Tests of linear hypotheses in the ANOVA under heteroscedasticity

It is often of interest to undertake a general linear hypothesis testing (GLHT) problem in the one-way ANOVA without assuming the equality of the group 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 does not take the group variance heteroscedasticity into account. To our knowledge, little work has been done for this heteroscedastic GLHT problem except for some special cases. In this paper, we propose a simple approximate Hotelling T 2 (AHT) test. We show that the AHT test is invariant under affine-transformations, different choices of the coefficient matrix used to define the same hypothesis, and different labeling schemes of the group means. Simulations and real data applications indicate that the AHT test is comparable with or outperforms some well-known approximate solutions proposed for the k-sample Behrens-Fisher problem which is a special case of the heteroscedastic GLHT problem.