C.R. Henderson

A simple method for computing the inverse of a numerator relationship matrix used in prediction of breeding values

The inverse of a numerator relationship matrix is needed for best linear unbiased prediction of breeding values. The purpose of this paper to is present a rapid and simple method for computation of the elements of this inverse without computing the relationship matrix itself. The method is particularly useful in noninbred populations but is much faster than the conventional method in the presence of inbreeding.

Theoretical Basis and Computational Methods for a Number of Different Animal Models

Abstract The animal model is actually a set of many different models. Their common feature is that all animals in a herd or region or nation are evaluated jointly in contrast, for example, to separate male and female evaluations. An animal model can account for repeated records, multiple traits, nonadditive genetic effects, litter effects, and, in addition, a number of environmental effects, both fixed and random. Some of these possible models are described, some computational aspects are presented, and where possible, corresponding reduced animal model methods are presented.

Analysis of covariance in mixed models with unequal subclass numbers

Blue of estimable linear functions and exact tests of hypotheses concerning such functions usually do not exist in the covariance model with random factors having unknown variances. This is true even in the equal subclass numbers case. This paper suggests alternative methods for finding linear unbiased estimators and presents methods for computing sampling variances which are linear functions of the unknown parameter variances. Also, higher level covariates are defined and nonestimability problems resulting from association of such covariates with fixed factors are discussed.

The Invariance and Calculation of Method 2 for Estimating Variance Components

Three methods of estimating variance components from unbalanced data are given in Henderson [1953]. Method 2, designed to circumvent biasedness that results from using MIethod 1 on mixed models, is characterized more generally in Searle [1968] as a special way of executing what is described there as a simplified form of a Generalized Method 2. In that description 2 limitations are discussed: (i) It is proven that the simplified form of the Generalized M\l[ethod 2 can be used only when there are no interactions between fixed and random factors. (ii) It is suggested that the Generalized Method 2 is not uniquely specified. There is no doubt about limitation (i): Hendersons Method 2 can be used only when the model has no interactions between fixed and random effects. However, we show here for a wide class of -models that Hendersons Method 2, despite being one of the methods collectively called Generalized Method 2, does not suffer from limitation (ii). Indeed, it is a generalization of limitation (i) that nullifies (ii); i.e., for a wide class of models, Method 2 is well defined. One comment on limitation (i) is appropriate. Models that include any nesting of fixed and random factors within each other are also excluded from Method 2, because such models are tantamount to having interactions between fixed aind random factors.