SIMPLE ESTIMATIONS OF THE VARIANCE COMPONENTS AND THE FIXED AND RANDOM EFFECTS IN MIXED, THREE-STAGE, HIERARCHAL MODELS

Abstract

FRR (Fixed, Random, Random) hierarchal models in which the first-stage" elements are fixed and the second and third-stage elements are random, are used in analyses of comparative experiments and, extensively, in animal breeding contexts where, in the latter, estimates of the second-stage elements and of combinations of them with first-stage elements, are of practical interest. The two procedures, i) empirical BLUP (Best Linear Unbiased Prediction) and ii) a Bayesian approach, used when the ratio of the within-second-stages and the within-third-stages variances is unknown are 'computationally intensive'. When the ratio of the second- to the third-stage variances is large, an alternative and computationally simpler procedure considered here is applicable. This approach provides estimates, including those of the variance components, which jointly maximize likelihood. Another simple method is proposed for further investigation. In all the procedures, estimates of the second-stage elements are obtained by centering or shrinkage translations from the observed means. It is shown that the validity of these adjustments is critically dependent on the distributional assumption made for the second-stage elements. The adjustments will not be centering unless the distribution is Gaussian, in particular, or 'centri-modal' in general.