ABSTRACT
The relationship between affine and projective planes can be generalized to other block designs. If B0 is any block of a pairwise balanced design with index λ, then any two treatments that do not belong to B0 must occur together in λ of the remaining blocks, while any two members of B0 must be together in λ−1 of the remaining blocks. It follows that the blocks B\B0 form a pairwise balanced design of index λ when B ranges through the remaining blocks, and the blocks B ∩ B0 form a pairwise balanced design of index λ − 1. We shall refer to these as the residual and derived designs of the original design with respect to the block B0. In general the replication number of a residual or derived design is not constant, and the block sizes follow no particular pattern. To solve the first of these problems it suffices to make the original design equireplicate: If a design has constant frequency r, its residual and derived designs also have constant frequencies, r and r − 1 respectively. The block pattern can be predicted if the size of B ∩ B0 is constant; the derived blocks will be of that constant size, and the blocks of the residual design will be |B ∩ B0| smaller than the blocks of the original. In particular, if we start with a symmetric balanced incomplete block design, we obtain constant frequencies and block sizes.
