##
**Optimum experimental designs.**
*(English)*
Zbl 0829.62070

Oxford Statistical Science Series. 8. Oxford: Clarendon Press. xv, 328 p. (1992).

The fundamental idea behind this book is the importance of the model relating the responses observed in an experiment to the experimental factors. The purpose of the experiment is to find out about the model, including its adequacy. The model can be very general: one receiving attention in several chapters is that for response surfaces in which the response is a smoothly varying function of the settings of the experimental variables. Experiments can then be designed to answer a variety of questions about the model. Often interest is in obtaining estimates of the parameters and using the fitted model for prediction. The variances of parameter estimates and predictions depend upon the experimental design and should be as small as possible. The unnecessarily large variances and imprecise predictions resulting from a poorly designed experiment waste resources.

The tool that we use to design experiments is the theory of optimum experimental design. The great power of this theory is that it leads to algorithms for the construction of designs which, as we show, can be applied in a wide range of circumstances. One purpose of the book is to describe enough of the theory to make apparent the overall pattern. However, as the title of the book makes clear, the emphasis is on the designs themselves. We are concerned both with their properties and with methods for their construction. The Appendix includes a Fortran program for the construction of designs, including many described in the examples.

The material has been divided into two main parts. The first eight chapters, ‘Fundamentals’, discuss the advantages of the statistical approach to the design of experiments and introduce many of the models and examples which are used in later chapters. The examples are, in the main, drawn from science and engineering. The same principles are applicable to agricultural experimentation, but the methods of design construction described here are not the most efficient for agricultural field trials. However, whatever the area of experimentation, the ideas of Part I are fundamental. These include an introduction to the ideas of models and least squares fitting. The ideas of optimum experimental design are introduced through the comparison of the variances of parameter estimates and the variance of the predicted response from a variety of designs and models. In Part II the relationship between these two sets of variances leads to the General Equivalence Theorem which, in turn, leads to algorithms for designs. As well as these ideas, Part I includes, in Chapter 7, a description of standard designs. In order to keep the book to a reasonable length, there is rather little material on the analysis of experimental results. However, Part I concludes, in Chapter 8, with some examples of such analyses both numerical and graphical.

Part II opens with a discussion of the general theory of optimum design, followed by a discussion of a variety of criteria that may be appropriate for designing an experiment. Of these the most often used is \(D\)- optimality, which is the subject of Chapter 11. Succeeding chapters describe experiments with mixtures and designs for extensions to response surface models to include qualitative factors. Examples include the batch of a raw material or the particular design of a chemical reactor in addition to quantitative factors such as time and temperature. Chapter 15 is concerned with algorithms for the construction of designs, one of which is implemented in the Appendix.

Each chapter is intended to cover a self-contained topic. As a result, the chapters are of varying lengths. One of the shortest is Chapter 16 which covers designs for arbitrary design regions: despite the shortness of the chapter, the ability to calculate good designs under such non- standard conditions is a strong argument in favour of the application of optimum design theory. Chapter 18 describes the extension of the methods to nonlinear regression models. The resulting designs depend upon prior estimates of the parameter values. Chapter 19 describes ways of incorporating uncertain prior knowledge into designs. Chapters 20 and 21 extend the discussion to multi-purpose designs and to those for discrimination between models.

In the last chapter we gather together a number of important topics. These include designs for off-line quality control (often known as Taguchi methods) which provide systematic methods for developing products which behave well under a broad range of conditions of use. Related ideas are used to design products which are insensitive to manufacturing fluctuations. This can be thought of as extending the statistical techniques of off-line quality control to the prevention, rather than mere removal, of substandard products. Other topics included in this chapter are designs for sequential clinical trials, for generalized linear models, and for computer simulation experiments.

The tool that we use to design experiments is the theory of optimum experimental design. The great power of this theory is that it leads to algorithms for the construction of designs which, as we show, can be applied in a wide range of circumstances. One purpose of the book is to describe enough of the theory to make apparent the overall pattern. However, as the title of the book makes clear, the emphasis is on the designs themselves. We are concerned both with their properties and with methods for their construction. The Appendix includes a Fortran program for the construction of designs, including many described in the examples.

The material has been divided into two main parts. The first eight chapters, ‘Fundamentals’, discuss the advantages of the statistical approach to the design of experiments and introduce many of the models and examples which are used in later chapters. The examples are, in the main, drawn from science and engineering. The same principles are applicable to agricultural experimentation, but the methods of design construction described here are not the most efficient for agricultural field trials. However, whatever the area of experimentation, the ideas of Part I are fundamental. These include an introduction to the ideas of models and least squares fitting. The ideas of optimum experimental design are introduced through the comparison of the variances of parameter estimates and the variance of the predicted response from a variety of designs and models. In Part II the relationship between these two sets of variances leads to the General Equivalence Theorem which, in turn, leads to algorithms for designs. As well as these ideas, Part I includes, in Chapter 7, a description of standard designs. In order to keep the book to a reasonable length, there is rather little material on the analysis of experimental results. However, Part I concludes, in Chapter 8, with some examples of such analyses both numerical and graphical.

Part II opens with a discussion of the general theory of optimum design, followed by a discussion of a variety of criteria that may be appropriate for designing an experiment. Of these the most often used is \(D\)- optimality, which is the subject of Chapter 11. Succeeding chapters describe experiments with mixtures and designs for extensions to response surface models to include qualitative factors. Examples include the batch of a raw material or the particular design of a chemical reactor in addition to quantitative factors such as time and temperature. Chapter 15 is concerned with algorithms for the construction of designs, one of which is implemented in the Appendix.

Each chapter is intended to cover a self-contained topic. As a result, the chapters are of varying lengths. One of the shortest is Chapter 16 which covers designs for arbitrary design regions: despite the shortness of the chapter, the ability to calculate good designs under such non- standard conditions is a strong argument in favour of the application of optimum design theory. Chapter 18 describes the extension of the methods to nonlinear regression models. The resulting designs depend upon prior estimates of the parameter values. Chapter 19 describes ways of incorporating uncertain prior knowledge into designs. Chapters 20 and 21 extend the discussion to multi-purpose designs and to those for discrimination between models.

In the last chapter we gather together a number of important topics. These include designs for off-line quality control (often known as Taguchi methods) which provide systematic methods for developing products which behave well under a broad range of conditions of use. Related ideas are used to design products which are insensitive to manufacturing fluctuations. This can be thought of as extending the statistical techniques of off-line quality control to the prevention, rather than mere removal, of substandard products. Other topics included in this chapter are designs for sequential clinical trials, for generalized linear models, and for computer simulation experiments.

### MSC:

62K05 | Optimal statistical designs |

62-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics |

62Kxx | Design of statistical experiments |