On the Selection of Designs for Comparative Experiments

The problem considered here is one which was posed by Pearce [1974] and may be stated as follows: Comparative experiments are frequently conducted to obtain information on particular contrasts among treatments, but all contrasts are not equally important. In these circumstances, how can the most suitable design be selected for a trial, subject to the usual constraints on time, money and material? Pearce gave a formal method of assigning importance to contrasts by using a weighting vector w and this idea will be adopted here. Thus, in an experiment with t treatments we consider a t X t matrix D, whose columns represent a set of orthogonal contrasts di that may be estimated in the experiment; we then look for a minimum of the function wi Zvi var(di). The variances depend on the experimental design so the problem is to select from all designs satisfying external constraints that which gives the smallest value to this function. Among the contrasts di, the tth will represent an estimate of the general mean, but for i # t, di' = 0 where 1 is the unit vector. Further, it will be convenient to take D'D = I, the unit matrix, as any adjustments can be made in the weighting vector w. Again, for convenience, we may write w'1 = 1 and take wZ, the last element of w, as zero since the estimation of the general mean is of no interest in a comparative trial. The matrix D will not be the same as the matrix C of basic contrasts of Pearce, Calin'ski and Marshall [1974] although C also has the elements in each column except the last summing to zero. However, C'r-'C = I, where r is a vector of replications, r' is this vector expressed as a diagonal matrix and r-' is its inverse. There is no useful relation between C and D, and C will not be considered further here. If Qo2 is the variance-covariance matrix of the design the diagonal terms in D'QD0-2 give the variances of the contrasts di . Weighting these variances by means of the weights in the vector w, we then have the condition that the trace of the matrix w'D'QD has to be as small as possible in order to get a suitable design. Since tr (w6D'CID) = tr (QDw'D'), it is easiest in practice to find Dw'D' for any required weighting of the variances and then multiply it by various possible matrices Q for different designs. This procedure will now be exemplified for Pearce's example of a block design in which t here are four blocks of five plots each and three treatments representing levels 0, 1 and 2