Constructions of Three Associate Constant Block-sum PBIB Designs

Abstract

Partially Balanced Incomplete Block (PBIB) designs find extensive use in various fields, including agriculture, animal studies, and biomedical research. In biomedical research, especially with animals, there is often a need to account for carryover effects from previous experiments. This necessitates the development of designs that ensure uniform cumulative exposure to treatment levels. Constant block-sum designs have been introduced in the literature to address this requirement. In such designs, treatment levels are treated as quantitative, and the goal is to ensure that, by the end of the experiments, all experimental units have received the same constant cumulative dose. In this article, we delve into three distinct three-associate class association schemes, utilizing the Petersen Graph, Pappus Graph, and Hexagon Graph. We provide detailed insights into the construction methods for constant block-sum PBIB(3) designs, along with illustrative examples to facilitate understanding and practical application.