On Partially Linked Block Designs

Abstract

Youden [8] introduced a new class of designs called Linked Block (LB) designs, where a LB design is defined as that in which any two blocks of the design have the same number of treatments in common. Later, the same author together with Connor [9] presented a set of designs called `chain block designs' useful in the field of Physical Sciences. These designs can be considered as `not fully linked block designs'. In this paper we shall extend the idea of LB designs in yet another way by defining Partially Linked Block (PLB) designs. The basic motive in introducing PLB designs is to generalise the notion of LB designs and also to see how far the existing designs yield new PBIB designs. Consider an arrangement of $v$ treatments in $b$ blocks of $k$ plots each $(k < v)$, such that each treatment occurs at most once in any block and altogether in $r$ blocks. This is called an incomplete block design and is denoted by $D(v, b, k, r)$. We have $bk = vr$. When $b = v$, the design becomes symmetric. The incidence matrix $N = (n_{ij}) (i = 1, 2, \cdots, v; j = 1, 2, \cdots, b)$, where $n_{ij} = 1$ or 0 according as the $i$th treatment occurs or does not occur in the $j$th block characterises such a design completely.