Multivariate Exploratory Data Analysis: A Perspective on Exploratory Factor Analysis

Description

In an exciting return to the roots of factor analysis, Allen Yates reviews its early history to clarify original objectives created by its discoverers and early developers. He then shows how computers can be used to accomplish the goals established by these early visionaries, while taking into account modern developments in the field of statistics that legitimize exploratory data analysis as a technique of discovery. The book presents a unique perspective on all phases of exploratory factor analysis. In doing so, the popular objectives of the method are literally turned upside down both at the stage where the model is being fitted to data and in the subsequent stage of simple structure transformation for meaningful interpretation. What results is a fully integrated approach to exploratory analysis of associations among observed variables, revealing underlying structure in a totally new and much more invariant manner than ever before possible. Multivariate Exploratory Data Analysis CONTENTS TABLES FIGURES PREFACE PROLOGUE PART I INTRODUCTORY ALGEBRA, GEOMETRY, AND TERMINOLOGY 1 The Factor Analysis Model Factor Analysis as a Special Case of Multivariate Multiple Regression The Geometry of Factor Analysis in the Case of an Ideal Simple Structure PART II TRANSFORMATION OF PRIMARY FACTORS ON THE BASIS OF THURSTONE'S ORIGINAL MATHEMATICAL CRITERIO... 2 Simple Structure Transformation Thurstone's Mathematical Criterion for a Best-Fitting Simple Structure Direct Geomin Transformation for Primary Factor Pattern Simplification The Geometry of Primary Factor Rotation. The Algebra of Factor Pair Rotation. 3 Practical Application of Direct Geomin Transformationi A Comparison of Direct Geomin and Direct Quartimin Hard vs. Soft Squeeze Direct Geomin Transformation Strategies Resistant Fitting Aspects of Direct Geomin Transformation. Sensitivity of Direct Geomin Transformation to Characteristics of the Starting Configuration. PART III BOUNDING THE TEST VECTOR CONFIGURATION WITH HYPERPLANES THROUGH SIMPLE STRUCTURE TRANSFORMA... 4 Simple Structure Reconsidered Beyond Factorial Purity and Ease of Interpretation The Positive Manifold Assumption. A Source of Prior Information About Test Vector Configuration Location 5 Bounding Hyperplane Simple Structure Transformation Toward Bounding the Test Vector Configuration with Hyperplanes in the Identification of Primary Fact... Distinguishability Weighting. Complexity Weighting and the Associated Factor Contribution Matrix. Factor Size Scaling. Direct Geoplane Transformation 6 A Global Strategy for Bounding Hyperplane Simple Structure Transformation Locating the Central Axis of a Polyhedral Convex Cone of Test Vectors Iterative Recomputation of Distinguishability and Complexity Weights. Iterative Reweighting and Factor Size Scaling. An Orthogonal Initial Configuration for Direct Geoplane Transformation 7 Factorial Invariance and the Global Direct Geoplane Transformation Strategy Analysis of Thurstone's Invariant 26-Variable Box Problem Assessing Factorial Invariance Through Extended Vectors Projection of Direct Geoplane Factors PART IV DISTINGUISHING INVARIANT MAJOR COMMON FACTORS FROM UNSTABLE MINOR FACTORS OF GROUP OVERLAP 8 Alternative Approaches to the Problem of Extracting Common Factors The Problem of Minor and Doublet Factors in Fitting the Common Factor Model The Classical Statistical Approach to Fitting the Factor Analysis Model A Perspective on the Maximum Likelihood Method of Factor Extraction. Alternatives to the Classical Confirmatory Approach to Fitting the Unrestricted Common Factor Model Gauss-Seidel and Multidimensional Sectioning Methods of Fitting the Factor Analysis Model. Collinearity-Resistant Fitting of the Common Factor Model in the Presence of Local Dependency Outlie... 9 Resistant Firring and the Number of Factors Problem Roward Fully Exploratory Factor Analysis Fitting Lord's Highly Clustered Data Practical Comparison of Collinearity-Resistant Fitting and Direct Geoplane Transformation With Avail... Achieving Solution Invariance Through Dimensionality Reduction in Fully Exploratory Factor Analysis.... An alternative to Second Order Factoring. Accommodating Observed Data of High Dimensionality. Resolution of the Psychometric Vs. Statistical Conflict in Fitting the Factor Analysis Model PART V FULLY EXPLORATORY FACTOR ANALYSIS 10 Historical Perspective, Practical Application, and Future Possibilities A Synthesis of the Classical Common Factor Theories of Spearman and Thurstone via Multivariate Explo... A Second Look at the Primary Mental Abilities Toward an Understanding of Higher Integrative Mental Functioning EPILOGUE: RELEASE FROM THE BURDEN OF CONFIRMATION IN FULLY EXPLORATORY MULTIVARIATE DATA ANALYSIS APPENDIX Summary of Computational Steps BIBLIOGRAPHY REFERENCE NOTES REFERENCES INDEX A B C D E F G H I J K L M N O P Q R S T V W Y