Applied Multivariate Analysis

Neil H. Timm

Springer Science & Business Media, Jun 21, 2007 - Mathematics - 695 pages

Univariate statistical analysis is concerned with techniques for the analysis of a single random variable. This book is about applied multivariate analysis. It was written to p- vide students and researchers with an introduction to statistical techniques for the ana- sis of continuous quantitative measurements on several random variables simultaneously. While quantitative measurements may be obtained from any population, the material in this text is primarily concerned with techniques useful for the analysis of continuous obser- tions from multivariate normal populations with linear structure. While several multivariate methods are extensions of univariate procedures, a unique feature of multivariate data an- ysis techniques is their ability to control experimental error at an exact nominal level and to provide information on the covariance structure of the data. These features tend to enhance statistical inference, making multivariate data analysis superior to univariate analysis. While in a previous edition of my textbook on multivariate analysis, I tried to precede a multivariate method with a corresponding univariate procedure when applicable, I have not taken this approach here. Instead, it is assumed that the reader has taken basic courses in multiple linear regression, analysis of variance, and experimental design. While students may be familiar with vector spaces and matrices, important results essential to multivariate analysis are reviewed in Chapter 2. I have avoided the use of calculus in this text.

Contents

• Introduction: 1
• 13 Scope of the Book: 3
• Vectors and Matrices: 7
• b Vector Spaces: 8
• c Vector Subspaces: 9
• 23 Bases Vector Norms and the Algebra of Vector Spaces: 12
• a Bases: 13
• c GramSchmidt Orthogonalization Process: 15
• d Orthogonal Spaces: 17
• e Vector Inequalities Vector Norms and Statistical Distance: 21
• 24 Basic Matrix Operations: 25
• a Equality Addition and Multiplication of Matrices: 26
• b Matrix Transposition: 28
• c Some Special Matrices: 29
• d Trace and the Euclidean Matrix Norm: 30
• e Kronecker and Hadamard Products: 32
• f Direct Sums: 35
• 25 Rank Inverse and Determinant: 41
• b Generalized Inverses: 47
• c Determinants: 50
• 26 Systems of Equations Transformations and Quadratic Forms: 55
• b Linear Transformations: 61
• c Projection Transformations: 63
• d Eigenvalues and Eigenvectors: 67
• e Matrix Norms: 71
• f Quadratic Forms and Extrema: 72
• g Generalized Projectors: 73
• 27 Limits and Asymptotics: 76
• Multivariate Distributions and the Linear Model: 79
• 33 The Multivariate Normal MVN Distribution: 84
• a Properties of the Multivariate Normal Distribution: 86
• b Estimating ft and S: 88
• c The Matrix Normal Distribution: 90
• 34 The ChiSquare and Wishart Distributions: 93
• b The Wishart Distribution: 96
• 35 Other Multivariate Distributions: 99
• c The Beta Distribution: 101
• d Multivariate t F and χ2 Distributions: 104
• 36 The General Linear Model: 106
• a Regression ANOVA and ANCOVA Models: 107
• b Multivariate Regression MANOVA and MANCOVA Models: 110
• c The Seemingly Unrelated Regression SUR Model: 114
• d The General MANOVA Model GMANOVA: 115
• 37 Evaluating Normality: 118
• 38 Tests of Covariance Matrices: 133
• c Testing for a Specific Covariance Matrix: 137
• d Testing for Compound Symmetry: 138
• e Tests of Sphericity: 139
• f Tests of Independence: 143
• g Tests for Linear Structure: 145
• 39 Tests of Location: 149
• b TwoSample Case: 156
• c TwoSample Case Nonnormality: 160
• e Profile Analysis Two Groups: 165
• f Profile Analysis₁ ₂: 175
• 310 Univariate Profile Analysis: 181
• a Univariate OneGroup Profile Analysis: 182
• Multivariate Regression Models: 185
• 42 Multivariate Regression: 186
• b Multivariate Regression Estimation and Testing Hypotheses: 187
• c Multivariate Influence Measures: 193
• d Measures of Association Variable Selection and LackofFit Tests: 197
• e Simultaneous Confidence Sets for a New Observation ynew and the Elements ofB: 204
• Mean Squared Error of Prediction in Multivariate Regression: 206
• 43 Multivariate Regression Example: 212
• 44 OneWay MANOVA and MANCOVA: 218
• b OneWay MANCOVA: 225
• c Simultaneous Test Procedures STPfor OneWay MANOVA MANCOVA: 230
• 45 OneWay MANOVAMANCOVA Examples: 234
• b MANCOVA Example 452: 239
• 46 MANOVAMANCOVA with Unequal i or Nonnormal Data: 245
• 47 OneWay MANOVA with Unequal i Example: 246
• b Additive TwoWay MAN OVA: 252
• c TwoWay MANCOVA: 256
• 49 TwoWay MANOVAMANCOVA Example: 257
• b TwoWay MANCOVA Example 492: 261
• 410 Nonorthogonal TwoWay MANOVA Designs: 264
• a Nonorthogonal TwoWay MANOVA Designs with andWithout Empty Cells and Interaction: 265
• b Additive TwoWay MANOVA Designs With Empty Cells: 268
• 412 Higher Ordered Fixed Effect Nested and Other Designs: 273
• 413 Complex Design Examples: 276
• b Latin Square Design Example 4132: 279
• 414 Repeated Measurement Designs: 282
• b Extended Linear Hypotheses: 286
• 415 Repeated Measurements and Extended Linear Hypotheses Example: 294
• b Extended Linear Hypotheses Example 4152: 298
• 416 Robustness and Power Analysis for MR Models: 301
• 417 Power CalculationsPowersas: 304
• 418 Testing for Mean Differences with Unequal Covariance Matrices: 307
• Seemingly Unrelated Regression Models: 310
• 52 The SUR Model: 312
• b Prediction: 314
• 53 Seeming Unrelated Regression Example: 316
• 54 The CGMANOVA Model: 318
• 55 CGMANOVA Example: 319
• 56 The GMANOVA Model: 320
• b Estimation and Hypothesis Testing: 321
• c Test of Fit: 324
• e GMANOVA vs SUR: 326
• 57 GMANOVA Example: 327
• a One Group Design Example 571: 328
• b Two Group Design Example 572: 330
• 58 Tests of Nonadditivity: 333
• 59 Testing for Nonadditivity Example: 335
• 511 Sum of Profile Designs: 338
• 512 The Multivariate SUR MSUR Model: 339
• 513 Sum of Profile Example: 341
• 514 Testing Model Specification in SUR Models: 344
• 515 Miscellanea: 348
• Multivariate Random and Mixed Models: 351
• 62 Random Coefficient Regression Models: 352
• b Estimating the Parameters: 353
• c Hypothesis Testing: 355
• 63 Univariate General Linear Mixed Models: 357
• b Covariance Structures and Model Fit: 359
• c Model Checking: 361
• d Balanced Variance Component Experimental Design Models: 366
• e Multilevel Hierarchical Models: 367
• f Prediction: 368
• 64 Mixed Model Examples: 369
• a Random Coefficient Regression Example 641: 371
• b Generalized Randomized Block Design Example 642: 376
• c Repeated Measurements Example 643: 380
• d HLM Model Example 644: 381
• 65 Mixed Multivariate Models: 385
• a Model Specification: 386
• b Hypothesis Testing: 388
• c Evaluating Expected Mean Square: 391
• d Estimating the Mean: 392
• 66 Balanced Mixed Multivariate Models Examples: 394
• a Twoway Mixed MANOVA: 395
• 67 Double Multivariate Model DMM: 400
• 68 Double Multivariate Model Examples: 403
• a Double Multivariate MAN OVA Example 681: 404
• b SplitPlot Design Example 682: 407
• 69 Multivariate Hierarchical Linear Models: 415
• 610 Tests of Means with Unequal Covariance Matrices: 417
• Discriminant and Classification Analysis: 418
• 72 Two Group Discrimination and Classification: 420
• a Fishers Linear Discriminant Function: 421
• b Testing Discriminant Function Coefficients: 422
• c Classification Rules: 424
• d Evaluating Classification Rules: 427
• 73 Two Group Discriminant Analysis Example: 429
• b Brain Size Example 732: 432
• 74 Multiple Group Discrimination and Classification: 434
• b Testing Discriminant Functions for Significance: 435
• c Variable Selection: 437
• d Classification Rules: 438
• e Logistic Discrimination and Other Topics: 439
• 75 Multiple Group Discriminant Analysis Example: 440
• Principal Component Canonical Correlation and Exploratory Factor Analysis: 445
• a Population Model for PCA: 446
• b Number of Components and Component Structure: 449
• c Principal Components with Covariates: 453
• d Sample PCA: 455
• e Plotting Components: 458
• 83 Principal Component Analysis Examples: 460
• b Semantic Differential Ratings Example 832: 461
• c Performance Assessment Program Example 833: 465
• 84 Statistical Tests in Principal Component Analysis: 468
• b Tests Using a Correlation Matrix: 472
• 85 Regression on Principal Components: 474
• a GMANOVA Model: 475
• 86 Multivariate Regression on Principal Components Example: 476
• 87 Canonical Correlation Analysis: 477
• b Sample CCA: 482
• c Tests of Significance: 483
• d Association and Redundancy: 485
• e Partial Part and Bipartial Canonical Correlation: 487
• f Predictive Validity in Multivariate Regression using CCA: 490
• g Variable Selection and Generalized Constrained CCA: 491
• a Rohwer CCA Example 881: 492
• b Partial and Part CCA Example 882: 494
• 89 Exploratory Factor Analysis: 496
• a Population Model for EFA: 497
• b Estimating Model Parameters: 502
• c Determining Model Fit: 506
• d Factor Rotation: 507
• e Estimating Factor Scores: 509
• f Additional Comments: 510
• 810 Exploratory Factor Analysis Examples: 511
• b Di Vesta and Walls Example 8102: 512
• Cluster Analysis and Multidimensional Scaling: 515
• 92 Proximity Measures: 516
• b Similarity Measures: 519
• c Clustering Variables: 522
• a Agglomerative Hierarchical Clustering Methods: 523
• b Nonhierarchical Clustering Methods: 530
• c Number of Clusters: 531
• d Additional Comments: 533
• a Protein Consumption Example 941: 534
• b Nonhierarchical Method Example 942: 536
• c Teacher Perception Example 943: 538
• d Cedar Project Example 944: 541
• a Classical Metric Scaling: 542
• b Nonmetric Scaling: 544
• c Additional Comments: 547
• 96 Multidimensional Scaling Examples: 548
• a Classical Metric Scaling Example 961: 549
• b Teacher Perception Example 962: 550
• c Nation Example 963: 553
• Structural Equation Models: 556
• 102 Path Diagrams Basic Notation and the General Approach: 558
• 103 Confirmatory Factor Analysis: 567
• 104 Confirmatory Factor Analysis Examples: 575
• b Performance Assessment 5Factor Model Example 1042: 578
• 105 Path Analysis: 580
• 106 Path Analysis Examples: 586
• b Nonrecursive Model Example 1062: 590
• 107 Structural Equations with Manifest and Latent Variables: 594
• 108 Structural Equations with Manifest and Latent Variables Example: 595
• 109 Longitudinal Analysis with Latent Variables: 600
• 1010 Exogeniety in Structural Equation Models: 604
• Appendix A: 609
• References: 625
• Author Index: 667
• Subject Index: 675

About the author

"This book is more than an up-to-date textbook on multivariate analysis. It could enable SAS users to take full and informed advantage of the many options offered in the SAS procedures. For non-SAS users, the clear statement of the models should enable them to fit and interpret them with other software."

ISI Short Book Reviews, Vol. 23/2, August 2003

Bibliographic information

• Title: Applied Multivariate Analysis; Springer Texts in Statistics
• Author: Neil H. Timm
• Edition: illustrated
• Publisher: Springer Science & Business Media, 2007
• ISBN: 0387227717, 9780387227719
• Length: 695 pages
• Subjects: Business & Economics / Statistics; Mathematics / Calculus; Mathematics / Mathematical Analysis; Mathematics / Probability & Statistics / General; Mathematics / Probability & Statistics / Stochastic Processes; Mathematics / Vector Analysis; Social Science / Research; Social Science / Statistics