Analysis of a Class of PBIB Designs with More than Two Associate Classes

Abstract

The use of 2-associate PBIB design is fairly common in experimental work. However, PBIB designs with more than two associate classes are not widely used because of the complicated nature of the analysis and construction involved. Recently, in an interesting paper [2], Shah constructed a number of 3-associate PBIB designs by what may be called the matrix substitution method. In this method, the incidence matrix of the 3-associate PBIB design is constructed by replacing the integers of a balanced matrix in $S$ integers (for example, the matrix might be the incidence matrix of a BIB design, that is, a balanced matrix in two integers) by the incidence matrices of $S$ associable BIB designs. The present author [1] and Shah [3] have shown that the above method may be used to construct a PBIB design with $2m + 1$ associate classes by replacing the integers of a PBIB design with $m$ associate classes by the incidence matrices of two associable BIB designs. Shah [3] has also given a simple method of analysis for PBIB designs with $2^n - 1$ associate classes constructed by the matrix substitution method. Thus, Shah's method of analysis may be used to analyze PBIB designs with 3 and 7 associate classes (corresponding to $n = 2$ and 3) which are of practical interest. In the present paper, simple methods of analysis for a class of PBIB designs with 3 and 5 associate classes are given. The method given here for PBIB designs with three associate classes provides an alternate method, and it is hoped that this method is more simple and direct than that given by Shah.