Methods of Multivariate Analysis

by Rencher, Alvin C.; Christensen, William F.

• ISBN13: 9780470178966
• ISBN10: 0470178965
• Edition: 3rd
• Format: Hardcover
• Copyright: 2012-07-10
• Publisher: Wiley

Summary

This new edition, now with a co-author, offers a complete and up-to-date examination of the field. The authors have streamlined previously tedious topics, such as multivariate regression and MANOVA techniques, to add newer, more timely content. Each chapter contains exercises, providing readers with the opportunity to test and extend their understanding. The new edition also presents several expanded topics in Kronecker product; prediction errors; maximum likelihood estimation; and selective key, but accessible proofs. This resource meets the needs of both statistics majors and those of students and professionals in other fields.

Author Biography

ALVIN C. RENCHER is Professor Emeritus in the Department of Statistics at Brigham Young University. A Fellow of the American Statistical Association, he is the author of Linear Models in Statistics, Second Edition and Multivariate Statistical Inference and Applications, both published by Wiley.

WILLIAM F. CHRISTENSEN is Professor in the Department of Statistics at Brigham Young University. He has been published extensively in his areas of research interest, which include multivariate analysis, resampling methods, and spatial and environmental statistics.

Table of Contents

Preface	p. xvii
Acknowledgments	p. xxi
Introduction	p. 1
Why Multivariate Analysis?	p. 1
Prerequisites	p. 3
Objectives	p. 3
Basic Types of Data And Analysis	p. 4
Matrix Algebra	p. 7
Introduction	p. 7
Notation and Basic Definitions	p. 8
Operations	p. 11
Partitioned Matrices	p. 22
Rank	p. 23
Inverse	p. 25
Positive Definite Matrices	p. 26
Determinants	p. 28
Trace	p. 31
Orthogonal Vectors and Matrices	p. 31
Eigenvalues and Eigenvectors	p. 32
Kronecker and VEC Notation	p. 37
Problems	p. 39
Characterizing and Displaying Multivariate Data	p. 47
Mean and Variance of a Univariate Random Variable	p. 47
Covariance and Correlation Of Bivariate Random Variables	p. 49
Scatter Plots of Bivariate Samples	p. 55
Graphical Displays for Multivariate Samples	p. 56
Dynamic Graphics	p. 58
Mean Vectors	p. 63
Covariance Matrices	p. 66
Correlation Matrices	p. 69
Mean Vectors and Covariance Matrices for Subsets of Variables	p. 71
Two Subsets	p. 71
Three or More Subsets	p. 73
Linear Combinations of Variables	p. 75
Sample Properties	p. 75
Population Properties	p. 81
Measures of Overall Variability	p. 81
Estimation of Missing Values	p. 82
Distance Between Vectors	p. 84
Problems	p. 85
The Multivariate Normal Distribution	p. 91
Multivariate Normal Density Function	p. 91
Properties of Multivariate Normal Random Variables	p. 94
Estimation in the Multivariate Normal	p. 99
Assessing Multivariate Normality	p. 101
Transformations to Normality	p. 108
Outliers	p. 111
Problems	p. 117
Tests on One or Two Mean Vectors	p. 125
Multivariate Versus Univariate Tests	p. 125
Tests on µ With ??Known	p. 126
Tests on µ When ??is Unknown	p. 130
Comparing two Mean Vectors	p. 134
Tests on Individual Variables Conditional on Rejection of H0 by the T2-test	p. 139
Computation of T2	p. 143
Paired Observations Test	p. 145
Test for Additional Information	p. 149
Profile Analysis	p. 152
Profile Analysis	p. 154
Problems	p. 161
Multivariate Analysis of Variance	p. 169
One-way Models	p. 169
Comparison of the Four Manova Test Statistics	p. 189
Contrasts	p. 191
Tests on Individual Variables Following Rejection of H0 by the Overall Manova Test	p. 195
Two-Way Classification	p. 198
Other Models	p. 207
Checking on the Assumptions	p. 210
Profile Analysis	p. 211
Repeated Measures Designs	p. 215
Growth Curves	p. 232
Tests on a Subvector	p. 241
Problems	p. 244
Tests on Covariance Matrices	p. 259
Introduction	p. 259
Testing a Specified Pattern for ∑	p. 259
Tests Comparing Covariance Matrices	p. 265
Tests of Independence	p. 269
Problems	p. 276
Discriminant Analysis: Description of Group Separation	p. 281
Introduction	p. 281
The Discriminant Function for two Groups	p. 282
Relationship Between two-group Discriminant Analysis and Multiple Regression	p. 286
Discriminant Analysis for Several Groups	p. 288
Standardized Discriminant Functions	p. 292
Tests of Significance	p. 294
Interpretation of Discriminant Functions	p. 298
Scatter Plots	p. 301
Stepwise Selection of Variables	p. 303
Problems	p. 306
Classification Analysis: Allocation of Observations to Groups	p. 309
Introduction	p. 309
Classification into two Groups	p. 310
Classification into Several Groups	p. 314
Estimating Misclassification Rates	p. 318
Improved Estimates of Error Rates	p. 320
Subset Selection	p. 322
Nonparametric Procedures	p. 326
Problems	p. 336
Multivariate Regression	p. 339
Introduction	p. 339
Multiple Regression: Fixed X’s	p. 340
Multiple Regression: Random X’s	p. 354
Multivariate Multiple Regression: Estimation	p. 354
Multivariate Multiple Regression: Hypothesis Tests	p. 364
Multivariate Multiple Regression: Prediction	p. 370
Measures of Association Between the Y’s and the X’s	p. 372
Subset Selection	p. 374
Multivariate Regression: Random X’s	p. 380
Problems	p. 381
Canonical Correlation	p. 385
Introduction	p. 385
Canonical Correlations and Canonical Variates	p. 385
Properties of Canonical Correlations	p. 390
Tests of Significance	p. 391
Interpretation	p. 395
Relationships of Canonical Correlation Analysis to Other Multivariate Problems	p. 402
Principal Component Analysis	p. 405
Introduction	p. 405
Geometric and Algebraic Bases of Principal Components	p. 406
Principal Components and Perpendicular Regression	p. 412
Plotting of Principal Components	p. 414
Principal Components from the Correlation Matrix	p. 419
Deciding How Many Components to Retain	p. 423
Information in the Last Few Principal Components	p. 427
Interpretation of Principal Components	p. 427
Selection of Variables	p. 430
Problems	p. 432
Exploratory Factor Analysis	p. 435
Introduction	p. 435
Orthogonal Factor Model	p. 437
Estimation of Loadings and Communalities	p. 442
Choosing the Number of Factors, m	p. 453
Rotation	p. 457
Factor Scores	p. 466
Validity of the Factor Analysis Model	p. 470
Relationship of Factor Analysis to Principal Component Analysis	p. 475
Problems	p. 476
Confirmatory Factor Analysis	p. 479
Introduction	p. 479
Model Specification and Identification	p. 480
Parameter Estimation and Model Assessment	p. 487
Inference for Model Parameters	p. 492
Factor Scores	p. 495
Problems	p. 496
Cluster Analysis	p. 501
Introduction	p. 501
Measures of Similarity or Dissimilarity	p. 502
Hierarchical Clustering	p. 505
Nonhierarchical Methods	p. 531
Choosing the Number of Clusters	p. 544
Cluster Validity	p. 546
Clustering Variables	p. 547
Problems	p. 548
Graphical Procedures	p. 555
Multidimensional Scaling	p. 555
Correspondence Analysis	p. 565
Biplots	p. 580
Problems	p. 588
Tables	p. 597
Answers and Hints to Problems	p. 637
Data Sets and SAS Files	p. 727
References	p. 729
Index	p. 747
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