A-optimal block designs for comparing test treatments with control treatment(s) - an algorithmic approach

Abstract

There are many experimental situations in which the interest of the experimenter lies in comparing a set of new treatments called test treatments with one or more standard or control treatments. In the presence of one nuisance factor, block designs are advisable. Before conducting the experiment, a block design should be so chosen in a class of all connected block designs that it minimizes the average variance of the test versus control(s) contrasts. Such a design is called A-optimal design for the problem under study. There are several work available establishing A-optimality of block designs for tests versus control(s) comparisons and several A-optimal block designs are also available in literature. In this project, we have dealt with the problem of obtaining A-optimal block designs for tests versus control(s) comparison through algorithmic approach. First Chapter of the project report gives the introduction, review of literature related to block designs for tests versus control comparisons, motivation and objectives of the project. In the second Chapter, we have proposed a an algorithm to obtain A-optimal balanced treatment incomplete block designs. Construction of an A-optimal BTIB design using the proposed algorithm is illustrated with an example. A catalogue of A-optimal BTIB design in a restricted parametric range is also provided. In the third Chapter we have presented an algorithm to construct A-optimal group divisible treatment designs. A list of A-optimal GDT designs are given in the Chapter. The fourth Chapter deals with the problem of construction of A-optimal balanced bipartite block designs. An algorithm is presented to this end and an example illustrating the construction of A-optimal BBPB design is also given. A catalogue of A-optimal BBPB designs in a restricted parametric range is also provided. In the fifth Chapter, an algorithm is presented for obtaining weighted A-optimal block designs for test verus test and test versus control comparisons. A list of weighted A-optimal BTIB designs in a restricted parametric range is also provided. The proposed algorithms mainly have two parts. In the first part, they obtain certain other parametric values from given number of treatments, number of blocks and block size utilizing the sufficient conditions of A-optimality and necessary parametric conditions of existence of these classes of designs. Then in the second part, the algorithms utilize linear integer programming approach to obtain the incidence matrix of A-optimal BTIB design. All the proposed methods have been implemented using R language. An R package called Aoptbdtvc has been developed and is available on cran.r-project.org/web/packages/Aoptbdtvc/index.html. A number of A-optimal BTIB designs, GDT designs, BBPB designs and weighted A-optimal BTIB designs are obtained in limited restricted parametric ranges and catalogues of these designs are prepared. The layouts of the designs are available on Design Resources Server at http://iasri.res.in/design/. These catalogues will serve as ready reckoner for experimenters as well as statisticians.