Bounds for the efficiency factor of row-column designs

Abstract

SUMMARY Upper bounds for the average efficiency factor of single block and resolvable row-column designs, based on the first two moments of the canonical efficiency factors, are obtained. In 245 out of the 379 cases considered these bounds are tighter than a bound given previously by Eccleston & McGilchrist (1985). Considerable work has been carried out to find good upper bounds for the efficiency factor of block designs, both resolvable and non-resolvable. For a review of this work see Jarrett (1989); for a more recent bound see Tjur (1990). Apart from a bound by Eccleston & McGilchrist (1985), little attention has been given to row-column designs. The purpose of this paper is to derive upper bounds for row-column designs, using some of the results available for block designs, and to compare these bounds with the one given by Eccleston & McGilchrist (1985). A single block row-column design has v treatments set out in an array of p rows and q columns with each treatment replicated r times, so that vr = pq. Latin and Youden squares are examples of such designs. A resolvable row-column design has v treatments in r replicates, with each replicate set out in an array of p rows and q columns. Each treatment is replicated once in each replicate, so that v = pq. An example of a resolvable row-column design for v = 12, p = 4, q = 3 and r =2 is given in Table 1. Other examples are given by lattice squares. The terminology row-column design will be used to include a design of either type. It will be assumed that p ? q.