Linear Models in Statistics

ISBN-10: 0471754986

ISBN-13: 9780471754985

Edition: 2nd 2008

Authors: Alvin C. Rencher, G. Bruce Schaalje

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Linear Models in Statistics, Second Edition discusses classical linear models from a matrix algebra perspective, making the subject easily accessible to readers encountering linear models for the first time. It provides a solid foundation from which to explore the literature and interpret correctly the output of computer packages. It brings together a number of approaches to regression and analysis of variance from which more experienced practitioners will also benefit. With an emphasis on broad coverage of essential topics, this book clearly and carefully develops the basic theory of regression and analysis of variance, illustrating it with examples from a wide range of disciplines.
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Book details

List price: $182.00
Edition: 2nd
Copyright year: 2008
Publisher: John Wiley & Sons, Incorporated
Publication date: 1/2/2008
Binding: Hardcover
Pages: 688
Size: 6.25" wide x 9.25" long x 1.50" tall
Weight: 2.706
Language: English

Alvin C. Rencher, PhD, is Professor of Statistics at Brigham Young University. Dr. Rencher is a Fellow of the American Statistical Association and the author of Methods of Multivariate Analysis and Multivariate Statistical Inference and Applications, both published by Wiley.G. Bruce Schaalje, PhD, is Professor of Statistics at Brigham Young University. He has authored over 120 journal articles in his areas of research interest, which include mixed linear models, small sample inference, and design of experiments.

Preface
Introduction
Simple Linear Regression Model
Multiple Linear Regression Model
Analysis-of-Variance Models
Matrix Algebra
Matrix and Vector Notation
Matrices, Vectors, and Scalars
Matrix Equality
Transpose
Matrices of Special Form
Operations
Sum of Two Matrices or Two Vectors
Product of a Scalar and a Matrix
Product of Two Matrices or Two Vectors
Hadamard Product of Two Matrices or Two Vectors
Partitioned Matrices
Rank
Inverse
Positive Definite Matrices
Systems of Equations
Generalized Inverse
Definition and Properties
Generalized Inverses and Systems of Equations
Determinants
Orthogonal Vectors and Matrices
Trace
Eigenvalues and Eigenvectors
Definition
Functions of a Matrix
Products
Symmetric Matrices
Positive Definite and Semidefinite Matrices
Idempotent Matrices
Vector and Matrix Calculus
Derivatives of Functions of Vectors and Matrices
Derivatives Involving Inverse Matrices and Determinants
Maximization or Minimization of a Function of a Vector
Random Vectors and Matrices
Introduction
Means, Variances, Covariances, and Correlations
Mean Vectors and Covariance Matrices for Random Vectors
Mean Vectors
Covariance Matrix
Generalized Variance
Standardized Distance
Correlation Matrices
Mean Vectors and Covariance Matrices for Partitioned Random Vectors
Linear Functions of Random Vectors
Means
Variances and Covariances
Multivariate Normal Distribution
Univariate Normal Density Function
Multivariate Normal Density Function
Moment Generating Functions
Properties of the Multivariate Normal Distribution
Partial Correlation
Distribution of Quadratic Forms in y
Sums of Squares
Mean and Variance of Quadratic Forms
Noncentral Chi-Square Distribution
Noncentral F and t Distributions
Noncentral F Distribution
Noncentral t Distribution
Distribution of Quadratic Forms
Independence of Linear Forms and Quadratic Forms
Simple Linear Regression
The Model
Estimation of [beta subscript 0], [beta subscript 1], and [sigma superscript 2]
Hypothesis Test and Confidence Interval for [beta subscript 1]
Coefficient of Determination
Multiple Regression: Estimation
Introduction
The Model
Estimation of [beta] and [sigma superscript 2]
Least-Squares Estimator for [beta]
Properties of the Least-Squares Estimator [beta]
An Estimator for [sigma superscript 2]
Geometry of Least-Squares
Parameter Space, Data Space, and Prediction Space
Geometric Interpretation of the Multiple Linear Regression Model
The Model in Centered Form
Normal Model
Assumptions
Maximum Likelihood Estimators for [beta] and [sigma superscript 2]
Properties of [beta] and [sigma superscript 2]
R[superscript 2] in Fixed-x Regression
Generalized Least-Squares: cov(y) = [sigma superscript 2]V
Estimation of [beta] and [sigma superscript 2] when cov(y) = [sigma superscript 2]V
Misspecification of the Error Structure
Model Misspecification
Orthogonalization
Multiple Regression: Tests of Hypotheses and Confidence Intervals
Test of Overall Regression
Test on a Subset of the [beta] Values
F Test in Terms of R[superscript 2]
The General Linear Hypothesis Tests for H[subscript 0]: C[beta] = 0 and H[subscript 0]: C[beta] = t
The Test for H[subscript 0]: C[beta] = 0
The Test for H[subscript 0]: C[beta] = t
Tests on [beta subscript j] and a' [beta]
Testing One [beta subscript j] or One a' [beta]
Testing Several [beta subscript j] or a'[subscript i beta] Values
Confidence Intervals and Prediction Intervals
Confidence Region for [beta]
Confidence Interval for [beta subscript j]
Confidence Interval for a'[beta]
Confidence Interval for E(y)
Prediction Interval for a Future Observation
Confidence Interval for [sigma superscript 2]
Simultaneous Intervals
Likelihood Ratio Tests
Multiple Regression: Model Validation and Diagnostics
Residuals
The Hat Matrix
Outliers
Influential Observations and Leverage
Multiple Regression: Random x's
Multivariate Normal Regression Model
Estimation and Testing in Multivariate Normal Regression
Standardized Regression Coefficients
R[superscript 2] in Multivariate Normal Regression
Tests and Confidence Intervals for R[superscript 2]
Effect of Each Variable on R[superscript 2]
Prediction for Multivariate Normal or Nonnormal Data
Sample Partial Correlations
Multiple Regression: Bayesian Inference
Elements of Bayesian Statistical Inference
A Bayesian Multiple Linear Regression Model
A Bayesian Multiple Regression Model with a Conjugate Prior
Marginal Posterior Density of [beta]
Marginal Posterior Densities of [tau] and [sigma superscript 2]
Inference in Bayesian Multiple Linear Regression
Bayesian Point and Interval Estimates of Regression Coefficients
Hypothesis Tests for Regression Coefficients in Bayesian Inference
Special Cases of Inference in Bayesian Multiple Regression Models
Bayesian Point and Interval Estimation of [sigma superscript 2]
Bayesian Inference through Markov Chain Monte Carlo Simulation
Posterior Predictive Inference
Analysis-of-Variance Models
Non-Full-Rank Models
One-Way Model
Two-Way Model
Estimation
Estimation of [beta]
Estimable Functions of [beta]
Estimators
Estimators of [lambda]'[beta]
Estimation of [sigma superscript 2]
Normal Model
Geometry of Least-Squares in the Overparameterized Model
Reparameterization
Side Conditions
Testing Hypotheses
Testable Hypotheses
Full-Reduced-Model Approach
General Linear Hypothesis
An Illustration of Estimation and Testing
Estimable Functions
Testing a Hypothesis
Orthogonality of Columns of X
One-Way Analysis-of-Variance: Balanced Case
The One-Way Model
Estimable Functions
Estimation of Parameters
Solving the Normal Equations
An Estimator for [sigma superscript 2]
Testing the Hypothesis H[subscript 0]: [mu subscript 1] = [mu subscript 2] = ... = [mu subscript k]
Full-Reduced-Model Approach
General Linear Hypothesis
Expected Mean Squares
Full-Reduced-Model Approach
General Linear Hypothesis
Contrasts
Hypothesis Test for a Contrast
Orthogonal Contrasts
Orthogonal Polynomial Contrasts
Two-Way Analysis-of-Variance: Balanced Case
The Two-Way Model
Estimable Functions
Estimators of [lambda]'[beta] and [sigma superscript 2]
Solving the Normal Equations and Estimating [lambda]'[beta]
An Estimator for [sigma superscript 2]
Testing Hypotheses
Test for Interaction
Tests for Main Effects
Expected Mean Squares
Sums-of-Squares Approach
Quadratic Form Approach
Analysis-of-Variance: The Cell Means Model for Unbalanced Data
Introduction
One-Way Model
Estimation and Testing
Contrasts
Two-Way Model
Unconstrained Model
Constrained Model
Two-Way Model with Empty Cells
Analysis-of-Covariance
Introduction
Estimation and Testing
The Analysis-of-Covariance Model
Estimation
Testing Hypotheses
One-Way Model with One Covariate
The Model
Estimation
Testing Hypotheses
Two-Way Model with One Covariate
Tests for Main Effects and Interactions
Test for Slope
Test for Homogeneity of Slopes
One-Way Model with Multiple Covariates
The Model
Estimation
Testing Hypotheses
Analysis-of-Covariance with Unbalanced Models
Linear Mixed Models
Introduction
The Linear Mixed Model
Examples
Estimation of Variance Components
Inference for [beta]
An Estimator for [beta]
Large-Sample Inference for Estimable Functions of [beta]
Small-Sample Inference for Estimable Functions of [beta]
Inference for the a[subscript i] Terms
Residual Diagnostics
Additional Models
Nonlinear Regression
Logistic Regression
Loglinear Models
Poisson Regression
Generalized Linear Models
Answers and Hints to the Problems
References
Index
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