Browsing by Author "Santos, Carla"
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- Inference for types and structured families of commutative orthogonal block structuresPublication . Carvalho, Francisco; Mexia, João T.; Santos, Carla; Nunes, CéliaModels 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.
- 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.