Some Row and Column Designs for Two Sets of Treatments

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Abstract
Some new simple designs are provided for the simultaneous estimation of the effects of 2 non-interacting sets of treatments, with 2-way elimination of heterogeneity. The designs are suitable for when a new set of treatments is to be applied to experimental material which may still be affected by an earlier set. One type of design, obtained by omitting a row or rows from a Greco-Latin square of order t, has each set arranged in a t x k Youden square design, t >k; this type includes series of designs for which t is a prime of the form (4n + 3) and k = 1/2(t[plus or minus]1). Closely related t x l/2(t[plus or minus]1) row-and column designs for which t is a prime of the form (4n + 1), and for which treatments of each set are partially balanced with respect to columns and with respect to the other set, are also discussed. A further type of row-and-column design is for one set of t treatments arranged in a t x k Youden square design, and one set of k treatments orthogonal to the first set and to columns, and totally balanced with respect to rows; 7 x 3 and 11 x 5 designs of this type are given for the first time.

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