The square array design

TL;DR

This paper introduces a new class of augmented square array designs for unreplicated trials, extending incomplete-block design representations to optimize variance properties and flexibility in experimental setups.

Contribution

It presents a novel method for constructing augmented square array designs using auxiliary block designs, including cyclic and non-cyclic types, with improved variance minimization.

Findings

Designs minimize average variances of paired comparisons.

Equireplicate cyclic designs offer flexible construction options.

Properties related to connectedness and randomization are analyzed.

Abstract

This paper is about the construction of augmented row-column designs for unreplicated trials. The method uses the representation of a $k \times t$ equireplicate incomplete-block design with $t$ treatments in $t$ blocks of size $k$, termed an auxiliary block design, as a $t \times t$ square array design with $k$ controls, where $k<t$. This can be regarded as an extension of the representation of a Youden square as a partial latin square for unreplicated trials. Properties of the designs, in particular in relation to connectedness and randomization, are explored. Particular attention is given to square array designs which minimize the average variances of the estimates of paired comparisons between test lines and controls and between test-line and test-line effects. The use of equireplicate cyclic designs as auxiliary block designs is highlighted. These provide a flexible and workable family of augmented row-column square array designs. Designs whose auxiliary block designs are not cyclic are also covered.