Efficient partially replicated block designs with each replication number one or two

TL;DR

This paper develops efficient partially replicated block designs where each treatment is replicated once or twice, allowing variable block sizes and providing new bounds and design strategies for resource-constrained experiments.

Contribution

It introduces a flexible framework for partially replicated block designs without fixed block sizes, deriving sharp bounds and exploring various design classes for improved efficiency.

Findings

Derived sharp lower bounds on A- and MV-criteria for these designs.

Explored linked block designs and their applications.

Provided illustrative examples demonstrating design efficiency.

Abstract

We investigate block designs, under the A- and MV-criteria, when each treatment can have only one or two replications due to resource constraints, as can happen, for example, in early generation varietal trials. While these are commonly known as partially replicated designs, a key new feature of the present work is that no restriction about a constant block size is imposed on the subdesign consisting of the twice replicated treatments. This makes the derivation more challenging but allows us to entertain a wider class of competing designs and hence increases the flexibility of the results. Considering all treatments as equally important, design-independent, sharp lower bounds on the A- and MV-criteria are derived, so as to find highly efficient designs over this wider class. The roles of (a) linked block designs, (b) designs in an online catalog designtheory.org, and (c) partially balanced incomplete block designs, or duals thereof, as adapted to our setup, are explored at length. Illustrative examples are presented.