Shayle R. Searle

Population Marginal Means in the Linear Model: An Alternative to Least Squares Means

Abstract The parameter concept in the term least squares mean is defined and given the more meaningful name population marginal mean; and its estimation is discussed.

The vec-permutation matrix, the vec operator and Kronecker products: a review

The vec-permutation matrix I m,n is defined by the equation vec A m × n = I m,n vecA′, Where vec is the vec operator such that vecA is the vector of columns of A stacked one under the other. The variety of definitions, names and notations for I m,n are discussed, and its properties are developed by simple proofs in contrast to certain lengthy proofs in the literature that are based on descriptive definitions. For example, the role of I m,n in reversing the order of Kronecker products is succinctly derived using the vec operator. The matrix M m,n is introduced as M m,n = I m,n M; it is the matrix having as rows,every nth row starting with the first, then every nth row starting with the second, and so on. Special cases of M m,n are discussed.

Vec and vech operators for matrices, with some uses in jacobians and multivariate statistics

The vec of a matrix X stacks columns of X one under another in a single column; the vech of a square matrix X does the same thing but starting each column at its diagonal element. The Jacobian of a one-to-one transformation X → Y is then ∣∣∂(vecX)/∂(vecY) ∣∣ when X and Y each have functionally independent elements; it is ∣∣ ∂(vechX)/∂(vechY) ∣∣ when X and Y are symmetric; and there is a general form for when X and Y are other patterned matrices. Kronecker product properties of vec(ABC) permit easy evaluation of this determinant in many cases. The vec and vech operators are also very convenient in developing results in multivariate statistics.

Minimum Variance Quadratic Unbiased Estimation (MIVQUE) of Variance Components

Minimum variance quadratic unbiased estimators (MIVQUEs) of variance components from unbalanced data are obtained for the one-way classification random model under normality. Explicit, computable expressions are given for the estimators, their variances, and their covariance. The variance expressions provide readily-calculated lower bounds for the variances of any quadratic unbiased estimators of the variance components. For unbalanced data, the estimators are functions of the data and of constants σ ao 2 and σ e 2, taken as a priori estimates of the variance components σ a 2 and σ e 2. The estimators are, for unbalanced data, only locally minimum variance, i.e., they are only minimum variance when σ ao 2 = σ a 2 and σ eo 2 = σ e 2. However, numerical results suggest that the “MIVQUE” of σ a 2 may have much smaller variance than the usual ANOVA estimator with unbalanced data, even when σ ao 2 and σ eo 2 deviate considerably from σ a 2 and σ e 2 respectively. In contrast, the ANOVA estimator of σ e 2 seem...

Another Look at Henderson's Methods of Estimating Variance Components

Three methods of estimating variance components given by Henderson [1953] are critically reviewed and reformulated in matrix theory. Some modifications, as well as a fourth method, are also considered. Some conclusions are: 1. Method 1 is the easiest method to compute but it is inappropriate for mixed models. 2. The Generalized Method 2 contains elements of arbitrariness and is not uniquely defined: the Simplified Generalized Method 2 cannot be used if there are interactions between fixed and random effects, neither when they are considered fixed nor random. Hendersons Method 2 is but one form of the Simplified Generalized Method 2. 3. Method 3 is the most suitable method for mixed models, and for models involving covariances between sets of random effects. However, it can involve matrices of very large order. 4. Method 4 is a variant of Method 2, involving somewhat simpler calculations but still restricted by the lack of uniqueness in Method 2.

On the History of the Kronecker Product

History reveals that what is today called the Kronecker product should be called the Zehfuss product.

Some Computational and Model Equivalences in Analyses of Variance of Unequal-Subclass-Numbers Data

Abstract Available methodologies for calculating sums of squares in analyses of variance of unequal-subclass-numbers (unbalanced) data include (i) full rank reparameterized models, (ii) “indirect” methods, (iii) the R(· | ·) notation, (iv) weighted squares of means, and (v) numerator sums of squares for testing hypotheses. These techniques are described, and relationships between them explained and illustrated. Numerical illustrations are given in the Appendix.

An Overview of Variance Component Estimation

Variance components estimation originated with estimating error variance in analysis of variance by equating error mean square to its expected value. This equating procedure was then extended to random effects models, first for balanced data (for which minimum variance properties were subsequently established) and later for unbalanced data. Unfortunately, this ANOVA methodology yields no optimum properties (other than unbiasedness) for estimation from unbalanced data. Today it is being replaced by maximum likelihood (ML) and restricted maximum likelihood (REML) based on normality assumptions and involving nonlinear equations that have to be solved numerically. There is also minimum norm quadratic unbiased estimation (MINQUE) which is closely related to REML but with fewer advantages.

A Note on Estimating Covariance Components

Abstract A well-known formula for expressing a covariance in terms of variances is shown to hold true for estimating components of covariance.

THE MATRIX HANDLING OF BLUE AND BLUP IN THE MIXED LINEAR MODEL

The mixed model of analysis of variance is a linear model in which some terms that would otherwise be unknown constants are, in fact, unobservable realizations of random variables. Estimation procedures for the constants and for the realized random variables are reviewed, with emphasis on their matrix features.