A Note on Incomplete Block Designs with Row Balance

Abstract

With the balanced incomplete block designs [1] $r$ replicates of each of $v$ treatments are arranged in $b$ `blocks', and the number of `plots', $k$, in each block is smaller than $v$. In order to eliminate systematic block differences from the comparison of treatment means and to obtain treatment comparisons of equal precision the well known conditions of balance specify (a) that no treatment should occur more than once in any block, (b) that the number of blocks in which any two particular treatments are both applied should be a constant number of blocks ($\lambda$ blocks) for all possible treatment pairs. When these designs are used in this form, the position of treatments within each block is not specified and can normally be regarded as unimportant. Accordingly the treatments in each block are randomized. Situations, however, arise in which the `plots' occupy certain characteristic positions in each block. Thus, if in a field experiment the blocks are vertical columns the plots would fall into $k$ horizontal rows which may also have systematic effects on the yields. In this case it will often be advantageous to balance the design with regard to rows (as well as with regard to columns) in a manner similar to the Latin square. Such an arrangement was first developed by Youden [2] who used the particular incomplete block designs with $b = v$ and in these rearranged the treatments in each block in such a way that every treatment occurred precisely once in each row. More recently one of us (S.S.S.) has carried out similar rearrangements for the other incomplete block designs with $b > v, v \leqq 10, r \leqq 10$ (ie., for those tabulated in standard tables and book), and has used a definition of balance resulting in at most two different precisions for treatment comparisons. In this note we show that (i) balancing with regard to rows resulting in an equal precision of all treatment comparisons is possible if $b = mv$ ($m$ integral), (ii) in all incomplete block designs with $r = mk \pm 1$ a row balance is possible resulting in treatment comparisons of two different precisions. One of us (W. B. T.) has prepared complete tables of double balanced designs suitable for practical use which it is hoped to publish together with the analysis of variance procedure with recovery of interblock information.*