Experiments with mixtures. Designs, models, and the analysis of mixture data.
2nd ed.

*(English)*Zbl 0732.62069
Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics Section. New York etc.: John Wiley & Sons Ltd. xix, 632 p. £54.30 (1990).

Since the publication of the first edition from 1981, see the review Zbl 0597.62087, about a decade ago, many developments in the area of experiments with mixtures have taken place and scores of research papers have been printed. This timely revised and expanded edition containing a considerable amount of new material is going to be of great use to chemical industries, agriculture, statisticians, and to industries as a whole. The author has written extensively in this area, and is well-known for his contributions to its theoretical and practical aspects. The updating of this book has also included rearrangement and reshuffling of topics from the earlier edition.

The book is self-contained in the sense that the background material (viz, matrix algebra, analysis of variance, principle of least squares, etc. etc.) is all included in chapter 9 now as opposed to chapter 7 in the earlier edition. Each chapter ends with a useful summary, with a set of exercises, and a list of references and recommended readings. An updated bibliography of more than 150 items from recent literature obtained from experiments with mixtures is given at the end of the book. An attractive feature of the book is the inclusion of a large number of solved examples, most of which are from real-life situations. Besides other changes in this revised edition, the reader will find in chapters 7, 8 and 10 new entries.

After presenting some basic concepts and a clear distinction between a factorial experiment and a mixture experiment in chapter 1, the next chapter includes the material on fitting mathematical equations to model the response surface over the entire simplex factor space, and to test the adequacy of the fitted model. It also has a discussion on the simplex-lattice designs and the associated polynomial models to be fitted to data collected at the points of these designs. The use of independent variables is taken in chapter 3 which has also has a new formula for obtaining the radius of the largest sphere with center at the point of main interest which fits inside the simplex region. Chapter 4 is thoroughly revised. Besides considering only lower bound pseudo components (known as L-pseudo components), this edition also includes upper bound pseudo-components (called U-pseudo-components). In the situations where both upper and lower bounds are considered, the constrained region is a convex polyhedron which is more complex. In case of inconsistent constraints, it is shown how to adjust these to consistent constraints. Formulae to enumerate the number of extreme vertices, faces, and edges are provided. Various algorithms for calculating the coordinates of the extreme vertices are also given. The last three sections of this chapter are on the topic of combining categories of components.

Chapter 5 is on techniques used in analyzing mixture data and these techniques are developed around the fitting of Scheffé polynomials. Besides including formulae for determining the component effects, one finds material on model reduction by eliminating the nonsignificant terms in the model. The next chapter is set aside to consider systems for which functions other than polynomials are more appropriate and suitable. Also included here is a new section on measuring additivity and interaction in the component blending properties by fitting log contrast models.

Chapter 7 is on the inclusion of process variables which are factors whose levels, when changed, could affect the blending properties of the ingredients but these process variables do not form any portion of the mixture. Two approaches to include such variables in the design are discussed - the first one deals with mixture components while the second one with mathematically independent variables called mixture-related variables. The next chapter deals with some special techniques which can be employed to supplement the designs and methods of analysis presented in earlier chapters. Some of the topics discussed are blocking designs, the generation of optimal designs, collinearity problems which arise when Scheffé-type models are fitted to data obtained from highly constrained regions, improving upon the accuracy of the estimates of the coefficients in the Scheffé-type models, also including an illustration of fitting segmented Scheffé-type models to freezing point data exhibiting a eutectic point. In the last chapter, the author presents a collection of real data sets, each set has an exercise in model fitting and each fitted model is provided with some useful summary statistics. In some cases complete analysis is presented and the reader is called upon to explain the roles of separate components, while in other cases one is to make recommendations and suggestions for trying additional blends for use in a follow-up experiment.

This is a well-written book on a subject of great interest to students, experimenters, and researchers in academic institutions and industry. The author has done a great job in presenting the vital information on experiments with mixtures in a lucid and readable style, has provided numerous useful and informative diagrams, and has extensively used numerical illustrations to get across otherwise difficult concepts to readers with a minimal background in mathematics. Many of the mathematical derivations (not of interest to practitioners and experimenters) are given in the appendixes at the end of chapters. A careful balance has been kept between theory and practical aspects throughout the book. It is aimed at the specialists as well as the nonspecialist, and combines both the features of being a reference as well as a textbook. It is indeed a very informative, interesting, and useful book on an important statistical topic.

The book is self-contained in the sense that the background material (viz, matrix algebra, analysis of variance, principle of least squares, etc. etc.) is all included in chapter 9 now as opposed to chapter 7 in the earlier edition. Each chapter ends with a useful summary, with a set of exercises, and a list of references and recommended readings. An updated bibliography of more than 150 items from recent literature obtained from experiments with mixtures is given at the end of the book. An attractive feature of the book is the inclusion of a large number of solved examples, most of which are from real-life situations. Besides other changes in this revised edition, the reader will find in chapters 7, 8 and 10 new entries.

After presenting some basic concepts and a clear distinction between a factorial experiment and a mixture experiment in chapter 1, the next chapter includes the material on fitting mathematical equations to model the response surface over the entire simplex factor space, and to test the adequacy of the fitted model. It also has a discussion on the simplex-lattice designs and the associated polynomial models to be fitted to data collected at the points of these designs. The use of independent variables is taken in chapter 3 which has also has a new formula for obtaining the radius of the largest sphere with center at the point of main interest which fits inside the simplex region. Chapter 4 is thoroughly revised. Besides considering only lower bound pseudo components (known as L-pseudo components), this edition also includes upper bound pseudo-components (called U-pseudo-components). In the situations where both upper and lower bounds are considered, the constrained region is a convex polyhedron which is more complex. In case of inconsistent constraints, it is shown how to adjust these to consistent constraints. Formulae to enumerate the number of extreme vertices, faces, and edges are provided. Various algorithms for calculating the coordinates of the extreme vertices are also given. The last three sections of this chapter are on the topic of combining categories of components.

Chapter 5 is on techniques used in analyzing mixture data and these techniques are developed around the fitting of Scheffé polynomials. Besides including formulae for determining the component effects, one finds material on model reduction by eliminating the nonsignificant terms in the model. The next chapter is set aside to consider systems for which functions other than polynomials are more appropriate and suitable. Also included here is a new section on measuring additivity and interaction in the component blending properties by fitting log contrast models.

Chapter 7 is on the inclusion of process variables which are factors whose levels, when changed, could affect the blending properties of the ingredients but these process variables do not form any portion of the mixture. Two approaches to include such variables in the design are discussed - the first one deals with mixture components while the second one with mathematically independent variables called mixture-related variables. The next chapter deals with some special techniques which can be employed to supplement the designs and methods of analysis presented in earlier chapters. Some of the topics discussed are blocking designs, the generation of optimal designs, collinearity problems which arise when Scheffé-type models are fitted to data obtained from highly constrained regions, improving upon the accuracy of the estimates of the coefficients in the Scheffé-type models, also including an illustration of fitting segmented Scheffé-type models to freezing point data exhibiting a eutectic point. In the last chapter, the author presents a collection of real data sets, each set has an exercise in model fitting and each fitted model is provided with some useful summary statistics. In some cases complete analysis is presented and the reader is called upon to explain the roles of separate components, while in other cases one is to make recommendations and suggestions for trying additional blends for use in a follow-up experiment.

This is a well-written book on a subject of great interest to students, experimenters, and researchers in academic institutions and industry. The author has done a great job in presenting the vital information on experiments with mixtures in a lucid and readable style, has provided numerous useful and informative diagrams, and has extensively used numerical illustrations to get across otherwise difficult concepts to readers with a minimal background in mathematics. Many of the mathematical derivations (not of interest to practitioners and experimenters) are given in the appendixes at the end of chapters. A careful balance has been kept between theory and practical aspects throughout the book. It is aimed at the specialists as well as the nonspecialist, and combines both the features of being a reference as well as a textbook. It is indeed a very informative, interesting, and useful book on an important statistical topic.

Reviewer: D.V.Chopra (Wichita)

##### MSC:

62Kxx | Design of statistical experiments |

62K15 | Factorial statistical designs |

62K99 | Design of statistical experiments |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

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