Multivariate Statistical Inference and Applications

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

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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

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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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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 1Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation

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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

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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


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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

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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


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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

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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



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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

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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


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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

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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


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6.8. Logistic and Probit Classification254


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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

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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


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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

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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



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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

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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



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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

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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

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page_549

< previous page page_549 next page >Page 549

Index

A

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

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page_549

Apparent error rate, see Error rate(s)

Apple data, 197

Association, measures of, 289

B

Beetle data, 118

Behrens-Fisher problem, 99104


univariate, 99100

Biochemical data

full, 199

partial, 192

Bonferroni critical values, 78, 144

table, 421422

C

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

D

Data

continuous, 2

discrete, 2

missing, 2327.

See also Missing data

Data matrix (Y), 7

Density estimation, 263

Density function, 3739

Diabetes data, 17

Diagonal matrix, 11, 400

Determinant, 409

Discriminant analysis (descriptive), 201229.

See also Discriminant function(s)

assumptions, 205206

and canonical correlation, 312


page_551

and classification analysis, 201

influence, 206

ridge discriminant analysis, 221222

robust discriminant analysis, 223227

several groups, 202206

subset selection

all possible subsets, 218

bias, 219221

in higher order designs, 218219

using discriminant functions, 217

stepwise discriminant analysis, 217221


two groups, 201202

for two-way designs, 210

Discriminant analysis (predictive), see Classification analysis

Discriminant function(s) (descriptive),

see also Discriminant analysis (descriptive)

confidence intervals for, 216

interpretation of, 210215

correlations (structure coefficients), 211215

partial F-tests, 211


invariance of, 202

for one group, 74

plotting, 216

ridge estimator, 221222

robust discriminant functions, 223227

for several groups (MANOVA), 128, 151, 202206

properties, 203


page_551

other estimators when Σ is near singular, 222


for two groups, 92, 201202

effect of each variable, 206207

other estimators when Σ is near singular, 221222

for unbalanced data, 157

Dispersion matrix, see Covariance matrix

Distance between two vectors (Mahalanobis), 2223

E

E matrix, 122, 203

Eigenvalues, 411414

and canonical correlations, 314316, 323

and discriminant functions, 128, 203205, 209, 212215, 222226

and factor analysis, 381383

and MANOVA, 131

and principal components, 338, 341

Eigenvectors, 411414

and discriminant functions, 203204, 210, 212213, 222225


and principal components, 338

Ellipsoid, constant density, 4041

Elliptically contoured distributions, 56, 112113

EM algorithm, 2526

Error rate(s), 240247

actual error rate, 240

apparent correct classification rate, 243245

apparent error rate, 243245

bias, correction for, 244247


page_551

bootstrap estimator, 245

comparison of methods, 245247


holdout method, 244

leaving-one-out-method, 244

conditional error rate, 240

expected actual error rate, 240

experimentwise, 2, 82, 95

maximum likelihood estimator of, 242

optimum error rate, 240242

plug-in estimator of error rate, 240243

resubstitution, 243

true error rate, 240

Estimation

least squares, 267268, 281282

likelihood function, 49

maximum likelihood, 4952

Estimator

least squares, 267268, 281282Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation



page_552


Estimator (continued)


unbiased, 23

Expected value

of random matrix, 10

of random vector [E(y)], 8

of sample covariance [E(sxy)],6

of sample covariance matrix [E(S)], 11

of sample mean [E( )], 2

of sample mean vector [E( )], 8

of sample variance [E(s2)], 3

of sum or product of random variables, 5

of univariate random variable [E(y)], 2

Experimentwise error rates, 2

F

F-test(s)

partial F-test for discriminant functions, 211

partial F-test in MANOVA, 110

and T2, 67

tables of power of the F-test, 423426

Factor analysis, 377398

applicability of the model, 392393

assumptions, 378379

communality, 379, 381

of correlation matrix, 379380, 383, 385386, 389, 392

of covariance matrix, 377388, 391395

eigenvalues, 381383


page_552

eigenvectors, 381

factor scores, 391392

factors, 377379

interpretation of, 378, 381, 390391

fit of the model, 392393

and grouped data, 393394

loadings, 378, 381385

estimation of, 381385

comparison of methods, 385386

iterated principal factor method, 384

maximum likelihood method, 384385

principal axis method, 383384

principal component method, 381383

principal factor method, 383384

model, 378379

scale invariance of, 379380

nonlinear models, 394

number of factors to retain, 383, 386387

and principal components, 361, 377

and regression, 394395

robustness to nonnormality, 394

rotation of loadings in the model, 380

rotation of sample loadings and factors, 387390

interpretation of factors, 390391

oblique rotation, 389390

pattern matrix, 390

structure matrix, 390

orthogonal rotation, 388389

equamax, 389


page_552

quartimax, 389

varimax, 388

specific variance, 379


three-mode factor analysis, 394

total variance, 382

Fisher's LSD test, 134

Fracture data, 171

Function of a random variable, distribution of, 3840

G

Gauss-Markov theorem


univariate, 268

Generalized linear models, 266

Generalized quadratic form, 5455

Generalized variance, 89, 40

Glucose data, 35

H

H matrix, 122, 203

Height-weight data, 4

Hierarchical linear models, 306

Hotelling's T2, see T2-tests

Hypothesis tests, see Tests of hypotheses

I

Idempotent matrix, 414

Identity matrix, 400

Imputation, see Missing data

Independence


page_552

two random variables, 5

two random vectors, 46

and zero covariance, 5, 4647

Influence

and canonical correlation, 333

and canonical variates, 333


and discriminant analysis (descriptive), 206

and multivariate regression, 301303


Intercorrelation, index of, 2022

condition number, 20

variance inflation factor, 21

Inverse matrix, 407

J

j vector and J matrix, 400

Jackknife, 327

Jacobian, 39

L

Largest root test, see Roy's test

Latent root regression, 363, 370Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation



page_553


Latent roots, see Eigenvalues

Lawley-Hotelling test, 130131, 292, 295296

F-approximations, 130131


Least squares estimators, 267268, 281282

Length of a vector, 402

Library data, 397

Likelihood function, 49

Likelihood ratio, 6265

Likelihood ratio test

in canonical correlation, 321326

comparing covariance matrices, 138

for discriminant functions, 209

generalized likelihood ratio, 262

in multivariate regression, 289299

one-sample t-test, 6265

one-sample T2test, 7172

one-way MANOVA, 122126

in principal components, 349351, 357359

random effects MANOVA, 151

in random-x regression, 279

two-sample t-test, 8586

twosample T2-test, 9192

Linear classification functions, 231232, 236

Linear classification rule, 231232, 236

Linear function(s) of random variables, 1420

confidence intervals for, 74, 9294


page_553

correlation of two linear combinations, 15, 20

covariance of two linear combinations, 15, 19

covariance matrix for several linear combinations, 1516, 20

covariance matrix for two sets of linear combinations, 16, 20

linear combinations of subvectors, 1617, 20

mean of a single linear combination, 14, 19

variance of a single linear combination, 15, 19

Linear hypotheses

in ANOVA and MANOVA, 163171


one sample, 7274, 7980, 8385

for H0: µl = µ2 = . . . = µp, 8385

two samples, 9495

in univariate regression, 272273

Log linear models, 2

Logistic classification, 254259

comparison with linear classification, 256257

quadratic logistic classification, 258

several groups, 258

Logistic regression, 254255

Longley data, 368

LSD test, 134

M

Mahalanobis distance, 2223

MANOVA, 121. See also Multivariate analysis of variance

Matrix (matrices)

addition of, 401

algebra of, 399416

bilinear form, 404


page_553

centering matrix, 910

characteristic equation, 411

Cholesky decomposition, 42, 408

correlation matrix, 1112. See also Correlation matrix

covariance matrix, 89. See also Covariance matrix

positive definite, 910

data matrix, 7

definition, 399

determinant, 409

diagonal matrix, 11, 400

diagonal of a matrix, 400

differentiation, 5051, 414416

eigenvalues, 411414. See also Eigenvalues

characteristic equation, 411

and determinant, 411

of inverse, 411

of positive definite matrix, 412

of product, 411

of symmetric matrix, 412

and trace, 411

eigenvectors, 411414. See also Eigenvectors

factoring of, 402

idempotent matrix, 414

identity, 400

inverse, 407

of partitioned matrix, 407

of product, 407

j vector, 400

J matrix, 400


page_553

multiplication of, 401

nonsingular matrix, 407

O (zero matrix), 401

operations with, 401406

distributive law, 401

factoring, 402

product, 401406

with diagonal matrix, 404

of matrix and vector, 405

as linear combination, 405

noncommutativity of, 401

of vectors, 401402

sum, 401402

commutativity of, 401

orthogonal matrices, 410

partitioned matrices, 404406Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation



page_554


Matrix (continued)

Perron-Frobenius theorem, 414

positive definite matrix, 408

positive semidefinite matrix, 408

products, 401, 404

quadratic form, 4849, 404

generalized quadratic form, 5455

random matrix, expected value of, 10

rank, 406

full rank, 406

singular matrix, 407, 408

size of a matrix, 399

spectral decomposition, 412

square root matrix, 413

trace, 410

triangular matrix, 400

vectors, See Vector(s)

zero matrix (O) and zero vector (0), 401

Maximum likelihood estimators, 4952

of covariance matrix, 4951

invariance property of, 5152


of mean vector, 4951

multivariate normal, 49

in regression, 286289

Mean, see also Expected value

population, 2


page_554

sample, 2

Mean deviation, 28

Mean vector

partitioned, 1213

population, 8

sample, 7

Misclassification rates, see Error rate(s)

Missing data, 2327, 299300


estimation of missing values (imputation), 2427

and hypothesis tests, 299300

listwise deletion, 23

missing at random, 23

missing completely at random, 23

pairwise deletion, 24

Models

generalized linear models, 2

log linear models, 2

Moment generating function, 4345

Moments, 4243

Multicolinearity, 40, 327, 364, 368

Multidimensional scaling, 373

Multiple correlation, 275. See also R2

Multiple regression, see Regression, multiple

Multivariate analysis, 1

Multivariate analysis of covariance. See Analysis of covariance, multivariate

Multivariate analysis of variance (MANOVA), 121200

additional information, test for, 174178

approximate tests


page_554

based on M-estimators, 194

based on ranks, 194

and canonical correlation, 312, 333

contrasts, 145148, 150151

orthogonal, 146

simultaneous, 146148

discriminant functions, 128, 151. See also Discriminant analysis; Discriminant functions

and eigenvalues, 131

Fisher's LSD test, 134

H and E matrices, 122, 124

higher order models, 151

individual variables, tests on, 134

Lawley-Hotelling test, 130131



likelihood ratio test, see Wilks' Λ

mixed models, 151

one-way classification, 121134

contrasts, 145148

unbalanced data, 155160

Pillai's test, 130131



power of the tests, 137138, 140142

tables of power of the F-test, 423426

robustness of test statistics, 137138

Roy's test (union-intersection), 127129


selection of variables, stepwise, 177


page_554

step-down test, 177


test statistics

comparison of, 132, 135138


power of, 137138, 140142

robustness of, 137138

tests on individual variables, 134

experimentwise error rate, 134

protected tests, 134

tests on a subvector, 174178

two-way classification, 148151

contrasts, 150151

discriminant functions, 151

interaction 148149

main effects, 148149Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation



page_555


model, 148

test statistics, 149

unbalanced data, 152, 155160, 168174

contrasts, 156160

Bonferroni confidence intervals, 157

simultaneous confidence intervals, 156157






union-intersection test, see Roy's test

Wilks' Λ (likelihood ratio), 122126

Chi-square approximation, 126


F-approximation, 125126



transformations to exact F, 125126

Multivariate central limit theorem, 53

Multivariate data, 12, 7

Multivariate descriptive procedures, 1

Multivariate inferential procedures, 12

Multivariate normal distribution, 3759

central limit theorem, 53

conditional distribution, 4748

constant density ellipsoid, 4041


page_555

contour plot, 4042

estimation of µ and Σ, 4951

properties of estimators, 5256

estimation of Pp, 5152

generating data from, 42

independence and zero covariance, 4647


linear functions of, 4546

marginal distribution, 46

moment generating function of, 4447

moments of, 4243

properties of, 4349

and quadratic form, 4849

tests for multivariate normality, 43

transformations to achieve multivariate normality. 5657

and Wishart distribution, 5356

Multivariate normality, tests for, 43

Multivariate regression, see Regression, multivariate

Multivariate t-distribution, 56

N

Nonparametric tests

for several mean vectors, 194

for two mean vectors, 113

Normal distribution

multivariate normal, see Multivariate normal distribution

univariate normal, 3738

standard normal, 37

Normality, tests for, 43

O


page_555

O matrix, 401

Orthogonal matrix, 410

Orthogonal vectors, 410

Outliers, 43, 301303, 373

P

Paired observation test, 9799

Partial F, 177, 211, 214215, 217, 219, 248249, 252

Perron-Frobenius theorem, 414

Physiological data, 224

Pillai's test, 130131, 292, 295296



Positive definite matrix, 408

Power of a test, 104108

noncentrality parameter, 104

Principal component regression, 363370

Principal components, 337376

common principal components, 371372

from correlation matrix (R), 342, 344349, 351, 353359, 364, 367370


correlations of, 341342

and correspondence analysis, 373

from covariance matrix (S), 338344, 348355, 357363,

definition, 338

discarding components, 347352

average eigenvalue, 348

bootstrap, 352


jackknife, 352


page_555

percent of variance, 347

rank of Σ, 352

scree graph, 348349


and eigenvalues, 338, 341

and eigenvectors, 338

empirical orthogonal functions, 337

and factor analysis, 361, 377

generalized principal components, 340

with grouped data, 371372

common principal components, 371372

comparing principal components, 371372

influence of each observation, 372373

interpretation, 353363

correlations, 361363Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation



page_556


Principal components (continued)

rotation, 359361

special patterns in S or R, 353359

equal correlations, 354, 358359

equal variances and covariances, 354, 357358

sensitivity analysis, 355

size and shape, 355

Karhunen-Loeve expansion, 337

large variance of a variable, effect of, 342

last few principal components, 352353

leverage of each observation, 372

and maximum variance, 338

and minimum variance, 339

and multidimensional scaling, 373

number of components to retain, see Principal components, discarding components

orthogonality of, 339

outliers, 373

percent of variance, 341

plotting of, 340

principal component regression, 363370

principal curves, 373

principal points, 373

projection pursuit, 340

and projections, 340


proportion of variance, 341

and regression, 363370


page_556

biased regression, 363

latent root regression, 363, 370

principal component regression, 363370

using correlation matrix, 364, 367370

using covariance matrix, 364367

ridge regression, 363

robust principal components, 373

as rotation of axes, 343344

scale invariance, lack of, 342

sensitivity analysis, 355

of singular matrix, 342

size and shape, 355


variance


large variance, effect of, 342

of principal component, 341

proportion of, 341

total, 341

Principal curves, 373

Principal points, 373

Probe word data, 22

Projection pursuit, 340

Projections, 340

Q

Quadratic classification rule, 232233

Quadratic form, 4849, 404


R


page_556

R2 (squared multiple correlation), 275, 278279

Rabbit data, 157

Rainfall data, 311

Random matrix

expected value of, 10

Random variables

bivariate, 46

correlation (population), 6

covariance (population), 5

independence of, 5

sample covariance, 6

sample correlation, 6

bias, 6

continuous, 2

discrete, 2

linear functions of, 1420

moments of, 4243

multivariate, see Random vectors

univariate, 24

mean (expected value), 2


sample mean, 23

sample variance, 3

Random vectors, 1

covariance matrix of one random vector, 810

covariance matrix of two random vectors, 13

independence of, 7

linear combinations of, 1420

mean vector (expected value), 8


page_556


sample mean vector, 7

standardized vector, 45

sum of two random vectors, 14

covariance matrix of, 14

expected value of, 14

Rank of a matrix, 406

Redundancy analysis, 331333

Regression, multiple (one y and several x's), 266279

beta weights, 278

confidence intervals

Bonferroni intervals for individual βj's, 273274

for an individual βj, 273

simultaneous intervals for a' β, 274

estimation of β

centered x's, 269271

covariances, 270271

least squares, 267268

properties of estimator, 268Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation



page_557


estimation of σ2, 268269


fixed x's, 267276

Gauss-Markov theorem, 268

model, 267

assumptions, 267

corrected for means, 269271

validation, 275276

multiple correlation, 275


R2 (squared multiple correlation), 275, 278279

random x's, 276279

confidence intervals, 279

estimation of the βj's and σ2, 277278

correlations, 278

beta weights, 278

model, 276

R2 (squared multiple correlation), 278279

tests of hypotheses, 279

ridge regression, 363

SSE, 269

SSR, 271


Bonferroni tests for individual βj's, 273274

general linear hypothesis H0:Cβ= 0, 272273

overall regression test, 271

simultaneous tests for individual βj's, 274


page_557

subset of the β's, 271272

test for an individual βj, 273

Regression, multivariate (several y's and several x's)

association, measures of, 289

confidence interval(s), 297299, 305

for a'Bb, 298

for an individual βjk, 297

simultaneous intervals for βjkand a'Bb, 298

correlated response model, 305306

estimation of B, 281284, 286289

with centered x's, 288289

with covariances, 288289

least squares, 281282


properties of estimators, 283284, 287288

estimation of Σ, 285288

properties of estimators, 285, 287288

Gauss-Markov theorem, 284285

hierarchical linear models, 306

measurement errors, 303

model, 279281

assumptions, 281

in centered form, 288289

normal model, 286288

validation, 300303

influence, 301303

lack-of-fit tests, 300

leverage, 301

outliers, 301303


page_557

residuals, 300301

random x's, 303305

confidence intervals, 305

estimation of the βj's and Σ, 304305

beta weights, 305

correlations, 305

model, 303304



seemingly unrelated regressions, 306

subset selection, 307

tests of hypotheses, 289300, 305

E matrix, 285, 286, 290

general linear hypotheses H0:CB = O and H0: CBM = O, 295297

H matrix, 290, 293, 296

for an individual βjk, 297

Lawley-Hotelling test, 292, 295, 296

likelihood ratio test, see Wilks' Λ

and missing data, 299300

overall regression test, 289293

in terms of canonical correlations, 323

power of tests, 292293

Pillai's test, 292, 295, 296

Roy's test, 291292, 295, 296

subset of the x's, 293295

union-intersection test, see Roy's test

Wilks' Λ, 290291, 294, 296

Repeated measures, 121, 183, 246, 296

Robust estimation


page_557

affine equivariance of (scale invariance), 30

breakdown point, 30

of canonical variates, 333

in classification analysis, 235

contaminated normal, 28

of correlation matrices, 2930

of covariance matrices, 2931

of discriminant functions, 223227

mean deviation, 28

of mean vectors, 2831

of means, 28

of principal components, 373

in regression, 307

Rotation, see Factor analysis

Roy's test (union-intersection), 127129, 291296


S

Seemingly unrelated regressions, 306

Selection of variables, 111, 177, 217221Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation



page_558


Selection of variables (continued)

all possible subsets, 218

bias, 219221

in classification analysis, 247251

using discriminant functions, 217

in higher order designs, 218219

in regression, 307


Singular value decomposition, 2627, 315316

Size and shape, 355

Soils data, 103

Specific variance, 379

Spectral decomposition, 412

Squared multiple correlation, see R2

SSE, 269

SSR, 271

Standardized vector, 45

Stepdown test, 110111

Stepwise discriminant analysis, 177178, 217221. See also Selection of variables

Subset selection, see Selection of variables

Subvector(s), 1214

conditional distribution, 47

covariance matrix of, 13

distribution of, 4648

independence of, 14

mean of, 13

paired observation, 116


page_558

sum of, 14

tests on, 108112, 174178

Sufficient statistic, 5253

T

t-tests

definition of t-statistic, 61

likelihood ratio, 6265, 86

one sample, 6165

properties of, 62

two samples, 8586

comparing means when variances are unequal, 99100

T2-tests, 60120

additional information, test for, 108112

approximate tests

nonparametric tests, 113

when Σ1 ≠ Σ2, 99104

Behrens-Fisher problem, 99104


univariate, 99100

degrees of freedom, 65

and discriminant function, 74, 92, 109

effect of each variable, 6770, 8791

for elliptically contoured distributions, 112113

and F-distribution, 67

formal definition of T2, 66

likelihood ratio test, 7172, 9192

matched pairs, 9799

and multivariate quality control, 114115

one mean vector


page_558

Σ known, 6061

Σ unknown, 6574

paired observation test, 9799

power, 104108

tables, 106, 423426

properties of, 70, 91

and quality control, 114115

robust versions of T2-tests, 114

robustness of T2-tests

to nonnormality, 9697

to unequal covariance matrices, 96

selection of variables, 111

on a subvector, 108112

covariates, 110

stepdown test, 110111


transformation to F, 67

two mean vectors

Σ1 = Σ2, 8592

Σ1≠ Σ2, 100104

union-intersection test, 7274, 92

Tests of hypotheses

t-tests, see t-tests

T2-tests, see T2-tests

accepting H0, 65

for additional information, 108110

analysis of covariance

multivariate, see Analysis of covariance, multivariate

univariate, see Analysis of covariance, univariate


page_558

canonical correlation, 320326

covariance matrices, equality of, 138140

Box's M-test, 138140


in discriminant analysis, 207210

for elliptically contoured distributions, 112113

equality of covariance matrices, 138140

experimentwise error rate, 82, 95

individual variables, 8083, 95

Bonferroni tests for, 8083, 95

modifications of, 8082


protected tests, 95

simultaneous tests for, 80, 94

likelihood ratio test, see Likelihood ratio test

for linear combinations

in ANOVA and MANOVA, 163171

in multivariate regression, 295297Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation



page_559


one sample, 7272, 7980, 8385

for H0: µ1= µ2 = . . . = µp, 8385

two samples, 9495

in univariate regression, 272273

MANOVA, see Multivariate analysis of variance mean vectors

likelihood ratio test, 7172, 9192

one mean vector

Σ known, 6061

Σ unknown, 6574

two mean vectors

Σ1= Σ2, 8592

Σ1≠Σ2, 100104

several mean vectors, 121134

nonparametric tests

for several mean vectors, 194

for two mean vectors, 113

paired observation test, 9799

partial F-test, 110

for principal components, 349351

protected tests, 95

in regression


univariate, 271274

robust versions of T2-tests, 114

simultaneous tests, 80, 94, 146148, 150, 274

stepdown test, 110111

on a subvector, 108112, 174178


page_559

union-intersection test, 7274

Total variance, 89, 341, 382

Trace of a matrix, 410

Transformation of a random variable, distribution of, 3840

Jacobian, 39

Two-sample test for mean vectors, 8792

comparing mean vectors when Σ1≠Σ2, 100104

U

Unbalanced data, 152, 155160, 168174

contrasts, 156160

Bonferroni confidence intervals, 157

simultaneous confidence intervals, 156157






Univariate analysis of variance, see Analysis of variance, univariate

Univariate normal distribution, 3738

V

Variables, see also Random variables

categorical, 2

commensurate, 1

correlated, 1

standardized, 11

Variance

generalized, 89

population, 3


page_559

sample, 3

unbiased, 3

total, 8, 341, 382

Variance inflation factor, 21

Variance matrix, see Covariance matrix

Variance-covariance matrix, see Covariance matrix

Varimax rotation, 388

Vector(s)

0 vector, 401

definition, 399

differentiation, 414415

distance between two vectors, 2223

distance from origin, 402

j vector, 400

length of, 402

linear independence and dependence of, 406

normalized, 402

orthogonal vectors, 410

partitioned, 1214

product of, 401402

random, 1. See also Random vectors

standardized vector, 45

subvectors, 1214. See also Subvector(s)

zero vector (0), 401

W

Wilks' Λ test statistic, 122126

in canonical correlation, 321326

Chi-square approximation, 126



page_559

F-approximation, 125126




transformations to exact F, 125126

Wishart distribution, 5356, 123124


independence, 55

and multivariate normal, 5356

and Wilks' Λ, 123124

Z

Zero matrix (O) and zero vector (0), 401Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation



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