The lattice structure of orthogonal linear models and orthogonal variance component models

Tjur (1984) showed that an orthogonal (=balanced) analysis of variance (ANOVA) design may be described and analysed in terms of an associated factor structure diagram. In this paper an extended class of orthogonal designs is defined and studied, the class of geometrically orthogonal designs of linear regression models, which includes all well-behaved ANOVA and regressions designs. It is shown that such designs may be characterized and analysed most naturally in terms of the lattice structure of L, the family of regression subspaces in the design. Any such design may be extended in a natural way to a family of canonical variance component models, called a geometrically orthogonal variance component design