Variety trials in two-dimensional layouts

Abstract

This thesis shows that efficient resolvable row-column designs can be constructed quickly by using properties of both the contraction and the dual design. The method of constructing the contraction of a r-replicate resolvable row-column design is described and is an extension of the research undertaken by Bailey and Patterson (1991) and Jarrett, Piper and Wild (1997). The structural properties of the contraction are investigated, including the row and column incidence matrices. It is shown that the information matrix for the dual design can be expressed in terms of the row and column incidence matrices for the contraction. A connection is made between the eigenvalues of the resolvable row-column design and the dual design. This relationship can be used to enable the average efficiency factor of the resolvable row-column design to be expressed in terms of the canonical efficiency factors of the dual design. Existing optimisation algorithms available for the construction of resolvable row-column designs work well for small experiments. For experiments with a large number of treatments they are computationally expensive. The dimensionality of the problem of constructing resolvable row-column designs is reduced by working with the dual design. By expressing the information matrix for the dual design in terms of the contraction, the result is a computationally faster algorithm. For large designs it is shown to be quicker to update properties of the contraction than those of the resolvable row-column design. The computational effort required for generating resolvable row-column designs with a large number of treatments can be reduced further by using the (M,S)-optimality criterion. When generating resolvable row-column designs using an optimisation algorithm, it is helpful to have a tight upper bound for the average efficiency factor. By comparing the average efficiency factor with the upper bound it is possible to determine when a good design has been obtained and a decision can then be made on whether to terminate the algorithm. Upper bounds for resolvable row-column designs are known not to be tight and a new upper bound is developed by exploiting some of the properties of the contraction and dual design.