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Foundations of factor analysis:

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Main Author:	Mulaik, Stanley A. (Author)
Format:	Book
Language:	English
Published:	Boca Raton [u.a.] CRC Press 2010
Edition:	2. ed.
Series:	Chapman & Hall/CRC Statistics in the social and behavioral sciences series
Subjects:	Factor analysis Faktorenanalyse
Online Access:	Inhaltsverzeichnis Klappentext
Physical Description:	XXIII, 524 S. graph. Darst.
ISBN:	9781420099614 1420099612

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adam_text	Contents Preface to the Second Edition ............................................................................xiii Preface to the First Edition .................................................................................xix 1 Introduction .....................................................................................................1 1.1 Factor Analysis and Structural Theories .............................................1 1.2 Brief History of Factor Analysis as a Linear Model ...........................3 1.3 Example of Factor Analysis .................................................................12 2 Mathematical Foundations for Factor Analysis .....................................17 2.1 Introduction ...........................................................................................17 2.2 Scalar Algebra ........................................................................................17 2.2.1 Fundamental Laws of Scalar Algebra ..................................18 2.2.1.1 Rules of Signs ............................................................18 2.2.1.2 Rules for Exponents .................................................19 2.2.1.3 Solving Simple Equations .......................................19 2.3 Vectors ....................................................................................................20 2.3.1 п -Tuples as Vectors ..................................................................22 2.3.1.1 Equality of Vectors ....................................................22 2.3.2 Scalars and Vectors ..................................................................23 2.3.3 Multiplying a Vector by a Scalar ...........................................23 2.3.4 Addition of Vectors .................................................................24 2.3.5 Scalar Product of Vectors .......................................................24 2.3.6 Distance between Vectors ......................................................25 2.3.7 Length of a Vector ...................................................................26 2.3.8 Another Definition for Scalar Multiplication ......................27 2.3.9 Cosine of the Angle between Vectors ...................................27 2.3.10 Projection of a Vector onto Another Vector .........................29 2.3.11 Types of Special Vectors .........................................................30 2.3.12 Linear Combinations ..............................................................31 2.3.13 Linear Independence ..............................................................32 2.3.14 Basis Vectors .............................................................................32 2.4 Matrix Algebra ......................................................................................32 2.4.1 Definition of a Matrix .............................................................32 2.4.2 Matrix Operations ...................................................................33 2.4.2.1 Equality ......................................................................34 2.4.2.2 Multiplication by a Scalar .......................................34 2.4.2.3 Addition .....................................................................34 2.4.2.4 Subtraction ................................................................35 2.4.2.5 Matrix Multiplication ..............................................35 vi Contents 2.4.3 Identity Matrix.........................................................................37 2.4.4 Scalar Matrix ............................................................................38 2.4.5 Diagonal Matrix ......................................................................39 2.4.6 Upper and Lower Triangular Matrices ................................39 2.4.7 Null Matrix ...............................................................................40 2.4.8 Transpose Matrix .....................................................................40 2.4.9 Symmetric Matrices ................................................................41 2.4.10 Matrix Inverse ..........................................................................41 2.4.11 Orthogonal Matrices ...............................................................42 2.4.12 Trace of a Matrix ......................................................................43 2.4.13 Invariance of Traces under Cyclic Permutations ................43 2.5 Determinants .........................................................................................44 2.5.1 Minors of a Matrix ..................................................................46 2.5.2 Rank of a Matrix ......................................................................47 2.5.3 Cofactors of a Matrix ..............................................................47 2.5.4 Expanding a Determinant by Cofactors ..............................48 2.5.5 Adjoint Matrix .........................................................................48 2.5.6 Important Properties of Determinants .................................49 2.5.7 Simultaneous Linear Equations ............................................50 2.6 Treatment of Variables as Vectors .......................................................51 2.6.1 Variables in Finite Populations .............................................51 2.6.2 Variables in Infinite Populations ...........................................53 2.6.3 Random Vectors of Random Variables .................................56 2.7 Maxima and Minima of Functions .....................................................58 2.7.1 Slope as the Indicator of a Maximum or Minimum ...........59 2.7.2 Index for Slope .........................................................................59 2.7.3 Derivative of a Function .........................................................60 2.7.4 Derivative of a Constant ........................................................62 2.7.5 Derivative of Other Functions ...............................................62 2.7.6 Partial Differentiation .............................................................64 2.7.7 Maxima and Minima of Functions of Several Variables.... 65 2.7.8 Constrained Maxima and Minima .......................................67 Composite Variables and Linear Transformations ................................69 3.1 Introduction ...........................................................................................69 3.1.1 Means and Variances of Variables ........................................69 3.1.1.1 Correlation and Causation ......................................71 3.2 Composite Variables .............................................................................72 3.3 Unweighted Composite Variables ......................................................73 3.3.1 Mean of an Unweighted Composite ....................................73 3.3.2 Variance of an Unweighted Composite ...............................73 3.3.3 Covariance and Correlation between Two Composites ......................................................................77 3.3.4 Correlation of an Unweighted Composite with a Single Variable .............................................................78 Contents vii 3.3.5 Correlation between Two Unweighted Composites .................................................................................80 3.3.6 Summary Concerning Unweighted Composites ..................83 3.4 Differentially Weighted Composites ..................................................83 3.4.1 Correlation between a Differentially Weighted Composite and Another Variable ...........................................83 3.4.2 Correlation between Two Differentially Weighted Composites ...............................................................84 3.5 Matrix Equations ...................................................................................84 3.5.1 Random Vectors, Mean Vectors, Variance-Covariance Matrices, and Correlation Matrices ........................................84 3.5.2 Sample Equations ......................................................................86 3.5.3 Composite Variables in Matrix Equations .............................88 3.5.4 Linear Transformations ............................................................89 3.5.5 Some Special, Useful Linear Transformations ......................91 4 Multiple and Partial Correlations .............................................................93 4.1 Multiple Regression and Correlation .................................................93 4.1.1 Minimizing the Expected Squared Difference between a Composite Variable and an External Variable ...................93 4.1.2 Deriving the Regression Weight Matrix for Multivariate Multiple Regression ...........................................95 4.1.3 Matrix Equations for Multivariate Multiple Regression .....97 4.1.4 Squared Multiple Correlations ................................................98 4.1.5 Correlations between Actual and Predicted Criteria ...........99 4.2 Partial Correlations .............................................................................100 4.3 Determinantal Formulas ....................................................................102 4.3.1 Multiple-Correlation Coefficient ...........................................103 4.3.2 Formulas for Partial Correlations .........................................104 4.4 Multiple Correlation in Terms of Partial Correlation ....................104 4.4.1 Matrix of Image Regression Weights ....................................105 4.4.2 Meaning of Multiple Correlation ..........................................107 4.4.3 Yule s Equation for the Error of Estimate ............................109 4.4.4 Conclusions ..............................................................................110 5 Multivariate Normal Distribution ..........................................................113 5.1 Introduction .........................................................................................113 5.2 Univariate Normal Density Function ..............................................113 5.3 Multivariate Normal Distribution ....................................................114 5.3.1 Bivariate Normal Distribution ..............................................115 5.3.2 Properties of the Multivariate Normal Distribution ..........116 5.4 Maximum-Likelihood Estimation .....................................................118 5.4.1 Notion of Likelihood ..............................................................118 5.4.2 Sample Likelihood ..................................................................119 viii Contents 5.4.3 Maximum-Likelihood Estimates ..........................................119 5.4.4 Multivariate Case ....................................................................124 5.4.4.1 Distribution of y and S ............................................128 6 Fundamental Equations of Factor Analysis ..........................................129 6.1 Analysis of a Variable into Components .........................................129 6.1.1 Components of Variance ........................................................132 6.1.2 Variance of a Variable in Terms of Its Factors .....................133 6.1.3 Correlation between Two Variables in Terms of Their Factors ........................................................134 6.2 Use of Matrix Notation in Factor Analysis ......................................135 6.2.1 Fundamental Equation of Factor Analysis ..........................135 6.2.2 Fundamental Theorem of Factor Analysis ..........................136 6.2.3 Factor-Pattern and Factor-Structure Matrices .....................137 7 Methods of Factor Extraction ...................................................................139 7.1 Rationale for Finding Factors and Factor Loadings .......................139 7.1.1 General Computing Algorithm for Finding Factors ..................................................................140 7.2 Diagonal Method of Factoring ..........................................................145 7.3 Centroid Method of Factoring ..........................................................147 7.4 Principal-Axes Methods .....................................................................147 7.4.1 Hotelling s Iterative Method .................................................151 7.4.2 Further Properties of Eigenvectors and Eigenvalues ..............................................................................154 7.4.3 Maximization of Quadratic Forms for Points on the Unit Sphere ..................................................................156 7.4.4 Diagonalizing the R Matrix into Its Eigenvalues ...............158 7.4.5 Jacobi Method ..........................................................................159 7.4.6 Powers of Square Symmetric Matrices ................................164 7.4.7 Factor-Loading Matrix from Eigenvalues and Eigenvectors .....................................................................165 8 Common-Factor Analysis ..........................................................................167 8.1 Preliminary Considerations ...............................................................167 8.1.1 Designing a Factor Analytic Study .......................................168 8.2 First Stages in the Factor Analysis ....................................................169 8.2.1 Concept of Minimum Rank ...................................................170 8.2.2 Systematic Lower-Bound Estimates of Communalities ....................................................................175 8.2.3 Congruence Transformations ................................................176 8.2.4 Sylvester s Law of Inertia ......................................................176 8.2.5 Eigenvector Transformations ................................................177 8.2.6 Guttman s Lower Bounds for Minimum Rank ...................177 8.2.7 Preliminary Theorems for Guttman s Bounds ....................178 Contents ix 8.2.8 Proof of the First Lower Bound ...........................................181 8.2.9 Proof of the Third Lower Bound .........................................181 8.2.10 Proof of the Second Lower Bound ......................................184 8.2.11 Heuristic Rules of Thumb for the Number of Factors .....185 8.2.11.1 Kaiser s Eigenvalues-Greater- Than-OneRule ......................................................186 8.2.11.2 Cattell s Scree Criterion .......................................186 8.2.11.3 Parallel Analysis ...................................................188 8.3 Fitting the Common-Factor Model to a Correlation Matrix .........192 8.3.1 Least-Squares Estimation of the Exploratory Common-Factor Model ........................................................193 8.3.2 Assessing Fit ..........................................................................197 8.3.3 Example of Least-Squares Common-Factor Analysis ...................................................................................197 8.3.4 Maximum-Likelihood Estimation of the Exploratory Common-Factor Model ..................................199 8.3.4.1 Maximum-Likelihood Estimation Obtained Using Calculus ......................................................202 8.3.5 Maximum-Likelihood Estimates ........................................206 8.3.6 Fletcher-Powell Algorithm ..................................................207 *8.3.7 Applying the Fletcher-Powell Algorithm to Maximum-Likelihood Exploratory Factor Analysis .........210 8.3.8 Testing the Goodness of Fit of the Maximum-Likelihood Estimates ........................................212 8.3.9 Optimality of Maximum-Likelihood Estimators ..............214 8.3.10 Example of Maximum-Likelihood Factor Analysis ......................................................................215 Other Models of Factor Analysis .............................................................217 9.1 Introduction .........................................................................................217 9.2 Component Analysis ..........................................................................217 9.2.1 Principal-Components Analysis .........................................219 9.2.2 Selecting Fewer Components than Variables ....................220 9.2.3 Determining the Reliability of Principal Components ....222 9.2.4 Principal Components of True Components .....................224 9.2.5 Weighted Principal Components ........................................226 9.3 Image Analysis ....................................................................................230 9.3.1 Partial-Image Analysis .........................................................231 9.3.2 Image Analysis and Common-Factor Analysis ................237 9.3.3 Partial-Image Analysis as Approximation of Common-Factor Analysis .....................................................244 9.4 Canonical-Factor Analysis .................................................................245 9.4.1 Relation to Image Analysis ..................................................249 9.4.2 Kaiser s Rule for the Number of Harris Factors ...............253 9.4.3 Quickie, Single-Pass Approximation for Common-Factor Analysis .....................................................253 Contents 9.5 Problem of Doublet Factors............................................................... 253 9.5.1 Butler s Descriptive-Factor-Analysis Solution .................254 9.5.2 Model That Includes Doublets Explicitly ..........................258 9.6 Metric Invariance Properties .............................................................262 9.7 Image-Factor Analysis ........................................................................263 9.7.1 Testing Image Factors for Significance ...............................264 9.8 Psychometric Inference in Factor Analysis .....................................265 9.8.1 Alpha Factor Analysis ..........................................................270 9.8.2 Communality in a Universe of Tests ..................................271 9.8.3 Consequences for Factor Analysis ......................................274 10 Factor Rotation ............................................................................................275 10.1 Introduction .......................................................................................275 10.2 Thurstone s Concept of a Simple Structure ...................................276 10.2.1 Implementing the Simple-Structure Concept .................280 10.2.2 Question of Correlated Factors .........................................282 10.3 Oblique Graphical Rotation ............................................................286 11 Orthogonal Analytic Rotation .................................................................301 11.1 Introduction .......................................................................................301 11.2 Quartimax Criterion .........................................................................302 11.3 Varimax Criterion ..............................................................................310 11.4 Transvarimax Methods .....................................................................312 11.4.1 Parsimax ...............................................................................313 11.5 Simultaneous Orthogonal Varimax and Parsimax .......................315 11.5.1 Gradient Projection Algorithm ..........................................323 12 Oblique Analytic Rotation .......................................................................325 12.1 General ...............................................................................................325 12.1.1 Distinctness of the Criteria in Oblique Rotation .............325 12.2 Oblimin Family .................................................................................326 12.2.1 Direct Oblimin by Planar Rotations .................................328 12.3 Harris-Kaiser Oblique Transformations .......................................332 12.4 Weighted Oblique Rotation .............................................................336 12.5 Oblique Procrustean Transformations ...........................................341 12.5.1 Promax Oblique Rotation ..................................................342 12.5.2 Rotation to a Factor-Pattern Matrix Approximating a Given Target Matrix .........................................................343 12.5.3 Promaj ...................................................................................343 12.5.4 Promiň ..................................................................................345 12.6 Gradient-Projection-Algorithm Synthesis .....................................348 12.6.1 Gradient-Projection Algorithm .........................................348 12.6.2 Jennrich s Use of the GPA ..................................................351 12.6.2.1 Gradient-Projection Algorithm ........................353 12.6.2.2 Quartimin ............................................................353 Contents xi 12.6.2.3 Oblimin Rotation................................................354 12.6.2.4 Least-Squares Rotation to a Target Matrix .....357 12.6.2.5 Least-Squares Rotation to a Partially Specified Target Pattern Matrix ........................357 12.6.3 Simplimax ............................................................................357 12.7 Rotating Using Component Loss Functions .................................360 12.8 Conclusions ........................................................................................366 13 Factor Scores and Factor Indeterminacy ................................................369 13.1 Introduction .......................................................................................369 13.2 Scores on Component Variables .....................................................370 13.2.1 Component Scores in Canonical-Component Analysis and Image Analysis ............................................373 13.2.1.1 Canonical-Component Analysis .......................373 13.2.1.2 Image Analysis ....................................................374 13.3 Indeterminacy of Common-Factor Scores .....................................375 13.3.1 Geometry of Correlational Indeterminacy ......................377 13.4 Further History of Factor Indeterminacy ......................................380 13.4.1 Factor Indeterminacy from 1970 to 1980..........................384 13.4.1.1 Infinite Domain Position ...............................392 13.4.2 Researchers with Well-Defined Concepts of Their Domains ................................................................395 13.4.2.1 Factor Indeterminacy from 1980 to 2000.........397 13.5 Other Estimators of Common Factors ...........................................399 13.5.1 Least Squares .......................................................................400 13.5.2 Bartlett s Method .................................................................401 13.5.3 Evaluation of Estimation Methods ...................................403 14 Factorial Invariance ....................................................................................405 14.1 Introduction .......................................................................................405 14.2 Invariance under Selection of Variables ........................................405 14.3 Invariance under Selection of Experimental Populations ..........408 14.3.1 Effect of Univariate Selection ............................................408 14.3.2 Multivariate Case ................................................................412 14.3.3 Factorial Invariance in Different Experimental Populations ..........................................................................414 14.3.4 Effects of Selection on Component Analysis ...................418 14.4 Comparing Factors across Populations .........................................419 14.4.1 Preliminary Requirements for Comparing Factor Analyses ...................................................................420 14.4.2 Inappropriate Comparisons of Factors ............................421 14.4.3 Comparing Factors from Component Analyses .............422 14.4.4 Contrasting Experimental Populations across Factors .......................................................................423 14.4.5 Limitations on Factorial Invariance ..................................424 xii Contents 15 Confirmatory Factor Analysis ..................................................................427 15.1 Introduction .......................................................................................427 15.1.1 Abduction, Deduction, and Induction ...........................428 15.1.2 Science as the Knowledge of Objects .............................429 15.1.3 Objects as Invariants in the Perceptual Field ................431 15.1.4 Implications for Factor Analysis .....................................433 15.2 Example of Confirmatory Factor Analysis ....................................434 15.3 Mathematics of Confirmatory Factor Analysis .............................440 15.3.1 Specifying Hypotheses .....................................................440 15.3.2 Identification ......................................................................441 15.3.3 Determining Whether Parameters and Models Are Identified ......................................................444 15.3.4 Identification of Metrics ...................................................450 15.3.5 Discrepancy Functions .....................................................452 15.3.6 Estimation by Minimizing Discrepancy Functions ......454 15.3.7 Derivatives of Elements of Matrices ...............................454 15.3.8 Maximum-Likelihood Estimation in Confirmatory Factor Analysis .........................................457 15.3.9 Least-Squares Estimation .................................................461 15.3.10 Generalized Least-Squares Estimation ..........................463 15.3.11 Implementing the Quasi-Newton Algorithm ...............463 15.3.12 Avoiding Improper Solutions ..........................................465 15.3.13 Statistical Tests ...................................................................466 15.3.14 What to Do When Chi-Square Is Significant .................467 15.3.15 Approximate Fit Indices ...................................................469 15.4 Designing Confirmatory Factor Analysis Models ........................473 15.4.1 Restricted versus Unrestricted Models ..........................473 15.4.2 Use for Unrestricted Model .............................................475 15.4.3 Measurement Model .........................................................476 15.4.4 Four-Step Procedure for Evaluating a Model ...............477 15.5 Some Other Applications .................................................................477 15.5.1 Faceted Classification Designs ........................................477 15.5.2 Multirater-Multioccasion Studies ..................................478 15.5.3 Multitrait-Multimethod Covariance Matrices ..............483 15.6 Conclusion .........................................................................................489 References ...........................................................................................................493 Author Index .......................................................................................................505 Subject Index ......................................................................................................509 Statistics Providing a practical, thorough understanding of how factor analysis works, Foundations of Factor Analysis, Second Edition discusses the assumptions underlying the equations and procedures of this method. It also explains the options in commercial computer programs for performing factor analysis and structural equation modeling. This long-awaited edition takes into account the various developments that have occurred since the publication of the original edition. New to the Second Edition • A new chapter on the multivariate normal distribution, its general properties, and the concept of maximum-likelihood estimation • More complete coverage of descriptive factor analysis and doublet factor analysis • A rewritten chapter on analytic oblique rotation that focuses on the gradient projection algorithm and its applications • Discussions on the developments of factor score indeterminacy • A revised chapter on confirmatory factor analysis that addresses philosophy of science issues, model specification and identification, parameter estimation, and algorithm derivation Presenting the mathematics only as needed to understand the derivation of an equation or procedure, this book enables readers to choose the proper factor analytic procedure, make modifications to the procedure, and produce new results. It covers several models of factor analysis, including common factor analysis, component analysis, image analysis, and canonical factor analysis models. The author discusses how factor analysis can yield objective results, explains the rationale behind rotation routines and how to write simple programs to perform them, and chronicles the history of factor indeterminacy and the various attempts to resolve this problem. He also shows how to formulate and test models through confirmatory factor analysis.
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isbn	9781420099614 1420099612
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-017634039
oclc_num	255899515
open_access_boolean
owner	DE-945 DE-703 DE-634 DE-188 DE-19 DE-BY-UBM DE-M347
owner_facet	DE-945 DE-703 DE-634 DE-188 DE-19 DE-BY-UBM DE-M347
physical	XXIII, 524 S. graph. Darst.
publishDate	2010
publishDateSearch	2010
publishDateSort	2010
publisher	CRC Press
record_format	marc
series2	Chapman & Hall/CRC Statistics in the social and behavioral sciences series
spelling	Mulaik, Stanley A. Verfasser aut Foundations of factor analysis Stanley A. Mulaik 2. ed. Boca Raton [u.a.] CRC Press 2010 XXIII, 524 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Chapman & Hall/CRC Statistics in the social and behavioral sciences series Factor analysis Faktorenanalyse (DE-588)4016338-6 gnd rswk-swf Faktorenanalyse (DE-588)4016338-6 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017634039&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017634039&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext
spellingShingle	Mulaik, Stanley A. Foundations of factor analysis Factor analysis Faktorenanalyse (DE-588)4016338-6 gnd
subject_GND	(DE-588)4016338-6
title	Foundations of factor analysis
title_auth	Foundations of factor analysis
title_exact_search	Foundations of factor analysis
title_full	Foundations of factor analysis Stanley A. Mulaik
title_fullStr	Foundations of factor analysis Stanley A. Mulaik
title_full_unstemmed	Foundations of factor analysis Stanley A. Mulaik
title_short	Foundations of factor analysis
title_sort	foundations of factor analysis
topic	Factor analysis Faktorenanalyse (DE-588)4016338-6 gnd
topic_facet	Factor analysis Faktorenanalyse
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017634039&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017634039&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT mulaikstanleya foundationsoffactoranalysis

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