Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface to the First Edition
Preface to the Second Edition
1. Review of Vector and Matrix Algebra
1.1. Notation
1.2. Basic properties of vectors
1.3. Basic properties of matrices
1.4. R Code
Exercises
2. Properties of Special Matrices
2.1. Partitioned matrices
2.2. Algorithms for matrix factorization
2.3. Symmetric and idempotent matrices
2.4. Nonnegative definite quadratic forms and matrices
2.5. Simultaneous diagonalization of matrices
2.6. Geometrical perspectives
2.7. Vector and matrix differentiation
2.8. Special operations on matrices
2.9. R Code
Exercises
3. Generalized Inverses and Solutions to Linear Systems
3.1. Generalized inverses
3.2. Solutions to linear systems
3.3. Linear optimization
3.3.1. Unconstrained minimization
3.3.2. Constrained minimization
3.4. R Code
Exercises
4. General Linear Model
4.1. Model definition and examples
4.2. Least squares approach
4.3. Estimable functions
4.4. Gauss-Markov theorem
4.5. Generalized least squares
4.6. Estimation subject to linear constraints
4.6.1. Method of Lagrangian multipliers
4.6.1.1. Case I: A'B is estimable
4.6.1.2. Case II: A'B is not estimable
4.6.2. Method of orthogonal projections
Exercises
5. Multivariate Normal and Related Distributions
5.1. Intergral evaluation theorems
5.2. Multivariate normal distribution and properties
5.3. Some noncentral distributions
5.4. Distributions of quadratic forms
5.5. Remedies for non-normality
5.5.1. Transformations to normality
5.5.1.1. Univariate transformations
5.5.1.2. Multivariate transformations
5.5.2. Alternatives to multivariate normal distribution
5.5.2.1. Mixture of normals
5.5.2.2. Spherical distributions
5.5.2.3. Elliptical distributions
5.6. R Code
Exercises
6. Sampling from the Multivariate Normal Distribution
6.1. Distribution of sample mean and covariance
6.2. Distributions related to correlation coefficients
6.3. Assessing the normality assumption
6.4. R Code
Exercises
7. Inference for the General Linear Model-I
7.1. Properties of least squares solutions
7.2. General linear hypotheses
7.2.1. Derivation of and motivation for the F-test
7.2.2. Power of the F-test
7.2.3. Testing independent and orthogonal contrasts
7.3. Restricted and reduced models
7.3.1. Estimation space and estimability under constraints
7.3.2. Nested sequence of models or hypotheses
7.4. Confidence intervals
7.4.1. Joint and marginal confidence intervals
7.4.2. Simultaneous confidence intervals
7.4.2.1. Scheffe intervals
7.4.2.2. Bonferroni t-intervals
Exercises
8. Inference for the General Linear Model-II
8.1. Likelihood-based approaches
8.1.1. Maximum likelihood estimation under normality
8.1.2. Model selection criteria
8.2. Departures from model assumptions
8.2.1. Graphical procedures
8.2.2. Heteroscedasticity
8.2.3. Serial correlation
8.3. Diagnostics for the GLM
8.3.1. Further properties of the projection matrix
8.3.2. Types of residuals
8.3.3. Outliers and high leverage observations
8.3.4. Diagnostic measures based on influence functions
8.4. Prediction intervals and calibration
Exercises
9. Multiple Linear Regression Models
9.1. Variable selection in regression
9.1.1. Graphical assessment of variables
9.1.2. Criteria-based variable selection
9.1.3. Variable selection based on significance tests
9.1.3.1. Sequential and partial F-tests
9.1.3.2. Stepwise regression and variants
9.2. Orthogonal and collinear predictors
9.2.1. Orthogonality in regression
9.2.2. Multicollinearity
9.2.3. Ridge regression
9.2.4. Principal components regression
9.3. Dummy variables in regression
Exercises
10. Fixed-Effects Linear Models
10.1. Inference for unbalanced ANOVA models
10.1.1. One-way cell means model
10.1.2. Higher-order overparametrized models
10.1.2.1. Two-factor additive models
10.1.2.2. Two-factor models with interaction
10.1.2.3. Nested or hierarchical models
10.2. Nonparametric procedures
10.3. Analysis of covariance
10.4. Multiple hypothesis testing
10.4.1. Error rates
10.4.2. Procedures for Controlling Type I Errors
10.4.2.1. FWER control
10.4.2.2. FDR control
10.4.2.3. Plug-in approach to estimate the FDR
10.4.3. Multiple comparison procedures
10.5. Generalized Gauss-Markov theorem
Exercises
11. Random- and Mixed-Effects Models
11.1. Setup and examples of mixed-effects linear models
11.2. Inference for mixed-effects linear models
11.2.1. Extended Gauss-Markov theorem
11.2.2. GLS estimation of fixed effects
11.2.3. ANOVA method for estimation
11.2.4. Method of maximum likelihood
11.2.5. REML estimation
11.2.6. MINQUE estimation
11.3. One-factor random-effects model
11.4. Two-factor randomand mixed-effects models
Exercises
12. Generalized Linear Models
12.1. Components of GLIM
12.2. Estimation approaches
12.2.1. Score and Fisher information for GLIM
12.2.2. Maximum likelihood estimation - Fisher scoring
12.2.3. Iteratively reweighted least squares
12.2.4. Quasi-likelihood estimation
12.3. Residuals and model checking
12.3.1. GLIM residuals
12.3.2. Goodness of fit measures
12.3.3. Hypothesis testing and model comparisons
12.3.3.1. Wald Test
12.3.3.2. Likelihood ratio test
12.3.3.3. Drop-in-deviance test
12.4. Binary and binomial response models
12.5. Count Models
Exercises
13. Special Topics
13.1. Multivariate general linear models
13.1.1. Model definition
13.1.2. Least squares estimation
13.1.3. Estimable functions and Gauss-Markov theorem
13.1.4. Maximum likelihood estimation
13.1.5. Likelihood ratio tests for linear hypotheses
13.1.6. Confidence region and Hotelling T2 distribution
13.1.7. One-factor multivariate analysis of variance model
13.2. Longitudinal models
13.2.1. Multivariate model for longitudinal data
13.2.2. Two-stage random-effects models
13.3. Elliptically contoured linear model
13.4. Bayesian linear models
13.4.1. Bayesian normal linear model
13.4.2. Hierarchical normal linear model
13.4.3. Bayesian model assessment and selection
13.5. Dynamic linear models
13.5.1. Kalman filter equations
13.5.2. Kalman smoothing equations
Exercises
14. Miscellaneous Topics
14.1. Robust regression
14.1.1. Least absolute deviations regression
14.1.2. M-regression
14.2. Nonparametric regression methods
14.2.1. Regression splines
14.2.2. Additive and generalized additive models
14.2.3. Projection pursuit regression
14.2.4. Multivariate adaptive regression splines
14.2.5. Neural networks regression
14.3. Regularized regression
14.3.1. L0-regularization
14.3.2. L1-regularization and Lasso
14.3.2.1. Geometry of Lasso
14.3.2.2. Fast forward stagewise selection
14.3.2.3. Least angle regression
14.3.3. Elastic net
14.4. Missing data analysis
14.4.1. EM algorithm
A. Multivariate Probability Distributions
B. Common Families of Distributions
C. Some Useful Statistical Notions
D. Solutions to Selected Exercises
Bibliography
Author Index
Subject Index