The statistical design and analysis of experiments and combinatorial design

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The proposal consists of a range of interrelated studies having potential applications in clinical trials, the manufacturing industry and agriculture, and aims to validate the most promising "loose ends" of statistical investigation in the applicant's distinguished research career. The focus is the statistical study of the design of experiments dealing especially but not exclusively with situations in which the treatments under comparison have a relatively complex structure. The issues to be studied are: (i) a special type of design, an incomplete factorial design, to be used in clinical trials in medicine with prognostic factors derived from the condition of the patient at entry to the trial. Also "weighted" factors will be considered in order for extreme combinations of factors to be included, thus permitting the designs to be judged to be "ethical"; (ii) extensions to a particular type of design called a plaid-square design, for use chiefly in medical applications and their link to two-phase designs. In addition, the plaid-square designs may be natural in industry for studies on reliability of products in manufacturing processes. The important point is to provide design or data-collection guidance which balances optimal estimation of covariate effects with sufficient statistical power to detect differences between various treatment effects if they were present; (iii) the determination of relationships between incomplete block designs and the mathematical theory of graphs which will add insight into the degree of robustness of a design, i.e. what conclusions can be made when some observations are missing or spurious; and (iv) the relationship of nearest-neighbour designs (configurations) and the generalized Sudoku square giving the possibility also of data compression. In the 9x9 Sudoku square, only seventeen numbers produce a unique square of 81 numbers. Practical applications might include data compression or encryption. Although the Sudoku puzzle was originally developed as a form of entertainment, it may shed light on some statistical design of experiment problems and be of itself an intriguing variation.