Relations among partitions

Abstract

Abstract Combinatorialists often consider a balanced incomplete-block design to consist of a set of points, a set of blocks, and an incidence relation between them which satisfies certain conditions. To a statistician, such a design is a set of experimental units with two partitions, one into blocks and the other into treatments; it is the relation between these two partitions which gives the design its properties. The most common binary relations between partitions that occur in statistics are refinement, orthogonality and balance. When there are more than two partitions, the binary relations may not suffice to give all the properties of the system. I shall survey work in this area, including designs such as double Youden rectangles. Introduction Many combinatorialists think of a balanced incomplete-block design (BIBD) as a set P of points together with a collection B of subsets of P , called blocks , which satisfy various conditions. For example, see [52]. Some papers, such as [16, 65, 201], call a BIBD simply a design . Others think of it as the pair of sets P and B with a binary incidence relation between their elements. These views are both rather different from that of a statistician who is involved in designing experiments. The following examples introduce the statistical point of view, as well as serving as a basis for the combinatorial ideas in this paper. Example 1.1 A horticultural enthusiast wants to compare three varieties of lettuce for people to grow in their own gardens. He enlists twelve people in his neighbourhood. Each of these prepares three patches in their vegetable garden, and grows one of the lettuce varieties on each patch, so that each gardener grows all three varieties. Here the patches of land are experimental units. There may be some differences between the gardeners, so the three patches in a single garden form what is called a block . Each variety occurs just once in each block, and so the blocks are said to be complete . Complete-block designs were advocated by Fisher in [78], and are frequently used in practice.