Showing papers on "Mixed model published in 1980"

Necessary and Sufficient Conditions for Explicit Solutions in the Multivariate Normal Estimation Problem for Patterned Means and Covariances

Abstract: The problem of finding maximum likelihood estimates for patterned means and covariance matrices in multivariate analysis is considered. Necessary and sufficient conditions are presented for the existence of explicit solutions and the obtaining of these explicit solutions in one iteration of the scoring equations from any positive definite starting point. Cases in which averaging yields the explicit maximum likelihood estimates are discussed. These results can be applied to the problems of finding maximum likelihood estimates for the parameters in the complete, compound and circular symmetry patterns; mixed models in the analysis of variance; and for finding asymptotic distributions of likelihood ratio statistics when the parameters under the null hypothesis have explicit maximum likelihood estimates.

Circular balanced repeated measurements designs

Abstract: The concept of a circular design is defined and when proper balance for various effects is assumed, its universal optimality is proved over the class of all designs with the same set of parameters, Such designs are shown to minimize the variance of the best linear unbiased estimators of contrasts of residual and direct effects over the class of equireplicated designs. All models assume first order residual effects and are of a circular nature. The proofs are presented in a unified manner for several models at a time. They are based on certain matrix domination which occurs when parameters are eliminated from a linear modelj this latter fact is proved for a general linear model.

A Method of Sire Evaluation for Dichotomies

Abstract: In this paper, the following problems associated with genetic evaluation of categorical traits by linear models are discussed: (1) Scores are arbitrarily assigned to response categories. (2) Mixed model solutions do not incorporate the restriction in the estimation space that the sum of response probabilities must total 1 across categories (3) The variance in the observed scale is not constant and depends on the genotypic value of the candidates for selection. (4) The additive genetic variance in the observed scale depends on the mean incidence of the character in the subpopulations considered in the model. (5) Nonadditive genetic variation is present in the observed scale. (6) Linear relationships fail outside a restricted range of the data. (7) Ranking optimality of best linear predictors is lacking when the conditional expectation of the predictand given the data is not linear. A method of sire evaluation of dichotomies based on a log-linear model is introduced, its properties are discussed and examples are presented.

A Characterization of Best Linear Unbiased Estimators in the General Linear Model

Abstract: A characterization of best linear unbiased estimators is given in the case of the general linear model. In addition necessary and sufficient conditions are derived for a given estimable function to have a best linear unbiased estimator. In particular models for which each estimable function has a best linear unbiased estimator are characterized. The conditions stated are given in a computational atractive form. The problems are discussed from the coordinate-free point of view. This is important for the results can be easily adopted in the case of estimation of variance components. The problem of estimation of either treatment or block effects in a mixed model serves as an example which illustrates the applicability of the results.

Explicit Maximum Likelihood Estimates from Balanced Data in the Mixed Model of the Analysis of Variance

Abstract: A result of Szatrowski, giving necessary and sufficient conditions for the existence of explicit maximum likelihood estimates for multivariate normal means and covariances with linear structure, can be applied to the problem of obtaining explicit maximum likelihood estimates in the mixed model of the analysis of variance. This application yields a simple procedure for checking whether or not explicit maximum likelihood estimates exist for the parameters in the balanced mixed model of the analysis of variance. Examples of this procedure as well as a discussion on finding the explicit maximum likelihood estimates are given.

A Multivariate Approach to a Mixed Linear Model

Abstract: The usual treatment of the two-way mixed model parallels that of the fixed models, with the usual sums of squares appearing in the F ratios. In this article, we argue in favor of a multivariate approach. We propose a new criterion for testing a general hypothesis under general conditions and offer an expression for the asymptotic expansion of the distribution function.

Confidence intervals for ratios of linear functions of mixed models with reference to animal breeding data.

Abstract: Summary Mixed linear models are widely used in animal breeding experiments from which estimates of ratios are desired. This paper presents methods for the development of exact confidence intervals for ratios of linear functions of fixed effects in mixed models. Approximate confidence intervals for ratios involving random effects can also be obtained by these methods. Additional problems considered include pooled ratio estimation from multiple data sets with constant variance and the case of unequal error variance with two sources of heterogeneity. Confidence intervals obtained here are expected to be larger than those based on approximations of the variance of a ratio through Taylor's series, since the latter underestimate the true variance under certain conditions.