Multivariate Statistical Inference and Applications

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

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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 1 Start 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. Introduction

1.2. Univariate and Bivariate Random Variables

1.2.1. Univariate Random Variables

1.2.2. Bivariate Random Variables

1.3. Mean Vectors and Covariance Matrices for Random Vectors

1.4. Correlation Matrices

1.5. Partitioned Mean Vectors and Covariance Matrices

1.6. Linear Functions of Random Variables

1.6.1. Sample Means, Variances, and Covariances

1.6.2. Population Means, Variances, and Covariances

1.7. Measuring Intercorrelation

1.8. Mahalanobis Distance

1.9. Missing Data

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1.10. Robust Estimators of µ and Σ

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2. The Multivariate Normal Distribution

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2.1. Univariate and Multivariate Normal Density Functions

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2.1.1. Univariate Normal

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2.1.2. Multivariate Normal

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2.1.3. Constant Density Ellipsoids

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2.1.4. Generating Multivariate Normal Data

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2.1.5. Moments

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2.2. Properties of Multivariate Normal Random Vectors

2.3. Estimation of Parameters in the Multivariate Normal Distribution

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2.3.1. Maximum Likelihood Method

2.3.2. Properties of y *and S

2.3.3. Wishart Distribution

2.4. Additional Topics 3. Hotelling's T2-Tests

3.1. Introduction

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3.2. Test for H0: µ = µ0 with Σ Known

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

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3.3.1. Univariate t-Test for H0: µ = µ0 with σ2 Unknown 3.3.2. Likelihood Ratio Method of Test Construction

3.3.3. One-Sample T2-Test

3.3.4. Formal Definition of T2 and Relationship to F

3.3.5. Effect on T2 of Adding a Variable

3.3.6. Properties of the T2-Test

3.3.7. Likelihood Ratio Test

3.3.8. Union-Intersection Test

3.4. Confidence Intervals and Tests for Linear Functions of µ

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3.4.1. Confidence Region for µ

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

3.4.3. Simultaneous Confidence Intervals for µj and a'µ

3.4.4. Bonferroni Confidence Intervals for µj and a'µ

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3.4.5. Tests for Ho: a'µ = a'µ0and H0: µj= µ0j

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3.4.6. Tests for H0:Cµ = 0

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3.5. Tests of H0: µ1 = µ2 Assuming Σ1= Σ2 3.5.1. Review of Univariate Likelihood Ratio Test for H0: µ1 = µ2 When σ2/1 = σ2/2

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3.5.2. Test for H0: µ1= µ2 When Σ1 = Σ2

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3.5.3. Effect on T2 of Adding a Variable

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3.5.4. Properties of the Two-Sample T2-Statistic

3.5.5. Likelihood Ratio and Union-Intersection Tests

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

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3.6.1. Confidence Region for µ1 µ2

3.6.2. Simultaneous Confidence Intervals for a'(µ1 µ2) and µ1j µ2j

3.6.3. Bonferroni Confidence Intervals for a'(µ1 µ2) and µ1j - µ2j

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3.6.4. Tests for H0: a'(µ1 µ2) = a'δ0and H0j: µ1j µ2j = 0

3.6.5. Test for H0: C(µ1 µ2) = 0

3.7. Robustness of the T2-test 3.7.1. Robustness to Σ1≠ Σ2 3.7.2. Robustness to Nonnormality

3.8. Paired Observation Test 3.9. Testing H0: µ1 = µ2When Σ1≠Σ2 3.9.1. Univariate Case

3.9.2. Multivariate Case

3.10. Power and Sample Size

3.11. Tests on a Subvector

3.11.1. Two-Sample Case

3.11.2. Step-Down Test

3.11.3. Selection of Variables

3.11.4. One-Sample Case

3.12. Nonnormal Approaches to Hypothesis Testing

3.12.1. Elliptically Contoured Distributions

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3.12.2 NonparametricTests

3.12.3 Robust Versions of T2

3.13. Application of T2 in Multivariate Quality Control 4. Multivariate Analysis of Variance

4.1. One-Way Classification

4.1.1. Model for One-Way Multivariate Analysis of Variance

4.1.2 Wilks' Likelihood Ratio Test

4.1.3. Roy's Union-Intersection Test

4.1.4. The Pillai and Lawley-Hotelling Test Statistics

4.1.5. Summary of the Four Test Statistics 4.1.6. Effect of an Additional Variable on Wilks' Λ 4.1.7. Tests on Individual Variables

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

4.3. Tests for Equality of Covariance Matrices

4.4. Power and Sample Size for the Four MANOVA Tests

4.5. Contrasts Among Mean Vectors

4.5.1. Univariate Contrasts

4.5.2. Multivariate Contrasts 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 Variance

4.7. Higher Order Models

4.8. Unbalanced Data

4.8.1. Introduction

4.8.2. Univariate One-Way Model

4.8.3. Multivariate One-Way Model

4.8.4. Univariate Two-Way Model

4.8.5. Multivariate Two-Way Model

4.9. Tests on a Subvector

4.9.1. Testing a Single Subvector

4.9.2. Step-Down Test

4.9.3. Stepwise Selection of Variables

4.10. Multivariate Analysis of Covariance

4.10.1. Introduction

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

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

4.10.4. Additional Topics in Univariate Analysis of Covariance

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4.10.5. Multivariate Analysis of Covariance

4.11. Alternative Approaches to Testing H0: µ1 = µ2 = · · · = µk 5. Discriminant Functions for Descriptive Group Separation

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5.1. Introduction

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5.2. Two Groups

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5.3. Several Groups

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5.3.1. Discriminant Functions

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5.3.2. Assumptions

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5.4. Standardized Coefficients

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5.5. Tests of Hypotheses

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5.5.1. Two Groups

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5.5.2. Several Groups

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5.6. Discriminant Analysis for Higher Order Designs

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5.7. Interpretation of Discriminant Functions

5.7.1. Standardized Coefficients and Partial F-Values

5.7.2. Correlations between Variables and Discriminant Functions

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5.7.3. Other Approaches

5.8. Confidence Intervals

5.9. Subset Selection

5.9.1. Discriminant Function Approach to Selection

5.9.2. Stepwise Selection

5.9.3. All Possible Subsets

5.9.4. Selection in Higher Order Designs

5.10. Bias in Subset Selection

5.11. Other Estimators of Discriminant Functions

5.11.1. Ridge Discriminant Analysis and Related Techniques

5.11.2. Robust Discriminant Analysis 6. Classification of Observations into Groups

6.1. Introduction

6.2. Two Groups

6.2.1. Equal Population Covariance Matrices

6.2.2. Unequal Population Covariance Matrices

6.2.3. Unequal Costs of Misclassification

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6.2.4. Posterior Probability Approach

6.2.5. Robustness to Departures from the Assumptions

6.2.6. Robust Procedures

6.3. Several Groups

6.3.1. Equal Population Covariance Matrices

6.3.2. Unequal Population Covariance Matrices

6.3.3. Use of Linear Discriminant Functions for Classification

6.4. Estimation of Error Rates

6.5. Correcting for Bias in the Apparent Error Rate

6.5.1. Partitioning the Sample

6.5.2. Holdout Method

6.5.3. Bootstrap Estimator

6.5.4. Comparison of Error Estimators

6.6. Subset Selection

6.6.1. Selection Based on Separation of Groups

6.6.2. Selection Based on Allocation

6.6.3. Selection in the Heteroscedastic Case

6.7. Bias in Stepwise Classification Analysis

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

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6.8.1. The Logistic Model for Two Groups with Σl = Σ2 6.8.2. Comparison of Logistic Classification with Linear Classification Functions 6.8.3. Quadratic Logistic Functions When Σl≠ Σ2 6.8.4. Logistic Classification for Several Groups

6.8.5. Additional Topics in Logistic Classification

6.8.6. Probit Classification

6.9. Additional Topics in Classification 7. Multivariate Regression

7.1. Introduction

7.2. Multiple Regression: Fixed x's

7.2.1. Least Squares Estimators and Properties 7.2.2. An Estimator for σ2 7.2.3. The Model in Centered Form

7.2.4. Hypothesis Tests and Confidence Intervals

7.2.5. R2 in Fixed-x Regression

7.2.6. Model Validation

7.3. Multiple Regression: Random x's

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7.3.1. Model for Random x's

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7.3.2. Estimation of β0, βl, and σ2

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7.3.3. R2 in Random-x Regression

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7.3.4. Tests and Confidence Intervals

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

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7.4.1. The Multivariate Model

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7.4.2. Least Squares Estimator for B

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7.4.3. Properties of B *

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7.4.4. An Estimator for Σ

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7.4.5. Normal Model for the yi's

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7.4.6. The Multivariate Model in Centered Form

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7.4.7. Measures of Multivariate Association

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

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7.5.1. Test for Significance of Regression

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7.5.2. Test on a Subset of the Rows of B

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

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

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7.5.5. Simultaneous Tests and Confidence Intervals for the βjk's and Bilinear Functions a'Bb 7.5.6. Tests in the Presence of Missing Data

7.6. Multivariate Model Validation: Fixed x's

7.6.1. Lack-of-Fit Tests

7.6.2. Residuals

7.6.3. Influence and Outliers

7.6.4. Measurement Errors

7.7. Multivariate Regression: Random x's

7.7.1. Multivariate Normal Model for Random x's 7.7.2. Estimation of β0, B1, and Σ 7.7.3. Tests and Confidence Intervals in the Multivariate Random-x Case

7.8. Additional Topics

7.8.1. Correlated Response Methods

7.8.2. Categorical Data

7.8.3. Subset Selection

7.8.4. Other Topics 8. Canonical Correlation

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8.1. Introduction

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8.2. Canonical Correlations and Canonical Variates

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8.3. Properties of Canonical Correlations and Variates

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8.3.1. Properties of Canonical Correlations

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8.3.2. Properties of Canonical Variates

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8.4. Tests of Significance for Canonical Correlations

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8.4.1. Tests of Independence of y and x

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

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8.5. Validation

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8.6. Interpretation of Canonical Variates

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8.6.1. Standardized Coefficients

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8.6.2. Rotation of Canonical Variate Coefficients

8.6.3. Correlations between Variables and Canonical Variates

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8.7. Redundancy Analysis

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

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9. Principal Component Analysis

9.1. Introduction

9.2. Definition and Properties of Principal Components

9.2.1. Maximum Variance Property

9.2.2. Principal Components as Projections

9.2.3. Properties of Principal Components

9.3. Principal Components as a Rotation of Axes

9.4. Principal Components from the Correlation Matrix

9.5. Methods for Discarding Components

9.5.1. Percent of Variance

9.5.2. Average Eigenvalue

9.5.3. Scree Graph

9.5.4. Significance Tests

9.5.5. Other Methods

9.6. Information in the Last Few Principal Components

9.7. Interpretation of Principal Components

9.7.1. Special Patterns in S or R

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9.7.2. Testing H0: Σ = σ2 [(1 ρ)I + ρJ] and Pρ = (1 ρ)I + ρJ 9.7.3. Additional Rotation

9.7.4. Correlations between Variables and Principal Components

9.8. Relationship Between Principal Components and Regression

9.8.1. Principal Component Regression

9.8.2. Latent Root Regression

9.9. Principal Component Analysis with Grouped Data

9.10. Additional Topics 10. Factor Analysis

10.1. Introduction

10.2. Basic Factor Model

10.2.1. Model and Assumptions

10.2.2. Scale Invariance of the Model

10.2.3. Rotation of Factor Loadings in the Model

10.3. Estimation of Loadings and Communalities

10.3.1. Principal Component Method

10.3.2. Principal Factor Method

10.3.3. Iterated Principal Factor Method 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 Method

10.3.5. Other Methods

10.3.6. Comparison of Methods

10.4. Determining the Number of Factors, m

10.5. Rotation of Factor Loadings

10.5.1. Introduction

10.5.2. Orthogonal Rotation

10.5.3. Oblique Rotations

10.5.4. Interpretation of the Factors

10.6. Factor Scores

10.7. Applicability of the Factor Analysis Model

10.8. Factor Analysis and Grouped Data

10.9. Additional Topics Appendix A. Review of Matrix Algebra

A.1. Introduction

A. 1.1. Basic Definitions

A. 1.2. Matrices with Special Patterns

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A.2. Properties of Matrix Addition and Multiplication

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A.3. Partitioned Matrices

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A.4. Rank of Matrices

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A.5. Inverse Matrices

A.6. Positive Definite and Positive Semidefinite Matrices

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A.7. Determinants

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A.8. Trace of a Matrix

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A.9. Orthogonal Vectors and Matrices

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A.10. Eigenvalues and Eigenvectors

A. 11. Eigenstructure of Symmetric and Positive Definite Matrices

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A.12. Idempotent Matrices

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A. 13. Differentiation Appendix B. Tables

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Appendix C. Answers and Hints to Selected Problems

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Appendix D. About the Diskette

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Bibliography

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Index

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

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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, 328 Start 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, 1213 Start 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 file:///G|/page_551.html (2 of 4)7/11/2009 2:12:28 PM

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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, 281282 Start 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 file:///G|/page_552.html (1 of 4)7/11/2009 2:12:29 PM

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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 file:///G|/page_552.html (2 of 4)7/11/2009 2:12:29 PM

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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, 370 Start 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 file:///G|/page_553.html (3 of 4)7/11/2009 2:12:29 PM

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

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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, 148149 Start 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 file:///G|/page_555.html (2 of 4)7/11/2009 2:12:30 PM

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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 file:///G|/page_555.html (3 of 4)7/11/2009 2:12:30 PM

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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, 361363 Start 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 file:///G|/page_556.html (3 of 4)7/11/2009 2:12:30 PM

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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, 268 Start 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 file:///G|/page_557.html (2 of 4)7/11/2009 2:12:30 PM

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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 file:///G|/page_557.html (3 of 4)7/11/2009 2:12:30 PM

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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, 217221 Start 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 file:///G|/page_558.html (3 of 4)7/11/2009 2:12:31 PM

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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, 295297 Start 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), 401 Start 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 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 Probability and Statistics ANDERSON · An Introduction to Multivariate Statistical Analysis, Second Edition *ANDERSON · The Statistical Analysis of Time Series ARNOLD, BALAKRISHNAN, and NAGARAJA · A First Course in Order Statistics BACCELLI, COHEN, OLSDER, and QUADRAT · Synchronization and Linearity: An Algebra for Discrete Event Systems BARTOSZYNSKI and NIEWIADOMSKA-BUGAJ · Probability and Statistical Inference BERNARDO and SMITH · Bayesian Statistical Concepts and Theory BHATTACHARYYA and JOHNSON · Statistical Concepts and Methods BILLINGSLEY · Convergence of Probability Measures BILLINGSLEY · Probability and Measure, Second Edition BOROVKOV · Asymptotic Methods in Queuing Theory BRANDT, FRANKEN, and LISEK · Stationary Stochastic Models CAINES · Linear Stochastic Systems CAIROLI and DALANG · Sequential Stochastic Optimization CHEN · Recursive Estimation and Control for Stochastic Systems CONSTANTINE · Combinatorial Theory and Statistical Design COOK and WEISBERG · An Introduction to Regression Graphics COVER and THOMAS · Elements of Information Theory CSORGO and HORVATH · Weighted Approximations in Probability Statistics *DOOB · Stochastic Processes DUDEWICZ and MISHRA · Modem Mathematical Statistics

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Probability and Statistics (Continued) JUREK and MASON · Operator-Limit Distributions in Probability Theory KASS and VOS · Geometrical Foundations of Asymptotic Inference KAUFMAN and ROUSSEEUW · Finding Groups in Data: An Introduction to Cluster Analysis LAMPERTI · Probability: A Survey of the Mathematical Theory, Second Edition LARSON · Introduction to Probability Theory and Statistical Inference, Third Edition LESSLER and KALSBEEK · Nonsampling Error in Surveys LINDVALL · Lectures on the Coupling Method McFADDEN · Management of Data in Clinical Trials MANTON, WOODBURY, and TOLLEY · Statistical Applications Using Fuzzy Sets MARDIA · The Art of Statistical Science: A Tribute to G. S. Watson MORGENTHALER and TUKEY · Configural Polysampling: A Route to Practical Robustness MUIRHEAD · Aspects of Multivariate Statistical Theory OLIVER and SMITH · Influence Diagrams, Belief Nets and Decision Analysis *PARZEN · Moder Probability Theory and Its Applications PRESS · Bayesian Statistics: Principles, Models, and Applications PUKELSHEIM · Optimal Experimental Design PURI and SEN · Nonparametric Methods in General Linear Models PURI, VILAPLANA, and WERTZ · New Perspectives in Theoretical and Applied Statistics RAO · Asymptotic Theory of Statistical Inference RAO · Linear Statistical Inference and Its Applications, Second Edition *RAO and SHANBHAG · Choquet-Deny Type Functional Equations with Applications to Stochastic Models RENCHER · Methods of Multivariate Analysis ROBERTSON, WRIGHT, and DYKSTRA · Order Restricted Statistical Inference ROGERS and WILLIAMS · Diffusions, Markov Processes, and Martingales, Volume I: Foundations, Second Edition: Volume II: tto Calculus ROHATGI · An Introduction to Probability Theory and Mathematical Statistics ROSS · Stochastic Processes file:///G|/page_561.html (1 of 2)7/11/2009 2:12:31 PM

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RUBINSTEIN · Simulation and the Monte Carlo Method RUBINSTEIN and SHAPIRO · Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method RUZSA and SZEKELY · Algebraic Probability Theory SCHEFFE · The Analysis of Variance SEBER · Linear Regression Analysis SEBER · Multivariate Observations SEBER and WILD · Nonlinear Regression SERFLING · Approximation Theorems of Mathematical Statistics SHORACK and WELLNER · Empirical Processes with Applications to Statistics SMALL and McLEISH · Hilbert Space Methods in Probability and Statistical Inference STAPLETON · Linear Statistical Models STAUDTE and SHEATHER · Robust Estimation and Testing STOYANOV · Counterexamples in Probability STYAN · The Collected Papers of T. W. Anderson: 19431985 TANAKA · Time Series Analysis: Nonstationary and Noninvertible Distribution Theory THOMPSON and SEBER · Adaptive Sampling WELSH · Aspects of Statistical Inference WHITTAKER · Graphical Models in Applied Multivariate Statistics YANG · The Construction Theory of Denumerable Markov Processes *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 ABRAHAM and LEDOLTER · Statistical Methods for Forecasting AGRESTI · Analysis of Ordinal Categorical Data AGRESTI · Categorical Data Analysis AGRESTI · An Introduction to Categorical Data Analysis ANDERSON and LOYNES · The Teaching of Practical Statistics ANDERSON, AUQUIER, HAUCK, OAKES, VANDAELE, and WEISBERG · Statistical Methods for Comparative Studies ARMITAGE and DAVID (editors) · Advances in Biometry *ARTHANARI and DODGE · Mathematical Programming in Statistics ASMUSSEN · Applied Probability and Queues *BAILEY · The Elements of Stochastic Processes with Applications to the Natural Sciences BARNETT and LEWIS · Outliers in Statistical Data, Second Edition BARTHOLOMEW, FORBES, and McLEAN · Statistical Techniques for Manpower Planning, Second Edition BATES and WATTS · Nonlinear Regression Analysis and Its Applications BECHHOFER, SANTNER, and GOLDSMAN · Design and Analysis of Experiments for Statistical Selection, Screening, and Multiple Comparisons BELSLEY · Conditioning Diagnostics: Collinearity and Weak Data in Regression 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 file:///G|/page_562.html (1 of 2)7/11/2009 2:12:32 PM

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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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CUSTOMER NOTE: IF THIS BOOK IS ACCOMPANIED BY SOFTWARE, PLEASE READ THE FOLLOWING BEFORE OPENING THE PACKAGE. This software contains files to help you utilize the models described in the accompanying book. By opening the package, you are agreeing to be bound by the following agreement: This software product is protected by copyright and all rights are reserved by the author and John Wiley & Sons, Inc. You are licensed to use this software on a single computer. Copying the software to another medium or format for use on a single computer does not violate the U. S. Copyright Law. Copying the software for any other purpose is a violation of the U. S. Copyright Law. This software product is sold as is without warranty of any kind, either express or implied, including but not limited to the implied warranty of merchantability and fitness for a particular purpose. Neither Wiley nor its dealers or distributors assumes any liability of any alleged or actual damages arising from the use of or the inability V to use this software. (Some states do not allow the exclusion of implied warranties, so the exclusion may not apply to you.)

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