1. Introduction 1
1.1 Why Multivariate Analysis?, 1
1.2 Prerequisites, 3
1.3 Objectives, 3
1.4 Basic Types of Data and Analysis, 3
2. Matrix Algebra 5
2.1 Introduction, 5
2.2 Notation and Basic Definitions, 5
2.2.1 Matrices, Vectors, and Scalars, 5
2.2.2 Equality of Vectors and Matrices, 7
2.2.3 Transpose and Symmetric Matrices, 7
2.2.4 Special Matrices, 8
2.3 Operations, 9
2.3.1 Summation and Product Notation, 9
2.3.2 Addition of Matrices and Vectors, 10
2.3.3 Multiplication of Matrices and Vectors, 11
2.4 Partitioned Matrices, 20
2.5 Rank, 22
2.6 Inverse, 23
2.7 Positive Definite Matrices, 25
2.8 Determinants, 26
2.9 Trace, 30
2.10 Orthogonal Vectors and Matrices, 31
2.11 Eigenvalues and Eigenvectors, 32
2.11.1 Definition, 32
2.11.2 I + A and I − A,33
2.11.3 tr(A) and |A|,34
2.11.4 Positive Definite and Semidefinite Matrices, 34
2.11.5 The Product AB,35
2.11.6 Symmetric Matrix, 35
2.11.7 Spectral Decomposition, 35
2.11.8 Square Root Matrix, 36
2.11.9 Square Matrices and Inverse Matrices, 36
2.11.10 Singular Value Decomposition, 36
3. Characterizing and Displaying Multivariate Data 43
3.1 Mean and Variance of a Univariate Random Variable, 43
3.2 Covariance and Correlation of Bivariate Random Variables, 45
3.2.1 Covariance, 45
3.2.2 Correlation, 49
3.3 Scatter Plots of Bivariate Samples, 50
3.4 Graphical Displays for Multivariate Samples, 52
3.5 Mean Vectors, 53
3.6 Covariance Matrices, 57
3.7 Correlation Matrices, 60
3.8 Mean Vectors and Covariance Matrices for Subsets of
Va r i abl e s , 62
3.8.1 Two Subsets, 62
3.8.2 Three or More Subsets, 64
3.9 Linear Combinations of Variables, 66
3.9.1 Sample Properties, 66
3.9.2 Population Properties, 72
3.10 Measures of Overall Variability, 73
3.11 Estimation of Missing Values, 74
3.12 Distance between Vectors, 76
4. The Multivariate Normal Distribution 82
4.1 Multivariate Normal Density Function, 82
4.1.1 Univariate Normal Density, 82
4.1.2 Multivariate Normal Density, 83
4.1.3 Generalized Population Variance, 83
4.1.4 Diversity of Applications of the Multivariate Normal, 85
4.2 Properties of Multivariate Normal Random Variables, 85
4.3 Estimation in the Multivariate Normal, 90
4.3.1 Maximum Likelihood Estimation, 90
4.3.2 Distribution of y and S,91
4.4 Assessing Multivariate Normality, 92
4.4.1 Investigating Univariate Normality, 92
4.4.2 Investigating Multivariate Normality, 96
4.5 Outliers, 99
4.5.1 Outliers in Univariate Samples, 100
4.5.2 Outliers in Multivariate Samples, 101
5. Tests on One or Two Mean Vectors 112
5.1 Multivariate versus Univariate Tests, 112
5.2 Tests on with Known, 113
5.2.1 Review of Univariate Test for H0: µ = µ0
with σ Known, 113
5.2.2 Multivariate Test for H0: = 0 with Known, 114
5.3 Tests on When Is Unknown, 117
5.3.1 Review of Univariate t -Test for H0: µ = µ0 with σ
Unknown, 117
5.3.2 Hotelling’s T 2-Test for H0: = 0 with Unknown, 117
5.4 Comparing Two Mean Vectors, 121
5.4.1 Review of Univariate Two-Sample t -Test, 121
5.4.2 Multivariate Two-Sample T 2-Test, 122
5.4.3 Likelihood Ratio Tests, 126
5.5 Tests on Individual Variables Conditional on Rejection of H0 by
the T 2-Test, 126
5.6 Computation of T 2, 130
5.6.1 Obtaining T 2 from a MANOVA Program, 130
5.6.2 Obtaining T 2 from Multiple Regression, 130
5.7 Paired Observations Test, 132
5.7.1 Univariate Case, 132
5.7.2 Multivariate Case, 134
5.8 Test for Additional Information, 136
5.9 Profile Analysis, 139
5.9.1 One-Sample Profile Analysis, 139
5.9.2 Two-Sample Profile Analysis, 141
6. Multivariate Analysis of Variance 156
6.1 One-Way Models, 156
6.1.1 Univariate One-Way Analysis of Variance (ANOVA), 156
6.1.2 Multivariate One-Way Analysis of Variance Model
(MANOVA), 158
6.1.3 Wilks’ Test Statistic, 161
6.1.4 Roy’s Test, 164
6.1.5 Pillai and Lawley–Hotelling Tests, 166
6.1.6 Unbalanced One-Way MANOVA, 168
6.1.7 Summary of the Four Tests and Relationship to T 2, 168
6.1.8 Measures of Multivariate Association, 173
6.2 Comparison of the Four Manova Test Statistics, 176
6.3 Contrasts, 178
6.3.1 Univariate Contrasts, 178
6.3.2 Multivariate Contrasts, 180
6.4 Tests on Individual Variables Following Rejection of H0 by the
Overall MANOVA Test, 183
6.5 Two-Way Classification, 186
6.5.1 Review of Univariate Two-Way ANOVA, 186
6.5.2 Multivariate Two-Way MANOVA, 188
6.6 Other Models, 195
6.6.1 Higher Order Fixed Effects, 195
6.6.2 Mixed Models, 196
6.7 Checking on the Assumptions, 198
6.8 Profile Analysis, 199
6.9 Repeated Measures Designs, 204
6.9.1 Multivariate vs. Univariate Approach, 204
6.9.2 One-Sample Repeated Measures Model, 208
6.9.3 k-Sample Repeated Measures Model, 211
6.9.4 Computation of Repeated Measures Tests, 212
6.9.5 Repeated Measures with Two Within-Subjects
Factors and One Between-Subjects Factor, 213
6.9.6 Repeated Measures with Two Within-Subjects
Factors and Two Between-Subjects Factors, 219
6.9.7 Additional Topics, 221
6.10 Growth Curves, 221
6.10.1 Growth Curve for One Sample, 221
6.10.2 Growth Curves for Several Samples, 229
6.10.3 Additional Topics, 230
6.11 Tests on a Subvector, 231
6.11.1 Test for Additional Information, 231
6.11.2 Stepwise Selection of Variables, 233
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