Rencher A.C. Methods of multivariate analysis

John Wiley & Sons, Inc., 2002. — 732 p. — ISBN 0-471-41889-7.A textbook on multivariate statistical analysis covering a fairly standard set of topics: matrix algebra, the concept of multivariate data and the simplest methods for analyzing them, multivariate normal distribution, comparison of means for vectors, multivariate analysis of variance, comparison of covariance matrices, discriminant analysis, classification, multivariate regression, canonical correlation, principal component analysis, factor analysis, cluster analysis, graphical methods.
For students and teachers of the course of mathematical statistics (rather for non-mathematical specialties) and specialists using statistical methods.Why Multivariate Analysis?PrerequisitesObjectives
Basic Types of Data and Analysis
Matrix AlgebraNotation and Basic Definitions
Operations
Partitioned Matrices
Rank
Inverse
Positive Definite Matrices
Determinants
Trace
Orthogonal Vectors and Matrices
Eigenvalues and Eigenvectors
Characterizing and Displaying Multivariate Data
Mean and Variance of a Univariate Random Variable
Covariance and Correlation of Bivariate Random Variables
Scatter Plots of Bivariate Samples
Graphical Displays for Multivariate Samples
Mean Vectors
Covariance Matrices
Correlation Matrices
Mean Vectors and Covariance Matrices for Subsets of Variables
Linear Combinations of Variables
Measures of Overall Variability
Estimation of Missing Values
Distance between Vectors
The Multivariate Normal Distribution
Multivariate Normal Density Function
Properties of Multivariate Normal Random Variables
Estimation in the Multivariate Normal
Outliers
Tests on One or Two Mean Vectors
Multivariate versus Univariate Tests
Tests on μ with Σ Known
Tests on μ When Σ Is Unknown
Comparing Two Mean Vectors
Tests on Individual Variables Conditional on Rejection of H₀ by the T₂-Test
Computation of T₂
Paired Observations Test
Test for Additional Information
Profile Analysis
Multivariate Analysis of Variance
One-Way Models
Comparison of the Four Manova Test Statistics
Contrasts
Tests on Individual Variables Following Rejection of H0 by the Overall MANOVA Test
Two-Way Classification
Other Models
Checking on the Assumptions
Profile Analysis
Repeated Measures Designs
Growth Curves
Tests on a Subvector
Tests on Covariance MatricesTesting a Specified Pattern for Σ
Tests Comparing Covariance Matrices
Tests of Independence
Discriminant Analysis: Description of Group SeparationThe Discriminant Function for Two Groups
Relationship between Two-Group Discriminant Analysis and Multiple Regression
Discriminant Analysis for Several Groups
Standardized Discriminant Functions
Tests of Significance
Interpretation of Discriminant Functions
Scatter Plots
Stepwise Selection of Variables
Classification Analysis: Allocation of Observations to GroupsClassification into Two Groups
Classification into Several Groups
Estimating Misclassification Rates
Improved Estimates of Error Rates
Subset Selection
Nonparametric Procedures
Multivariate RegressionMultiple Regression: Fixed x’s
Multiple Regression: Random x’s
Multivariate Multiple Regression: Estimation
Multivariate Multiple Regression: Hypothesis Tests
Measures of Association between the y’s and the x’s
Subset Selection
Multivariate Regression: Random x’s
Canonical CorrelationCanonical Correlations and Canonical Variates
Properties of Canonical Correlations
Tests of Significance
Interpretation
Relationships of Canonical Correlation Analysis to Other Multivariate Techniques
Principal Component AnalysisGeometric and Algebraic Bases of Principal Components
Principal Components and Perpendicular Regression
Plotting of Principal Components
Principal Components from the Correlation Matrix
Deciding How Many Components to Retain
Information in the Last Few Principal Components
Interpretation of Principal Components
Selection of Variables
Factor AnalysisOrthogonal Factor Model
Estimation of Loadings and Communalities
Choosing the Number of Factors, m
Rotation
Factor Scores
Validity of the Factor Analysis Model
The Relationship of Factor Analysis to Principal Component Analysis
Cluster AnalysisMeasures of Similarity or Dissimilarity
Hierarchical Clustering
Nonhierarchical Methods
Choosing the Number of Clusters
Cluster Validity
Clustering Variables
Graphical Procedures
Multidimensional Scaling
Correspondence Analysis
Biplots
Tables
Answers and Hints to Problems
Data Sets and SAS Files