Commutative Orthogonal Block Structure Research Articles

Addressing the algebraic structure of a linear mixed model with balanced design towards model extension

Linear mixed models provide a general and versatile approach for analyzing data collected in experiments, suitable for modeling repeated, longitudinal, or clustered observations. Important results for estimation can be obtained when subclasses of these mixed models are considered, based on some characteristics of their algebraic structure. The class of the models with commutative orthogonal block structure, for which least squares estimators are the best linear unbiased estimators, is of great interest. In an approach based on the algebraic structure of the models, and availing ourselves of U matrices, we study the possibility of extending a balanced mixed model, which could lead to a model with commutative orthogonal block structure.

Designs for half-diallel experiments with commutative orthogonal block structure

In some experiments, the experimental units are all pairs of individuals who have to undertake a given task together. The set of such pairs forms a triangular association scheme. Appropriate randomization then gives two non-trivial strata. The design is said to have commutative orthogonal block structure (COBS) if the best linear unbiased estimators of treatment contrasts do not depend on the stratum variances. There are precisely three ways in which such a design can have COBS. We give a complete description of designs for which all treatment contrasts are in the same stratum. Then we give a very general construction for designs with COBS which have some treatment contrasts in each stratum.

Joining Iso-Structured Models with Commutative Orthogonal Block Structure

In this work, we focus on a special class of mixed models, named models with commutative orthogonal block structure (COBS), whose covariance matrix is a linear combination of known pairwise orthogonal projection matrices that add to the identity matrix, and for which the orthogonal projection matrix on the space spanned by the mean vector commutes with the covariance matrix. The COBS have least squares estimators giving the best linear unbiased estimators for estimable vectors. Our approach to COBS relies on their algebraic structure, based on commutative Jordan algebras of symmetric matrices, which proves to be advantageous as it leads to important results in the estimation. Specifically, we are interested in iso-structured COBS, applying to them the operation of models joining. We show that joining iso-structured COBS gives COBS and that the estimators for the joint model may be obtained from those for the individual models.

Models with commutative orthogonal block structure: a general condition for commutativity

A linear mixed model whose variance-covariance matrix is a linear combination of known pairwise orthogonal projection matrices that add to the identity matrix, is a model with orthogonal block structure (OBS). OBS have estimators with good behavior for estimable vectors and variance components, moreover it may be interesting that the least squares estimators give the best linear unbiased estimators, for estimable vectors. We can achieve it, requiring commutativity between the orthogonal projection matrix, on the space spanned by the mean vector, and the orthogonal projection matrices involved in the expression of the variance-covariance matrix. This commutativity condition defines a more restrict class of OBS, named COBS (model with commutative orthogonal block structure). With this work we aim to present a commutativity condition, resorting to a special class of matrices, named U-matrices.

Joining models with commutative orthogonal block structure

Mixed linear models are a versatile and powerful tool for analysing data collected in experiments in several areas. A mixed model is a model with orthogonal block structure, OBS, when its variance–covariance matrix is of all the positive semi-definite linear combinations of known pairwise orthogonal orthogonal projection matrices that add up to the identity matrix. Models with commutative orthogonal block structure, COBS, are a special case of OBS in which the orthogonal projection matrix on the space spanned by the mean vector commutes with the variance–covariance matrix.Using the algebraic structure of COBS, based on Commutative Jordan algebras of symmetric matrices, and the Cartesian product we build up complex models from simpler ones through joining, in order to analyse together models obtained independently. This commutativity condition of COBS is a necessary and sufficient condition for the least square estimators, LSE, to be best linear unbiased estimators, BLUE, whatever the variance components. Since joining COBS we obtain new COBS, the good properties of estimators hold for the joined models.

Estimation and Orthogonal Block Structure

Estimators with good behaviors for estimable vectors and variance components are obtained for a class of models that contains the well known models with orthogonal block structure, OBS, see [15], [16] and [1], [2]. The study observations of these estimators uses commutative Jordan Algebras, CJA, and extends the one given for a more restricted class of models, the models with commutative orthogonal block structure, COBS, in which the orthogonal projection matrix on the space spanned by the means vector commute with all variance-covariance matrices, see [7].

Inference for types and structured families of commutative orthogonal block structures

Models with commutative orthogonal block structure, COBS, have orthogonal block structure, OBS, and their least square estimators for estimable vectors are, as it will be shown, best linear unbiased estimator, BLUE. Commutative Jordan algebras will be used to study the algebraic structure of the models and to define special types of models for which explicit expressions for the estimation of variance components are obtained. Once normality is assumed, inference using pivot variables is quite straightforward. To illustrate this class of models we will present unbalanced examples before considering families of models. When the models in a family correspond to the treatments of a base design, the family is structured. It will be shown how, under quite general conditions, the action of the factors in the base design on estimable vectors, can be studied.

Commutative orthogonal block structure and error orthogonal models

A model has orthogonal block structure, OBS, if it has variance-covariance matrix that is a linear combination of known pairwise orthogonal orthogonal projection matrices that sum to the identity matrix. These models were introduced by Nelder is 1965, and continue to play an important part in randomized block designs. Two important types of OBS are related, and necessary and sufficient conditions for model of one type belonging to the other are determined. The first type, is that of models with commutative orthogonal block structure in which T, the orthogonal projection matrix on the space spanned by the mean vector, commutes with the orthogonal projection matrices in the expression of the variance-covariance matrix. The second type, is that of error orthogonal models. These results open the possibility of deepening the study of the important class of models with OBS.

Estimation in Models with Commutative Orthogonal Block Structure

Abstract A model with variance-covariance matrix V = Σ v i=1 σ2 iPi , where P 1,…,Pv are known pairwise orthogonal orthogonal projection matrices, will have Orthogonal Block Structure with variance components σ2 1,…,σ2 v . Moreover, if matrices P 1,…,Pv commute with the orthogonal projection matrix T on the space spanned by the mean vector, the model will have Commutative Orthogonal Block Structure (COBS). In this paper we will use Commutative Jordan Algebras to study the algebraic properties of these models as well as optimal estimators. We show that once normality is assumed, sufficient complete statistics are obtained and estimators are Uniformly Minimum Variance Unbiased Estimators.

Inference in normal models with commutative orthogonal block structure

Linear mixed normal models are studied in this paper. Using commutative Jordan algebras, the algebraic properties of these models are studied, as well as optimal estimators, hypothesis tests and confidence regions for fixed and random effects. Model crossing andnesting is then presented and analyzed.

Relevant statistics for models with commutative orthogonal block structure and unbiased estimator for variance components

A model has orthogonal block structure, OBS, when it’s covariance matrix is a linear combination of pairwise orthogonal projection matrices P 1, … , Pl . If one of this models has mean vector μ n = ΔT v and the P 1, … , P l commute with the orthogonal projection matrix on the range space of ΔT, the model will have commutative orthogonal block structure, COBS. We obtain relevant statistics for these models as well as BLUE for estimable vectors and unbiased estimators for variance components. When we assume the normality, we show that these statistics will be sufficient and completes. These statistics will be used to get confidence regions and tests of hypotheses for variance components and estimable vectors. We will also show how to induce probabilities measures in parametric spaces and how to adjust confidence ellipsoids for estimable vectors. Lastly we characterize mixed models with COBS.

Canonic inference and commutative orthogonal block structure

It is shown how to define the canonic formulation for orthogonal models associated to commutative Jordan algebras. This canonic formulation is then used to carry out inference. The case of models with commutative orthogonal block structures is stressed out.