Variances of Variance Components: II. The Unbalanced Single Classification

Abstract

The variance of the usual estimate of between variance components in an unbalanced single classification has been found for arbitrary infinite populations by Hammersley [1], who found it necessary to use rather heavy algebra. The methods of polykays are here applied to a family of weighted estimates to obtain the variances and covariances of the estimates of between and within variance components. These apply to arbitrary finite populations. Weighting column means equally seems to give a better estimate than the classical proportional weighting for the between variance component as soon as (i) the between component exceeds $\frac{1}{2}$ of the within component in a moderately unbalanced design, or (ii) the between component exceeds the within component in a substantially unbalanced design. Slight further gains come from intermediate weighting. Numerical examples are given. While pooling mean squares instead of sums of squares across columns loses accuracy, notably for the within variance component, doing the same in calculating the between variance component seems to have a minor effect. If the within contributions are sufficiently non-normal, this effect will be favorable.