Estimation of variance components for mixed models with cell means

Abstract

Statistical methods of estimating variance components have long enjoyed use in many fields of application, especially in agricultural, biological, industrial types of experiment. For more than twenty years prior to 1967, the specific methods available were all based on the same theme, but in succeeding years several new methods, in particular ML, REML, and MINQUE have been developed that depart quite radically from this theme. These approaches have many attractive features and involve a considerable corpus of matrix algebra and other mathematics. Although these methods possess good properties, they have not been used in practice because effective computational algorithms are not readily available to practitioners. The main purpose of this thesis is to give more efficient computational algorithms for producing ML, REML, and MINQUE methods of variance components based on the same approach of W transformation. The W transformation was suggested by Hemmerle and Hartley (1973) for ML estimation of the parameters of the Mixed analysis of variance model. They reduce the problems to one requiring the inversion of a smaller m x m matrix, where m is the total number of random levels in the mixed model. In this thesis, the W-matrix can be reconstructed as the terms of balanced design matrices haveing one and only one observation in each cell, the vector of cell means, and error sum of squares. Also this reformation of W-matrix not only eliminates the need for the explicit computation of the $N\times{N}$ inverse matrix H but permits handling the iterative calculations such that they do not depend upon the number of observations N. The method described here requires inversions of an $n \times n$ matrix at each iteration, where n is the number of nonempty cells. Obviously, n is smaller than m for crossed design with all interaction terms present, for nested designs, and for designs with many empty cells. But there exist designs, for example, which are additive or include onl...