Multivariate Statistical Inference and Applications

transcript

cover next page >

title :author :

publisher :isbn10 | asin :print isbn13 :

ebook isbn13 :language :

subject publication date :

lcc :ddc :

subject :

cover next page >

file:///G|/cover.html7/11/2009 2:11:55 PM

page_i

< previous page page_i next page >Page i

Multivariate Statistical Inference and ApplicationsStart of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation

< previous page page_i next page >

file:///G|/page_i.html7/11/2009 2:11:56 PM

page_ii

< previous page page_ii next page >Page ii

WILEY SERIES IN PROBABILITY AND STATISTICS TEXTS AND REFERENCES SECTION

Established by WALTER A. SHEWHART and SAMUEL S. WILKS

Editors: Vic Barnett, Ralph A. Bradley, Noel A. C. Cressie, Nicholas I. Fisher, lain M. Johnstone, J. B. Kadane, David G. Kendall, David W. Scott,

Bernard W. Silverman, Adrian F. M. Smith, JozefL. Teugels, Geoffrey S. Watson; J. Stuart Hunter, Emeritus

A complete list of the titles in this series appears at the end of this volume.Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation

< previous page page_ii next page >

file:///G|/page_ii.html7/11/2009 2:11:56 PM

page_iii

< previous page page_iii next page >Page iii

Multivariate Statistical Inference and Applications

Alvin C. Rencher

Department of Statistics Brigham Young University

Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation

< previous page page_iii next page >

file:///G|/page_iii.html7/11/2009 2:11:57 PM

page_iv

< previous page page_iv next page >Page iv

Disclaimer:

This netLibrary eBook does not include the ancillary media that was packaged with the original printed version of the book.

This text is printed on acid-free paper.

Published simultaneously in Canada.

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (508) 750-8400, fax (508) 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ@WILEY.COM.

Library of Congress Cataloging-in-Publication Data:

Rencher, Alvin C., 1934 Multivariate statistical inference and applications / Alvin C. Rencher. p. cm. (Wiley series in probability and statistics. Texts and references section) "A Wiley-Interscience publication." Includes bibliographical references and index. ISBN 0-471-57151-2 (cloth : alk. paper) 1. Multivariate analysis. I. Title. II. Series. QA278.R453 1998 519.5'35dc21 97-5255 CIP

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation

< previous page page_iv next page >

file:///G|/page_iv.html7/11/2009 2:11:57 PM

page_v

< previous page page_v next page >Page v

Contents

1. Some Properties of Random Vectors and Matrices 1

1.1. Introduction1

1.2. Univariate and Bivariate Random Variables2

1.2.1. Univariate Random Variables2

1.2.2. Bivariate Random Variables4

1.3. Mean Vectors and Covariance Matrices for Random Vectors6

1.4. Correlation Matrices11

1.5. Partitioned Mean Vectors and Covariance Matrices12

1.6. Linear Functions of Random Variables14

1.6.1. Sample Means, Variances, and Covariances14

1.6.2. Population Means, Variances, and Covariances19

1.7. Measuring Intercorrelation20

1.8. Mahalanobis Distance22

1.9. Missing Data23

1.10. Robust Estimators of µ and Σ27

2. The Multivariate Normal Distribution 37

file:///G|/page_v.html (1 of 2)7/11/2009 2:11:57 PM

page_v

2.1. Univariate and Multivariate Normal Density Functions37

2.1.1. Univariate Normal37

2.1.2. Multivariate Normal38

2.1.3. Constant Density Ellipsoids40

2.1.4. Generating Multivariate Normal Data42

2.1.5. Moments42

2.2. Properties of Multivariate Normal Random Vectors43

2.3. Estimation of Parameters in the Multivariate Normal Distribution49

< previous page page_v next page >

file:///G|/page_v.html (2 of 2)7/11/2009 2:11:57 PM

page_vi

< previous page page_vi next page >Page vi

2.3.1. Maximum Likelihood Method49

2.3.2. Properties of y *and S52

2.3.3. Wishart Distribution53

2.4. Additional Topics56

3. Hotelling's T2-Tests 60

3.1. Introduction60

3.2. Test for H0: µ = µ0 with Σ Known60

3.3. Hotelling's T2-test for Ho: µ = µ0 with Σ Unknown61

3.3.1. Univariate t-Test for H0: µ = µ0 with σ2 Unknown61

3.3.2. Likelihood Ratio Method of Test Construction62

3.3.3. One-Sample T2-Test65

3.3.4. Formal Definition of T2 and Relationship to F66

3.3.5. Effect on T2 of Adding a Variable67

3.3.6. Properties of the T2-Test70

3.3.7. Likelihood Ratio Test71

3.3.8. Union-Intersection Test72

3.4. Confidence Intervals and Tests for Linear Functions of µ74

file:///G|/page_vi.html (1 of 2)7/11/2009 2:11:58 PM

page_vi

3.4.1. Confidence Region for µ74

3.4.2. Confidence Interval for a Single Linear Combination a'µ74

3.4.3. Simultaneous Confidence Intervals for µj and a'µ74

3.4.4. Bonferroni Confidence Intervals for µj and a'µ77

3.4.5. Tests for Ho: a'µ = a'µ0and H0: µj= µ0j79

3.4.6. Tests for H0:Cµ = 083

3.5. Tests of H0: µ1 = µ2 Assuming Σ1= Σ285

3.5.1. Review of Univariate Likelihood Ratio Test for H0: µ1 = µ2 When σ2/1 = σ2/285

3.5.2. Test for H0: µ1= µ2 When Σ1 = Σ287

3.5.3. Effect on T2 of Adding a Variable87

3.5.4. Properties of the Two-Sample T2-Statistic91

3.5.5. Likelihood Ratio and Union-Intersection Tests91

3.6. Confidence Intervals and Tests for Linear Functions of Two Mean Vectors92

3.6.1. Confidence Region for µ1 µ293

3.6.2. Simultaneous Confidence Intervals for a'(µ1 µ2) and µ1j µ2j93

3.6.3. Bonferroni Confidence Intervals for a'(µ1 µ2) and µ1j - µ2j94

< previous page page_vi next page >

file:///G|/page_vi.html (2 of 2)7/11/2009 2:11:58 PM

page_vii

< previous page page_vii next page >Page vii

3.6.4. Tests for H0: a'(µ1 µ2) = a'δ0and H0j: µ1j µ2j = 094

3.6.5. Test for H0: C(µ1 µ2) = 095

3.7. Robustness of the T2-test96

3.7.1. Robustness to Σ1≠ Σ296

3.7.2. Robustness to Nonnormality96

3.8. Paired Observation Test97

3.9. Testing H0: µ1 = µ2When Σ1≠Σ299

3.9.1. Univariate Case99

3.9.2. Multivariate Case100

3.10. Power and Sample Size104

3.11. Tests on a Subvector108

3.11.1. Two-Sample Case108

3.11.2. Step-Down Test110

3.11.3. Selection of Variables111

3.11.4. One-Sample Case112

3.12. Nonnormal Approaches to Hypothesis Testing112

3.12.1. Elliptically Contoured Distributions112

file:///G|/page_vii.html (1 of 3)7/11/2009 2:11:58 PM

page_vii

3.12.2 NonparametricTests113

3.12.3 Robust Versions of T2114

3.13. Application of T2 in Multivariate Quality Control114

4. Multivariate Analysis of Variance 121

4.1. One-Way Classification121

4.1.1. Model for One-Way Multivariate Analysis of Variance121

4.1.2 Wilks' Likelihood Ratio Test122

4.1.3. Roy's Union-Intersection Test127

4.1.4. The Pillai and Lawley-Hotelling Test Statistics130

4.1.5. Summary of the Four Test Statistics131

4.1.6. Effect of an Additional Variable on Wilks' Λ132

4.1.7. Tests on Individual Variables134

4.2. Power and Robustness Comparisons for the Four MANOVA Test Statistics135

4.3. Tests for Equality of Covariance Matrices138

4.4. Power and Sample Size for the Four MANOVA Tests140

4.5. Contrasts Among Mean Vectors142

4.5.1. Univariate Contrasts142

4.5.2. Multivariate Contrasts145

page_vii

< previous page page_vii next page >

page_viii

< previous page page_viii next page >Page viii

4.6. Two-Way Multivariate Analysis of Variance148

4.7. Higher Order Models151

4.8. Unbalanced Data152

4.8.1. Introduction152

4.8.2. Univariate One-Way Model153

4.8.3. Multivariate One-Way Model155

4.8.4. Univariate Two-Way Model160

4.8.5. Multivariate Two-Way Model168

4.9. Tests on a Subvector174

4.9.1. Testing a Single Subvector174

4.9.2. Step-Down Test177

4.9.3. Stepwise Selection of Variables177

4.10. Multivariate Analysis of Covariance178

4.10.2. Univariate Analysis of Covariance: One-Way Model with One Covariate179

4.10.3. Univariate Analysis of Covariance: Two-Way Model with One Covariate183

4.10.4. Additional Topics in Univariate Analysis of Covariance186

file:///G|/page_viii.html (1 of 2)7/11/2009 2:11:58 PM

page_viii

4.10.5. Multivariate Analysis of Covariance187

4.11. Alternative Approaches to Testing H0: µ1 = µ2 = · · · = µk194

5. Discriminant Functions for Descriptive Group Separation 201

5.1. Introduction201

5.2. Two Groups201

5.3. Several Groups202

5.3.1. Discriminant Functions202

5.3.2. Assumptions205

5.4. Standardized Coefficients206

5.5. Tests of Hypotheses207

5.5.1. Two Groups207

5.5.2. Several Groups209

5.6. Discriminant Analysis for Higher Order Designs210

5.7. Interpretation of Discriminant Functions210

5.7.1. Standardized Coefficients and Partial F-Values211

5.7.2. Correlations between Variables and Discriminant Functions211

< previous page page_viii next page >

file:///G|/page_viii.html (2 of 2)7/11/2009 2:11:58 PM

page_ix

< previous page page_ix next page >Page ix

5.7.3. Other Approaches215

5.8. Confidence Intervals216

5.9. Subset Selection216

5.9.1. Discriminant Function Approach to Selection217

5.9.2. Stepwise Selection217

5.9.3. All Possible Subsets218

5.9.4. Selection in Higher Order Designs218

5.10. Bias in Subset Selection219

5.11. Other Estimators of Discriminant Functions221

5.11.1. Ridge Discriminant Analysis and Related Techniques222

5.11.2. Robust Discriminant Analysis223

6. Classification of Observations into Groups 230

6.2. Two Groups230

6.2.1. Equal Population Covariance Matrices230

6.2.2. Unequal Population Covariance Matrices232

6.2.3. Unequal Costs of Misclassification233

file:///G|/page_ix.html (1 of 3)7/11/2009 2:11:58 PM

page_ix

6.2.4. Posterior Probability Approach234

6.2.5. Robustness to Departures from the Assumptions234

6.2.6. Robust Procedures235

6.3. Several Groups236

6.3.1. Equal Population Covariance Matrices236

6.3.2. Unequal Population Covariance Matrices237

6.3.3. Use of Linear Discriminant Functions for Classification239

6.4. Estimation of Error Rates240

6.5. Correcting for Bias in the Apparent Error Rate244

6.5.1. Partitioning the Sample244

6.5.2. Holdout Method244

6.5.3. Bootstrap Estimator245

6.5.4. Comparison of Error Estimators245

6.6. Subset Selection247

6.6.1. Selection Based on Separation of Groups247

6.6.2. Selection Based on Allocation249

6.6.3. Selection in the Heteroscedastic Case250

6.7. Bias in Stepwise Classification Analysis251

page_ix

6.8. Logistic and Probit Classification254

< previous page page_ix next page >

page_x

< previous page page_x next page >Page x

6.8.1. The Logistic Model for Two Groups with Σl = Σ2254

6.8.2. Comparison of Logistic Classification with Linear Classification Functions256

6.8.3. Quadratic Logistic Functions When Σl≠ Σ2258

6.8.4. Logistic Classification for Several Groups258

6.8.5. Additional Topics in Logistic Classification259

6.8.6. Probit Classification259

6.9. Additional Topics in Classification261

7. Multivariate Regression 266

7.2. Multiple Regression: Fixed x's267

7.2.1. Least Squares Estimators and Properties267

7.2.2. An Estimator for σ2268

7.2.3. The Model in Centered Form269

7.2.4. Hypothesis Tests and Confidence Intervals271

7.2.5. R2 in Fixed-x Regression275

7.2.6. Model Validation275

7.3. Multiple Regression: Random x's276

file:///G|/page_x.html (1 of 2)7/11/2009 2:11:59 PM

page_x

7.3.1. Model for Random x's276

7.3.2. Estimation of β0, βl, and σ2277

7.3.3. R2 in Random-x Regression278

7.3.4. Tests and Confidence Intervals279

7.4. Estimation in the Multivariate Multiple Regression Model: Fixed x's279

7.4.1. The Multivariate Model279

7.4.2. Least Squares Estimator for B281

7.4.3. Properties of B *283

7.4.4. An Estimator for Σ285

7.4.5. Normal Model for the yi's286

7.4.6. The Multivariate Model in Centered Form288

7.4.7. Measures of Multivariate Association289

7.5. Hypothesis Tests in the Multivariate Multiple Regression Model: Fixed x's289

7.5.1. Test for Significance of Regression289

7.5.2. Test on a Subset of the Rows of B293

7.5.3. General Linear Hypotheses CB = O and CBM = O295

7.5.4. Tests and Confidence Intervals for a Single bjkand a Bilinear Function a'Bb297

< previous page page_x next page >

file:///G|/page_x.html (2 of 2)7/11/2009 2:11:59 PM

page_xi

< previous page page_xi next page >Page xi

7.5.5. Simultaneous Tests and Confidence Intervals for the βjk's and Bilinear Functions a'Bb298

7.5.6. Tests in the Presence of Missing Data299

7.6. Multivariate Model Validation: Fixed x's300

7.6.1. Lack-of-Fit Tests300

7.6.2. Residuals300

7.6.3. Influence and Outliers301

7.6.4. Measurement Errors303

7.7. Multivariate Regression: Random x's303

7.7.1. Multivariate Normal Model for Random x's303

7.7.2. Estimation of β0, B1, and Σ304

7.7.3. Tests and Confidence Intervals in the Multivariate Random-x Case305

7.8.1. Correlated Response Methods305

7.8.2. Categorical Data306

7.8.3. Subset Selection307

7.8.4. Other Topics307

8. Canonical Correlation 312

file:///G|/page_xi.html (1 of 2)7/11/2009 2:11:59 PM

page_xi

8.2. Canonical Correlations and Canonical Variates312

8.3. Properties of Canonical Correlations and Variates318

8.3.1. Properties of Canonical Correlations318

8.3.2. Properties of Canonical Variates320

8.4. Tests of Significance for Canonical Correlations320

8.4.1. Tests of Independence of y and x320

8.4.2. Test of Dimension of Relationship between the y's and the x's324

8.5. Validation326

8.6. Interpretation of Canonical Variates328

8.6.1. Standardized Coefficients328

8.6.2. Rotation of Canonical Variate Coefficients328

8.6.3. Correlations between Variables and Canonical Variates329

8.7. Redundancy Analysis331

< previous page page_xi next page >

file:///G|/page_xi.html (2 of 2)7/11/2009 2:11:59 PM

page_xii

< previous page page_xii next page >Page xii

9. Principal Component Analysis 337

9.2. Definition and Properties of Principal Components338

9.2.1. Maximum Variance Property338

9.2.2. Principal Components as Projections340

9.2.3. Properties of Principal Components341

9.3. Principal Components as a Rotation of Axes343

9.4. Principal Components from the Correlation Matrix344

9.5. Methods for Discarding Components347

9.5.1. Percent of Variance347

9.5.2. Average Eigenvalue348

9.5.3. Scree Graph348

9.5.4. Significance Tests349

9.5.5. Other Methods352

9.6. Information in the Last Few Principal Components352

9.7. Interpretation of Principal Components353

9.7.1. Special Patterns in S or R353

file:///G|/page_xii.html (1 of 3)7/11/2009 2:11:59 PM

page_xii

9.7.2. Testing H0: Σ = σ2 [(1 ρ)I + ρJ] and Pρ = (1 ρ)I + ρJ357

9.7.3. Additional Rotation359

9.7.4. Correlations between Variables and Principal Components361

9.8. Relationship Between Principal Components and Regression363

9.8.1. Principal Component Regression364

9.8.2. Latent Root Regression370

9.9. Principal Component Analysis with Grouped Data371

10. Factor Analysis 377

10.2. Basic Factor Model377

10.2.1. Model and Assumptions377

10.2.2. Scale Invariance of the Model379

10.2.3. Rotation of Factor Loadings in the Model380

10.3. Estimation of Loadings and Communalities381

10.3.1. Principal Component Method381

10.3.2. Principal Factor Method383

10.3.3. Iterated Principal Factor Method384

page_xii

< previous page page_xii next page >

page_xiii

< previous page page_xiii next page >Page xiii

10.3.4. Maximum Likelihood Method384

10.3.5. Other Methods385

10.3.6. Comparison of Methods385

10.4. Determining the Number of Factors, m386

10.5. Rotation of Factor Loadings387

10.5.2. Orthogonal Rotation388

10.5.3. Oblique Rotations389

10.5.4. Interpretation of the Factors390

10.6. Factor Scores391

10.7. Applicability of the Factor Analysis Model392

10.8. Factor Analysis and Grouped Data393

Appendix A. Review of Matrix Algebra 399

A.1. Introduction399

A. 1.1. Basic Definitions399

A. 1.2. Matrices with Special Patterns400

file:///G|/page_xiii.html (1 of 2)7/11/2009 2:12:00 PM

page_xiii

A.2. Properties of Matrix Addition and Multiplication401

A.3. Partitioned Matrices404

A.4. Rank of Matrices406

A.5. Inverse Matrices407

A.6. Positive Definite and Positive Semidefinite Matrices408

A.7. Determinants409

A.8. Trace of a Matrix410

A.9. Orthogonal Vectors and Matrices410

A.10. Eigenvalues and Eigenvectors411

A. 11. Eigenstructure of Symmetric and Positive Definite Matrices412

A.12. Idempotent Matrices414

A. 13. Differentiation414

Appendix B. Tables 417

Appendix C. Answers and Hints to Selected Problems 449

Appendix D. About the Diskette 505

Bibliography 507

Index 549 Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation

< previous page page_xiii next page >

file:///G|/page_xiii.html (2 of 2)7/11/2009 2:12:00 PM

page_xv

< previous page page_xv next page >Document

< previous page page_xv next page >

file:///G|/page_xv.html7/11/2009 2:12:00 PM

page_xvi

< previous page page_xvi next page >Document

< previous page page_xvi next page >

file:///G|/page_xvi.html7/11/2009 2:12:00 PM

page_xvii

< previous page page_xvii next page >Document

< previous page page_xvii next page >

file:///G|/page_xvii.html7/11/2009 2:12:00 PM

page_xviii

< previous page page_xviii next page >Document

< previous page page_xviii next page >

file:///G|/page_xviii.html7/11/2009 2:12:00 PM

page_xix

< previous page page_xix next page >Document

< previous page page_xix next page >

file:///G|/page_xix.html7/11/2009 2:12:00 PM

page_xx

< previous page page_xx next page >Document

< previous page page_xx next page >

file:///G|/page_xx.html7/11/2009 2:12:00 PM

page_1

< previous page page_1 next page >Document

< previous page page_1 next page >

file:///G|/page_1.html7/11/2009 2:12:00 PM

page_2

page_3

page_4

page_5

page_6

page_7

page_8

page_9

page_10

page_11

page_12

page_13

page_14

page_15

page_16

page_17

page_18

page_19

page_20

page_21

page_22

page_23

page_24

page_25

page_26

page_27

page_28

page_29

page_30

page_31

page_32

page_33

page_34

page_35

page_36

page_37

page_38

page_39

page_40

page_41

page_42

page_43

page_44

page_45

page_46

page_47

page_48

page_49

page_50

page_51

page_52

page_53

page_54

page_55

page_56

page_57

page_58

page_59

page_60

page_61

page_62

page_63

page_64

page_65

page_66

page_67

page_68

page_69

page_70

page_71

page_72

page_73

page_74

page_75

page_76

page_77

page_78

page_79

page_80

page_81

page_82

page_83

page_84

page_85

page_86

page_87

page_88

page_89

page_90

page_91

page_92

page_93

page_94

page_95

page_96

page_97

page_98

page_99

page_100

page_101

page_102

page_103

page_104

page_105

page_106

page_107

page_108

page_109

page_110

page_111

page_112

page_113

page_114

page_115

page_116

page_117

page_118

page_119

page_120

page_121

page_122

page_123

page_124

page_125

page_126

page_127

page_128

page_129

page_130

page_131

page_132

page_133

page_134

page_135

page_136

page_137

page_138

page_139

page_140

page_141

page_142

page_143

page_144

page_145

page_146

page_147

page_148

page_149

page_150

page_151

page_152

page_153

page_154

page_155

page_156

page_157

page_158

page_159

page_160

page_161

page_162

page_163

page_164

page_165

page_166

page_167

page_168

page_169

page_170

page_171

page_172

page_173

page_174

page_175

page_176

page_177

page_178

page_179

page_180

page_181

page_182

page_183

page_184

page_185

page_186

page_187

page_188

page_189

page_190

page_191

page_192

page_193

page_194

page_195

page_196

page_197

page_198

page_199

page_200

page_201

page_202

page_203

page_204

page_205

page_206

page_207

page_208

page_209

page_210

page_211

page_212

page_213

page_214

page_215

page_216

page_217

page_218

page_219

page_220

page_221

page_222

page_223

page_224

page_225

page_226

page_227

page_228

page_229

page_230

page_231

page_232

page_233

page_234

page_235

page_236

page_237

page_238

page_239

page_240

page_241

page_242

page_243

page_244

page_245

page_246

page_247

page_248

page_249

page_250

page_251

page_252

page_253

page_254

page_255

page_256

page_257

page_258

page_259

page_260

page_261

page_262

page_263

page_264

page_265

page_266

page_267

page_268

page_269

page_270

page_271

page_272

page_273

page_274

page_275

page_276

page_277

page_278

page_279

page_280

page_281

page_282

page_283

page_284

page_285

page_286

page_287

page_288

page_289

page_290

page_291

page_292

page_293

page_294

page_295

page_296

page_297

page_298

page_299

page_300

page_301

page_302

page_303

page_304

page_305

page_306

page_307

page_308

page_309

page_310

page_311

page_312

page_313

page_314

page_315

page_316

page_317

page_318

page_319

page_320

page_321

page_322

page_323

page_324

page_325

page_326

page_327

page_328

page_329

page_330

page_331

page_332

page_333

page_334

page_335

page_336

page_337

page_338

page_339

page_340

page_341

page_342

page_343

page_344

page_345

page_346

page_347

page_348

page_349

page_350

page_351

page_352

page_353

page_354

page_355

page_356

page_357

page_358

page_359

page_360

page_361

page_362

page_363

page_364

page_365

page_366

page_367

page_368

page_369

page_370

page_371

page_372

page_373

page_374

page_375

page_376

page_377

page_378

page_379

page_380

page_381

page_382

page_383

page_384

page_385

page_386

page_387

page_388

page_389

page_390

page_391

page_392

page_393

page_394

page_395

page_396

page_397

page_398

page_399

page_400

page_401

page_402

page_403

page_404

page_405

page_406

page_407

page_408

page_409

page_410

page_411

page_412

page_413

page_414

page_415

page_416

page_417

page_419

page_420

page_421

page_422

page_423

page_424

page_425

page_426

page_427

page_428

page_429

page_430

page_431

page_432

page_433

page_434

page_435

page_436

page_437

page_438

page_439

page_440

page_441

page_442

page_443

page_444

page_445

page_446

page_447

page_449

page_450

page_451

page_452

page_453

page_454

page_455

page_456

page_457

page_458

page_459

page_460

page_461

page_462

page_463

page_464

page_465

page_466

page_467

page_468

page_469

page_470

page_471

page_472

page_473

page_474

page_475

page_476

page_477

page_478

page_479

page_480

page_481

page_482

page_483

page_484

page_485

page_486

page_487

page_488

page_489

page_490

page_491

page_492

page_493

page_494

page_495

page_496

page_497

page_498

page_499

page_500

page_501

page_502

page_503

page_505

page_506

page_507

page_508

page_509

page_510

page_511

page_512

page_513

page_514

page_515

page_516

page_517

page_518

page_519

page_520

page_521

page_522

page_523

page_524

page_525

page_526

page_527

page_528

page_529

page_530

page_531

page_532

page_533

page_534

page_535

page_536

page_537

page_538

page_539

page_540

page_541

page_542

page_543

page_544

page_545

page_546

page_547

page_549

< previous page page_549 next page >Page 549

Additional information, test on, see Tests of hypotheses, on a subvector

Analysis of covariance, 110, 178194

multivariate, 187194

and canonical correlations, 190

one-way model, 187

two-way model, 188

univariate, 178187

assumptions, 178179

one-way model, 179183

two-way model, 183186, 191

unbalanced model, 186187

Analysis of variance, multivariate, see Multivariate analysis of variance

Analysis of variance, univariate (ANOVA)

contrasts, 142145

Bonferroni procedure, 144

orthogonal, 144

Scheffe procedure, 144145

unbalanced data, 152155, 160168

cell means model, 152, 160161

contrasts, 153155, 163165

two-way model, 160168

constrained model, 165168

ANOVA, see Analysis of variance, univariate

file:///G|/page_549.html (1 of 3)7/11/2009 2:12:28 PM

page_549

Apparent error rate, see Error rate(s)

Apple data, 197

Association, measures of, 289

Beetle data, 118

Behrens-Fisher problem, 99104

univariate, 99100

Biochemical data

full, 199

partial, 192

Bonferroni critical values, 78, 144

table, 421422

Calcium data, 69

Canonical correlation(s), 190, 312336

canonical variates, see Canonical variates

definition of, 313

and discriminant analysis, 312

and eigenvalues, 314316, 323

influence, 333

and MANOVA, 312, 333

and measures of association, 289

and multiple correlation, 314, 319

properties of, 317320

redundancy analysis, 331333

robust estimators, 333

singular value decomposition, 315316

page_549

tests of significance, 320326

Lawley-Hotelling test, 323

likelihood ratio test (Wilks' A), 321323

Pillai's test, 323

Roy's test, 323

test of independence, 320321

and test of overall regression, 321, 323

subset of canonical correlations, 324326

union-intersection test (Roy's), 324

validation, 326327

cross validation, 326327

jackknife, 327

Canonical variates, 313, 317320, 326333

canonical ridge weights, 327

common canonical variates, 333

definition of, 313

and eigenvectors, 313320, 327

influence, 333

interpretation, 328331

correlations (structure coefficients), 329331

rotation, 328Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation

page_550

Canonical variates (continued)

standardized coefficients, 328

nonlinear canonical variates, 333

properties of, 320

redundancy analysis, 331333

scaling, 317

Categorical data, 255, 262, 266, 306307

Centering matrix, 910

Central limit theorem (multivariate), 53

Characteristic roots, see Eigenvalues

Chi-square distribution, 4849, 5354, 60

Cholesky decomposition, 42, 408

Classification analysis (allocation), 230265

assigning a sampling unit to a group, 230

assumptions, 234

robustness to departures from, 234

for categorical data, 262

correct classification rates, 240

costs of misclassification, 233

density estimation, 263

and discriminant analysis, 201

discriminant functions used in classification, 232, 239

error rates, 240247. See also Error rate(s)

influence of individual observations, 262

logistic classification, 254259

page_550

comparison with linear classification, 256257

quadratic logistic classification, 258

for several groups, 258

missing data, 262263

nearest neighbor method, 263

posterior probabilities, 234, 236237

prior probabilities, 230, 236, 238

probit classification, 259261

ridge classification, 261

robust classifications procedures, 235

several groups, 236239

asymptotic optimality, 236

comparison of linear and quadratic rules, 238239

linear classification function, 236

maximum likelihood rule, 236

optimal classification rule, 236

quadratic classification function, 237238

regularized discriminant (classification) analysis, 239

subset selection, 247251

using error rates, 249250

using stepwise discriminant analysis, 247249

bias, 251254

with unequal covariance matrices, 250251

two groups, 230235

asymptotic optimality, 232

linear classification rule, 231232

maximum likelihood rule, 230

optimal classification rule, 230

quadratic classification rule, 232233

page_550

Coefficient of determination, see R2

Communality, see Factor analysis, communality

Condition number, 20

Confidence interval(s), 7479

Bonferroni intervals, 7779, 94

for linear combination(s), 74, 9294

for regression coefficients, 273274

simultaneous intervals, 7576, 9395

Confidence region

for µ, 74

for µ1 µ2, 93

Constant density ellipsoid, 4041

Contaminated normal, 28

Contour plot, 4042

Contrast(s), 84, 95, 142148, 150151, 153160

Bonferroni procedure, 144

orthogonal, 144, 154

Scheffe procedure, 144145

simultaneous, 150

with unbalanced data, 153160

Contrast matrix, 84, 163, 166, 169170

Correct classification rate, 240

Correlation

bias, 6

canonical, see Canonical correlation(s)

of two linear combinations, 15

of two random variables, 6

Correlation matrix

bias, 12

page_550

and factor analysis, 379380, 383, 385386, 389, 392

of linear combinations of variables, 16

population, 1112

and principal components, 342, 344347, 349, 351, 353359, 364, 367370

relationship to covariance matrix, 1112

sample, 11

as standardized covariance matrix, 1

test comparing two covariance matrices, 138140

Correspondence analysis, 2, 373

Covariance

of two linear combinations, 15

of two random variables, 56

Covariance matrix (matrices)

for one random vector, 810

partitioned, 1213Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation

page_551

pooled, 87

population, 10

positive definite, 910

relationship to correlation matrix, 1112

sample, 89

distribution of, 55

unbiased, 11

test for equality of, 138140

table of critical values, 446447

for two random vectors, 13

Cross validation, 244, 326327

continuous, 2

discrete, 2

missing, 2327.

Multivariate Statistical Inference and Applications

Documents