Robust Optimum Invariant Tests in One-Way Unbalanced and Two-Way Balanced Models

摘要

In one-way random effects unbalanced model the locally best invariant test for the equality of the treatment effects is derived. Surprisingly, this is different from the widely used familiar F-test. In the balanced case, however the two tests coincide and represent the uniformly most powerful invariant tests, For two-way random effects and mixed effects balanced models, the uniformly most powerful invariant test for the equality of the treatment effects is derived both with and without interaction, and shown to be equivalent to the usual F-tests under fixed effects models. The optimum invariant tests derived here are shown not to depend on the assumption of normality. Different aspects of null, nonnull and optimality robustness of these tests (Kariya and Sinha, Annals of Statistics, 1985) are studied. In the unbalanced two-way models however unlike in the fixed effects model providing a UMPI test, both random and mixed effects models present a difficulty which is pointed out.