Variances of Variance-Component Estimators for the Unbalanced 2-Way Cross Classification with Application to Balanced Incomplete Block Designs

Abstract

"Best" estimators of variance components for the unbalanced cases of random-effects models are not known. In fact, even for the very simplest of the unbalanced "designs", the balanced incomplete block designs, the question of the existence of minimum variance unbiased estimators remains open (Kapadia and Weeks [5]). The traditional approach to the derivation of variance-component estimators for unbalanced cases has been to pick several quadratic functions of the data, set these functions equal to their expectations, and then solve the resulting system of equations for the variance components. Two of the estimators derived in this fashion for the variance components associated with the unbalanced two-way cross classification are those referred to as the Methods-1 and -3 estimators of Henderson [4]. Method-1 utilizes quadratics analogous to the sums of squares in a balanced analysis of variance. The quadratics employed in Method-3 represent differences between reductions in sums of squares due to fitting different models. Since in Method-3 more differences between reductions are available than one has variance components to estimate, the method is not uniquely defined. Here, the Method-3 estimators of the components associated with the two-way classification are taken to be those in Harville [3], which are the ones most commonly used. Searle [9] obtained algebraic expressions for the sampling variances of the Method-1 estimators of the "two-way" components. Low [6] gave similar expressions for the Method-3 estimators for the zero-interaction case. Their results were obtained by applying well-known formulas for the variances and covariances of quadratic functions of multivariate-normal random variables. These formulas state that if $\mathbf{y}$ is a random vector having the multivariate normal distribution with mean $\mathbf{u}$ and variance-covariance matrix $\mathbf{V}$ and if $\mathbf{A}$ and $\mathbf{B}$ are square symmetric matrices of appropriate dimension having fixed elements, then \begin{align}\tag{1}\operatorname{var} \lbrack\mathbf{y}'\mathbf{Ay}\rbrack &= 4\mathbf{u}'\mathbf{AV}\mathbf{Au} + 2\mathrm{tr}(\mathbf{VA})^2\end{align}and\begin{align}\tag{2} \operatorname{cov} \lbrack\mathbf{y}'\mathbf{Ay},\mathbf{y}' \mathbf{B}y\rbrack &= 4\mathbf{u}'\mathbf{AVBu} + 2 \mathrm{tr} (\mathbf{VAVB})\end{align} Searle [8], [10] and Mahamunulu [7] have also used these formulas to obtain algebraic expressions for the variances of commonly-used estimators of the components of variance associated with other unbalanced classifications. In the present paper, results (supplementary to those of Searle) are given which lead to expressions for the sampling variances of Method-3 estimators of the variance components associated with the unbalanced two-way cross classification with interaction. By using these results in combination with those of Searle, the variances of Method-1 and Method-3 estimators can be directly compared for a given set of subclass numbers. The results are shown to simplify when the "unbalancedness" is of the type associated with a balanced incomplete block design. Neither estimator of any component is uniformly better than the other for any such design. (Except for the estimators of the residual component which are identically equal.)