Methods of Multivariate Analysis

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  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 2012-07-10
  • Publisher: Wiley

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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

Prefacep. xvii
Acknowledgmentsp. xxi
Introductionp. 1
Why Multivariate Analysis?p. 1
Prerequisitesp. 3
Objectivesp. 3
Basic Types of Data And Analysisp. 4
Matrix Algebrap. 7
Introductionp. 7
Notation and Basic Definitionsp. 8
Operationsp. 11
Partitioned Matricesp. 22
Rankp. 23
Inversep. 25
Positive Definite Matricesp. 26
Determinantsp. 28
Tracep. 31
Orthogonal Vectors and Matricesp. 31
Eigenvalues and Eigenvectorsp. 32
Kronecker and VEC Notationp. 37
Problemsp. 39
Characterizing and Displaying Multivariate Datap. 47
Mean and Variance of a Univariate Random Variablep. 47
Covariance and Correlation Of Bivariate Random Variablesp. 49
Scatter Plots of Bivariate Samplesp. 55
Graphical Displays for Multivariate Samplesp. 56
Dynamic Graphicsp. 58
Mean Vectorsp. 63
Covariance Matricesp. 66
Correlation Matricesp. 69
Mean Vectors and Covariance Matrices for Subsets of Variablesp. 71
Two Subsetsp. 71
Three or More Subsetsp. 73
Linear Combinations of Variablesp. 75
Sample Propertiesp. 75
Population Propertiesp. 81
Measures of Overall Variabilityp. 81
Estimation of Missing Valuesp. 82
Distance Between Vectorsp. 84
Problemsp. 85
The Multivariate Normal Distributionp. 91
Multivariate Normal Density Functionp. 91
Properties of Multivariate Normal Random Variablesp. 94
Estimation in the Multivariate Normalp. 99
Assessing Multivariate Normalityp. 101
Transformations to Normalityp. 108
Outliersp. 111
Problemsp. 117
Tests on One or Two Mean Vectorsp. 125
Multivariate Versus Univariate Testsp. 125
Tests on With ??Knownp. 126
Tests on When ??is Unknownp. 130
Comparing two Mean Vectorsp. 134
Tests on Individual Variables Conditional on Rejection of H0 by the T2-testp. 139
Computation of T2p. 143
Paired Observations Testp. 145
Test for Additional Informationp. 149
Profile Analysisp. 152
Profile Analysisp. 154
Problemsp. 161
Multivariate Analysis of Variancep. 169
One-way Modelsp. 169
Comparison of the Four Manova Test Statisticsp. 189
Contrastsp. 191
Tests on Individual Variables Following Rejection of H0 by the Overall Manova Testp. 195
Two-Way Classificationp. 198
Other Modelsp. 207
Checking on the Assumptionsp. 210
Profile Analysisp. 211
Repeated Measures Designsp. 215
Growth Curvesp. 232
Tests on a Subvectorp. 241
Problemsp. 244
Tests on Covariance Matricesp. 259
Introductionp. 259
Testing a Specified Pattern for ∑p. 259
Tests Comparing Covariance Matricesp. 265
Tests of Independencep. 269
Problemsp. 276
Discriminant Analysis: Description of Group Separationp. 281
Introductionp. 281
The Discriminant Function for two Groupsp. 282
Relationship Between two-group Discriminant Analysis and Multiple Regressionp. 286
Discriminant Analysis for Several Groupsp. 288
Standardized Discriminant Functionsp. 292
Tests of Significancep. 294
Interpretation of Discriminant Functionsp. 298
Scatter Plotsp. 301
Stepwise Selection of Variablesp. 303
Problemsp. 306
Classification Analysis: Allocation of Observations to Groupsp. 309
Introductionp. 309
Classification into two Groupsp. 310
Classification into Several Groupsp. 314
Estimating Misclassification Ratesp. 318
Improved Estimates of Error Ratesp. 320
Subset Selectionp. 322
Nonparametric Proceduresp. 326
Problemsp. 336
Multivariate Regressionp. 339
Introductionp. 339
Multiple Regression: Fixed X’sp. 340
Multiple Regression: Random X’sp. 354
Multivariate Multiple Regression: Estimationp. 354
Multivariate Multiple Regression: Hypothesis Testsp. 364
Multivariate Multiple Regression: Predictionp. 370
Measures of Association Between the Y’s and the X’sp. 372
Subset Selectionp. 374
Multivariate Regression: Random X’sp. 380
Problemsp. 381
Canonical Correlationp. 385
Introductionp. 385
Canonical Correlations and Canonical Variatesp. 385
Properties of Canonical Correlationsp. 390
Tests of Significancep. 391
Interpretationp. 395
Relationships of Canonical Correlation Analysis to Other Multivariate Problemsp. 402
Principal Component Analysisp. 405
Introductionp. 405
Geometric and Algebraic Bases of Principal Componentsp. 406
Principal Components and Perpendicular Regressionp. 412
Plotting of Principal Componentsp. 414
Principal Components from the Correlation Matrixp. 419
Deciding How Many Components to Retainp. 423
Information in the Last Few Principal Componentsp. 427
Interpretation of Principal Componentsp. 427
Selection of Variablesp. 430
Problemsp. 432
Exploratory Factor Analysisp. 435
Introductionp. 435
Orthogonal Factor Modelp. 437
Estimation of Loadings and Communalitiesp. 442
Choosing the Number of Factors, mp. 453
Rotationp. 457
Factor Scoresp. 466
Validity of the Factor Analysis Modelp. 470
Relationship of Factor Analysis to Principal Component Analysisp. 475
Problemsp. 476
Confirmatory Factor Analysisp. 479
Introductionp. 479
Model Specification and Identificationp. 480
Parameter Estimation and Model Assessmentp. 487
Inference for Model Parametersp. 492
Factor Scoresp. 495
Problemsp. 496
Cluster Analysisp. 501
Introductionp. 501
Measures of Similarity or Dissimilarityp. 502
Hierarchical Clusteringp. 505
Nonhierarchical Methodsp. 531
Choosing the Number of Clustersp. 544
Cluster Validityp. 546
Clustering Variablesp. 547
Problemsp. 548
Graphical Proceduresp. 555
Multidimensional Scalingp. 555
Correspondence Analysisp. 565
Biplotsp. 580
Problemsp. 588
Tablesp. 597
Answers and Hints to Problemsp. 637
Data Sets and SAS Filesp. 727
Referencesp. 729
Indexp. 747
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