Block designs for comparing dual with single treatments

This investigation concerns the design of experiments whose purpose is to compare the joint effects of two factors A and B at n and m levels respectively with the effect of the individual factor. The experiments are subject to the constraint that one particular treatment combination cannot be used. An example is a medical trial to investigate the joint effects of two drugs, each of which is either absent or given at a number of predefined dose levels, in which it is unethical to administer a double placebo. This type of clinical trial has practical application in the quest for treatments of acute conditions, such as severe hypertension when the improvement produced by a single drug might be inadequate. The aim of this investigation is to find efficient designs in the sense of having small variance for the estimators of the contrasts of interest. The criterion employed for design choice is the A-criterion. The methods used include finding a lower bound on the total of the variances of the estimators of the contrasts and identifying a class of designs containing many efficient designs. For m&#61; n&#61; 2, the problem is a special case of the test treatments versus a control problem, for which series of A-optimal and near A-optimal designs are already available. These known results are used to find series of new A-optimal and near A-optimal designs to fill the gaps in a practical range of parameter values. For any n and m&#61; 2, the class of PBDS designs is identified and shown to contain very efficient designs. Methods of constructing such designs are developed and overall A-optimal and efficient designs are tabulated. For n and m both greater than 2, a generalization of the PBDS class is developed and shown to include highly efficient designs by comparison with the bound and, for small experiments, computer generated designs. Further issues on which results are given include the design of completely randomized experiments, efficient designs for estimating certain contrasts more accurately than others and the estimation of factorial effects. Finally, a method is developed of identifying designs efficient for estimating specific contrasts, C&tau;, through linking the structure of the intra-block information matrix to the structure of C'C.</p