Common Treatments Between Blocks of Certain Partially Balanced Incomplete Block Designs

Abstract

The study of common treatments between the blocks of statistical designs is not new. The block structure of BIB designs was most exhaustively studied by Connor [4], Shrikhande, Trehan, Chakrabarti [5] and others. Connor [3] extended this search to symmetrical Group Divisible (GD) designs restricting himself mainly to regular and semi-regular cases. Roy and Laha [6] were interested in partially balanced designs which are of the LB type. This paper deals with the study of treatments common to blocks of certain PBIB designs which are Kronecker product of certain known designs. The credit for showing that the Kronecker product of two BIB designs has the association scheme of the rectangular type goes to Vartak [8]. In Section 3 of this paper we have obtained the matrices of characteristic vectors of the treatment characteristic matrix $NN'$ of designs having rectangular association scheme. These are then used to set up bounds for the number of treatments common to any two blocks belonging to a group of blocks of the designs which are PBIB with three and two associate classes obtained by taking the Kronecker product of two BIB designs. Also the limits to the number of treatments common to some two blocks of a singular GD are determined.