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.

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• Preface xvii Acknowledgments xxi1 Introduction 11.1 Why Multivariate Analysis? 11.2 Prerequisites 31.3 Objectives 31.4 Basic Types of Data And Analysis 42 Matrix Algebra 72.1 Introduction 72.2 Notation and Basic Definitions 82.3 Operations 112.4 Partitioned Matrices 222.5 Rank 232.6 Inverse 252.7 Positive Definite Matrices 262.8 Determinants 282.9 Trace 312.10 Orthogonal Vectors and Matrices 312.11 Eigenvalues and Eigenvectors 322.12 Kronecker and VEC Notation 37Problems 393 Characterizing and Displaying Multivariate Data 473.1 Mean and Variance of a Univariate Random Variable 473.2 Covariance and Correlation Of Bivariate Random Variables 493.3 Scatter Plots of Bivariate Samples 553.4 Graphical Displays for Multivariate Samples 563.5 Dynamic Graphics 583.6 Mean Vectors 633.7 Covariance Matrices 663.8 Correlation Matrices 693.9 Mean Vectors and Covariance Matrices for Subsets of Variables 713.9.1 Two Subsets 713.9.2 Three or More Subsets 733.10 Linear Combinations of Variables 753.10.1 Sample Properties 753.10.2 Population Properties 813.11 Measures of Overall Variability 813.12 Estimation of Missing Values 823.13 Distance Between Vectors 84Problems 854 The Multivariate Normal Distribution 914.1 Multivariate Normal Density Function 914.2 Properties of Multivariate Normal Random Variables 944.3 Estimation in the Multivariate Normal 994.4 Assessing Multivariate Normality 1014.5 Transformations to Normality 1084.6 Outliers 111Problems 1175 Tests on One or Two Mean Vectors 1255.1 Multivariate Versus Univariate Tests 1255.2 Tests on µ With ??Known 1265.3 Tests on µ When ??is Unknown 1305.4 Comparing two Mean Vectors 1345.5 Tests on Individual Variables Conditional on Rejection of H0 by the T2-test1395.6 Computation of T2 1435.7 Paired Observations Test 1455.8 Test for Additional Information 1495.9 Profile Analysis 152Profile Analysis 154Problems 1616 Multivariate Analysis of Variance 1696.1 One-way Models 1696.2 Comparison of the Four Manova Test Statistics 1896.3 Contrasts 1916.4 Tests on Individual Variables Following Rejection of H0 by the Overall Manova Test 1956.5 Two-Way Classification 1986.6 Other Models 2076.7 Checking on the Assumptions 2106.8 Profile Analysis 2116.9 Repeated Measures Designs 2156.10 Growth Curves 2326.11 Tests on a Subvector 241Problems 2447 Tests on Covariance Matrices 2597.1 Introduction 2597.2 Testing a Specified Pattern for ∑ 2597.3 Tests Comparing Covariance Matrices 2657.4 Tests of Independence 269Problems 2768 Discriminant Analysis: Description of Group Separation 2818.1 Introduction 2818.2 The Discriminant Function for two Groups 2828.3 Relationship Between two-group Discriminant Analysis and Multiple Regression 2868.4 Discriminant Analysis for Several Groups 2888.5 Standardized Discriminant Functions 2928.6 Tests of Significance 2948.7 Interpretation of Discriminant Functions 2988.8 Scatter Plots 3018.9 Stepwise Selection of Variables 303Problems 3069 Classification Analysis: Allocation of Observations to Groups3099.1 Introduction 3099.2 Classification into two Groups 3109.3 Classification into Several Groups 3149.4 Estimating Misclassification Rates 3189.5 Improved Estimates of Error Rates 3209.6 Subset Selection 3229.7 Nonparametric Procedures 326Problems 33610 Multivariate Regression 33910.1 Introduction 33910.2 Multiple Regression: Fixed X’s 34010.3 Multiple Regression: Random X’s 35410.4 Multivariate Multiple Regression: Estimation 35410.5 Multivariate Multiple Regression: Hypothesis Tests 36410.6 Multivariate Multiple Regression: Prediction 37010.7 Measures of Association Between the Y’s and the X’s 37210.8 Subset Selection 37410.9 Multivariate Regression: Random X’s 380Problems 38111 Canonical Correlation 38511.1 Introduction 38511.2 Canonical Correlations and Canonical Variates 38511.3 Properties of Canonical Correlations 39011.4 Tests of Significance 39111.5 Interpretation 39511.6 Relationships of Canonical Correlation Analysis to Other MultivariateProblems 40212 Principal Component Analysis 40512.1 Introduction 40512.2 Geometric and Algebraic Bases of Principal Components 40612.3 Principal Components and Perpendicular Regression 41212.4 Plotting of Principal Components 41412.5 Principal Components from the Correlation Matrix 41912.6 Deciding How Many Components to Retain 42312.7 Information in the Last Few Principal Components 42712.8 Interpretation of Principal Components 42712.9 Selection of Variables 430Problems 43213 Exploratory Factor Analysis 43513.1 Introduction 43513.2 Orthogonal Factor Model 43713.3 Estimation of Loadings and Communalities 44213.4 Choosing the Number of Factors, m 45313.5 Rotation 45713.6 Factor Scores 46613.7 Validity of the Factor Analysis Model 47013.8 Relationship of Factor Analysis to Principal Component Analysis 475Problems 47614 Confirmatory Factor Analysis 47914.1 Introduction 47914.2 Model Specification and Identification 48014.3 Parameter Estimation and Model Assessment 48714.4 Inference for Model Parameters 49214.5 Factor Scores 495Problems 49615 Cluster Analysis 50115.1 Introduction 50115.2 Measures of Similarity or Dissimilarity 50215.3 Hierarchical Clustering 50515.4 Nonhierarchical Methods 53115.5 Choosing the Number of Clusters 54415.6 Cluster Validity 54615.7 Clustering Variables 547Problems 54816 Graphical Procedures 55516.1 Multidimensional Scaling 55516.2 Correspondence Analysis 56516.3 Biplots 580Problems 588Appendix A: Tables 597Appendix B: Answers and Hints to Problems 637Appendix C: Data Sets and SAS Files 727References 729Index 747

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.

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• Preface xvii Acknowledgments xxi1 Introduction 11.1 Why Multivariate Analysis? 11.2 Prerequisites 31.3 Objectives 31.4 Basic Types of Data And Analysis 42 Matrix Algebra 72.1 Introduction 72.2 Notation and Basic Definitions 82.3 Operations 112.4 Partitioned Matrices 222.5 Rank 232.6 Inverse 252.7 Positive Definite Matrices 262.8 Determinants 282.9 Trace 312.10 Orthogonal Vectors and Matrices 312.11 Eigenvalues and Eigenvectors 322.12 Kronecker and VEC Notation 37Problems 393 Characterizing and Displaying Multivariate Data 473.1 Mean and Variance of a Univariate Random Variable 473.2 Covariance and Correlation Of Bivariate Random Variables 493.3 Scatter Plots of Bivariate Samples 553.4 Graphical Displays for Multivariate Samples 563.5 Dynamic Graphics 583.6 Mean Vectors 633.7 Covariance Matrices 663.8 Correlation Matrices 693.9 Mean Vectors and Covariance Matrices for Subsets of Variables 713.9.1 Two Subsets 713.9.2 Three or More Subsets 733.10 Linear Combinations of Variables 753.10.1 Sample Properties 753.10.2 Population Properties 813.11 Measures of Overall Variability 813.12 Estimation of Missing Values 823.13 Distance Between Vectors 84Problems 854 The Multivariate Normal Distribution 914.1 Multivariate Normal Density Function 914.2 Properties of Multivariate Normal Random Variables 944.3 Estimation in the Multivariate Normal 994.4 Assessing Multivariate Normality 1014.5 Transformations to Normality 1084.6 Outliers 111Problems 1175 Tests on One or Two Mean Vectors 1255.1 Multivariate Versus Univariate Tests 1255.2 Tests on µ With ??Known 1265.3 Tests on µ When ??is Unknown 1305.4 Comparing two Mean Vectors 1345.5 Tests on Individual Variables Conditional on Rejection of H0 by the T2-test1395.6 Computation of T2 1435.7 Paired Observations Test 1455.8 Test for Additional Information 1495.9 Profile Analysis 152Profile Analysis 154Problems 1616 Multivariate Analysis of Variance 1696.1 One-way Models 1696.2 Comparison of the Four Manova Test Statistics 1896.3 Contrasts 1916.4 Tests on Individual Variables Following Rejection of H0 by the Overall Manova Test 1956.5 Two-Way Classification 1986.6 Other Models 2076.7 Checking on the Assumptions 2106.8 Profile Analysis 2116.9 Repeated Measures Designs 2156.10 Growth Curves 2326.11 Tests on a Subvector 241Problems 2447 Tests on Covariance Matrices 2597.1 Introduction 2597.2 Testing a Specified Pattern for ∑ 2597.3 Tests Comparing Covariance Matrices 2657.4 Tests of Independence 269Problems 2768 Discriminant Analysis: Description of Group Separation 2818.1 Introduction 2818.2 The Discriminant Function for two Groups 2828.3 Relationship Between two-group Discriminant Analysis and Multiple Regression 2868.4 Discriminant Analysis for Several Groups 2888.5 Standardized Discriminant Functions 2928.6 Tests of Significance 2948.7 Interpretation of Discriminant Functions 2988.8 Scatter Plots 3018.9 Stepwise Selection of Variables 303Problems 3069 Classification Analysis: Allocation of Observations to Groups3099.1 Introduction 3099.2 Classification into two Groups 3109.3 Classification into Several Groups 3149.4 Estimating Misclassification Rates 3189.5 Improved Estimates of Error Rates 3209.6 Subset Selection 3229.7 Nonparametric Procedures 326Problems 33610 Multivariate Regression 33910.1 Introduction 33910.2 Multiple Regression: Fixed X’s 34010.3 Multiple Regression: Random X’s 35410.4 Multivariate Multiple Regression: Estimation 35410.5 Multivariate Multiple Regression: Hypothesis Tests 36410.6 Multivariate Multiple Regression: Prediction 37010.7 Measures of Association Between the Y’s and the X’s 37210.8 Subset Selection 37410.9 Multivariate Regression: Random X’s 380Problems 38111 Canonical Correlation 38511.1 Introduction 38511.2 Canonical Correlations and Canonical Variates 38511.3 Properties of Canonical Correlations 39011.4 Tests of Significance 39111.5 Interpretation 39511.6 Relationships of Canonical Correlation Analysis to Other MultivariateProblems 40212 Principal Component Analysis 40512.1 Introduction 40512.2 Geometric and Algebraic Bases of Principal Components 40612.3 Principal Components and Perpendicular Regression 41212.4 Plotting of Principal Components 41412.5 Principal Components from the Correlation Matrix 41912.6 Deciding How Many Components to Retain 42312.7 Information in the Last Few Principal Components 42712.8 Interpretation of Principal Components 42712.9 Selection of Variables 430Problems 43213 Exploratory Factor Analysis 43513.1 Introduction 43513.2 Orthogonal Factor Model 43713.3 Estimation of Loadings and Communalities 44213.4 Choosing the Number of Factors, m 45313.5 Rotation 45713.6 Factor Scores 46613.7 Validity of the Factor Analysis Model 47013.8 Relationship of Factor Analysis to Principal Component Analysis 475Problems 47614 Confirmatory Factor Analysis 47914.1 Introduction 47914.2 Model Specification and Identification 48014.3 Parameter Estimation and Model Assessment 48714.4 Inference for Model Parameters 49214.5 Factor Scores 495Problems 49615 Cluster Analysis 50115.1 Introduction 50115.2 Measures of Similarity or Dissimilarity 50215.3 Hierarchical Clustering 50515.4 Nonhierarchical Methods 53115.5 Choosing the Number of Clusters 54415.6 Cluster Validity 54615.7 Clustering Variables 547Problems 54816 Graphical Procedures 55516.1 Multidimensional Scaling 55516.2 Correspondence Analysis 56516.3 Biplots 580Problems 588Appendix A: Tables 597Appendix B: Answers and Hints to Problems 637Appendix C: Data Sets and SAS Files 727References 729Index 747