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Fractional factorial plans
Journal of Statistical Planning and Inference 91 (2000) 177–178
www.elsevier.com/locate/jspi
Book review Fractional factorial plans A. Dey and R. Mukerjee: Wiley, New York, 1999, price $94.95 paperback, ISBN 0-471-29414-4 This book is an excellent reference text covering recent research on optimality of fractional factorial plans, especially those plans formed from orthogonal arrays. The style of writing is mathematical with extensive internal crossreferencing, but for a reader with the appropriate mathematical expertise and knowledge of factorial experiments, the book is not dicult to read. It is unlikely to be suitable reading for a practical introduction to fractional factorial experiments, although it would make excellent supplementary reading for a course at Ph.D. level on the topic. There are eight chapters. The rst is an introduction and the following ve chapters deal with various aspects of optimality of fractional factorial plans. The last two chapters discuss further topics of practical importance. Chapter 2 starts with general issues of estimability of contrasts which span spaces corresponding to main e ects and interactions of up to f factors. The model assumes that all interactions involving t + 1 or more factors are negligible (t¿f): Fractions which allow estimability of the contrasts of interest are called Resolution (f; t) plans. Optimality is then discussed and this leads naturally to the use of orthogonal arrays. In this setting, Resolution (f; t) coincides with the traditional concept of Resolution (g + 1), where g = f + t is the strength of the array. Chapters 3 and 4 deal, respectively, with various methods of construction of symmetric and asymmetric orthogonal arrays. These chapters are presented as a guide to constructions for fractional factorial plans for a large number of practical settings and do not seek to provide a complete coverage of the topic. A summary list of orthogonal array construcitons is given in an Appendix for various numbers of factors, numbers of levels and up to 50 runs. The construction techniques discussed are based on a variety of concepts including Hadamard matrices, Galois elds, di erence matrices, collapsing levels and replacement of levels. The constructions are typically presented as proofs of theorems with very brief examples for illustration. It is a pity that the publishers did not allow more pages for this endeavor. With a large number of additional examples showing the elements in the orthogonal arrays, the book would have been accessible to a wider readership. Unfortunately, here, even when the arrays are presented, they have been transposed to save space, so that when the text refers to “the rst column”, the reader is actually looking at a row in the display (albeit with a0 to denote transpose). c 2000 Elsevier Science B.V. All rights reserved. 0378-3758/00/$ - see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 0 0 ) 0 0 1 3 1 - 2
178
Book review / Journal of Statistical Planning and Inference 91 (2000) 177–178
Chapter 5 deals with various existence issues, including bounds for the maximum number of factors that can be handled in an N -run orthogonal array. In Chapter 6, attention is turned to run sizes which do not admit existence of orthogonal arrays. A number of results are given concerning optimality of plans obtained from the addition of one or more treatment combinations to an orthogonal array. The E-optimality criterion leads to di erent designs from the A- and D-criteria, and is discussed separately. A nice summary is given in Remark 6.5.2 of known optimal main-e ect plans (i.e., Resolution (1,1)) for 2n factorials and all possible numbers of runs. The remainder of Chapter 6 deals with 2 or 3 factors with various numbers of levels and runs. Insights are given as to which designs should perform well when no optimality resutls are available. The nal two chapters discuss some special topics. First, the issue of experimentation under a time trend is considered, where the time trend is modeled by a polynomial of order s over equi-spaced time periods. Requirements on the ordering of the treatment combinations are given to ensure that an orthogonal array is sth-order trend free. The issue of minimization of cost is also raised brie y. Division of treatement combinations in an orthogonal array into blocks is done using one column of the array as an indicator variable for the blocks. In Chapter 9, regular fractions of symmetric mn factorial experiments are considered, where the treatment combinations are represented in traditional fashion as elements of a Galois eld GF(m). De ning contrasts and aliases are discussed in terms of pencils in the corresponding Euclidean Geometry EG(n; m). These ideas are then matched with those of the orthogonal arrays of the earlier chapters. The consideration of pencils leads to the concept of aberration of a design and the estimability of a subset of interaction e ects involving more than f factors, as well as blocking of fractions. The book ends with a brief but thorough summary of search designs and super-saturated designs. The writing style of the book is good and the material has been assembled in a thoughtful way. Although I found the internal cross-referencing intrusive, it serves the useful purpose of tying the material together so that the individual chapters read like part of a uni ed theory. This is certainly a text that I would want on my bookshelf as a reference and as a summary of recent research in optimality of fractional factorial plans. I would also consider using it as supplementary reading for an advanced course in factorial experiments and as required reading for a student wishing to do research in the area. A.M. Dean Department of Statistics, Ohio State Universtiy, Columbus, OH 43210, USA E-mail address: [email protected].
www.elsevier.com/locate/jspi
Book review Fractional factorial plans A. Dey and R. Mukerjee: Wiley, New York, 1999, price $94.95 paperback, ISBN 0-471-29414-4 This book is an excellent reference text covering recent research on optimality of fractional factorial plans, especially those plans formed from orthogonal arrays. The style of writing is mathematical with extensive internal crossreferencing, but for a reader with the appropriate mathematical expertise and knowledge of factorial experiments, the book is not dicult to read. It is unlikely to be suitable reading for a practical introduction to fractional factorial experiments, although it would make excellent supplementary reading for a course at Ph.D. level on the topic. There are eight chapters. The rst is an introduction and the following ve chapters deal with various aspects of optimality of fractional factorial plans. The last two chapters discuss further topics of practical importance. Chapter 2 starts with general issues of estimability of contrasts which span spaces corresponding to main e ects and interactions of up to f factors. The model assumes that all interactions involving t + 1 or more factors are negligible (t¿f): Fractions which allow estimability of the contrasts of interest are called Resolution (f; t) plans. Optimality is then discussed and this leads naturally to the use of orthogonal arrays. In this setting, Resolution (f; t) coincides with the traditional concept of Resolution (g + 1), where g = f + t is the strength of the array. Chapters 3 and 4 deal, respectively, with various methods of construction of symmetric and asymmetric orthogonal arrays. These chapters are presented as a guide to constructions for fractional factorial plans for a large number of practical settings and do not seek to provide a complete coverage of the topic. A summary list of orthogonal array construcitons is given in an Appendix for various numbers of factors, numbers of levels and up to 50 runs. The construction techniques discussed are based on a variety of concepts including Hadamard matrices, Galois elds, di erence matrices, collapsing levels and replacement of levels. The constructions are typically presented as proofs of theorems with very brief examples for illustration. It is a pity that the publishers did not allow more pages for this endeavor. With a large number of additional examples showing the elements in the orthogonal arrays, the book would have been accessible to a wider readership. Unfortunately, here, even when the arrays are presented, they have been transposed to save space, so that when the text refers to “the rst column”, the reader is actually looking at a row in the display (albeit with a0 to denote transpose). c 2000 Elsevier Science B.V. All rights reserved. 0378-3758/00/$ - see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 0 0 ) 0 0 1 3 1 - 2
178
Book review / Journal of Statistical Planning and Inference 91 (2000) 177–178
Chapter 5 deals with various existence issues, including bounds for the maximum number of factors that can be handled in an N -run orthogonal array. In Chapter 6, attention is turned to run sizes which do not admit existence of orthogonal arrays. A number of results are given concerning optimality of plans obtained from the addition of one or more treatment combinations to an orthogonal array. The E-optimality criterion leads to di erent designs from the A- and D-criteria, and is discussed separately. A nice summary is given in Remark 6.5.2 of known optimal main-e ect plans (i.e., Resolution (1,1)) for 2n factorials and all possible numbers of runs. The remainder of Chapter 6 deals with 2 or 3 factors with various numbers of levels and runs. Insights are given as to which designs should perform well when no optimality resutls are available. The nal two chapters discuss some special topics. First, the issue of experimentation under a time trend is considered, where the time trend is modeled by a polynomial of order s over equi-spaced time periods. Requirements on the ordering of the treatment combinations are given to ensure that an orthogonal array is sth-order trend free. The issue of minimization of cost is also raised brie y. Division of treatement combinations in an orthogonal array into blocks is done using one column of the array as an indicator variable for the blocks. In Chapter 9, regular fractions of symmetric mn factorial experiments are considered, where the treatment combinations are represented in traditional fashion as elements of a Galois eld GF(m). De ning contrasts and aliases are discussed in terms of pencils in the corresponding Euclidean Geometry EG(n; m). These ideas are then matched with those of the orthogonal arrays of the earlier chapters. The consideration of pencils leads to the concept of aberration of a design and the estimability of a subset of interaction e ects involving more than f factors, as well as blocking of fractions. The book ends with a brief but thorough summary of search designs and super-saturated designs. The writing style of the book is good and the material has been assembled in a thoughtful way. Although I found the internal cross-referencing intrusive, it serves the useful purpose of tying the material together so that the individual chapters read like part of a uni ed theory. This is certainly a text that I would want on my bookshelf as a reference and as a summary of recent research in optimality of fractional factorial plans. I would also consider using it as supplementary reading for an advanced course in factorial experiments and as required reading for a student wishing to do research in the area. A.M. Dean Department of Statistics, Ohio State Universtiy, Columbus, OH 43210, USA E-mail address: [email protected].