ANOVA estimates of variance components for quasi-balanced mixed models

Abstract

This article examines a simple and intuitively appealing way to estimate variance components in mixed models: perform as many ordinary least-squares regressions as there are variance components, using dummy variables for all but one random effect in each regression. By equating these residual sums of squares to their expectations, unbiased estimates of all of the variance components can be obtained. This approach is a special case of Method 3 of Henderson (Biometrics 9 (1953) 226–252). A quasi-balanced mixed model is defined here to be a model in which the indicator matrix for one of the random effects is essentially arbitrary, but the model obtained by omitting this random effect is nested and balanced. For quasi-balanced normal models the exact distributions of residual sums of squares are derived; they lead to the exact distributions of Henderson Method 3 estimates. An example involving strength data for a composite material is discussed in detail.