology 84: 231–236
Heatherton T.F., Kozlowski L.T., Frecker R.C., Fagerström K. (1991) The Fagerström test for nicotine dependence: A revision of the Fagerström tolerance Questionnaire. British Journal of Addiction 86: 1119–1127
Huber P.J. (1973) Robust regression: Asymptotics, conjectures and Monte Carlo. Annals of Statistics 1: 799–821
Ito P.K. (1969) On the effect of heteroscedasticity and non-normality upon some multivariate test procedures. In: Krishnaiah P.R. (eds) Multivariate analysis Vol. 2. Academic Press, New York, pp 87–120
Ito P.K. (1980) Robustness of ANOVA and MANOVA test procedures. In: Krishnaiah P.R. (eds) Handbook of statistics Vol. 1. North-Holland, Amsterdam, pp 199–236
James G.S. (1954) Test for linear hypothesis in univariate and multivariate analysis when ratios of the population variances are unknown. Biometrika 41: 19–43
Kakizawa Y. (2007) A test of equality of mean vectors of several heteroscedastic multivariate populations. Journal of Japan Statistical Society 37: 253–283
Kakizawa Y. (2009) Third-order power comparisons for a class of tests for multivariate linear hypothesis under general distributions. Journal of Multivariate Analysis 100: 473–496
Kakizawa Y., Iwashita T. (2008) Hotelling’s one-sample and two-sample T 2 tests and the multivariate Behrens–Fisher problem under nonnormality. Journal of Statistical Planning and Inference 138: 3379–3404
Kohout F.J., Berkman L.F., Evans D.A., Cornoni-Huntley J. (1993) Two shorter forms of the CES-D depression symptoms index. Journal of Aging and Health 5: 179–193
Levene H. (1960) Robust tests for equality of variances. In: Olkin I., Ghurye S.G., Hoeffding W., Madow W.G., Mann H.B. (eds) Contributions to probability and statistics: Essays in honor of Harold Hotelling. Stanford University Press, Palo Alto, pp 278–292
Magnus J.R., Neudecker H. (1979) The commutation matrix: Some properties and applications. Annals of Statistics 7: 381–394
Muirhead R.J. (1985) Aspects of multivariate statistical theory. Wiley, New York
Rao C.R. (1973) Linear statistical inference and its applications. Wiley, New York
Rao C.R., Kleffe J. (1989) Estimation of variance components and applications. Elsevier, Amsterdam
Serfling R.J. (1980) Approximation theorems of mathematical statistics. Wiley, New York
Siotani M., Hayakawa T., Fujikoshi Y. (1985) Modern multivariate statistical analysis: A graduate course and handbook. American Sciences Press, Syracuse
Wakaki H., Yanagihara H., Fujikoshi Y. (2002) Asymptotic expansions of the null distributions of statistics for multivariate linear hypothesis under non-normality. Hiroshima Mathematical Journal 32: 17–50
Wang H., Akritas M.G. (2004) Rank tests for ANOVA with large number of factor levels. Journal of Nonparametric Statistics 16: 563–589
Wang, H., Akritas, M. G. (2009). Asymptotically distribution free tests in heteroscedastic unbalanced high dimensional ANOVA. Technical Report, II-09-1, Department of Statistics, Kansas State University.
Wang L., Akritas M.G. (2006) Two-way heteroscedastic ANOVA when the number of levels is large. Statistica Sinica 16: 1387–1408
Wellman R.J., DiFranza J.R., Savageau J.A., Godiwala S., Friedman K., Hazelton J. (2005) Measuring adults’ loss of autonomy over cigarette use: The hooked on nicotine checklist. Nicotine and Tobacco 7: 157–161
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Harrar, S.W., Bathke, A.C. A modified two-factor multivariate analysis of variance: asymptotics and small sample approximations. Ann Inst Stat Math 64, 135–165 (2012). https://doi.org/10.1007/s10463-010-0299-0
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DOI: https://doi.org/10.1007/s10463-010-0299-0