Square Array Designs for Unreplicated Test Treatments with Replicated Controls

Abstract This paper is about the construction of augmented row–column designs for unreplicated trials. The method uses the representation of a $$k \times t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>×</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:math> equireplicate incomplete-block design with t treatments in t blocks of size k , termed an auxiliary block design, as a $$t \times t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>×</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:math> square array design with k controls, where $$k&lt;t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:math> . 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. Supplementary materials accompanying this paper appear on-line.