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Methods of Multivariate Analysis

Methods of Multivariate Analysis Second Edition Methods of Multivariate Analysis Second Edition ALVIN C. RENCHER. Brigham Young University A JOHN WILEY & SONS, INC. PUBLICATION. This book is printed on acid-free paper.. Copyright . c 2002 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, (978) 750-8400, fax (978).

3.3 Scatter Plots of Bivariate Samples, 50 3.4 Graphical Displays for Multivariate Samples, 52 ... 4.4 Assessing Multivariate Normality, 92 4.4.1 Investigating Univariate Normality, 92 4.4.2 Investigating Multivariate Normality, 96. ... 5.5 Tests on Individual Variables Conditional on Rejection of H0 by the T2-Test, 126

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Transcription of Methods of Multivariate Analysis

1 Methods of Multivariate Analysis Second Edition Methods of Multivariate Analysis Second Edition ALVIN C. RENCHER. Brigham Young University A JOHN WILEY & SONS, INC. PUBLICATION. This book is printed on acid-free paper.. Copyright . c 2002 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, (978) 750-8400, fax (978).

2 750-4744. 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: For ordering and customer service, call 1-800-CALL-WILEY. Library of Congress Cataloging-in-Publication Data Rencher, Alvin C., 1934 . Methods of Multivariate Analysis / Alvin C. Rencher. 2nd ed. p. cm. (Wiley series in probability and mathematical statistics). A Wiley-Interscience publication.. Includes bibliographical references and index. ISBN 0-471-41889-7 (cloth). 1. Multivariate Analysis . I. Title. II. Series. QA278 .R45 2001. 35 dc21 2001046735. Printed in the United States of America 10 9 8 7 6 5 4 3 2 1. Contents 1. Introduction 1.

3 Why Multivariate Analysis ?, 1. Prerequisites, 3. Objectives, 3. Basic Types of Data and Analysis , 3. 2. Matrix Algebra 5. Introduction, 5. Notation and Basic Definitions, 5. Matrices, Vectors, and Scalars, 5. Equality of Vectors and Matrices, 7. Transpose and Symmetric Matrices, 7. Special Matrices, 8. Operations, 9. Summation and Product Notation, 9. Addition of Matrices and Vectors, 10. Multiplication of Matrices and Vectors, 11. Partitioned Matrices, 20. Rank, 22. Inverse, 23. Positive Definite Matrices, 25. Determinants, 26. Trace, 30. Orthogonal Vectors and Matrices, 31. Eigenvalues and Eigenvectors, 32. Definition, 32. I + A and I A, 33. tr(A) and |A|, 34. Positive Definite and Semidefinite Matrices, 34. The Product AB, 35. Symmetric Matrix, 35.

4 V vi CONTENTS. Spectral Decomposition, 35. Square Root Matrix, 36. Square Matrices and Inverse Matrices, 36. Singular Value Decomposition, 36. 3. Characterizing and Displaying Multivariate Data 43. Mean and Variance of a Univariate Random Variable, 43. Covariance and Correlation of Bivariate Random Variables, 45. Covariance, 45. Correlation, 49. Scatter Plots of Bivariate Samples, 50. Graphical Displays for Multivariate Samples, 52. Mean Vectors, 53. Covariance Matrices, 57. Correlation Matrices, 60. Mean Vectors and Covariance Matrices for Subsets of Variables, 62. Two Subsets, 62. Three or More Subsets, 64. Linear Combinations of Variables, 66. Sample Properties, 66. Population Properties, 72. Measures of Overall Variability, 73. Estimation of Missing Values, 74.

5 Distance between Vectors, 76. 4. The Multivariate Normal Distribution 82. Multivariate Normal Density Function, 82. Univariate Normal Density, 82. Multivariate Normal Density, 83. Generalized Population Variance, 83. Diversity of Applications of the Multivariate Normal, 85. Properties of Multivariate Normal Random Variables, 85. Estimation in the Multivariate Normal, 90. Maximum Likelihood Estimation, 90. Distribution of y and S, 91. Assessing Multivariate normality , 92. Investigating Univariate normality , 92. Investigating Multivariate normality , 96. CONTENTS vii Outliers, 99. Outliers in Univariate Samples, 100. Outliers in Multivariate Samples, 101. 5. tests on One or Two Mean Vectors 112. Multivariate versus Univariate tests , 112. tests on with Known, 113.

6 Review of Univariate Test for H0 : = 0. with Known, 113. Multivariate Test for H0 : = 0 with Known, 114. tests on When Is Unknown, 117. Review of Univariate t-Test for H0 : = 0 with . Unknown, 117. Hotelling's T 2 -Test for H0 : = 0 with Unknown, 117. Comparing Two Mean Vectors, 121. Review of Univariate Two-Sample t-Test, 121. Multivariate Two-Sample T 2 -Test, 122. Likelihood Ratio tests , 126. tests on Individual Variables Conditional on Rejection of H0 by the T 2 -Test, 126. Computation of T 2 , 130. Obtaining T 2 from a MANOVA Program, 130. Obtaining T 2 from Multiple Regression, 130. Paired Observations Test, 132. Univariate Case, 132. Multivariate Case, 134. Test for Additional Information, 136. Profile Analysis , 139. One-Sample Profile Analysis , 139.

7 Two-Sample Profile Analysis , 141. 6. Multivariate Analysis of Variance 156. One-Way Models, 156. Univariate One-Way Analysis of Variance (ANOVA), 156. Multivariate One-Way Analysis of Variance Model (MANOVA), 158. Wilks' Test Statistic, 161. Roy's Test, 164. Pillai and Lawley Hotelling tests , 166. viii CONTENTS. Unbalanced One-Way MANOVA, 168. Summary of the Four tests and Relationship to T 2 , 168. Measures of Multivariate Association, 173. Comparison of the Four Manova Test Statistics, 176. Contrasts, 178. Univariate Contrasts, 178. Multivariate Contrasts, 180. tests on Individual Variables Following Rejection of H0 by the Overall MANOVA Test, 183. Two-Way Classification, 186. Review of Univariate Two-Way ANOVA, 186. Multivariate Two-Way MANOVA, 188.

8 Other Models, 195. Higher Order Fixed Effects, 195. Mixed Models, 196. Checking on the Assumptions, 198. Profile Analysis , 199. Repeated Measures Designs, 204. Multivariate vs. Univariate Approach, 204. One-Sample Repeated Measures Model, 208. k-Sample Repeated Measures Model, 211. Computation of Repeated Measures tests , 212. Repeated Measures with Two Within-Subjects Factors and One Between-Subjects Factor, 213. Repeated Measures with Two Within-Subjects Factors and Two Between-Subjects Factors, 219. Additional Topics, 221. Growth Curves, 221. Growth Curve for One Sample, 221. Growth Curves for Several Samples, 229. Additional Topics, 230. tests on a Subvector, 231. Test for Additional Information, 231. Stepwise Selection of Variables, 233.

9 7. tests on Covariance Matrices 248. Introduction, 248. Testing a Specified Pattern for , 248. Testing H0 : = 0 , 248. CONTENTS ix Testing Sphericity, 250. Testing H0 : = 2 [(1 )I + J], 252. tests Comparing Covariance Matrices, 254. Univariate tests of Equality of Variances, 254. Multivariate tests of Equality of Covariance Matrices, 255. tests of Independence, 259. Independence of Two Subvectors, 259. Independence of Several Subvectors, 261. Test for Independence of All Variables, 265. 8. Discriminant Analysis : Description of Group Separation 270. Introduction, 270. The Discriminant Function for Two Groups, 271. Relationship between Two-Group Discriminant Analysis and Multiple Regression, 275. Discriminant Analysis for Several Groups, 277. Discriminant Functions, 277.

10 A Measure of Association for Discriminant Functions, 282. Standardized Discriminant Functions, 282. tests of Significance, 284. tests for the Two-Group Case, 284. tests for the Several-Group Case, 285. Interpretation of Discriminant Functions, 288. Standardized Coefficients, 289. Partial F-Values, 290. Correlations between Variables and Discriminant Functions, 291. Rotation, 291. Scatter Plots, 291. Stepwise Selection of Variables, 293. 9. Classification Analysis : Allocation of Observations to Groups 299. Introduction, 299. Classification into Two Groups, 300. Classification into Several Groups, 304. Equal Population Covariance Matrices: Linear Classification Functions, 304. Unequal Population Covariance Matrices: Quadratic Classification Functions, 306.


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