MODERN FACTOR ANALYSIS
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1 MODERN FACTOR ANALYSIS Harry H. Harman «ö THE pigj UNIVERSITY OF CHICAGO PRESS
2 Contents LIST OF ILLUSTRATIONS GUIDE TO NOTATION xv xvi Parti Foundations of Factor Analysis 1. INTRODUCTION Brief History of Factor Analysis Applications of Factor Analysis Scientific Explanation and Choice 8 2. FACTOR ANALYSIS MODEL Introduction Basic Statistics Linear Model for a Statistical Variable Variance Components Factor Patterns and Structures Factor Patterns as Classical Regression Equations Statistical Fit of the Factor Model Indeterminateness of Factor Solutions MATRIX CONCEPTS ESSENTIAL TO FACTOR ANALYSIS Introduction Basic Concepts of Determinants and Matrices The Factor Model in Matrix Notation Solution of Systems of Linear Equations: Method of Substitution Solution of Systems of Linear Equations: Square Root Method Calculation of the Inverse of a Matrix GEOMETRIC CONCEPTS ESSENTIAL TO FACTOR ANALYSIS Introduction Geometry of N Dimensions Cartesian Coordinate System Linear Combination and Dependence Distance Formulas in Rectangular Coordinates Orthogonal Transformations Angular Separation between Two Lines Distance and Angle in General Cartesian Coordinates Geometrie Interpretation of Correlation Subspaces Employed in Factor Analysis 64 xi
3 xii 5. THE PROBLEM OF COMMTJNALITY Introduction Determination of the Common-Factor Space Conditions for One Common Factor Conditions for Two Common Factors Determination of Communality from Approximate Rank Numerical Example Employing Approximate Rank Theoretical Solution for Communality Arbitrary Approximations to Communality Complete Approximations to Communality Examples of Approximations to Communality Direct Factor Solution PROPERTIES OF DIFFERENT TYPES OF FACTOR SOLUTIONS Introduction Mathematical and Logical Criteria Square Root Solutions Solutions Not Requiring Communalities Preliminary Solutions Involving Communalities Multiple-Factor Solution and Simple Structure Principles Summary of Factor Solutions 114 Part II Direct Solutions 7. TWO-FACTOR SOLUTION Introduction Summation Method Method of Triads The Heywood Case BI-FACTOR SOLUTION Introduction Grouping of Variables General-Factor Coefficients Group-Factor Coefficients Adjustments to the Bi-Factor Solution Illustrative Examples Computing Procedures PRINCIPAL-FACTOR SOLUTION Introduction Derivation of Principal-Factor Method Additional Theory for Computing Applications Computing Procedures With Desk Calculator 164
4 xiii 9.5. Solutions Obtained With Desk Calculators Outline of Electronic Computer Program Solutions Obtained with Electronic Computers CENTROID SOLUTION Introduction Derivation of Centroid Method Computing Procedures Illustrative Examples Averoid Method MULTIPLE-GROUP SOLUTION Introduction Concepts and Notation The Oblique Solution The Orthogonal Solution Multiple-Group Factor Algorithm Numerical Illustration 227 Part III Derived Solutions 12. DIFFERENT SOLUTIONS IN COMMON-FACTOR SPACE Introduction Relationship between Two Known Solutions Graphical Procedures for Orthogonal Multiple- Factor Solution Numerical Illustrations of Orthogonal Multiple- Factor Solutions Other Problems of Relationships between Factor Solutions OBLIQUE MULTIPLE-FACTOR SOLUTIONS Introduction Geometrie Basis for an Oblique Solution Computing Procedures for Oblique Primary-Factor Solution Oblique Reference Solution Relationship between Two Types of Oblique Solutions Numerical Illustrations ANALYTICAL METHODS FOR THE MULTIPLE-FACTOR SOLUTION: ORTHOGONAL CASE Introduction Rationale for Analytical Methods Quartimax Method Varimax Method 301
5 xiv 15. ANALYTICAL METHODS FOR THE MULTIPLE-FACTOR SOLUTION: OBLIQUE CASE Introduction Oblimax Method Quartimin Method Oblimin Methods 324 Part IV Special Topics 16. MEASUREMENT OF FACTORS Introduction Direct Solution versus Estimation Complete Estimation Method Approximation Method Short Method Estimation by Minimizing Unique Factors Factor Measurements by Ideal Variables STATISTICAL TESTS OF HYPOTHESES IN FACTOR ANALYSIS Introduction Statistical Estimation Maximum-Likelihood Estimates of Factor Loadings Test of Significance for the Number of Common Factors Computing Procedures Numerical Illustrations Concluding Remarks 380 Part V Problems and Exercises PROBLEMS 387 ANSWERS 411 Appendix STATISTICAL TABLES 439 BIBLIOGRAPHY 445 INDEX 465
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