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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: 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
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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
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 = 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
Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation
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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
4.10.1. Introduction178
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
Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation
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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
6.1. Introduction230
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
7.1. Introduction266
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
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'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. Additional Topics305
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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8.1. Introduction312
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
8.8. Additional Topics333
Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation
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9. Principal Component Analysis 337
9.1. Introduction337
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
9.10. Additional Topics372
10. Factor Analysis 377
10.1. Introduction377
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
Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation
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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
10.5.1. Introduction387
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
10.9. Additional Topics394
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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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
one-way model, 153155
two-way model, 160168
constrained model, 165168
ANOVA, see Analysis of variance, univariate
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Apparent error rate, see Error rate(s)
Apple data, 197
Association, measures of, 289
B
Beetle data, 118
Behrens-Fisher problem, 99104
multivariate, 100104
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
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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
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Canonical variates (continued)
standardized coefficients, 328
nonlinear canonical variates, 333
properties of, 320
redundancy analysis, 331333
robust estimators, 333
scaling, 317
standardized coefficients, 319
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
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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
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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
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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
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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
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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
tests of significance, 207210
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
standardized coefficients, 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
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other estimators when Σ is near singular, 222
standardized coefficients, 206207
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 factor analysis, 381
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
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bootstrap estimator, 245
comparison of methods, 245247
cross validation, 244
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
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Estimator (continued)
maximum likelihood, 4952
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
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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
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quartimax, 389
varimax, 388
specific variance, 379
tests of significance, 386387
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
multivariate, 284285
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
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two random variables, 5
two random vectors, 46
and zero covariance, 5, 4647
Influence
and canonical correlation, 333
and canonical variates, 333
and classification analysis, 262
and discriminant analysis (descriptive), 206
and multivariate regression, 301303
and principal components, 372373
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
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Latent roots, see Eigenvalues
Lawley-Hotelling test, 130131, 292, 295296
F-approximations, 130131
table of critical values, 441445
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
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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
in multivariate regression, 295297
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
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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
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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
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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
likelihood function, 49
of mean vector, 4951
multivariate normal, 49
in regression, 286289
Mean, see also Expected value
population, 2
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sample, 2
Mean deviation, 28
Mean vector
partitioned, 1213
population, 8
sample, 7
Misclassification rates, see Error rate(s)
Missing data, 2327, 299300
and classification analysis, 262263
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
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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
F-approximations, 130131
table of critical values, 441445
likelihood ratio test, see Wilks' Λ
mixed models, 151
one-way classification, 121134
contrasts, 145148
unbalanced data, 155160
Pillai's test, 130131
F-approximations, 130
table of critical values, 435437
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
table of critical values, 435437
selection of variables, stepwise, 177
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step-down test, 177
stepwise discriminant analysis, 177178
test statistics
comparison of, 132, 135138
and eigenvalues, 131
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
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model, 148
test statistics, 149
unbalanced data, 152, 155160, 168174
contrasts, 156160
Bonferroni confidence intervals, 157
simultaneous confidence intervals, 156157
discriminant functions, 157
one-way model, 155160
tests on individual variables, 157
two-way model, 168174
constrained model, 169171
union-intersection test, see Roy's test
Wilks' Λ (likelihood ratio), 122126
Chi-square approximation, 126
effect of each variable, 132134
F-approximation, 125126
properties of, 124126
table of critical values, 427434
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
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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
likelihood function, 49
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
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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
F-approximations, 130
table of critical values, 435437
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
properties of, 344345
correlations of, 341342
and correspondence analysis, 373
from covariance matrix (S), 338344, 348355, 357363,
definition, 338
discarding components, 347352
average eigenvalue, 348
bootstrap, 352
cross validation, 352
jackknife, 352
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percent of variance, 347
rank of Σ, 352
scree graph, 348349
tests of significance, 349351
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
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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
properties of, 341343
proportion of variance, 341
and regression, 363370
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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
tests of significance, 349351
variance
and eigenvalues, 341
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
generalized quadratic form, 5455
R
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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
moment generating function of, 4344
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
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moment generating function of, 44
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
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estimation of σ2, 268269
and factor analysis, 394395
fixed x's, 267276
Gauss-Markov theorem, 268
model, 267
assumptions, 267
corrected for means, 269271
validation, 275276
multiple correlation, 275
and principal components, 363370
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
tests of hypotheses, 271274
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
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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
maximum likelihood, 286289
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
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residuals, 300301
random x's, 303305
confidence intervals, 305
estimation of the βj's and Σ, 304305
beta weights, 305
correlations, 305
model, 303304
tests of hypotheses, 305
robust estimators, 307
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
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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
table of critical values, 435437
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
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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
stepwise discriminant analysis, 217221
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
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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
multivariate, 100104
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
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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
table of critical values, 419420
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
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canonical correlation, 320326
covariance matrices, equality of, 138140
Box's M-test, 138140
table of critical values, 446447
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
table of critical values, 78
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
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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
multivariate, 289299
univariate, 271274
robust versions of T2-tests, 114
simultaneous tests, 80, 94, 146148, 150, 274
stepdown test, 110111
on a subvector, 108112, 174178
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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
discriminant functions, 157
one-way model, 155160
tests on individual variables, 157
two-way model, 168174
constrained model, 169171
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
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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
effect of each variable, 132134
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F-approximation, 125126
in multivariate regression, 289299
properties of, 124126
table of critical values, 427434
transformations to exact F, 125126
Wishart distribution, 5356, 123124
generalized quadratic form, 5455
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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BELSLEY, KUH, and WELSCH · Regression Diagnostics: Identifying Influential Data and Sources of Collinearity
BERRY · Bayesian Analysis in Statistics and Econometrics: Essays in Honor of Arnold Zellner
BERRY, CHALONER, and GEWEKE · Bayesian Analysis in Statistics and Econometrics: Essays in Honor of Arnold Zellner
BHAT · Elements of Applied Stochastic Processes, Second Edition
BHATTACHARYA and WAYMIRE · Stochastic Processes with Applications
BIEMER, GROVES, LYBERG, MATHIOWETZ, and SUDMAN · Measurement Errors in Surveys
BIRKES and DODGE · Alternative Methods of Regression
BLOOMFIELD · Fourier Analysis of Time Series: An Introduction
BOLLEN · Structural Equations with Latent Variables
BOULEAU · Numerical Methods for Stochastic Processes
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BOX · R. A. Fisher, the Life of a Scientist
BOX and DRAPER · Empirical Model-Building and Response Surfaces
BOX and DRAPER · Evolutionary Operation: A Statistical Method for Process Improvement
BOX, HUNTER, and HUNTER · Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building
BOX and LUCENO · Statistical Control by Monitoring and Feedback Adjustment
BROWN and HOLLANDER · Statistics: A Biomedical Introduction
BUCKLEW · Large Deviation Techniques in Decision, Simulation, and Estimation
BUNKE and BUNKE · Nonlinear Regression, Functional Relations and Robust Methods: Statistical Methods of Model Building
CHATTERJEE and HADI · Sensitivity Analysis in Linear Regression
CHATTERJEE and PRICE · Regression Analysis by Example, Second Edition
CLARKE and DISNEY · Probability and Random Processes: A First Course with Applications, Second Edition
COCHRAN · Sampling Techniques, Third Edition
*COCHRAN and COX · Experimental Designs, Second Edition
CONOVER · Practical Nonparametric Statistics, Second Edition
CONOVER and IMAN · Introduction to Modern Business Statistics
*Now available in a lower priced paperback edition in the Wiley Classics Library.Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation
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Applied Probability and Statistics (Continued)
CORNELL · Experiments with Mixtures, Designs, Models, and the Analysis of Mixture Data, Second Edition
COX · A Handbook of Introductory Statistical Methods
*COX · Planning of Experiments
COX, BINDER, CHINNAPPA, CHRISTIANSON, COLLEDGE, and KOTT · Business Survey Methods
CRESSIE · Statistics for Spatial Data, Revised Edition
DANIEL · Applications of Statistics to Industrial Experimentation
DANIEL · Biostatistics: A Foundation for Analysis in the Health Sciences, Sixth Edition
DAVID · Order Statistics, Second Edition
*DEGROOT, FIENBERG, and KADANE · Statistics and the Law
*DEMING · Sample Design in Business Research
DETTE and STUDDEN · The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis
DILLON and GOLDSTEIN · Multivariate Analysis: Methods and Applications
DODGE and ROMIG · Sampling Inspection Tables, Second Edition
DOWDY and WEARDEN · Statistics for Research, Second Edition
DRAPER and SMITH · Applied Regression Analysis, Second Edition
DUNN · Basic Statistics: A Primer for the Biomedical Sciences, Second Edition
DUNN and CLARK · Applied Statistics: Analysis of Variance and Regression, Second Edition
ELANDT-JOHNSON and JOHNSON · Survival Models and Data Analysis
EVANS, PEACOCK, and HASTINGS · Statistical Distributions, Second Edition
FISHER and VAN BELLE · Biostatistics: A Methodology for the Health Sciences
FLEISS · The Design and Analysis of Clinical Experiments
FLEISS · Statistical Methods for Rates and Proportions, Second Edition
FLEMING and HARRINGTON · Counting Processes and Survival Analysis
FLURY · Common Principal Components and Related Multivariate Models
GALLANT · Nonlinear Statistical Models
GLASSERMAN and YAO · Monotone Structure in Discrete-Event Systems
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GNANADESIKAN · Methods for Statistical Data Analysis of Multivariate Observations, Second Edition
GREENWOOD and NIKULIN · A Guide to Chi-Squared Testing
GROSS and HARRIS · Fundamentals of Queueing Theory, Second Edition
GROVES · Survey Errors and Survey Costs
GROVES, BIEMER, LYBERG, MASSEY, NICHOLLS, and WAKSBERG · Telephone Survey Methodology
HAHN and MEEKER · Statistical Intervals: A Guide for Practitioners
HAND · Discrimination and Classification
*HANSEN, HURWITZ, and MADOW · Sample Survey Methods and Theory, Volume 1: Methods and Applications
*HANSEN, HURWITZ, and MADOW · Sample Survey Methods and Theory, Volume II: Theory
HEIBERGER · Computation for the Analysis of Designed Experiments
HELLER · MACSYMA for Statisticians
HINKELMAN and KEMPTHORNE: · Design and Analysis of Experiments, Volume I: Introduction to Experimental Design
HOAGLIN, MOSTELLER, and TUKEY · Exploratory Approach to Analysis of Variance
HOAGLIN, MOSTELLER, and TUKEY · Exploring Data Tables, Trends and Shapes
HOAGLIN, MOSTELLER, and TUKEY · Understanding Robust and Exploratory Data Analysis
HOCHBERG and TAMHANE · Multiple Comparison Procedures
HOCKING · Methods and Applications of Linear Models: Regression and the Analysis of Variables
*Now available in a lower priced paperback edition in the Wiley Classics Library.Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation
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Applied Probability and Statistics (Continued)
HOEL · Elementary Statistics, Fifth Edition
HOGG and KLUGMAN · Loss Distributions
HOLLANDER and WOLFE · Nonparametric Statistical Methods
HOSMER and LEMESHOW · Applied Logistic Regression
H0YLAND and RAUSAND · System Reliability Theory: Models and Statistical Methods
HUBERTY · Applied Discriminant Analysis
IMAN and CONOVER · Modem Business Statistics
JACKSON · A User's Guide to Principle Components
JOHN · Statistical Methods in Engineering and Quality Assurance
JOHNSON · Multivariate Statistical Simulation
JOHNSON and BALAKRISHNAN · Advances in the Theory and Practice of Statistics: A Volume in Honor of Samuel Kotz
JOHNSON and KOTZ · Distributions in Statistics Continuous Multivariate Distributions
JOHNSON, KOTZ, and BALAKRISHNAN · Continuous Univariate Distributions, Volume 1, Second Edition
JOHNSON, KOTZ, and BALAKRISHNAN · Continuous Univariate Distributions, Volume 2, Second Edition
JOHNSON, KOTZ, and BALAKRISHNAN · Discrete Multivariate Distributions
JOHNSON, KOTZ, and KEMP · Univariate Discrete Distributions, Second Edition
JUDGE, GRIFFITHS, HILL, LOTKEPOHL, and LEE · The Theory and Practice of Econometrics, Second Edition
JUDGE, HILL, GRIFFITHS, LOTKEPOHL, and LEE · Introduction to the Theory and Practice of Econometrics, Second Edition
JUREC KOVA and SEN · Robust Statistical Procedures: Aymptotics and Interrelations
KADANE · Bayesian Methods and Ethics in a Clinical Trial Design
KADANE AND SCHUM · A Probabilistic Analysis of the Sacco and Vanzetti Evidence
KALBFLEISCH and PRENTICE · The Statistical Analysis of Failure Time Data
KASPRZYK, DUNCAN, KALTON, and SINGH · Panel Surveys
KISH · Statistical Design for Research
*KISH · Survey Sampling
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LAD · Operational Subjective Statistical Methods: A Mathematical, Philosophical, and Historical Introduction
LANGE, RYAN, BILLARD, BRILLINGER, CONQUEST, and GREENHOUSE · Case Studies in Biometry
LAWLESS · Statistical Models and Methods for Lifetime Data
LE · Applied Survival Analysis
LEBART, MORINEAU, and WARWICK · Multivariate Descriptive Statistical Analysis: Correspondence Analysis and Related Techniques for Large Matrices
LEE · Statistical Methods for Survival Data Analysis, Second Edition
LEPAGE and BILLARD · Exploring the Limits of Bootstrap
LEVY and LEMESHOW · Sampling of Populations: Methods and Applications
LINHART and ZUCCHINI · Model Selection
LITTLE and RUBIN · Statistical Analysis with Missing Data
LYBERG · Survey Measurement
MAGNUS and NEUDECKER · Matrix Differential Calculus with Applications in Statistics and Econometrics
MAINDONALD · Statistical Computation
MALLOWS · Design, Data, and Analysis by Some Friends of Cuthbert Daniel
MANN, SCHAFER, and SINGPURWALLA · Methods for Statistical Analysis of Reliability and Life Data
MASON, GUNST, and HESS · Statistical Design and Analysis of Experiments with Applications to Engineering and Science
*Now available in a lower priced paperback edition in the Wiley Classics Library.Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation
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Applied Probability and Statistics (Continued)
McLACHLAN and KRISHNAN · The EM Algorithm and Extensions
McLACHLAN · Discriminant Analysis and Statistical Pattern Recognition
McNEIL · Epidemiological Research Methods
MILLER · Survival Analysis
MONTGOMERY and MYERS · Response Surface Methodology: Process and Product in Optimization Using Designed Experiments
MONTGOMERY and PECK · Introduction to Linear Regression Analysis, Second Edition
NELSON · Accelerated Testing, Statistical Models, Test Plans, and Data Analyses
NELSON · Applied Life Data Analysis
OCHI · Applied Probability and Stochastic Processes in Engineering and Physical Sciences
OKABE, BOOTS, and SUGIHARA · Spatial Tesselations: Concepts and Applications of Voronoi Diagrams
OSBORNE · Finite Algorithms in Optimization and Data Analysis
PANKRATZ · Forecasting with Dynamic Regression Models
PANKRATZ · Forecasting with Univariate Box-Jenkins Models: Concepts and Cases
PIANTADOSI · Clinical Trials: A Methodologic Perspective
PORT · Theoretical Probability for Applications
PUTERMAN · Markov Decision Processes: Discrete Stochastic Dynamic Programming
RACHEV · Probability Metrics and the Stability of Stochastic Models
RENCHER · Multivariate Statistical Inference and Applications
RENYI · A Diary on Information Theory
RIPLEY · Spatial Statistics
RIPLEY · Stochastic Simulation
ROSS · Introduction to Probability and Statistics for Engineers and Scientists
ROUSSEEUW and LEROY · Robust Regression and Outlier Detection
RUBIN · Multiple Imputation for Nonresponse in Surveys
RYAN · Modern Regression Methods
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RYAN · Statistical Methods for Quality Improvement
SCHOTT · Matrix Analysis for Statistics
SCHUSS · Theory and Applications of Stochastic Differential Equations
SCOTT · Multivariate Density Estimation: Theory, Practice, and Visualization
*SEARLE Linear Models
SEARLE · Linear Models for Unbalanced Data
SEARLE · Matrix Algebra Useful for Statistics
SEARLE, CASELLA, and McCULLOCH · Variance Components
SKINNER, HOLT, and SMITH · Analysis of Complex Surveys
STOYAN, KENDALL, and MECKE · Stochastic Geometry and Its Applications, Second Edition
STOYAN and STOYAN · Fractals, Random Shapes and Point Fields: Methods of Geometrical Statistics
THOMPSON · Empirical Model Building
THOMPSON · Sampling
TIERNEY · LISP-STAT: An Object-Oriented Environment for Statistical Computing and Dynamic Graphics
TIJMS · Stochastic Modeling and Analysis: A Computational Approach
TITTERINGTON, SMITH, and MAKOV · Statistical Analysis of Finite Mixture Distributions
UPTON and FINGLETON · Spatial Data Analysis by Example, Volume 1: Point Pattern and Quantitative Data
UPTON and FINGLETON · Spatial Data Analysis by Example, Volume II: Categorical and Directional Data
VAN RIJCKEVORSEL and DE LEEUW · Component and Correspondence Analysis
*Now available in a lower priced paperback edition in the Wiley Classics Library.Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation
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Applied Probability and Statistics (Continued)
WEISBERG · Applied Linear Regression, Second Edition
WESTFALL and YOUNG · Resampling-Based Multiple Testing: Examples and Methods for p-Value Adjustment
WHITTLE · Optimization Over Time: Dynamic Programming and Stochastic Control, Volume I and Volume II
WHITTLE · Systems in Stochastic Equilibrium
WONNACOTT and WONNACOTT · Econometrics, Second Edition
WONNACOTT and WONNACOTT · Introductory Statistics, Fifth Edition
WONNACOTT and WONNACOTT · Introductory Statistics for Business and Economics, Fourth Edition
WOODING · Planning Pharmaceutical Clinical Trials: Basic Statistical Principles
WOOLSON · Statistical Methods for the Analysis of Biomedical Data
*ZELLNER · An Introduction to Bayesian Inference in Econometrics
Tracts on Probability and Statistics
BILLINGSLEY · Convergence of Probability Measures
KELLY · Reversability and Stochastic Networks
TOUTENBURG · Prior Information in Linear Models
*Now available in a lower priced paperback edition in the Wiley Classics Library.Start of Citation[PU]John Wiley & Sons, Inc. (US)[/PU][DP]1998[/DP]End of Citation
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