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Multivariate Statistical Inference and Applications
Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation
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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.
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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
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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.
Copyright © 1998 by John Wiley & Sons, Inc. All rights reserved.
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: [email protected].
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 1
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Contents
1. Some Properties of Random Vectors and Matrices 1
1.1. Introduction 1
1.2. Univariate and Bivariate Random Variables 2
1.2.1. Univariate Random Variables 2
1.2.2. Bivariate Random Variables 4
1.3. Mean Vectors and Covariance Matrices for Random Vectors 6
1.4. Correlation Matrices 11
1.5. Partitioned Mean Vectors and Covariance Matrices 12
1.6. Linear Functions of Random Variables 14
1.6.1. Sample Means, Variances, and Covariances 14
1.6.2. Population Means, Variances, and Covariances 19
1.7. Measuring Intercorrelation 20
1.8. Mahalanobis Distance 22
1.9. Missing Data 23
1.10. Robust Estimators of µ and Σ 27
2. The Multivariate Normal Distribution 37
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2.1. Univariate and Multivariate Normal Density Functions 37
2.1.1. Univariate Normal 37
2.1.2. Multivariate Normal 38
2.1.3. Constant Density Ellipsoids 40
2.1.4. Generating Multivariate Normal Data 42
2.1.5. Moments 42
2.2. Properties of Multivariate Normal Random Vectors 43
2.3. Estimation of Parameters in the Multivariate Normal Distribution 49 Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation
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2.3.1. Maximum Likelihood Method 49
2.3.2. Properties of y *and S 52
2.3.3. Wishart Distribution 53
2.4. Additional Topics 56
3. Hotelling's T2-Tests 60
3.1. Introduction 60
3.2. Test for H0: µ = µ0 with Σ Known 60
3.3. Hotelling's T2-test for Ho: µ = µ0 with Σ Unknown 61
3.3.1. Univariate t-Test for H0: µ = µ0 with σ2 Unknown 61
3.3.2. Likelihood Ratio Method of Test Construction 62
3.3.3. One-Sample T2-Test 65
3.3.4. Formal Definition of T2 and Relationship to F 66
3.3.5. Effect on T2 of Adding a Variable 67
3.3.6. Properties of the T2-Test 70
3.3.7. Likelihood Ratio Test 71
3.3.8. Union-Intersection Test 72
3.4. Confidence Intervals and Tests for Linear Functions of µ 74
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3.4.1. Confidence Region for µ 74
3.4.2. Confidence Interval for a Single Linear Combination a'µ 74
3.4.3. Simultaneous Confidence Intervals for µj and a'µ 74
3.4.4. Bonferroni Confidence Intervals for µj and a'µ 77
3.4.5. Tests for Ho: a'µ = a'µ0and H0: µj= µ0j 79
3.4.6. Tests for H0:Cµ = 0 83
3.5. Tests of H0: µ1 = µ2 Assuming Σ1= Σ2 85
3.5.1. Review of Univariate Likelihood Ratio Test for H0: µ1 = µ2 When σ2/1 = σ2/2 85
3.5.2. Test for H0: µ1= µ2 When Σ1 = Σ2 87
3.5.3. Effect on T2 of Adding a Variable 87
3.5.4. Properties of the Two-Sample T2-Statistic 91
3.5.5. Likelihood Ratio and Union-Intersection Tests 91
3.6. Confidence Intervals and Tests for Linear Functions of Two Mean Vectors 92
3.6.1. Confidence Region for µ1 µ2 93
3.6.2. Simultaneous Confidence Intervals for a'(µ1 µ2) and µ1j µ2j 93
3.6.3. Bonferroni Confidence Intervals for a'(µ1 µ2) and µ1j - µ2j 94 Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation
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3.6.4. Tests for H0: a'(µ1 µ2) = a'δ0and H0j: µ1j µ2j = 0 94
3.6.5. Test for H0: C(µ1 µ2) = 0 95
3.7. Robustness of the T2-test 96
3.7.1. Robustness to Σ1≠ Σ2 96
3.7.2. Robustness to Nonnormality 96
3.8. Paired Observation Test 97
3.9. Testing H0: µ1 = µ2When Σ1≠Σ2 99
3.9.1. Univariate Case 99
3.9.2. Multivariate Case 100
3.10. Power and Sample Size 104
3.11. Tests on a Subvector 108
3.11.1. Two-Sample Case 108
3.11.2. Step-Down Test 110
3.11.3. Selection of Variables 111
3.11.4. One-Sample Case 112
3.12. Nonnormal Approaches to Hypothesis Testing 112
3.12.1. Elliptically Contoured Distributions 112
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3.12.2 NonparametricTests 113
3.12.3 Robust Versions of T2 114
3.13. Application of T2 in Multivariate Quality Control 114
4. Multivariate Analysis of Variance 121
4.1. One-Way Classification 121
4.1.1. Model for One-Way Multivariate Analysis of Variance 121
4.1.2 Wilks' Likelihood Ratio Test 122
4.1.3. Roy's Union-Intersection Test 127
4.1.4. The Pillai and Lawley-Hotelling Test Statistics 130
4.1.5. Summary of the Four Test Statistics 131
4.1.6. Effect of an Additional Variable on Wilks' Λ 132
4.1.7. Tests on Individual Variables 134
4.2. Power and Robustness Comparisons for the Four MANOVA Test Statistics 135
4.3. Tests for Equality of Covariance Matrices 138
4.4. Power and Sample Size for the Four MANOVA Tests 140
4.5. Contrasts Among Mean Vectors 142
4.5.1. Univariate Contrasts 142
4.5.2. Multivariate Contrasts 145
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4.6. Two-Way Multivariate Analysis of Variance 148
4.7. Higher Order Models 151
4.8. Unbalanced Data 152
4.8.1. Introduction 152
4.8.2. Univariate One-Way Model 153
4.8.3. Multivariate One-Way Model 155
4.8.4. Univariate Two-Way Model 160
4.8.5. Multivariate Two-Way Model 168
4.9. Tests on a Subvector 174
4.9.1. Testing a Single Subvector 174
4.9.2. Step-Down Test 177
4.9.3. Stepwise Selection of Variables 177
4.10. Multivariate Analysis of Covariance 178
4.10.1. Introduction 178
4.10.2. Univariate Analysis of Covariance: One-Way Model with One Covariate 179
4.10.3. Univariate Analysis of Covariance: Two-Way Model with One Covariate 183
4.10.4. Additional Topics in Univariate Analysis of Covariance 186
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4.10.5. Multivariate Analysis of Covariance 187
4.11. Alternative Approaches to Testing H0: µ1 = µ2 = · · · = µk 194
5. Discriminant Functions for Descriptive Group Separation 201
5.1. Introduction 201 5.2. Two Groups 201 5.3. Several Groups 202 5.3.1. Discriminant Functions 202 5.3.2. Assumptions 205 5.4. Standardized Coefficients 206 5.5. Tests of Hypotheses 207 5.5.1. Two Groups 207 5.5.2. Several Groups 209
5.6. Discriminant Analysis for Higher Order Designs 210
5.7. Interpretation of Discriminant Functions 210
5.7.1. Standardized Coefficients and Partial F-Values 211
5.7.2. Correlations between Variables and Discriminant Functions 211 Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation
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5.7.3. Other Approaches 215
5.8. Confidence Intervals 216
5.9. Subset Selection 216
5.9.1. Discriminant Function Approach to Selection 217
5.9.2. Stepwise Selection 217
5.9.3. All Possible Subsets 218
5.9.4. Selection in Higher Order Designs 218
5.10. Bias in Subset Selection 219
5.11. Other Estimators of Discriminant Functions 221
5.11.1. Ridge Discriminant Analysis and Related Techniques 222
5.11.2. Robust Discriminant Analysis 223
6. Classification of Observations into Groups 230
6.1. Introduction 230
6.2. Two Groups 230
6.2.1. Equal Population Covariance Matrices 230
6.2.2. Unequal Population Covariance Matrices 232
6.2.3. Unequal Costs of Misclassification 233
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6.2.4. Posterior Probability Approach 234
6.2.5. Robustness to Departures from the Assumptions 234
6.2.6. Robust Procedures 235
6.3. Several Groups 236
6.3.1. Equal Population Covariance Matrices 236
6.3.2. Unequal Population Covariance Matrices 237
6.3.3. Use of Linear Discriminant Functions for Classification 239
6.4. Estimation of Error Rates 240
6.5. Correcting for Bias in the Apparent Error Rate 244
6.5.1. Partitioning the Sample 244
6.5.2. Holdout Method 244
6.5.3. Bootstrap Estimator 245
6.5.4. Comparison of Error Estimators 245
6.6. Subset Selection 247
6.6.1. Selection Based on Separation of Groups 247
6.6.2. Selection Based on Allocation 249
6.6.3. Selection in the Heteroscedastic Case 250
6.7. Bias in Stepwise Classification Analysis 251
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6.8. Logistic and Probit Classification 254
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6.8.1. The Logistic Model for Two Groups with Σl = Σ2 254
6.8.2. Comparison of Logistic Classification with Linear Classification Functions 256
6.8.3. Quadratic Logistic Functions When Σl≠ Σ2 258
6.8.4. Logistic Classification for Several Groups 258
6.8.5. Additional Topics in Logistic Classification 259
6.8.6. Probit Classification 259
6.9. Additional Topics in Classification 261
7. Multivariate Regression 266
7.1. Introduction 266
7.2. Multiple Regression: Fixed x's 267
7.2.1. Least Squares Estimators and Properties 267
7.2.2. An Estimator for σ2 268
7.2.3. The Model in Centered Form 269
7.2.4. Hypothesis Tests and Confidence Intervals 271
7.2.5. R2 in Fixed-x Regression 275
7.2.6. Model Validation 275
7.3. Multiple Regression: Random x's 276
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7.3.1. Model for Random x's 276
7.3.2. Estimation of β0, βl, and σ2 277
7.3.3. R2 in Random-x Regression 278
7.3.4. Tests and Confidence Intervals 279
7.4. Estimation in the Multivariate Multiple Regression Model: Fixed x's 279
7.4.1. The Multivariate Model 279
7.4.2. Least Squares Estimator for B 281
7.4.3. Properties of B * 283
7.4.4. An Estimator for Σ 285
7.4.5. Normal Model for the yi's 286
7.4.6. The Multivariate Model in Centered Form 288
7.4.7. Measures of Multivariate Association 289
7.5. Hypothesis Tests in the Multivariate Multiple Regression Model: Fixed x's 289
7.5.1. Test for Significance of Regression 289
7.5.2. Test on a Subset of the Rows of B 293
7.5.3. General Linear Hypotheses CB = O and CBM = O 295
7.5.4. Tests and Confidence Intervals for a Single bjkand a Bilinear Function a'Bb 297 Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation
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7.5.5. Simultaneous Tests and Confidence Intervals for the βjk's and Bilinear Functions a'Bb 298
7.5.6. Tests in the Presence of Missing Data 299
7.6. Multivariate Model Validation: Fixed x's 300
7.6.1. Lack-of-Fit Tests 300
7.6.2. Residuals 300
7.6.3. Influence and Outliers 301
7.6.4. Measurement Errors 303
7.7. Multivariate Regression: Random x's 303
7.7.1. Multivariate Normal Model for Random x's 303
7.7.2. Estimation of β0, B1, and Σ 304
7.7.3. Tests and Confidence Intervals in the Multivariate Random-x Case 305
7.8. Additional Topics 305
7.8.1. Correlated Response Methods 305
7.8.2. Categorical Data 306
7.8.3. Subset Selection 307
7.8.4. Other Topics 307
8. Canonical Correlation 312
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8.1. Introduction 312
8.2. Canonical Correlations and Canonical Variates 312
8.3. Properties of Canonical Correlations and Variates 318
8.3.1. Properties of Canonical Correlations 318
8.3.2. Properties of Canonical Variates 320
8.4. Tests of Significance for Canonical Correlations 320
8.4.1. Tests of Independence of y and x 320
8.4.2. Test of Dimension of Relationship between the y's and the x's 324
8.5. Validation 326
8.6. Interpretation of Canonical Variates 328
8.6.1. Standardized Coefficients 328
8.6.2. Rotation of Canonical Variate Coefficients 328
8.6.3. Correlations between Variables and Canonical Variates 329
8.7. Redundancy Analysis 331
8.8. Additional Topics 333
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9. Principal Component Analysis 337
9.1. Introduction 337
9.2. Definition and Properties of Principal Components 338
9.2.1. Maximum Variance Property 338
9.2.2. Principal Components as Projections 340
9.2.3. Properties of Principal Components 341
9.3. Principal Components as a Rotation of Axes 343
9.4. Principal Components from the Correlation Matrix 344
9.5. Methods for Discarding Components 347
9.5.1. Percent of Variance 347
9.5.2. Average Eigenvalue 348
9.5.3. Scree Graph 348
9.5.4. Significance Tests 349
9.5.5. Other Methods 352
9.6. Information in the Last Few Principal Components 352
9.7. Interpretation of Principal Components 353
9.7.1. Special Patterns in S or R 353
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9.7.2. Testing H0: Σ = σ2 [(1 ρ)I + ρJ] and Pρ = (1 ρ)I + ρJ 357
9.7.3. Additional Rotation 359
9.7.4. Correlations between Variables and Principal Components 361
9.8. Relationship Between Principal Components and Regression 363
9.8.1. Principal Component Regression 364
9.8.2. Latent Root Regression 370
9.9. Principal Component Analysis with Grouped Data 371
9.10. Additional Topics 372
10. Factor Analysis 377
10.1. Introduction 377
10.2. Basic Factor Model 377
10.2.1. Model and Assumptions 377
10.2.2. Scale Invariance of the Model 379
10.2.3. Rotation of Factor Loadings in the Model 380
10.3. Estimation of Loadings and Communalities 381
10.3.1. Principal Component Method 381
10.3.2. Principal Factor Method 383
10.3.3. Iterated Principal Factor Method 384
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10.3.4. Maximum Likelihood Method 384
10.3.5. Other Methods 385
10.3.6. Comparison of Methods 385
10.4. Determining the Number of Factors, m 386
10.5. Rotation of Factor Loadings 387
10.5.1. Introduction 387
10.5.2. Orthogonal Rotation 388
10.5.3. Oblique Rotations 389
10.5.4. Interpretation of the Factors 390
10.6. Factor Scores 391
10.7. Applicability of the Factor Analysis Model 392
10.8. Factor Analysis and Grouped Data 393
10.9. Additional Topics 394
Appendix A. Review of Matrix Algebra 399
A.1. Introduction 399
A. 1.1. Basic Definitions 399
A. 1.2. Matrices with Special Patterns 400
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A.2. Properties of Matrix Addition and Multiplication 401
A.3. Partitioned Matrices 404
A.4. Rank of Matrices 406
A.5. Inverse Matrices 407
A.6. Positive Definite and Positive Semidefinite Matrices 408
A.7. Determinants 409
A.8. Trace of a Matrix 410
A.9. Orthogonal Vectors and Matrices 410
A.10. Eigenvalues and Eigenvectors 411
A. 11. Eigenstructure of Symmetric and Positive Definite Matrices 412
A.12. Idempotent Matrices 414
A. 13. Differentiation 414
Appendix B. Tables 417
Appendix C. Answers and Hints to Selected Problems 449
Appendix D. About the Diskette 505
Bibliography 507
Index 549
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