On a Generalisation of the Kronecker Product Designs

Abstract

The use of incomplete block designs for estimating and judging the significance of the difference of treatment effects is now standard. Fisher and Yates [2] have provided a complete table of balanced incomplete block designs (BIB) for the value of $r, k \leqq 10$. In this paper a method of constructing a special class of partially balanced incomplete block designs (PBIB) from the known BIB designs is given. Vartak [6] and Sillito [4] have used the Kronecker product of matrices to construct statistical designs. Their method and the method given in this paper differ only in the fact that in their method two distinct elements of a matrix are replaced by two distinct matrices, while in the method considered in this paper, two or more distinct elements of a matrix are replaced by different matrices. All the PBIB designs considered in this paper are with three associate classes, and the rectangular association scheme for $\nu = pq$ treatments are written in $p$ rows and $q$ columns so that treatments in the same row are the first associates, treatments in the same column are the second associates and the rest are the third associates.