Definition of OBIBD and related designs.

OBIBD
Suppose a set of s BIBDs (with identical parameters) superimposed one upon the other, in order, so that a single treatment at a given plot is replaced by a vector of s treatments. Further, if the incidence matrix of the ith treatment set w.r.t blocks is ni0 and w.r.t. the jth is nij then nij- ni0n'j0 = 0.

The notation adopted is OBIBD(v,b,r,k,λ;s) or some shorter version thereof.

Perfect Graeco-Latin BIBD (pergola)
An OBIBD in which nij, i > 0 and j > 0, is of the form aI + bJ, i.e. is the matrix of an SBIBD or the matrix of a symmetric balanced supercomplete design.
Correlated Block Design (CBD)
Suppose a set of s BIBDs (with identical parameters) superimposed one upon the other, in order, so that a single block is replaced by a vector of s blocks. Further, if the incidence matrix of the ith treatment set w.r.t blocks is ni0, then ni0n'j0 is of the form aI+bJ.

This is the same as dropping the matching of individual treatments at a plot from a pergola. There may be some interest in the designs obtained by doing the same to more general OBIBDs, but clearly just superimposing BIBDs is not of much interest.

These constitute a sub-category of a general class of nested designs studied by Fuji_Hara et al.

Returning to OBIBDs, the definition means:

  1. Within a given treatment set, each treatment appears within a block equally often with every other treatment
  2. Across treatment sets, suppose that treatment A of the ith set is superimposed on treatment B of the jth in m blocks. Then over all the blocks, treatment A of the ith must occur in the same block as treatment B of the jthon mk occasions (including the m superimpositions).

Three main types of OBIBD have been investigated to date.

  1. Those in which each treatment of a given set occurs an equal number of times (usually 1) with every treatment of every other set, except itself (or, rather, its analogue).
  2. The second type is similar, save that now each treatment in a given set is allowed to occur with its analogue in every other set.
  3. The third kind has each treatment of one set occurring with a subset of the other set, different treatments having different subsets, but any pair of subsets having a fixed number of treatments in common.

In all cases, the design relating first to second designs is balanced. In the original definition of designs of this type, due to Preece, this feature was part of the specification, and these designs are here called pergolas (from PERfect GraecO-LAtinBIBD) as defined above: the current definition and terminology (of OBIBD) is essentially due to Morgan & Uddin,

All three (of the above categories of pergola) can be summarised by saying that nijn'ij = aI + bJ, where I is the identity of order v, and J is the matrix of order v comprising all 1's. In the first case, n is the incidence matrix of the unreduced design for v treatments in blocks of (v-1); in the second case, the same design is augmented by occurrences of the missing treatment, and in the third case n is the incidence matrix of a non-trivial symmetric BIBD.