The construction of augmented designs in square arrays

TL;DR

This paper explores the relationship between augmented designs in square arrays and their contractions, establishing a formal connection that enables the construction of optimal augmented designs based on contraction properties.

Contribution

It formally links augmented designs in square arrays to their contractions, allowing for the creation of optimal designs through contraction analysis and computer search.

Findings

Established the connection between augmented designs and contractions.

Updated the cyclic contraction table for optimality.

Provided methods to find near-optimal augmented designs when cyclic contractions are suboptimal.

Abstract

Augmented designs are typically used in early-stage breeding programs to compare single replicates of test entries by combining them with replicated check varieties. One or two dimensional incomplete blocking can be incorporated in the design to accommodate possible site variation. An augmented design in a square array can be derived from a smaller row-column design (the contraction). In a recent paper Bailey and Haines (2025) investigated the link between an augmented design in a square array and its contraction. Here we formally establish this connection by expressing the average efficiency factor of the augmented design in terms of that of its contraction. A consequence of this is that an optimal contraction can be used to construct an optimal augmented design. The table of cyclic contractions presented by Bailey and Haines (2025) is updated in terms of optimality. Specifically, in cases where a cyclic contraction is not optimal, an augmented design with optimal or near-optimal efficiency can be obtained via computer search.