Applied Multiway Data Analysis

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