Weighted A-efficient block designs for comparing test treatments with controls with unequal precision
by
V.K. Gupta
IASRI, Library Avenue, New Delhi - 110 012, India

In a plant improvement programme for development of new varieties or strains, it is often desired to include some controls (check varieties) in the experiment to investigate the performance of the new varieties against the check varieties. These check varieties may be a local variety, a national check, a disease resistant variety, etc. The comparisons of each of the new varieties with the local variety or with the disease resistant variety may be more meaningful and hence important to the experimenter and, therefore, would be required to be given higher precision than the other comparisons. Similar situations also occur in clinical trials for evaluation of new drugs where two treatments namely a placebo and an existing active drug are taken as control treatments. For regulatory purposes, it often is necessary to demonstrate the magnitude of the activity of the new drug, and therefore the comparison with the placebo is more important. It is not always necessary to demonstrate to the regulatory agency that the new drug is more effective than the existing drug. But for the purposes of the pharmaceutical company's marketing efforts, in fact, the second comparison is likely to be the more important.

To deal with such experimental situations, the problem of obtaining optimal block designs for comparing w test treatments with u controls in b blocks of size k has been considered. Conditions under which a design is weighted A-optimal for estimating test treatments vs controls contrasts with unequal precision are derived, the weights being given according to the relative importance of the controls. A new class of designs called Generalised Balanced Treatment Incomplete Block (GBTIB) designs has been introduced. A general method of construction of GBTIB designs for two controls is given. Catalogues of weighted A-optimal designs for two controls and weighted A-efficient GBTIB designs are given. A method of construction of GBTIB designs using resolvable balanced incomplete block (BIB) designs has also been given and illustrated through an example. These designs are also useful for On-farm trials.

It is not always possible to obtain designs that are efficient in a global class through algebraic minimization. In such situations, recourse is made to an intelligent computer algorithm that is based on exchange and interchange of treatments. Using the algorithm many efficient designs have been obtained. A catalogue of the designs obtained has been prepared for the use of practicing statisticians and experimenters.

Date received: July 14, 2003