On Construction of Doubly Nested Partially Balanced Incomplete Block Designs

Background: A doubly nested partially balanced incomplete block (DNPBIB) design is defined as an arrangement of v treatments each replicated r times in three systems of blocks if, each block of the first system contains m1 blocks of second system and each block of the second system contains m² blocks of the third system. Each stage of DNPBIB design, treated independently, is a PBIB design. 

Methods: Three component designs viz., (i) ignoring the first and second system of blocks, a PBIB design with b₃ blocks each of size k₃ (< v) units with, λ_3i concurrences of any pair of treatments which are ith associates of each other, (ii) ignoring first and third system of blocks, another PBIB design with b₂ blocks each of size k₂(< v) units with λ_3i concurrence of two treatments which are i^th associates of each other and (iii) ignoring the second and third system of blocks, a third PBIB design with b₁ block each of size k₁ (< v) units with, λ_1i concurrences of two treatments within first associate of each other, are obtained. 

Result: This study provides two new general methods of constructing DNPBIB designs, where the component designs are PBIB designs with two or three associate classes. Group divisible and rectangular association schemes have been used for developing such designs. Methods are illustrated with appropriate examples.