A Method of Estimating Variance Components in Unbalanced Designs

Abstract

Another method of estimating variance components in the unbalanced case is presented for a general model with unbalanced population structure and unequal variances over the population. The method utilizes unweighted means and can be shown to give unbiased estimates when the numbers of observations are sampled and certain to be > 1. When the design is balanced the estimators coincide with the usual anova-estimators. The construction of estimators is exemplified for four situations with factors crossed and nested in different ways. In balanced designs it is easy to estimate the variance components. The estimators are certain linear combinations of the usual anova-mean-squares. Rules, all of which (except some modifications to avoid negative estimates) give the same result, can be found in most modern text books discussing the analysis of variance. When on the other hand, the design is unbalanced (but complete), there exist several different methods, and new methods (see (1) and (2)) and modifications of old ones (3) are still appearing. It is difficult to say that one method is generally better than the other. Numerical comparisons between variances of variance components estimated according to different methods have been presented for special cases by Bush and Andersson (4) and Hirotsu (5). All the methods are constructed for the case where the numbers of observations are considered as constants, fixed beforehand. In practice, however, the numbers are often determined in a random manner as, for example, in two-stage sampling when the numbers of sampled secondary units are proportional to the total numbers of secondary units per primary unit or as in an experiment with many missing observations. One of the few published results of research in this area is given by Harville (6), who studied the expectations of some estimators for the one-way classification. The model used in this paper is a bit more general than those usually treated in the literature on variance components. Consideration is mainly given to unbalance in the population structure and to the possibility of unequal variances in different levels of a nesting factor. Unbalance in the population has not been given so much attention in the literature as has unbalance in the