Methods of Multivariate Analysis
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Acknowledgments xxi
1 Introduction 1
1.1 Why Multivariate Analysis? 1
1.2 Prerequisites 3
1.3 Objectives 3
1.4 Basic Types of Data And Analysis 4
2 Matrix Algebra 7
2.1 Introduction 7
2.2 Notation and Basic Definitions 8
2.3 Operations 11
2.4 Partitioned Matrices 22
2.5 Rank 23
2.6 Inverse 25
2.7 Positive Definite Matrices 26
2.8 Determinants 28
2.9 Trace 31
2.10 Orthogonal Vectors and Matrices 31
2.11 Eigenvalues and Eigenvectors 32
2.12 Kronecker and VEC Notation 37
Problems 39
3 Characterizing and Displaying Multivariate Data 47
3.1 Mean and Variance of a Univariate Random Variable 47
3.2 Covariance and Correlation Of Bivariate Random Variables 49
3.3 Scatter Plots of Bivariate Samples 55
3.4 Graphical Displays for Multivariate Samples 56
3.5 Dynamic Graphics 58
3.6 Mean Vectors 63
3.7 Covariance Matrices 66
3.8 Correlation Matrices 69
3.9 Mean Vectors and Covariance Matrices for Subsets of Variables 71
3.9.1 Two Subsets 71
3.9.2 Three or More Subsets 73
3.10 Linear Combinations of Variables 75
3.10.1 Sample Properties 75
3.10.2 Population Properties 81
3.11 Measures of Overall Variability 81
3.12 Estimation of Missing Values 82
3.13 Distance Between Vectors 84
Problems 85
4 The Multivariate Normal Distribution 91
4.1 Multivariate Normal Density Function 91
4.2 Properties of Multivariate Normal Random Variables 94
4.3 Estimation in the Multivariate Normal 99
4.4 Assessing Multivariate Normality 101
4.5 Transformations to Normality 108
4.6 Outliers 111
Problems 117
5 Tests on One or Two Mean Vectors 125
5.1 Multivariate Versus Univariate Tests 125
5.2 Tests on µ With ??Known 126
5.3 Tests on µ When ??is Unknown 130
5.4 Comparing two Mean Vectors 134
5.5 Tests on Individual Variables Conditional on Rejection of H0 by the T2-test
139
5.6 Computation of T2 143
5.7 Paired Observations Test 145
5.8 Test for Additional Information 149
5.9 Profile Analysis 152
Profile Analysis 154
Problems 161
6 Multivariate Analysis of Variance 169
6.1 One-way Models 169
6.2 Comparison of the Four Manova Test Statistics 189
6.3 Contrasts 191
6.4 Tests on Individual Variables Following Rejection of H0 by the Overall Manova Test 195
6.5 Two-Way Classification 198
6.6 Other Models 207
6.7 Checking on the Assumptions 210
6.8 Profile Analysis 211
6.9 Repeated Measures Designs 215
6.10 Growth Curves 232
6.11 Tests on a Subvector 241
Problems 244
7 Tests on Covariance Matrices 259
7.1 Introduction 259
7.2 Testing a Specified Pattern for ¿ 259
7.3 Tests Comparing Covariance Matrices 265
7.4 Tests of Independence 269
Problems 276
8 Discriminant Analysis: Description of Group Separation 281
8.1 Introduction 281
8.2 The Discriminant Function for two Groups 282
8.3 Relationship Between two-group Discriminant Analysis and Multiple Regression 286
8.4 Discriminant Analysis for Several Groups 288
8.5 Standardized Discriminant Functions 292
8.6 Tests of Significance 294
8.7 Interpretation of Discriminant Functions 298
8.8 Scatter Plots 301
8.9 Stepwise Selection of Variables 303
Problems 306
9 Classification Analysis: Allocation of Observations to Groups309
9.1 Introduction 309
9.2 Classification into two Groups 310
9.3 Classification into Several Groups 314
9.4 Estimating Misclassification Rates 318
9.5 Improved Estimates of Error Rates 320
9.6 Subset Selection 322
9.7 Nonparametric Procedures 326
Problems 336
10 Multivariate Regression 339
10.1 Introduction 339
10.2 Multiple Regression: Fixed X's 340
10.3 Multiple Regression: Random X's 354
10.4 Multivariate Multiple Regression: Estimation 354
10.5 Multivariate Multiple Regression: Hypothesis Tests 364
10.6 Multivariate Multiple Regression: Prediction 370
10.7 Measures of Association Between the Y's and the X's 372
10.8 Subset Selection 374
10.9 Multivariate Regression: Random X's 380
Problems 381
11 Canonical Correlation 385
11.1 Introduction 385
11.2 Canonical Correlations and Canonical Variates 385
11.3 Properties of Canonical Correlations 390
11.4 Tests of Significance 391
11.5 Interpretation 395
11.6 Relationships of Canonical Correlation Analysis to Other Multivariate
Problems 402
12 Principal Component Analysis 405
12.1 Introduction 405
12.2 Geometric and Algebraic Bases of Principal Components 406
12.3 Principal Components and Perpendicular Regression 412
12.4 Plotting of Principal Components 414
12.5 Principal Components from the Correlation Matrix 419
12.6 Deciding How Many Components to Retain 423
12.7 Information in the Last Few Principal Components 427
12.8 Interpretation of Principal Components 427
12.9 Selection of Variables 430
Problems 432
13 Exploratory Factor Analysis 435
13.1 Introduction 435
13.2 Orthogonal Factor Model 437
13.3 Estimation of Loadings and Communalities 442
13.4 Choosing the Number of Factors, m 453
13.5 Rotation 457
13.6 Factor Scores 466
13.7 Validity of the Factor Analysis Model 470
13.8 Relationship of Factor Analysis to Principal Component Analysis 475
Problems 476
14 Confirmatory Factor Analysis 479
14.1 Introduction 479
14.2 Model Specification and Identification 480
14.3 Parameter Estimation and Model Assessment 487
14.4 Inference for Model Parameters 492
14.5 Factor Scores 495
Problems 496
15 Cluster Analysis 501
15.1 Introduction 501
15.2 Measures of Similarity or Dissimilarity 502
15.3 Hierarchical Clustering 505
15.4 Nonhierarchical Methods 531
15.5 Choosing the Number of Clusters 544
15.6 Cluster Validity 546
15.7 Clustering Variables 547
Problems 548
16 Graphical Procedures 555
16.1 Multidimensional Scaling 555
16.2 Correspondence Analysis 565
16.3 Biplots 580
Problems 588
Appendix A: Tables 597
Appendix B: Answers and Hints to Problems 637
Appendix C: Data Sets and SAS Files 727
References 729
Index 747
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