Factorial Designs. by B. L. Raktoe; A. Hedayat; W. T. FedererReview by: Raymond H. MyersJournal of the American Statistical Association, Vol. 79, No. 385 (Mar., 1984), p. 232Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2288371 .

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232 Journal of the American Statistical Association, March 1984

texts. Furthermore, tables of percentage points for many of the standard likelihood ratio statistics, presented on pages 595-649, should be useful for implementation of these tests.

Presumably because of limited space and the main emphasis on basic distribution theory, some other aspects of multivariate statistical anal- ysis have not been covered adequately. Among these are minimum norm quadratic estimation (MINQUE) theory, MANOVA of repeated meas- urements (the "general multivariate linear models"), and simultaneous inference for MANOVA models. There are intricate distributional re- sults associated with these problems, and an adequate description of the general theory would have been very helpful.

Discrete multivariate analysis deserves at least a short introduction. Since traditional normal theory MANOVA procedures are increasingly being adapted to such categorical data analysis, some broad comments about the applicability of the results in this book to a wider domain (including this discrete counterpart) would have been very enlightening.

There is a basic question about the multinormality assumption in MANOVA and related procedures. To what extent can this assumption be disregarded if the sample size is not small? How robust are the clas- sical procedures when there are outliers or gross errors in the data set? What modifications are needed to apply the MANOVA theory to a mixed model containing partly continuous variates and partly discrete or categorical ones? Some aspects of these issues, if treated adequately, would have been very informative and helpful to the audience.

Muirhead's book is a very welcome addition and should be adopted for standard graduate-level courses, especially if distribution theory is the main topic of such a course.

P.K. SEN University of North Carolina

Experiments With Mixtures: Designs, Models, and the Analysis of Mixture Data.

John A. Cornell. New York: John Wiley, 1981. xvii + 305 pp. $30.95.

This is the book to consult if you are interested in learning about mixture designs. Indeed, it is the only book on the subject. Mixture problems arise when independent variables are proportions of a com- position, such as the proportions of gin and vermouth in a drink. As Cornell points out, such problems arise naturally in the chemical and food industries. The principal change from ordinary studies is that the design space is now a simplex (since the proportions sum to 1.0). This presents interesting challenges in both design and analysis.

The book's seven chapters include an introduction (Chapter 1) and a review of matrix algebra and elementary regression and ANOVA (Chap- ter 7). Chapter 2 discusses Scheffd's canonical polynomial models and simplex-lattice designs. Chapter 3 treats transformations of the mixture variables to a linearly independent subset (basis), which allows appli- cation of standard factorial designs to mixture problems. It also permits the easy inclusion of nonmixture variables (e.g., time and temperature) in the designs. Chapter 4 describes several methods of handling con- straints on the mixture components. Chapter 5 is entitled "The Analysis of Mixture Data," but it also includes an important example of a screen- ing design. Chapter 6 presents a brief overview of other mixture models.

The book will be useful for practicing statisticians or research sci- entists who understand the principles of experimental design and are comfortable with regression analysis at the level of Weisberg (1980) or Draper and Smith (1981). For scientists with only basic knowledge of experimental design and regression, I believe the book will be much less useful. Furthermore, I feel several technical misconceptions may prove misleading. For example, a nonstandard method of forming pre- diction intervals is given on page 31, while on page 168 the author seems to forget the equivalence of the partial F-test and corresponding t-test for whether a regression coefficient is zero. I also differ with the author's interpretation of results in several places. Overall, my qualms are con- cerned almost exclusively with analysis, where weaknesses would not present a great obstacle to a thinking scientist with a thorough knowl- edge of regression methodology.

For later editions I suggest revising the text to accentuate the strong points of the book-namely, the ideas of mixture designs. I would par- ticularly like to see more material on designing for quadratic response surfaces in constrained mixtures, an area that has proven extremely useful in my consulting with industrial research chemists. Cornell's el-

lipsoidal regions of interest fall short of their needs. A deemphasis on analysis would not adversely affect the book.

In conclusion, I recommend this book as a starting point for the prac- ticing statistician who desires to learn about mixtures. With supple- mentation from the extensive and quite useful bibliography, a statistician will gain a solid knowledge of mixture problems.

KINLEY LARNTZ University of Minnesota

REFERENCES

DRAPER, N.R., and SMITH, H. (1981), Applied Regression Analysis (2nd ed.), New York: John Wiley.

WEISBERG, SANFORD (1980), Applied Linear Regression, New York: John Wiley.

Factorial Designs. B.L. Raktoe, A. Hedayat, and W.T. Federer. New York: John Wiley, 1981. xii + 209 pp. $29.95.

This book presents a unified approach to factorial design with a self- contained although somewhat unorthodox treatment of the theory of factorial design. It is a concise, well-organized treatise with combina- torial and other mathematical developments presented to support cri- teria for "quality" of factorial designs. Specific designs sprinkled throughout the text serve as illustrations of the concepts.

The authors begin with two brief chapters dealing with preliminaries. Definitions of factors, levels, and runs or assemblies are given along with a discussion of how selection of a design is linked to the linear model. A chapter entitled "Some Facets of Experimental Design" pro- vides a rather succinct discussion of such basic concepts as linear par- ametric functions, contrasts, orthogonal comparisons, and estimability. Least squares estimation is introduced, and the notion of the b.l.u.e. in the linear model is discussed. A succeeding chapter deals with the mathematical details associated with orthogonal polynomial models. In this development, use is made of Helmert or "contrast" matrices. Care is taken to show how the entries in the design matrix are calculated, and the results are properly linked with linear models estimation theory. The notion of optimal design is then discussed with motivation given for A, D, E, G, and Q optimality. Other areas of particular note include randomized factorial designs and factorial designs of resolutions III, IV, and V. There is a complete chapter on the rather new but intriguing notion of Search Factorial Designs. The concepts are based on the ideas that there is no fixed model but rather a family of competing linear models, and that the task for the experimenter entails a search for the most reasonable model followed by estimation and further inference. The last chapter discusses the construction of designs.

The major objective of this book is to bring together the theory of factorial design in a somewhat new and systematic way. This objective appears to have been fulfilled. The prospective reader should be made aware, however, that the major motivation here is the theory, and the appeal will clearly be to the student or researcher who is well versed in matrix theory and combinatorics. The authors point out that the book is the ". . place to which statisticians and mathematicians may turn in order to become acquainted with the subject...." Assuming "the sub- ject" is factorial design, one should hasten to point out that it is not the place where experimental statisticians should turn. In fact, the book is most appropriate for one who has already studied practical experimental design, who appreciates and understands linear models theory and the mathematical concepts involved, and who is looking for a treatment of some of the foundations of factorial designs. To be sure, the book is not appropriate for a first course in design for the statistics graduate student. At that level, the reader requires more breadth and a practical exposure to real data examples. The authors indicate in the Preface that the text began as lecture notes for an Advanced Design course at Cornell University. There is a rich list of references under "Selected Additional Reading," but the book is sorely lacking in exercises.

For the specialist this book can be a valuable reference. For the teacher of design at the advanced graduate level, it may supply at least a portion of the textual material.

RAYMOND H. MYERS Virginia Polytechnic Institute

and State University

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