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Exploratory and Confirmatory Factor Analysis: Understanding Concepts

and Applications Linda Reichwein Zientek a

a Department of Mathematics and Statistics, Sam Houston State University,

To cite this Article Zientek, Linda Reichwein(2008) 'Exploratory and Confirmatory Factor Analysis: Understanding Concepts and Applications', Structural Equation Modeling: A Multidisciplinary Journal, 15: 4, 729 — 734

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Structural Equation Modeling ,15:729–734,2008

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DOI:10.1080/10705510802339122

Exploratory and Confirmatory Factor Analysis:Understanding Concepts and Applications.Bruce Thompson.Washington,DC:American Psychological Association,2004,195pages,$49.95(hardcover).

Reviewed by Linda Reichwein Zientek

Department of Mathematics and Statistics

Sam Houston State University

Exploratory and Confirmatory Factor Analysis:Understanding Concepts and Applications by Bruce Thompson provides “the important concepts required for implementing two disciplines of factor analysis:exploratory factor analysis (EFA)and confirmatory factor analysis (CFA)”(p.ix).Thompson attempts to provide a balance “between accuracy and completeness versus overwhelming technical complexity”(p.ix)and writes as if he “were speaking directly to you,the student”(p.x).Thompson’s easy-to-read writing style,accompanied by his ability to present complex material in easy-to-understand terms,makes this book a must read for those interested in understanding EFA and CFA concepts.The book is also an excellent resource for those who wish to understand connections among EFA,CFA,and the general linear model (GLM).

The book is divided into 12chapters.Notation and glossary sections are included.Each chapter concludes with a “Major Concepts”section recapping the chapter to ensure that key ideas are not missed.Heuristic examples with the inclusion of a real-world data set,which can be downloaded from the author’s Web site,allow readers to explore the presented concepts and repeat the given analyses.

This book is most appropriate for graduate students or researchers interested in learning the fundamentals of EFA and CFA,and could serve well as a

Correspondence should be addressed to Linda Reichwein Zientek,Sam Houston State University,Mathematics and Statistics Department,P .O.Box 2206,Huntsville,TX 77341-2206.Email:lrzientek@shsu.edu

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textbook for a graduate course on EFA,or as one textbook among several in a multivariate statistics class.Thompson is successful in providing accurate and complete information without overwhelming the reader;however,the trade-off is minimal coverage of the mathematics involved in EFA,particularly geometric representations.Thompson’s book is least beneficial for the advanced researcher who already has a conceptual understanding of EFA and CFA and wishes to become an expert in EFA and CFA with an in-depth understanding of the related mathematics.For researchers looking to become experts on EFA,this book could serve as a support for the more mathematically oriented and detailed book by Gorsuch (1983).A comparison of Gorsuch’s and Thompson’s tables of contents shows that both books are aligned in terms of topics.

Chapter 1begins with a historical overview of the uses of factor analysis in the social sciences.Three main purposes and two major classes of factor analysis are introduced.Thompson explains which approach is applicable for various purposes and includes criticisms that have been directed toward EFA (Armstrong,1967;Cronkhite &Liska,1980).Thompson explains the helpfulness of understanding EFA to facilitate a deeper conceptualization of other statistical analyses,particularly CFA.These explanations give insight into why a large proportion of the book is devoted to EFA.The conclusion of chapter 1(a)contains a description of the organization and layout of the chapters and (b)emphasizes a focus of the book:establishing linkages among GLM,EFA,and CFA to facilitate understanding of statistical concepts.

Chapter 2begins with matrix algebra concepts that facilitate understanding of factor analysis concepts.Connections between factor analysis and the GLM are developed;terminology is introduced;explanations of how to interpret statistics (e.g.,communality coefficients,pattern coefficients,and structure coefficients)are presented;and sample size recommendations are provided.A heuristic ex-ample illustrates these basic factor analysis concepts.

Chapter 3presents the linear decision sequence in EFA and the considerations relevant to these decisions.Thompson emphasizes that decisions in EFA are not subject to a point-and-click mind-set where decisions magically appear on the screen,and the output should be carefully investigated.For example,he notes that when iterative methods are involved “computer programs print only innocuous-sounding warnings about failure to converge,and then still print full results”(p.37),and the unknowing researcher might mistakenly interpret these questionable results.

Decisions about choosing rotational methods are explained so that someone with even minimal statistical knowledge can make educated choices.The logics of the developers of these methods are introduced,and differences between prin-cipal components (PC)analysis and principle axes (PA)analysis are discussed.Thompson notes that techniques often become popular because they are the default in statistical packages,and not because they are necessarily always the D o w n l o a d e d B y : [E B S C O H o s t E J S C o n t e n t D i s t r i b u t i o n - S u p e r c e d e d b y 9127733] A t : 05:09 1 O c t o b e r 2010

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best methods (see Henson &Roberts,2006).For several sophisticated methods not automated in statistical packages (e.g.,parallel analysis),the necessary SPSS syntax or links to syntax are provided.Thompson emphasizes the need to utilize several strategies during the decision sequence “with the hope that different approaches to making this decision will corroborate each other”(p.31).He characterizes the purpose of EFA as isolating “factors that have simple structure or,in other words,are interpretable”(p.48).The reader leaves this chapter with the understanding that a myriad of decisions need to be made at each step of the linear decision sequence.

Chapter 4seeks to explain “the extent to which extraction choices may affect factor interpretations,and especially,to understand why and when differences may arise across methods for a given data set”(p.49).Three iterative extrac-tion methods (e.g.,alpha,maximum likelihood,and image factor analysis)are compared,with most of the discussion devoted to the impact of measurement error and sampling error on PC and PA methods.The fact that as the number of measured variables increases,the differences between extraction methods decreases is emphasized.

Chapter 5seeks “to facilitate deeper understanding of the factor score esti-mation choices and to elaborate additional differences in the two most common extraction methods”(p.57).Thompson accomplishes this goal by comparing three factor score methods (i.e.,regression,Bartlett,and Anderson–Rubin)across two factor extraction methods (i.e.,PC and PA analyses).The characteristics of communalities are developed by computing “R 2between the factor scores of n people with the scores of n people on the measured variables”(p.65).A concrete heuristic example is provided,and SPSS syntax is included.Factor scores are computed to aid in grasping “the nature of both structure coefficients and communality coefficients”(p.65).

Chapter 6examines oblique rotations and higher order factors,and the author emphasizes that in EFA “higher-order factors should be extracted whenever factors are correlated”(p.72).Thompson likens this to investigating both stan-dardized weights and structure coefficients in the GLM.Interpretation becomes more complex when variables or factors are correlated,resulting in the need to simultaneously interpret more parameter estimates.He emphasizes the im-portance of model parsimony.As noted by Thompson,our goal is to obtain a parsimonious model because if two models are relatively equal in the fit to our data,the more parsimonious model will be more likely to be true and “things that are true are also more likely to replicate in future research,and researchers usually do not want to be embarrassed by making discoveries that no one else can replicate”(p.71).Building on basic matrix concepts presented in chapter 2and syntax provided in chapter 6,a heuristic example is provided for using the Schmid and Leiman (1957)method for expressing the second-order factors in terms of measured variables.D o w n l o a d e d B y : [E B S C O H o s t E J S C o n t e n t D i s t r i b u t i o n - S u p e r c e d e d b y 9127733] A t : 05:09 1 O c t o b e r 2010

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Chapter 7introduces six two-mode techniques developed by Cattell (1966)for the analysis of two-dimensional data matrices,with a substantial portion of the discussion devoted to Q-technique.R-technique is the most commonly utilized two-mode method with people as rows and variables as columns in the raw data matrix.Q-technique is sometimes used in psychology and has people as columns and variables as rows in the data matrix,and thus factors people rather than variables.As noted by Thompson,“the computer does not know (or care)what data are being analyzed”(p.85).Decisions are subject to the researcher’s judgment.A heuristic example with SPSS software is provided.Chapter 8discusses various applications of methods for comparing two struc-tures analytically.Thompson notes that comparing factors across samples can be just as difficult as comparing unrotated factors.A procrustean matrix is utilized to empirically investigate the invariance of the varimax-rotated pattern and structure coefficients by comparing two structures and rotating one factor pattern matrix to a best fit position with the other factor pattern target matrix.For the heuristic example,a target matrix consisting of zeros off of the diagonal and the eigenvalues on the diagonal was used.A complete discussion for choosing a procrustean matrix can be found in Gorsuch’s (1983)book.More examples are provided on the applicability of using procrustean rotation methods,such as testing factor invariance across samples,internal replicability,and factor adequacy.An appendix contains the SPSS syntax for performing a best fit procrustean rotation.The chapter concludes with a discussion on salience of variables and the subsequent interpretation and naming of factors.He notes that all information (e.g.,both pattern and structure coefficients,rotational methods)should be provided in publications to allow readers to make their own interpre-tations and conduct secondary analyses.A warning is issued about accepting the naming of factors by others at face value.

Chapter 9begins with a discussion of three types of errors that influence all statistical methods:“sampling error,measurement error,and model specification error”(p.99).External and internal replication methods are introduced,and heuristic examples are provided for EFA cross-validation and the bootstrap method.It is explained that if internal replication cannot occur with a single sample,the possibility of external replication from a new sample is even less likely.The procrustean rotation methods introduced in chapter 8are utilized for the bootstrap method for the purpose of creating a common factor space within which to summarize results across numerous bootstrap resamples.

Chapters 10to 12cover CFA.However,because EFA is a building block for CFA,readers wanting to develop a more in-depth understanding of CFA should not limit their reading to chapters 10through 12.Chapter 10introduces the decision sequence used in CFA and elaborates on “the similarities between EFA and CFA as part of a single underlying GLM,while also highlighting the differences in the two sets of methods”(p.109).Graphical representations D o w n l o a d e d B y : [E B S C O H o s t E J S C o n t e n t D i s t r i b u t i o n - S u p e r c e d e d b y 9127733] A t : 05:09 1 O c t o b e r 2010

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are provided to present similarities and differences between EFA and CFA.Connections are made among sample size,regression,and CFA.The chapter sequentially goes through the preanalysis and postanalysis decision-making process of CFA and,when appropriate,makes comparisons to EFA.Testing rival models and using various matrix-of-association choices when the scaling of data is in question is endorsed.However,graphical representations would have been helpful for explaining when a model “will always perfectly reproduce or fit the data”(p.118;i.e.,a saturated model)and for explaining bivariate normality,especially for readers who have difficulty visualizing three-dimensional objects.

Chapter 11presents “aspects of CFA that are unique to CFA model tests”(p.133)and relates these aspects to analyses within the GLM (e.g.,standardized or unstandardized weights).Model identification choices and investigations of both pattern and structure coefficients are emphasized for correlated factors.Comparisons of model fit and fit statistics are discussed,and a heuristic example of higher order analyses is provided.The book concludes with chapter 12,which is devoted to using CFA to test model invariance to “seek factors that are invariant over samples and analyses”(p.159).

This book is an excellent resource for researchers or graduate students who want a conceptual understanding of EFA and CFA.The use of heuristic exam-ples,the inclusion of SPSS syntax,the conveying of complex topics in easy-to-understand terms,and ability to make the reader feel as if he or she were a student in Thompson’s classroom are unique characteristics of this book.The book’s intent is not to teach the mathematical details of EFA and CFA,and it is therefore least beneficial to the researcher who seeks an in-depth understanding of the mathematical computations conducted in EFA and CFA.However,he does provide brief explanations of the mathematics involved,and therefore could be used as a supplement to the other more technically oriented books (e.g.,Mulaik,1972).

Although the book discusses some modern topics (e.g.,standard errors for factor pattern or structure coefficients,EFA rotation toward a target matrix),references to other modern topics are omitted.It does not discuss dichotomous or categorical items or modern rotation techniques such as the Crawford–Ferguson family,dynamic factor analysis for longitudinal data,factor analysis with fi-nite latent mixtures,or two-level models for factor analysis of clustered data.However,two-level modes are discussed (e.g.,R-technique and Q-technique).

The book’s focus is on understanding EFA and CFA concepts and statistical ideas,rather than teaching concepts in isolation.Throughout the book,con-nections are made among the GLM,EFA,and CFA,thus facilitating further understanding of statistical analyses and their applications.The reader is left with the understanding that a multitude of decisions are made when conducting EFA or CFA and is equipped with the necessary tools to make educated de-cisions.This book serves as an important source for introducing and building D o w n l o a d e d B y : [E B S C O H o s t E J S C o n t e n t D i s t r i b u t i o n - S u p e r c e d e d b y 9127733] A t : 05:09 1 O c t o b e r 2010

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a foundation of EFA and CFA concepts and for connecting these concepts to the GLM.

REFERENCES

Armstrong,J.S.(1967).Derivation of theory by means of factor analysis or Tom Swift and his electric factor analysis machine.American Statistician,21(5),17–21.

Cattell,R.B.(1966).The data box:Its ordering of total resources in terms of possible relational systems.In R.B.Cattell (Ed.),Handbook of multivariate experimental psychology (pp.67–128).Chicago:Rand McNally.

Cronkhite,G.,&Liska,J.R.(1980).The judgment of communicant acceptability.In M.R.Roloff &G.R.Miller (Eds.),Persuasion:New directions in theory and research (pp.101–139).Beverly Hills,CA:Sage.

Gorsuch,R.L.(1983).Factor analysis (2nd ed.).Hillsdale,NJ:Erlbaum.

Henson,R.K.,&Roberts,J.K.(2006).Use of exploratory factor analysis in published research:Common errors and some comment on improved practice.Educational and Psychological Mea-surement,66,393–416.

Mulaik,S.(1972).The foundations of factor analysis.New York:McGraw-Hill.

Schmid,J.,&Leiman,J.(1957).The development of hierarchical factor solutions.Psychometrika,22,53–61.

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