Exact results for extensions of twofold nested anova for variance components

Abstract

Consider the standard twofold nested design for analysis of variance with random effects (a variance components model). The usual assumption of zero means and equal variances is made for each of the three types of random variables that occur in this balanced model, and the random variables are assumed to be mutually uncorrelated. The customary normality assumption is also made when tests or confidence regions are desired. This standard model is extended, in several ways, by adding one, two or three additional random terms (of an “error” nature) to the standard model. The extensions apply to very much more general situations than does the standard model. Exact procedures are obtained, however, for investigating all of the mean effect and the three types of variance components, and for investigating various subsets of these parameters. The generality level for an extended model depends on which of the parameters are investigated simultaneously, and is greater for a subset of a set of the parameters than for the set. Most of the investigation procedures are different from those customarily used for the standard model.