Estimation of Variance and Covariance Components

The theory of variance component analysis has been discussed recently by Crump (1946, 1951) and by Eisenhart (1947). These papers and, indeed, most of the published works on estimating variance components deal with the one-way classification, with nested classifications, and with factorial classifications having equal subclass numbers. Also most papers on this subject are concerned with what Eisenhart (1947) has called Model II; that is, all elements of the linear model save gi are regarded as random variables. In the above cases, estimation of variance components is usually accomplished by computing the mean squares in the standard analysis of variance, equating these mean squares to their expectations, and solving for the unknown variances. These techniques are described in many statistical textbooks. Unfortunately, research workers in some of those fields in which much use is made of variance component estimates are unable to obtain data which have the above described characteristics. This is particularly true in those fields in which survey data must be used or where, even in a well-planned experiment, the subclasses are of quite unequal size due, for example, to differences in litter numbers. Also,