On the derivation of complex linear models from simpler ones

Linear mixed models are useful in biology, genetics, medical research, agriculture, industry, and many other fields, providing a flexible approach in situations of correlated data. Based on the structure of the variance-covariance matrix, emerged a special class of linear mixed models, those of models with orthogonal block structure, which allows optimal estimation for variance components of blocks and contrasts of treatments. This approach triggered a more restrict class of mixed models, models with commutative orthogonal block structure, whose interest lies in the possibility of achieving least squares estimators giving best linear unbiased estimators for estimable vectors. Exploring the possibility of joint analysis of linear mixed models, obtained independently, and focusing on the approach based on the algebraic structure of the models, some authors have investigated the conditions in which the good properties of the estimators are preserved. In this work we intend to highlight the ideas underlying the techniques for the joint analysis of models, since these aspects were underexplored in the works where the theoretical formulation of the techniques were introduced. Given that these techniques were developed involving models with commutative orthogonal block structure, we provide a selective review of the literature focusing on the contributions addressing this special class of mixed linear models