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Foundations of factor analysis. 2nd ed. (English) Zbl 1188.62185

Chapman & Hall/CRC Statistics in the Social and Behavioral Sciences Series. Boca Raton, FL: CRC Press (ISBN 978-1-4200-9961-4/hbk). xxiii, 524 p. (2010).
This new edition, the first one appeared in 1972, gives a more complete coverage of descriptive factor analysis and doublet factor analysis. A chapter (Chapter 5) is added. Chapters on oblique analytic rotations, on factor scores and factor indeterminacy, and on confirmatory factor analysis have been revised. The book has 15 chapters.
Chapter 1 is an introduction, which briefly reviews factor analysis, structural theories, the history of factor analysis, and gives some examples on factor analysis. Chapter 2 provides a mathematical review of concepts of algebra and calculus and an introduction to vector spaces and matrix algebra. Chapter 3 is regarding composite variables and linear transformations. Chapter 4 covers multiple and partial correlations. Chapter 5 presents the multivariate normal distributions, their general properties, and the concept of maximum-likelihood estimation. Chapter 6 introduces fundamental equations of factor analysis. Chapter 7 discusses how to extract common factors. Chapter 8 provides common-factor analysis.
Chapter 9 examines several variants of factor analysis models, including principal components, weighted principal components, image analysis, canonical factor analysis, descriptive factor analysis, and alpha factor analysis. Chapter 10 regards factor rotations. Chapter 11 introduces orthogonal analytic rotations. Chapter 12 discusses analytic oblique rotations focusing on the gradient projection algorithm and its applications. Chapter 13 addresses factor scores and factor indeterminacy. Chapter 14 deals with factorial invariance. Chapter 15, a chapter on confirmatory factor analysis, addresses the philosophy of science issues, model specification and identification, parameter estimation, and derivation of algorithms.
This book provides the necessary mathematics for readers to understand the derivation of an equation or procedure. It can also serve as a preparation textbook for students to prepare for later courses on structural equations modeling. It enables them to choose the proper factor analytic procedures, to make modifications of the procedures, and to produce new results. Exercises are not contained in this book.

MSC:

62H25 Factor analysis and principal components; correspondence analysis
62-02 Research exposition (monographs, survey articles) pertaining to statistics
62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
62P15 Applications of statistics to psychology

Keywords:

structural theory; history of factor analysis; linear models; examples; mathematical foundations of factor analysis; scalar algebra; matrix algebra; determinants; treatment of variables as vectors; maxima and minima of functions; composite variables; linear transformations; unweighted composite variables; differentially weighted composites; matrix equations; multiple correlations; partial correlations; multiple regression; multivariate normal distributions; univariate normal density functions; maximum-likelihood estimation; fundamental equations of factor analysis; decomposition of variables into components; factor extraction; rationales for finding factors; factor loadings; diagonal method of factoring; centroid method of factoring; principal-axes methods; common-factors analysis; fitting common factor models to correlation matrices; component analysis; image analysis; canonical-factor analysis; problem of doublet factors; metric invariance properties; image-factor analysis; psychometric inference in factor analysis; factor rotation; Thurstone’s concept of simple structures; oblique graphical rotations; orthogonal analytic rotations; quartimax criterion; varimax criterion; transvarimax methods; simultaneous orthogonal varimax and parsimax; oblique analytic rotations; oblimin family; Harris-Kaiser oblique transformations; weighted oblique rotations; oblique procrustean transformations; gradient-projection-algorithm synthesis; rotating using component loss functions; factor scores and factor indeterminacy; scores on component variables; indeterminacy of common-factor scores; factorial invariance; invariance under selection of variables; invariance under selection of experimental populations; comparing factors across populations; confirmatory-factor analysis
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