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Applied Multivariate Statistical Analysis, Sixth Edition

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Aspects of Multivariate Analysis

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Applied Multivariate Statistical Analysis, Sixth Edition

by Richard A. Johnson; Dean W. Wichern

Publisher: Pearson

Copyright Year: 2008

Publishing Date: 2007/03/23

eText ISBN-10: 0-13-208437-6

eText ISBN-13: 978-0-13-208437-6

Print ISBN-10: 0-13-187715-1

Print ISBN-13: 978-0-13-187715-3

Course: Research Methods and Statistics, Business Statistics, Marketing Research

Pages: 800

eTextbook $67.33

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Table of Contents | Description

Instructor Resources | Student Resources

ABOUT THIS TITLE - TABLE OF CONTENTS

1. Aspects of Multivariate Analysis, 1

1.1. Introduction, 1

1.2. Applications of Multivariate Techniques, 3

1.3. The Organization of Data, 5

Descriptive Statistics, 6

Graphical Techniques, 11

1.4. Data Displays and Pictorial Representations, 19

Linking Multiple Two-Dimensional Scatter Plots, 20

Graphs of Growth Curves, 24

Chernoff Faces, 27

1.5. Distance, 30

1.6. Final Comments, 37

2. Matrix Algebra and Random Vectors, 49

2.1. Introduction, 49

2.2. Some Basics of Matrix and Vector Algebra, 49

2.3. Positive Definite Matrices, 60

2.4. A Square-Root Matrix, 65

2.5. Random Vectors and Matrices, 66

2.6. Mean Vectors and Covariance Matrices, 68

Partitioning the Covariance Matrix, 73

The Mean Vector and Covariance Matrix for Linear Combinations of Random Variables, 75

Partitioning the Sample Mean Vector and Covariance Matrix, 77

2.7. Matrix Inequalities and Maximization, 78

Supplement 2A. Vectors and Matrices: Basic Concepts, 82

References, 110

3. Sample Geometry and Random Sampling, 111

3.1. Introduction, 111

3.2. The Geometry of the Sample, 111

3.3. Random Samples and the Expected Values of the Sample Mean and Covariance Matrix, 119

3.4. Generalized Variance, 123

Situations in which the Generalized Sample Variance Is Zero, 129

Generalized Variance Determined by |R| and Its Geometrical Interpretation, 134

Another Generalization of Variance, 137

3.5. Sample Mean, Covariance, and Correlation As Matrix Operations, 137

3.6. Sample Values of Linear Combinations of Variables, 140

References, 148

4. The Multivariate Normal Distribution, 149

4.1. Introduction, 149

4.2. The Multivariate Normal Density and Its Properties, 149

Additional Properties of the Multivariate Normal Distribution, 156

4.3. Sampling from a Multivariate Normal Distribution and Maximum Likelihood Estimation, 168

The Multivariate Normal Likelihood, 168

Maximum Likelihood Estimation of μ and Σ, 170

Sufficient Statistics, 173

4.4. The Sampling Distribution of X and S, 173

Properties of the Wishart Distribution, 174

4.5. Large-Sample Behavior of X and S, 175

4.6. Assessing the Assumption of Normality, 177

Evaluating the Normality of the Univariate Marginal Distributions, 177

Evaluating Bivariate Normality, 182

4.7. Detecting Outliers and Cleaning Data, 187

Steps for Detecting Outliers, 189

4.8. Transformations to Near Normality, 192

Transforming Multivariate Observations, 195

References, 208

5. Inferences About a Mean Vector, 210

5.1. Introduction, 210

5.2. The Plausibility of μ_{0} as a Value for a Normal Population Mean, 210

5.3. Hotelling’s T^2 and Likelihood Ratio Tests, 216

General Likelihood Ratio Method, 219

5.4. Confidence Regions and Simultaneous Comparisons of Component Means, 220

Simultaneous Confidence Statements, 223

A Comparison of Simultaneous Confidence Intervals with One-at-a-Time Intervals, 229

The Bonferroni Method of Multiple Comparisons, 232

5.5. Large Sample Inferences about a Population Mean Vector, 234

5.6. Multivariate Quality Control Charts, 239

Charts for Monitoring a Sample of Individual Multivariate Observations for Stability, 241

Control Regions for Future Individual Observations, 247

Control Ellipse for Future Observations, 248

T^2-Chart for Future Observations, 248

Control Charts Based on Subsample Means, 249

Control Regions for Future Subsample Observations, 251

5.7. Inferences about Mean Vectors when Some Observations Are Missing, 251

5.8. Difficulties Due to Time Dependence in Multivariate Observations, 256

Supplement 5A. Simultaneous Confidence Intervals and Ellipses as Shadows of the p-Dimensional Ellipsoids, 258

References, 272

6. Comparisons of Several Multivariate Means, 273

6.1. Introduction, 273

6.2. Paired Comparisons and a Repeated Measures Design, 273

Paired Comparisons, 273

A Repeated Measures Design for Comparing Treatments, 279

6.3. Comparing Mean Vectors from Two Populations, 284

Assumptions Concerning the Structure of the Data, 284

Further Assumptions When n_{1} and n_{2} Are Small, 285

Simultaneous Confidence Intervals, 288

The Two-Sample Situation When Σ_{1} ≠ Σ_{2}, 291

An Approximation to the Distribution of T^2 for Normal Populations When Sample Sizes Are Not Large, 294

6.4. Comparing Several Multivariate Population Means (One-Way Manova), 296

Assumptions about the Structure of the Data for One-Way MANOVA, 296

A Summary of Univariate ANOVA, 297

Multivariate Analysis of Variance (MANOVA), 301

6.5. Simultaneous Confidence Intervals for Treatment Effects, 308

6.6. Testing for Equality of Covariance Matrices, 310

6.7. Two-Way Multivariate Analysis of Variance, 312

Univariate Two-Way Fixed-Effects Model with Interaction, 312

Multivariate Two-Way Fixed-Effects Model with Interaction, 315

6.8. Profile Analysis, 323

6.9. Repeated Measures Designs and Growth Curves, 328

6.10. Perspectives and a Strategy for Analyzing Multivariate Models, 332

References, 358

7. Multivariate Linear Regression Models, 360

7.1. Introduction, 360

7.2. The Classical Linear Regression Model, 360

7.3. Least Squares Estimation, 364

Sum-of-Squares Decomposition, 366

Geometry of Least Squares, 367

Sampling Properties of Classical Least Squares Estimators, 369

7.4. Inferences About the Regression Model, 370

Inferences Concerning the Regression Parameters, 370

Likelihood Ratio Tests for the Regression Parameters, 374

7.5. Inferences from the Estimated Regression Function, 378

Estimating the Regression Function at Z_{0}, 378

Forecasting a New Observation at Z_{0}, 379

7.6. Model Checking and Other Aspects of Regression, 381

Does the Model Fit?, 381

Leverage and Influence, 384

Additional Problems in Linear Regression, 384

7.7. Multivariate Multiple Regression, 387

Likelihood Ratio Tests for Regression Parameters, 395

Other Multivariate Test Statistics, 398

Predictions from Multivariate Multiple Regressions, 399

7.8. The Concept of Linear Regression, 401

Prediction of Several Variables, 406

Partial Correlation Coefficient, 409

7.9. Comparing the Two Formulations of the Regression Model, 410

Mean Corrected Form of the Regression Model, 410

Relating the Formulations, 412

7.10. Multiple Regression Models with Time Dependent Errors, 413

Supplement 7A. The Distribution of the Likelihood Ratio for the Multivariate Multiple Regression Model, 418

References, 428

8. Principal Components, 430

8.1. Introduction, 430

8.2. Population Principal Components, 430

Principal Components Obtained from Standardized Variables, 436

Principal Components for Covariance Matrices with Special Structures, 439

8.3. Summarizing Sample Variation by Principal Components, 441

The Number of Principal Components, 444

Interpretation of the Sample Principal Components, 448

Standardizing the Sample Principal Components, 449

8.4. Graphing the Principal Components, 454

8.5. Large Sample Inferences, 456

Large Sample Properties of λ_{i} and ê_{i}, 456

Testing for the Equal Correlation Structure, 457

8.6. Monitoring Quality with Principal Components, 459

Checking a Given Set of Measurements for Stability, 459

Controlling Future Values, 463

Supplement 8A. The Geometry of the Sample Principal Component Approximation, 466

The p-Dimensional Geometrical Interpretation, 468

The n-Dimensional Geometrical Interpretation, 469

References, 480

9. Factor Analysis and Inference for Structured Covariance Matrices, 481

9.1. Introduction, 481

9.2. The Orthogonal Factor Model, 482

9.3. Methods of Estimation, 488

The Principal Component (and Principal Factor) Method, 488

A Modified Approach—the Principal Factor Solution, 494

The Maximum Likelihood Method, 495

A Large Sample Test for the Number of Common Factors, 501

9.4. Factor Rotation, 504

Oblique Rotations, 512

9.5. Factor Scores, 513

The Weighted Least Squares Method, 514

The Regression Method, 516

9.6. Perspectives and a Strategy for Factor Analysis, 519

Supplement 9A. Some Computational Details for Maximum Likelihood Estimation, 527

Recommended Computational Scheme, 528

Maximum Likelihood Estimators of ρ = L_{Z}L′_{Z} + ψ_{Z}, 529

References, 538

10. Canonical Correlation Analysis, 539

10.1. Introduction, 539

10.2. Canonical Variates and Canonical Correlations, 539

10.3. Interpreting the Population Canonical Variables, 545

Identifying the Canonical Variables, 545

Canonical Correlations as Generalizations of Other Correlation Coefficients, 547

The First r Canonical Variables as a Summary of Variability, 548

A Geometrical Interpretation of the Population Canonical Correlation Analysis, 549

10.4. The Sample Canonical Variates and Sample Canonical Correlations, 550

10.5. Additional Sample Descriptive Measures, 558

Matrices of Errors of Approximations, 558

Proportions of Explained Sample Variance, 561

10.6. Large Sample Inferences, 563

References, 574

11. Discrimination and Classification, 575

11.1. Introduction, 575

11.2. Separation and Classification for Two Populations, 576

11.3. Classification with Two Multivariate Normal Populations, 584

Classification of Normal Populations When Σ_{1} = Σ_{2} = Σ, 584

Fisher’s Approach to Classification with Two Populations, 590

Is Classification a Good Idea?, 592

Classification of Normal Populations When Σ_{1} ≠ Σ_{2}, 593

11.4. Evaluating Classification Functions, 596

11.5. Classification with Several Populations, 606

The Minimum Expected Cost of Misclassification Method, 606

Classification with Normal Populations, 609

11.6. Fisher’s Method for Discriminating among Several Populations, 621

Using Fisher’s Discriminants to Classify Objects, 628

11.7. Logistic Regression and Classification, 634

Introduction, 634

The Logit Model, 634

Logistic Regression Analysis, 636

Classification, 638

Logistic Regression with Binomial Responses, 640

11.8. Final Comments, 644

Including Qualitative Variables, 644

Classification Trees, 644

Neural Networks, 647

Selection of Variables, 648

Testing for Group Differences, 648

Practical Considerations Regarding Multivariate Normality, 649

References, 669

12. Clustering, Distance Methods, and Ordination, 671

12.1. Introduction, 671

12.2. Similarity Measures, 673

Distances and Similarity Coefficients for Pairs of Items, 673

Similarities and Association Measures for Pairs of Variables, 677

Concluding Comments on Similarity, 678

12.3. Hierarchical Clustering Methods, 680

Single Linkage, 682

Complete Linkage, 685

Average Linkage, 690

Ward’s Hierarchical Clustering Method, 692

Final Comments—Hierarchical Procedures, 695

12.4. Nonhierarchical Clustering Methods, 696

K-means Method, 696

Final Comments—Nonhierarchical Procedures, 701

12.5. Clustering Based on Statistical Models, 703

12.6. Multidimensional Scaling, 706

The Basic Algorithm, 708

12.7. Correspondence Analysis, 716

Algebraic Development of Correspondence Analysis, 718

Interpretation in Two Dimensions, 726

Final Comments, 726

12.8. Biplots for Viewing Sampling Units and Variables, 726

Constructing Biplots, 727

12.9. Procrustes Analysis: A Method for Comparing Configurations, 732

Constructing the Procrustes Measure of Agreement, 733

Supplement 12A. Data Mining, 740

Introduction, 740

The Data Mining Process, 741

Model Assessment, 742

References, 755

Data Index, 764

Subject Index, 767

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ABOUT THIS TITLE - DESCRIPTION

For courses in Multivariate Statistics, Marketing Research, Intermediate Business Statistics, Statistics in Education, and graduate-level courses in Experimental Design and Statistics. Appropriate for experimental scientists in a variety of disciplines, this market-leading text offers a readable introduction to the statistical analysis of multivariate observations. Its primary goal is to impart the knowledge necessary to make proper interpretations and select appropriate techniques for analyzing multivariate data. Ideal for a junior/senior or graduate level course that explores the statistical methods for describing and analyzing multivariate data, the text assumes two or more statistics courses as a prerequisite.

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