Applied Factor Analysis in the Natural Sciences

Rating:
Summary: a good, but demanding introduction.
Review: The object of this book is to introduce the multivariate technique of factor analysis to students within the natural sciences. It is the revised and expanded version of a book by Joreskog, Klovan and Reyment, that first appeared in 1976. The first chapter provides a brief example of a factor analysis and an overview of the problems amenable to factor analysis. This overview provides the reader with an impression of the many and varied fields of scientific research that are subsumed under the heading of the natural sciences. The second chapter explains basic mathematical and statistical concepts that are needed in the subsequent chapters. This chapter covers a lot of ground in a relatively short space. As a result, certain concepts are treated rather poorly. For instance, the explanation of a determinant as 'a scalar derived from operations on a matrix' (p. 32) is unlikely to engender very much understanding. Fortunately, the authors do provide references to more extensive treatments of the subject matter, and certain subjects do re-emerge later in the chapter. The chapter ends with a treatment of the not-so-basic concepts of the eigenvalue decomposition, the Eckart-Young theorem, and finally, the canonical analysis of asymmetry. The canonical analysis of asymmetry receives a curt treatment. Chapter three introduces the aims, ideas, and models of factor analysis. This chapter starts with a good description of the exploratory model for R-mode factor analysis, i.e. the analysis of the relationship among variables. A matrix and scalar notation are used to explain the model both for the observations and for the dispersion matrix. Both principal component analysis and true factor analysis are distinguished and the major difference in objectives explain. The chapter ends with an example of a common factor exploratory (true) factor analysis followed by orthogonal and oblique rotations. The data are artificial, so the results are unambiguous. Chapter 4 is devoted entirely to R-model factor analysis. Again, true factor analysis and principal component analysis are discussed. The fact that PCA is also applied as an exhaustive, and informative transformation (e.g. in allometry), is mentioned briefly (p. 101), but, as is now to be expected, the emphasis is on PCA as a form of factor analysis. Many important subjects related to PCA are explained clearly, and illustrated. These include robust PCA, cross validation, sensitivity analysis and the analysis of compositional data. The illustration of robust PCA involves the analysis of a heterogeneous dataset. Apparently the data are a mixture of two multivariate normal distributions. This is could be useful to emphasize the importance of the distributional assumption of identicalness and independence, but is less than ideal in an illustration of robust PCA. Remarkably, the reader is warned against heterogeneity of samples later on in the book (p. 198). True factor analysis receives less attention, although here such issues as sensitivity and cross validation are also important. The Heywood case, an inadmissible solution due to a negative residual variance, is not treated. This is a pity, because Heywood cases are known to occur often in exploratory factor analysis, and are an important diagnostic of model mis-specification. This omission is all the more surprising, because the illustration of a true factor analysis on page 110 actually contains a Heywood case. The chapter ends with a brief discussion of path analysis and Wrightian factor analysis. The exact definition of 'Wrightian factor analysis' is lacking. Judging by the illustrative analysis of the Wright's famous leghorn data, Wrightian factor analysis is simply a confirmatory factor analysis using ULS estimation. The results of this illustrative analysis are obtained using LISREL and another program. On page 135 one reads 'Note the system LISREL is a convenient way of making a path analysis'. There is little doubt that this is true, but there is nothing in this the book to substantiate the remark. Chapter 5 is devoted to Q-mode methods. These are methods concerned with the relationship among objects, rather than among cases. Three Q-mode methods are treated: Imbrie's Q-mode factor analysis, Gower's principal coordinates, and later in the chapter, the analysis of asymmetry, which is also due to Gower. It is clear that Q-mode methods are more difficult than R-mode methods. They require more technical knowledge (e.g. distance measures) and are sometimes less intuitively appealing (e.g. analysis of asymmetry based on the factorization of a skew-symmetric matrix). However, the difficulty of the subject matter is somewhat compounded by details relating to the presentation. For instance, it is stated on page 137 that the 'main interest in doing a Q-mode analysis is graphical', but that Imbrie's Q-mode factor analysis 'is concerned with obtaining answers to several questions, of which the graphical appraisal of the data is only one (...)'. Later still, (p. 142) the graphical representation in Q-mode factor analysis is 'incidental to the analysis'. On page 140, the terms Euclidian distance and Pythagorean distance in two consecutive sentences, while they represent the same distance. The term duality is used repeatedly to indicate a certain similarity between techniques (e.g., p. 140), but never defined. These are just detail, of course, but they do not make the subject matter in this chapter any easier to follow. Still it should be said that the explanations of the Q-mode techniques are quite clear. The illustrations using artificial and real data are informative and work well to facilitate and deepen the understanding of the models. Q-R-mode analysis is the subject of chapter 6. The treatment of Q-R-mode analysis is fairly brief. The technique is illustrated. Subsequently, Gariel's biplots, and CANOCO, an amalgamation of canonical correlation analysis and correspondence analysis, are discussed. Again the techniques are illustrated. Chapter 7 concerns various practical aspects that one faces in carrying out an statistical analysis. Many of these are quite general in that they apply to any statistical analysis, not just to a factor analysis. Other aspects are inherent to exploratory factor analysis, such as the method of extraction, the choice of the number of factors to retain and the method or rotation. In view of the many techniques discussed, especially in chapters 5 and 6, a section containing a clear discussion of the issue of 'which technique to use when', would have been welcome. In chapter 8, a number of examples and two case histories are presented. These are useful and quite interesting to read. They do require specialized knowledge of the subject matter to fully appreciate the substantial aspects of the results. The appendix is written by Leslie F. Marcus. It is devoted entirely to the MATLAB programming language and scripts to carry out many of the illustrative analyses presented in the book. The inclusion of the appendix fits in well with general approach of the book and offers the reader an opportunity to gain first hand experience in carrying out the various analyses. One does require the MATLAB program, and an understanding of the MATLAB programming language. Possibilities, if any, to carry out analyses using multi-purpose statistical packages are not discussed. This is not a simple introduction to applied factor analysis. It requires a fair degree of mathematical sophistication to understand the various models, especially those based on Q-mode and Q-R-mode techniques. It requires careful reading to come to grips with the subtle distinctions between the various model. The presentation at times could be a little bit more accommodating. Still, with its extensive use of illustration and worked examples, and the availability of MATLAB m-files to use in one's own analyses, I think that this book is a useful introduction. The book contains a few typographical errors, all of which are innocuous.