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.