Methods of Multivariate Analysis by Alvin C. RencherReview by: Ruth MickeyJournal of the American Statistical Association, Vol. 93, No. 443 (Sep., 1998), p. 1239Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2669873 .

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Book Reviews 1239

Methods of Multivariate Analysis. Alvin C. RENCHER. New York: Wiley, 1995. ISBN 0-471-57152-0. xvi + 627 pp. $79.95.

A challenge for many instructors who teach multivariate analysis is that their students come from a variety of fields, have varying statistical back- grounds, and have different reasons for taking the course. Instructors must try to teach their course in such a way that the needs of their own statis- tics majors are met, while also servicing the needs of students from other quantitative disciplines. This book was written as a textbook for courses in applied multivariate analysis that have such a diverse group of students.

When writing the book, the author had three objectives for the students: "to gain a thorough understanding of the details of various multivariate techniques, their purposes, their assumptions, their limitations and so on ... to be able to select one or more appropriate techniques for a given multivariate data set ... to be able to interpret the results of a computer analysis of a multivariate data set" (p. 3).

The book is targeted toward students who have had two or more statistics methods courses. Matrix algebra is used throughout the text. Chapter 2 provides an overview of concepts and methods of matrix algebra that are used later on, such as positive definite matrices, determinants, eigenvalues, and eigenvectors. This material focuses on key concepts and is clearly written, so that students do not need a course in matrix algebra prior to using the book. No calculus is required.

After introductory chapters on matrix algebra, characterization and dis- play of multivariate data, and the multivariate normal distribution, the re- maining chapters deal with multivariate techniques that traditionally have been covered in texts for multivariate analysis: tests on one or two mean vectors, multivariate analysis of variance (MANOVA), tests on covari- ance matrices, discriminant analysis, classification analysis, multivariate regression, canonical correlation, principal components, and factor anal- ysis. More than 20% of the chapter material is devoted to MANOVA; one-way and two-way models, profile analysis, repeated measures, and growth curves are discussed. Other multivariate techniques, such as logis- tic regression, survival analysis, and multiway contingency tables, are not covered.

The author does not shy away from statistical terminology. He makes use of expected value, maximum likelihood and likelihood ratio tests and carefully distinguishes between underlying models, expressed in terms of population parameters, and the estimation of these models' parameters.

Some derivations, which make use of the matrix algebra introduced in Chapter 2, are presented. Expressing the covariance matrix of the original variables as a function of the matrix of factor loadings and the covari- ance of the unique factors in factor analysis is one such example (p. 449). Other derivations, which would require calculus, are avoided. Thus the co- efficients of the linear discriminant function are presented but not derived (p. 297).

A diskette that accompanies the text contains SAS programs and datasets for most of the examples in the book. The instructions for using the diskette are straightforward; students could almost immediately generate SAS output corresponding to what is presented in the text examples. The following SAS procedures are used: CANCORR, CANDISC, DISCRIM, FACTOR, GLM, PRINCOMP, REG, STEPDISC, and IML (for testing equality of mean vectors). Although no computer output is included in the text, the text was written with SAS in mind. For example, in the chapters on MANOVA and discriminant analysis, H and E indicate matrices of "between" and "within" sums of squares and cross-products, as in SAS. For hypothesis testing, the four statistics reported in SAS, based on Wilk's lambda, Roy's greatest root, Pillai's trace, and the Hotelling-Lawley trace, are discussed. However, for some procedures, the instructor would need to supplement what is given in the text. Students might find the output from CANDISC for multigroup discriminant analysis confusing, because so many different types of coefficients are generated, which are not de- scribed in the book. The coefficients that correspond to "raw canonical coefficients" are presented, although they are described only as eigenvec- tors of E-1H. A similar problem could arise with the SAS output for factor analysis; the pattern and structure matrices are only briefly men- tioned.

At the end of each chapter are numerous problems that can be used as student exercises. The first problems highlight some of the formulas presented in the chapter and require the students to show or verify some- thing about the expressions. Later problems require students to perform analyses for accompanying data. Thus the problems reflect the diversity of students' interests in the course. The book devotes 69 pages to either solu- tions or outlines of how to solve each of the problems. Students could find

this useful, although there is the danger that some might become overly dependent on "the answers in the back of the book."

Methods of Multivariate Analysis has much in common with Applied Multivariate Statistical Analysis (Johnson and Wichern 1988). Both books are directed toward students who have completed two or more statistics courses, contain numerous data examples drawn from a variety of fields, and make use of matrix algebra. My sense is that Rencher's book contains more text and fewer formulas, so that it might be more readable, partic- ularly for nonstatistics majors, and easier for the instructor for purposes of teaching if SAS were used. Another text, Computer-Aided Mutltivari- ate Analysis (Afifi and Clark 1990), similarly assumes a basic foundation in statistics but uses geometrical and graphical explanations, rather than matrix algebra, to describe multivariate techniques. Afifi and Clark (1990) provide additional practical information, such as discussion of data screen- ing, data editing, and transformations of variables; they also cover logistic regression and regression using survival data and cluster analysis, but do not cover multivariate regression or MANOVA, as Rencher does.

Recently, when I was teaching a course in multivariate analysis, I spotted Methocls of Multivariate Analysis on a colleague's bookshelf. Because its explanations seemed clear and the level at which the material is presented was appropriate for my class, I started using it as a reference. The students found it very readable and helpful; one-third of them ordered their own copies.

This book strikes a nice balance between meeting the needs of statistics majors and students in other fields. The discussion of each multivariate technique is straightforward and quite comprehensive. This textbook is likely to become a useful reference for students in their future work.

The author states in the Preface that his "objective ... has been to make the book accessible" and his "overriding goal ... has been clarity of exposition." He has achieved his objective and goal. Furthermore, students who use this text should achieve the three objectives, stated earlier, which he specifies for them.

Ruth MICKEY Uniiversity of Vermont

REFERENCES

Johnson, R., and Wichern, D. (1988), Applied Multivariate Statistical Anal- ysis, Englewood Cliffs, NJ: Prentice Hall.

Afifi, A., and Clark, V. (1990), Computer-Aided Mutltivariate Analysis, New York: Van Nostrand Reinhold.

Advances in Statistical Decision Theory and Applications. S. PANCHAPAKESAN and N. BALAKRISHNAN (eds.). Boston: Birkhauser, 1997. ISBN 0-8176-3965-9. xix + 448 pp. $79.95.

The classical approach in testing the homogeneity hypothesis was found to be inadequate to serve experimenters' needs, which in practice often involves selecting the best among several alternatives or ranking them. The attempt to statistically formulate this problem and to give answers to such important questions set the stage for the development of the theory of selection and ranking. Shanti S. Gupta has made pioneering contributions to ranking and selection theory, particularly to subset selection theory. Besides ranking and selection, his interests include order statistics and reliability theory. To honor Shanti S. Gupta, S. Panchapakesan and N. Balakrishnan have edited this book. The first editor's association with Professor Gupta goes back to 1965 when he came to Purdue to do his Ph.D. He has been a student, a colleague, and a long-standing collaborator of Professor Gupta. The second editor's association with Professor Gupta began in 1978 when he started his research in the area of order statistics. He has collaborated with Professor Gupta on several publications.

The book reviews some of the recent developments in statistical de- cision theory and applications, focuses and highlights some new results and discusses their applications, and indicates interesting possible direc- tions for further research. For this volume, 28 articles were written by a number of authors who form a representative group from former stu- dents, colleagues, and other professional associates of Shanti S. Gupta. They are also experts in different aspects of statistical decision theory and applications. The authors who contributed to this volume are (in alphabet- ical order) N. Balakrishnan, J. 0. Berger, R. L. Berger, B. Benzion, R. J. Carroll, N. R. Chaganty, J. Deely, K. Frankowski, K. S. Fritsch, T. Gastaldi, S. Ghosal, J. K. Ghosh, D. M. Goldsman, A. J. Hayter, J. C. Hsu, D. Y. Huang, W. T. Huang, J. T. G. Hwang, S. Jeyaratnam, W. 0. Johnson, K. H. Kang, T. M. Kastner, W. C. Kim, T. Liang, C. C. Lin, Y. Ma, E. C. Malthouse, G. C. McDonald, K. J. Miesche, D. S. Moore, I. Olkin, W. C. Palmer, G. Pan, S. Panchapakesan, B. U. Park, H. Park, B. L. S. Prakasa

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