An Approximate Hotelling T^2-Test for 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 T2 distribution with one parameter estimated from the data, resulting in the so-called approximate Hotelling T2 (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.

[1]  Jin-Ting Zhang.  Tests of Linear Hypotheses in the ANOVA under Heteroscedasticity , 2013 .

[2]  K. Krishnamoorthy,et al.  A parametric bootstrap solution to the MANOVA under heteroscedasticity , 2010 .

[3]  A. Belloni,et al.  On the Behrens–Fisher problem: A globally convergent algorithm and a finite-sample study of the Wald, LR and LM tests , 2008, 0811.0672.

[4]  K. Krishnamoorthy,et al.  A parametric bootstrap approach for ANOVA with unequal variances: Fixed and random models , 2007, Comput. Stat. Data Anal..

[5]  K. Krishnamoorthy,et al.  On Selecting Tests for Equality of Two Normal Mean Vectors , 2006, Multivariate behavioral research.

[6]  Ke-Hai Yuan,et al.  Three Approximate Solutions to the Multivariate Behrens–Fisher Problem , 2005 .

[7]  Jin-Ting Zhang.  Approximate and Asymptotic Distributions of Chi-Squared–Type Mixtures With Applications , 2005 .

[8]  K. Krishnamoorthy,et al.  Modified Nel and Van der Merwe test for the multivariate Behrens–Fisher problem , 2004 .

[9]  J. Algina,et al.  Performance of Four Multivariate Tests Under Variance-Covariance Heteroscedasticity. , 1993, Multivariate behavioral research.

[10]  Seong-Ju Kim.  A practical solution to the multivariate Behrens-Fisher problem , 1992 .

[11]  Ying Yao,et al.  An approximate degrees of freedom solution to the multivariate Behrens-Fisher problem* , 1965 .

[12]  G. S. James.  TESTS OF LINEAR HYPOTHESES IN UNIVERIATE AND MULTIVARIATE ANALYSIS WHEN THE RATIOS OF THE POPULATION VARIANCES ARE UNKNOWN , 1954 .

[13]  B. L. Welch.  ON THE COMPARISON OF SEVERAL MEAN VALUES: AN ALTERNATIVE APPROACH , 1951 .

[14]  Satterthwaite Fe.  An approximate distribution of estimates of variance components. , 1946 .

[15]  Jin-Ting Zhang.  An approximate degrees of freedom test for heteroscedastic two-way ANOVA , 2012 .

[16]  R. Bapat.  Tests of Linear Hypotheses , 2012 .

[17]  Changsong Deng,et al.  Statistics and Probability Letters , 2011 .

[18]  Anja Vogler,et al.  An Introduction to Multivariate Statistical Analysis , 2004 .

[19]  Thomas Mathew,et al.  Generalized p -values and generalized confidence regions for the multivariate Behrens-Fisher problem and MANOVA , 2004 .

[20]  William F. Christensen,et al.  A comparison of type i error rates and power levels for seven solutions to the multivariate behrens-fisher problem , 1997 .

[21]  D. G. Nel,et al.  A solution to the multivariate behrens-fisher problem , 1986 .

[22]  S. Johansen,et al.  The Welch - James approximation to the distribution of the residual sum of squares in a weighted lin , 1980 .

[23]  N. L. Johnson,et al.  Multivariate Analysis , 1958, Nature.

[24]  F. E. Satterthwaite.  An approximate distribution of estimates of variance components. , 1946, Biometrics.