On exact inference in linear models with two variance-covariance components

Abstract

Linear models with variance-covariance components are used in a
wide variety of applications. In most situations it is possible to partition the re-
sponse vector into a set of independent subvectors, such as in longitudinal models
where the response is observed repeatedly on a set of sampling units (see e.g.,
Laird & Ware 1982). Often the objective of inference is either a test of linear
hypotheses about the mean or both, the mean and the variance components.
Confidence intervals for parameters of interest can be constructed as an alter-
native to a test. These questions have kept many statisticians busy for several
decades. Even under the assumption that the response can be modeled by a mul-
tivariate normal distribution, it is not clear what test to recommend except in a
few settings such as balanced or orthogonal designs. Here we investigate statis-
tical properties, such as accuracy of p-values and powers of exact (Crainiceanu
& Ruppert 2004) tests and compare with properties of approximate asymptotic
tests. Simultaneous exact confidence regions for variance components and mean
parameters are constructed as well.