Combining Models

In this work we study a special class of linear mixed models - models with orthogonal block structure. Imposing a commutativity condition on them, we get a new class of mixed models, called models with commutative orthogonal block structure, COBS. 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. We present a review of three techniques that enable us to analyze complex models, designed from simpler ones, emphasizing the conditions of applicability of each of them, their limitations and advantages. The techniques, that consist in models crossing, models nesting and models joining, rests on the algebraic structure of the models and binary operations on commutative Jordan Algebras of symmetric matrices. Since crossing, nesting or joining COBS we obtain new COBS, the good properties of estimators hold for the resulting models.