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The models in structured families correspond to the treatments of a fixed effects base design \(\pi \). Then the action of factors in \(\pi \), on the fixed effects parameters of the models, is studied. Analyzing such a families enables the study of the action of nesting factors on the effects and interactions of nested factors. When \(\pi \) has an orthogonal structure, the family of models is said to be orthogonal. The models in the family can have one, two or more strata. Models with more than one stratum are obtained through nesting of one stratum models. A general treatment of the case in which the base design has orthogonal structure is presented and a special emphasis is given to the families of prime basis factorials models. These last models are, as it is well known, widely used in fertilization trials.

Given an orthogonal model $${{\bf \lambda}}=\sum_{i=1}^m{{{\bf X}}_i}{\boldsymbol{\alpha}}_i$$ an L model $${{\bf y}}={\bf L}\left(\sum_{i=1}^m{{{\bf X}}_i}{\boldsymbol{\alpha}}_i\right)+{\bf e}$$ is obtained, and the only restriction is the linear independency of the column vectors of matrix L. Special cases of the L models correspond to blockwise diagonal matrices L = D(L1, . . . , Lc). In multiple regression designs this matrix will be of the form $${\bf L}={\bf D}(\check{{\bf X}}_1,\ldots,\check{{\bf X}}_{c})$$ with \({\check{{\bf X}}_j, j=1,\ldots,c}\) the model matrices of the individual regressions, while the original model will have fixed effects. In this way, we overcome the usual restriction of requiring all regressions to have the same model matrix.