αₙ-Designs

This thesis defines a broad class of resolvable incomplete block designs for multifactor experiments, called αₙ-designs. A general methodology for their construction is described and is shown to be a natural extension of the method used by Patterson and Williams (1976) in their construction of α-designs. In fact, the class of α-designs are a special case of αₙ-designs with n = 1. The family of αₙ-designs was primarily developed for their use in factorial experiments. While they are particularly suitable for this purpose, for some combinations of design parameters they also provide more efficient designs than the best available α-designs. Algorithms for the generation of efficient designs require computationally expensive eigenvalue calculations. Therefore, considerable attention is devoted to the study of the structural properties of αₙ-designs and how these can be used to develop computationally fast algorithms. In this endeavour it is shown that the treatment concurrence and information matrices have an important property known as BⁿC-structure. This is used to derive a closed form mathematical expression for the average efficiency factor of any generalized interaction. It is further shown that this expression can be written recursively. These results, combined with the fact than an αₙ-design is completely specified by a small generating array, mean that computer search algorithms can be developed to quickly find highly efficient designs. A desirable feature in a factorial design is the property of orthogonal factorial structure. The family of αₙ-designs do not, in general, admit this property. However, another class of n-dimensional cyclic designs, known as n-cyclic designs, suitable for non-resolvable designs for factorial experiments are also considered in this thesis. They possess orthogonal factorial structure and include a sizeable subset of designs which are resolvable. These resolvable n-cyclic designs are shown to be αₙ-designs. The discovery and subsequent development of the family of αₙ-designs has stimulated a new set of interesting questions in relation to factorial experiments. These are discussed at the end of the thesis

A Parametric Control Chart for Monitoring Individual Batting Performances in Cricket

A control chart is developed specifically for monitoring batting performances in cricket. Issues associated with the need to retain extreme values and limited sampling opportunities are overcome by using a parametric approach to obtain theoretical quartiles from the distribution of individual batting scores. A mixed distribution, named the Ducks 'n' Runs distribution, is proposed. A beta distribution models zero scores (ducks) and a geometric distribution describes the distribution of non zero scores (runs). This suitability of this probability distribution model is demonstrated using data from New Zealand first class batsmen in domestic three day cricket over a four year period. Changes in the process (player performance) are detected using rules adapted from the supplementary rules for Shewhart control charts.