Orthogonal Partitions in Designed Experiments

A survey is given of the statistical theory of orthogonal partitions on a finite set. Orthogonality, closure under suprema, and one trivial partition give an orthogonal decomposition of the corresponding vector space into subspaces indexed by the partitions. These conditions plus uniformity, closure under infima and the other trivial partition give association schemes. Examples covered by the theory include Latin squares, orthogonal arrays, semilattices of subgroups, and partitions defined by the ancestral subsets of a partially ordered set (the poset block structures).Isomorphism, equivalence and duality are discussed, and the automorphism groups given in some cases. Finally, the ideas are illustrated by some examples of real experiments.Dedicated to Hanfried Lenz on the occasion of his 80th birthday.