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1．APPLIED MULTIVARIATE METHODS 1

1.1 An Overview of Multivariate Methods 1

Contents 1

Variable-and Individual-Directed Techniques 2

Creating New Variables 2

Principal Components Analysis 3

Factor Analysis 3

Discriminant Analysis 4

Cluster Analysis 5

Canonical Discriminant Analysis 5

Logistic Regression 5

Multivariate Analysis of Variance 6

Canonical Variates Analysis 7

Canonical Correlation Analysis 7

Where to Find the Preceding Topics 7

1.2 Two Examples 8

1.3 Types of Variables 11

Independence of Experimental Units 11

1.4 Data Matrices and Vectors 12

Variable Notation 13

Data Matrix 13

Data Vectors 13

Data Subscripts 14

1.5 The Multivariate Normal Distribution 15

Some Definitions 15

Summarizing Multivariate Distributions 16

Mean Vectors and Variance-Covariance Matrices 16

Correlations and Correlation Matrices 17

The Multivariate Normal Probability Density Function 19

Bivariate Normal Distributions 19

1.6 Statistical Computing 22

Cautions About Computer Usage 22

Missing Values 22

Removing Rows of the Data Matrix 23

Replacing Missing Values by Averages 23

Replacing Missing Values by Zeros 23

Sampling Strategies 24

Data Entry Errors and Data Verification 24

1.7 Multivariate Outliers 25

Locating Outliers 25

Dealing with Outliers 25

Outliers May Be Influential 26

1.8 Multivariate Summary Statistics 26

1.9 Standardized Data and/or Z Scores 27

Exercises 28

2．SAMPLE CORRELATIONS 35

2.1 Statistical Tests and Confidence Intervals 35

Are the Correlations Large Enough to Be Useful? 36

Confidence Intervals by the Chart Method 36

Confidence Intervals by Fisher's Approximation 38

Confidence Intervals by Ruben's Approximation 39

Variable Groupings Based on Correlations 40

Relationship to Factor Analysis 46

2.2 Summary 46

Exercises 47

3．MULTIVARIATE DATA PLOTS 55

3.1 Three-Dimensional Data Plots 55

3.2 Plots of Higher Dimensional Data 59

Chernoff Faces 61

Star Plots and Sun-Ray Plots 63

Andrews'Plots 65

Side-by-Side Scatter Plots 66

3.3 Plotting to Check for Multivariate Normality 67

Summary 73

Exercises 73

4．EIGENVALUES AND EIGENVECTORS 77

4.1 Trace and Determinant 77

Examples 78

4.2 Eigenvalues 78

4.3 Eigenvectors 79

Positive Definite and Positive Semidefinite Matrices 80

4.4 Geometric Descriptions（p=2） 82

Vectors 82

Bivariate Normal Distributions 83

4.5 Geometric Descriptions（p=3） 87

Vectors 87

Trivariate Normal Distributions 87

4.6 Geometric Descriptions（p>3） 90

Exercises 91

Summary 91

5．PRINCIPAL COMPONENTS ANALYSIS 93

5.1 Reasons for Using Principal Components Analysis 93

Data Screening 93

Clustering 95

Discriminant Analysis 95

Regression 95

5.3 Principal Components Analysis on the Variance-Covariance Matrix ∑ 96

5.2 Objectives of Principal Components Analysis 96

Principal Component Scores 98

Component Loading Vectors 98

5.4 Estimation of Principal Components 99

Estimation of Principal Component Scores 99

5.5 Determining the Number of Principal Components 99

Method 1 100

Method 2 100

5.6 Caveats 107

5.7 PCA on the Correlation Matrix P 109

Principal Component Scores 110

Component Correlation Vectors 110

Sample Correlation Matrix 110

Determining the Number of Principal Components 110

5.8 Testing for Independence of the Original Variables 111

5.9 Structural Relationships 111

SASR PRINCOMP Procedure 112

5.10 Statistical Computing Packages 112

Principal Components Analysis Using Factor Analysis Programs 118

PCA with SPSS's FACTOR Procedure 124

Summary 142

Exercises 142

6．FACTOR ANALYSIS 147

6.1 Objectives of Factor Analysis 147

6.3 Some History of Factor Analysis 148

6.2 Caveats 148

6.4 The Factor Analysis Model 150

Assumptions 150

Matrix Form of the Factor Analysis Model 151

Definitions of Factor Analysis Terminology 151

6.5 Factor Analysis Equations 151

Nonuniqueness of the Factors 152

6.6 Solving the Factor Analysis Equations 153

6.7 Choosing the Appropriate Number of Factors 155

Objective Criteria 156

Subjective Criteria 156

6.8 Computer Solutions of the Factor Analysis Equations 157

Principal Factor Method on R 158

Principal Factor Method with Iteration 159

6.9 Rotating Factors 170

Examples（m=2） 171

Rotation Methods 172

The Varimax Rotation Method 173

6.10 Oblique Rotation Methods 174

6.11 Factor Scores 180

Bartlett's Method or the Weighted Least-Squares Method 181

Thompson's Method or the Regression Method 181

Ad Hoc Methods 181

Summary 212

Exercises 213

7．DISCRIMINANT ANALYSIS 217

7.1 Discrimination for Two Multivariate Normal Populations 217

A Posterior Probability Rule 218

A Mahalanobis Distance Rule 218

The Linear Discriminant Function Rule 218

A Likelihood Rule 218

Sample Discriminant Rules 219

Estimating Probabilities of Misclassification 220

Resubstitution Estimates 220

Estimates from Holdout Data 220

Cross-Validation Estimates 221

7.2 Cost Functions and Prior Probabilities（Two Populations） 229

7.3 A General Discriminant Rule（Two Populations） 231

A Cost Function 232

Prior Probabilities 232

Average Cost of Misclassification 232

A Bayes Rule 233

Classification Functions 233

Unequal Covariance Matrices 233

Tricking Computing Packages 234

7.4 Discriminant Rules（More than Two Populations） 235

Basic Discrimination 238

7.5 Variable Selection Procedures 245

Forward Selection Procedure 245

Backward Elimination Procedure 246

Stepwise Selection Procedure 246

Recommendations 247

Caveats 247

7.6 Canonical Discriminant Functions 255

The First Canonical Function 256

A Second Canonical Function 257

Determining the Dimensionality of the Canonical Space 260

Discriminant Analysis with Categorical Predictor Variables 273

7.7 Nearest Neighbor Discriminant Analysis 275

7.8 Classification Trees 283

Summary 283

Exercises 283

8.1 Logistic Regression Model 287

8.2 The Logit Transformation 287

8．LOGISTIC REGRESSION METHODS 287

Model Fitting 288

8.3 Variable Selection Methods 296

8.4 Logistic Discriminant Analysis（More Than Two Populations） 301

Logistic Regression Models 301

Model Fitting 302

Another SAS LOGISTIC Analysis 314

Exercises 316

Ruler Distance 319

9．CLUSTER ANALYSIS 319

9.1 Measures of Similarity and Dissimilarity 319

Standardized Ruler Distance 320

A Mahalanobis Distance 320

Dissimilarity Measures 320

9.2 Graphical Aids in Clustering 321

Scatter Plots 321

9.3 Clustering Methods 322

Other Methods 322

Andrews'Plots 322

Using Principal Components 322

Nonhierarchical Clustering Methods 323

Hierarchical Clustering 323

Nearest Neighbor Method 323

A Hierarchical Tree Diagram 325

Other Hierarchical Clustering Methods 326

Verification of Clustering Methods 327

How Many Clusters? 327

Comparisons of Clustering Methods 327

Beale's F-Type Statistic 328

A Pseudo Hotelling's T2 Test 329

The Cubic Clustering Criterion 329

Clustering Order 334

Estimating the Number of Clusters 339

Principal Components Plots 348

Clustering with SPSS 355

SAS's FASTCLUS Procedure 369

9.4 Multidimensional Scaling 385

Exercises 395

10．MEAN VECTORS AND VARIANCE-COVARIANCE MATRICES 397

10.1 Inference Procedures for Variance-Covariance Matrices 397

A Test for a Specific Variance-Covariance Matrix 398

A Test for Sphericity 400

A Test for Compound Symmetry 403

A Test for the Huynh-Feldt Conditions 405

A Test for Independence 406

A Test for Independence of Subsets of Variables 407

A Test for the Equality of Several Variance-Covariance Matrices 408

10.2 Inference Procedures for a Mean Vector 408

Hotelling's T2 Statistic 409

Hypothesis Test for μ 409

Confidence Region for μ 409

A More General Result 411

Special Case—A Test of Symmetry 412

Fitting a Line to Repeated Measures 418

A Test for Linear Trend 418

Multivariate Quality Control 419

10.3 Two Sample Procedures 420

Repeated Measures Experiments 420

10.4 Profile Analyses 431

10.5 Additional Two-Group Analyses 432

Paired Samples 432

Small Sample Sizes 433

Large Sample Sizes 433

Unequal Variance-Covariance Matrices 433

Summary 434

Exercises 434

11．MULTIVARIATE ANALYSIS OF VARIANCE 439

11.1 MANOVA 439

MANOVA Assumptions 440

Test Statistics 440

Test Comparisons 441

Why Do We Use MANOVAs? 441

A Conservative Approach to Multiple Comparisons 442

11.2 Dimensionality of the Alternative Hypothesis 455

11.3 Canonical Variates Analysis 456

The First Canonical Variate 456

The Second Canonical Variate 457

Other Canonical Variates 457

11.4 Confidence Regions for Canonical Variates 458

Summary 485

Exercises 485

12.1 Multiple Regression 489

12．PREDICTION MODELS AND MULTIVARIATE REGRESSION 489

12.2 Canonical Correlation Analysis 494

Two Sets of Variables 494

The First Canonical Correlation 495

The Second Canonical Correlation 495

Number of Canonical Correlations 496

Estimates 496

Hypothesis Tests on the Canonical Correlations 497

Interpreting Canonical Functions 508

Canonical Correlation Analysis with SPSS 511

12.3 Factor Analysis and Regression 515

Summary 522

Exercises 522

APPENDIX A:MATRIX RESULTS 525

A.1 Basic Definitions and Rules of Matrix Algebra 525

A.2 Quadratic Forms 527

A.3 Eigenvalues and Eigenvectors 528

A.5 Miscellaneous Results 529

A.4 Distances and Angles 529

APPENDIX B:WORK ATTITUDES SURVEY 531

B.1 Data File Structure 536

B.2 SPSS Data Entry Commands 538

B.3 SAS Data Entry Commands 543

APPENDIX C:FAMILY CONTROL STUDY 547

REFERENCES 555

Index 563