Applied Multi–Way Data Analysis

Pieter M. Kroonenberg, PhD, is Professor of Multivariate Analysis in the Department of Education and Child Studies at Leiden University in the Netherlands. He has over thirty years of academic and consulting experience and has published over fifty articles on the subject of three-mode data analysis.

Foreword xv Preface xvii PART I DATA, MODELS, AND ALGORITHMS 1 Overture 3 1.1 Three-way and multiway data 4 1.2 Multiway data analysis 5 1.3 Before the arrival of three-mode analysis 6 1.4 Three-mode data-analytic techniques 7 1.5 Example: Judging Chopin's preludes 7 1.6 Birth of the Tucker model 12 1.7 Current status of multiway analysis 12 2 Overview 15 2.1 What are multiway data? 15 2.2 Why multiway analysis? 17 2.3 What is a model? 18 2.4 Some history 20 2.5 Multiway models and methods 24 2.6 Conclusions 24 3 Three-Way and Multiway Data 27 3.1 Chapter preview 27 3.2 Terminology 28 3.3 Two-way solutions to three-way data 30 3.4 Classification principles 31 3.5 Overview of three-way data designs 33 3.6 Fully crossed designs 33 3.7 Nested designs 38 3.8 Scaling designs 40 3.9 Categorical data 41 4 Component Models for Fully-Crossed Designs 43 4.1 Introduction 43 4.2 Chapter preview 45 4.3 Two-mode modeling of three-way data 45 4.4 Extending two-mode component models to three-mode models 47 4.5 Tucker models 51 4.6 Parafac models 57 4.7 ParaTuck2 model 63 4.8 Core arrays 64 4.9 Relationships between component models 66 4.10 Multiway component modeling under constraints 68 4.11 Conclusions 74 5 Algorithms for Multiway Models 77 5.1 Introduction 77 5.2 Chapter preview 78 5.3 Terminology and general issues 79 5.4 An example of an iterative algorithm 81 5.5 General behavior of multiway algorithms 84 5.6 The Parallel factor model - Parafac 85 5.7 The Tucker models 97 5.8 STATIS 105 5.9 Conclusions 106 PART II DATA HANDLING, MODEL SELECTION, AND INTERPRETATION 6 Preprocessing 109 6.1 Introduction 109 6.2 Chapter preview 112 6.3 General considerations 112 6.4 Model-based arguments for preprocessing choices 117 6.5 Content-based arguments for preprocessing choices 128 6.6 Preprocessing and specific multiway data designs 130 6.7 Centering and analysis-of-variance models: Two-way data 134 6.8 Centering and analysis-of-variance models: Three-way data 137 6.9 Recommendations 141 7 Missing Data in Multiway Analysis 143 7.1 Introduction 143 7.2 Chapter preview 147 7.3 Handling missing data in two-mode PCA 148 7.4 Handling missing data in multiway analysis 154 7.5 Multiple imputation in multiway analysis: Data matters 156 7.6 Missing data in multiway analysis: Practice 157 7.7 Example: Spanjer's Chromatography data 159 7.8 Example: NICHD Child care data 168 7.9 Further applications 172 7.10 Computer programs for multiple imputation 174 8 Model and Dimensionality Selection 175 8.1 Introduction 175 8.2 Chapter preview 176 8.3 Sample size and stochastics 176 8.4 Degrees of freedom 177 8.5 Selecting the dimensionality of a Tucker model 179 8.6 Selecting the dimensionality of a Parafac model 184 8.7 Model selection from a hierarchy 186 8.8 Model stability and predictive power 187 8.9 Example: Chopin prelude data 190 8.10 Conclusions 208 9 Interpreting Component Models 209 9.1 Chapter preview 209 9.2 General principles 210 9.3 Representations of component models 215 9.4 Scaling of components 218 9.5 Interpreting core arrays 225 9.6 Interpreting extended core arrays 231 9.7 Special topics 232 9.8 Validation 233 9.9 Conclusions 235 10 Improving Interpretation through Rotations 237 10.1 Introduction 237 10.2 Chapter preview 240 10.3 Rotating components 241 10.4 Rotating full core arrays 244 10.5 Theoretical simplicity of core arrays 254 10.6 Conclusions 256 11 Graphical Displays for Components 257 11.1 Introduction 257 11.2 Chapter preview 258 11.3 General considerations 259 11.4 Plotting single modes 260 11.5 Plotting different modes together 270 11.6 Conclusions 279 12 Residuals, Outliers, and Robustness 281 12.1 Introduction 281 12.2 Chapter preview 282 12.3 Goals 283 12.4 Procedures for analyzing residuals 284 12.5 Decision schemes for analyzing multiway residuals 287 12.6 Structured squared residuals 287 12.7 Unstructured residuals 292 12.8 Robustness: Basics 294 12.9 Robust methods of multiway analysis 297 12.10 Examples 301 12.1 1 Conclusions 307 PART III MULTIWAY DATA AND THEIR ANALYSIS 13 Modeling Multiway Profile Data 311 13.1 Introduction 311 13.2 Chapter preview 313 13.3 Example: Judging parents' behavior 313 13.4 Multiway profile data: General issues 320 13.5 Multiway profile data: Parafac in practice 322 13.6 Multiway profile data: Tucker analyses in practice 331 13.7 Conclusions 342 14 Modeling Multiway Rating Scale Data 345 14.1 Introduction 345 14.2 Chapter preview 346 14.3 Three-way rating scale data: Theory 346 14.4 Example: Coping at school 354 14.5 Analyzing three-way rating scales: Practice 360 14.6 Example: Differences within a multiple personality 361 14.7 Conclusions 370 15 Exploratory Multivariate Longitudinal Analysis 373 15.1 Introduction 373 15.2 Chapter preview 375 15.3 Overview of longitudinal modeling 375 15.4 Longitudinal three-mode modeling 378 15.5 Example: Organizational changes in Dutch hospitals 385 15.6 Example: Morphological development of French girls 394 15.7 Further reading 400 15.8 Conclusions 401 16 Three-Mode Clustering 403 16.1 Introduction 403 16.2 Chapter preview 405 16.3 Three-mode clustering analysis: Theory 405 16.4 Example: Identifying groups of diseased blue crabs 409 16.5 Three-mode cluster analysis: Practice 411 16.6 Example: Behavior of children in the Strange Situation 424 16.7 Extensions and special topics 430 16.8 Conclusions 432 17 Multiway Contingency Tables 433 17.1 Introduction 433 17.2 Chapter preview 434 17.3 Three-way correspondence analysis: Theory 435 17.4 Example: Sources of happiness 444 17.5 Three-way correspondence analysis: Practice 448 17.6 Example: Playing with peers 454 17.7 Conclusions 458 18 Three-Way Binary Data 459 18.1 Introduction 459 18.2 Chapter preview 460 18.3 A graphical introduction 460 18.4 Formal description of the Tucker-HICLAS models 462 18.5 Additional issues 465 18.6 Example: Hostile behavior in frustrating situations 465 18.7 Conclusion 467 19 From Three-Way Data to Four-Way Data and Beyond 469 19.1 Introduction 469 19.2 Chapter preview 471 19.3 Examples of multiway data 471 19.4 Multiway techniques: Theory 474 19.5 Example: Differences within a multiple personality 476 19.6 Example: Austrian aerosol particles 480 19.7 Further reading and computer programs 487 19.8 Conclusions 488 Appendix A: Standard Notation for Multiway Analysis 489 Appendix B: Biplots and Their Interpretation 491 B. 1 Introduction 492 B.2 Singular value decomposition 492 B.3 Biplots 494 B.4 Relationship with PCA 499 B.5 Basic vector geometry relevant to biplots 499 References 501 Glossary 527 Acronyms 543 Author Index 545 Subject Index 553