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**Combinatorics of experimental design.**
*(English)*
Zbl 0622.05001

Oxford Science Publications. Oxford: Clarendon Press. XIV, 400 p. (1987).

This book is likely to be of great value to statisticians as well as to combinatorialists as it unfolds how designs (known as incidence structures to combinatorialists, and experimental layouts to statisticians) provide a common ground rich with theoretical as well as practical challenging problems to both the groups. It includes material on the construction of different designs and on how these designs have been or are being used in the experimental setting. The main theme of this book is stated clearly as follows in the Preface, ”... members of each of these groups (i.e. statisticians and combinatorialists) are sometimes unaware of related developments and problems arising in the other area. We aim to bridge this gap by providing the background necessary to make the combinatorial aspects of statistical literature more easily accessible to combinatorialists, and vice versa”. The authors succeed remarkably well in their stated objectives.

The book has 15 chapters in all containing several topics on designs of interest in both statistics and combinatorics. The first chapter describes briefly the early history of designs, their use as sampling schemes, and their relation to linear models. The next three chapters include material on balanced incomplete block designs (BIBDS), their constructions by using difference sets, the structure of these designs as described by their automorphism groups, and their irreducibility. Chapters 5 through 7 are set aside for Latin squares, orthogonality in Latin squares, and the techniques leading to the construction of a set of mutually orthogonal Latin squares. Resolvable designs and their construction, affine geometries and their extensions to projective geometries are treated in chapter 8. The next two chapters deal with factorial designs - chapter 9 is on symmetrical factorial designs (in which every factor is at the same number of levels) based on the flats of a finite geometry, and chapter 10 is on the constructions for the single replicate case for symmetrical as well as asymmetrical designs. Partially balanced incomplete block designs (PBIBDS) and various association schemes are treated in chapter 11. The next two chapters derive and discuss some of the existence results for symmetrical balanced incomplete block designs, and for designs with index one and a given block size. The final two chapters are devoted to designs in which the arrangement of the treatments relative to each other, within each block or an array, is very crucial.

Every chapter is followed by informative and useful comments and references - providing the reader with access to additional interesting material on designs. It includes a useful subject index, and a valuable list of references. A set of exercises at the end of each chapter covers essentially every topic discussed in that chapter. These exercises should provide an ample supply suitable for a standard course in this area. The background material needed to go through this book includes linear algebra, number theory, finite fields, and some statistical concepts - of course the concepts on the last three topics are developed as needed in the text. A list of the areas of design theory not covered in this book is given in the preface. Over and above it, this reviewer is disappointed to find very little discussion of important topics such as asymmetrical factorial designs, fractional factorial designs, orthogonal and other arrays related to design theory.

The material in this book is carefully organized, and the treatment of each selected topic is thought out well. The material is presented in an easy-to-understand style, and the technical portions of the proofs are explained with meticulous care and in great detail. Numerous illustrative examples explaining away otherwise difficult and abstract concepts make it a very stimulating, interesting and enjoyable book. It can be used as a text for an advanced undergraduate and/or a graduate level course. It is a useful and interesting book, and comes highly recommended to students of statistics and combinatorics.

The book has 15 chapters in all containing several topics on designs of interest in both statistics and combinatorics. The first chapter describes briefly the early history of designs, their use as sampling schemes, and their relation to linear models. The next three chapters include material on balanced incomplete block designs (BIBDS), their constructions by using difference sets, the structure of these designs as described by their automorphism groups, and their irreducibility. Chapters 5 through 7 are set aside for Latin squares, orthogonality in Latin squares, and the techniques leading to the construction of a set of mutually orthogonal Latin squares. Resolvable designs and their construction, affine geometries and their extensions to projective geometries are treated in chapter 8. The next two chapters deal with factorial designs - chapter 9 is on symmetrical factorial designs (in which every factor is at the same number of levels) based on the flats of a finite geometry, and chapter 10 is on the constructions for the single replicate case for symmetrical as well as asymmetrical designs. Partially balanced incomplete block designs (PBIBDS) and various association schemes are treated in chapter 11. The next two chapters derive and discuss some of the existence results for symmetrical balanced incomplete block designs, and for designs with index one and a given block size. The final two chapters are devoted to designs in which the arrangement of the treatments relative to each other, within each block or an array, is very crucial.

Every chapter is followed by informative and useful comments and references - providing the reader with access to additional interesting material on designs. It includes a useful subject index, and a valuable list of references. A set of exercises at the end of each chapter covers essentially every topic discussed in that chapter. These exercises should provide an ample supply suitable for a standard course in this area. The background material needed to go through this book includes linear algebra, number theory, finite fields, and some statistical concepts - of course the concepts on the last three topics are developed as needed in the text. A list of the areas of design theory not covered in this book is given in the preface. Over and above it, this reviewer is disappointed to find very little discussion of important topics such as asymmetrical factorial designs, fractional factorial designs, orthogonal and other arrays related to design theory.

The material in this book is carefully organized, and the treatment of each selected topic is thought out well. The material is presented in an easy-to-understand style, and the technical portions of the proofs are explained with meticulous care and in great detail. Numerous illustrative examples explaining away otherwise difficult and abstract concepts make it a very stimulating, interesting and enjoyable book. It can be used as a text for an advanced undergraduate and/or a graduate level course. It is a useful and interesting book, and comes highly recommended to students of statistics and combinatorics.

Reviewer: D.V.Chopra