Genealogical Tree of Mixed Linear Models
by
Ricardo Covas
Polytechnical Institute of Tomar

Orthogonal Mixed Models have been increasingly approached, since the 1970's with the work of Seely (1970, 1971, 1972, 1977), trough the use of Jordan algebras, which were first introduced by Jordan in 1934.. These, nowadays constitute a natural framework for deriving best unbiased estimators and hypothesis tests.

We define an binary operation between Jordan algebras, the Restricted Kronecker Product (generalizing what was first introduced by Fonseca in 2006), which connects Jordan algebras with all crossing, nesting and replicates in linear mixed models. It is possible to trace back any model until we reach singular Jordan algebras (the algebra connected with the simplest linear model - the random sample), drawing a genealogical tree for a model. This procedure starts from very simple input and allows us to get not only the principal basis of the Jordan algebra associated to the model, but also the matrices from the variance-covariance matrix and the incidence matrices of the model.

The Genealogical Tree proves to be of great advantage to practitioners.

Date received: January 30, 2008

Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cavi-69.