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- Estimation and incommutativity in mixed modelsPublication . Ferreira, Dário; Ferreira, Sandra S.; Nunes, Célia; Fonseca, Miguel; Silva, Adilson; Mexia, João T.In this paper we present a treatment for the estimation of variance components and estimable vectors in linear mixed models in which the relation matrices may not commute. To overcome this difficulty, we partition the mixed model in sub-models using orthogonal matrices. In addition, we obtain confidence regions and derive tests of hypothesis for the variance components. A numerical example is included. There we illustrate the estimation of the variance components using our treatment and compare the obtained estimates with the ones obtained by the ANOVA method. Besides this, we also present the restricted and unrestricted maximum likelihood estimates.
- Optimal Estimators in Mixed Models with Orthogonal Block StructuresPublication . Ferreira, Dário; Ferreira, Sandra S.; Nunes, Célia; Mexia, João T.Mixed models whose variance–covariance matrices are the positive definite linear combinations of pairwise orthogonal orthogonal projection matrices have orthogonal block structure. Here, we will obtain uniformly minimum-variance unbiased estimators for the relevant parameters when normality is assumed and we show that those for estimable vectors are, in general, uniformly best linear unbiased estimators. This is, they are best linear unbiased estimators whatever the variance components.
- Inference forLorthogonal modelsPublication . Ferreira, Sandra S.; Ferreira, Dário; Moreira, Elsa E.; Mexia, João T.We generalize the class of linear mixed models when normality is assumed.
- Joining models with commutative orthogonal block structurePublication . Santos, Carla; Nunes, Célia; Dias, Cristina; Mexia, João T.Mixed linear models are a versatile and powerful tool for analysing data collected in experiments in several areas. Amixed model is a model with orthogonal block structure, OBS, when its variance–covariance matrix is ofall the positive semi-definite linear combinations of known pairwise orthogo-nal 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 Commuta-tive Jordan algebras of symmetric matrices, and the Carte-sian 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 esti-mators, 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 in mixed models through three step minimizationPublication . Ferreira, Dário; Ferreira, Sandra S.; Nunes, Célia; Mexia, João T.The aim of this article is to present an estimation procedure for both fixed effects and variance components in linear mixed models. This procedure consists of a maximum likelihood method which we call Three Step Minimization, TSM. The major contribution of this method is that when variances tend to be null standard algorithms behave badly, unlike the TSM method, which uses a grid search algorithm in a compact set. A numerical application with real and simulated data is provided.
- Tests and relevancies for the hypotheses of an orthogonal family in a model with orthogonal block structurePublication . Ferreira, Dário; Ferreira, Sandra S.; Nunes, Célia; Mexia, João T.A model has an orthogonal block structure if it has, as covariance matrix, a linear combination of pairwise orthogonal projection matrices, that add up to the iden- tity matrix. The range space of these matrices are associated to hypotheses of an orthogonal family. In this paper we show how to obtain tests for these hypotheses when normality is assumed and how to consider their relevance when normality is discarded. Besides the notion of relevance, we formulate hypotheses in a general way that may be applied to models with orthogonal block structure, whose factors may have xed and/or random e ects. The results are applied to prime basis factorial models and an example is presented.