• Encompasses a wide variety of topics in linear models that incorporate the classical approach and more recent trends and modeling techniques
• Emphasizes the central role matrices have played in the modern development of linear models
• Covers both balanced mixed-effects models and unbalanced linear models so that readers have a full understanding of how to analyze data under different modeling situations
• Presents a unified approach to modeling discrete and continuous response data
• Includes numerous examples and end-of-chapter exercises

Given the importance of linear models in statistical theory and experimental research, a good understanding of their fundamental principles and theory is essential. Supported by a large number of examples, Linear Model Methodology provides a strong foundation in the theory of linear models and explores the latest developments in data analysis.

After presenting the historical evolution of certain methods and techniques used in linear models, the book reviews vector spaces and linear transformations and discusses the basic concepts and results of matrix algebra that are relevant to the study of linear models. Although mainly focused on classical linear models, the next several chapters also explore recent techniques for solving well-known problems that pertain to the distribution and independence of quadratic forms, the analysis of estimable linear functions and contrasts, and the general treatment of balanced random and mixed-effects models. The author then covers more contemporary topics in linear models, including the adequacy of Satterthwaite’s approximation, unbalanced fixed- and mixed-effects models, heteroscedastic linear models, response surface models with random effects, and linear multiresponse models. The final chapter introduces generalized linear models, which represent an extension of classical linear models.

Linear models provide the groundwork for analysis of variance, regression analysis, response surface methodology, variance components analysis, and more, making it necessary to understand the theory behind linear modeling. Reflecting advances made in the last thirty years, this book offers a rigorous development of the theory underlying linear models.

Linear Models: Some Historical Perspectives

The Invention of Least Squares

The Gauss–Markov Theorem

Estimability

Maximum Likelihood Estimation

Analysis of Variance (ANOVA)

Quadratic Forms and Craig’s Theorem

The Role of Matrix Algebra

The Geometric Approach

Basic Elements of Linear Algebra

Introduction

Vector Spaces

Vector Subspaces

Bases and Dimensions of Vector Spaces

Linear Transformations

Basic Concepts in Matrix Algebra

Introduction and Notation

Some Particular Types of Matrices

Basic Matrix Operations

Partitioned Matrices

Determinants

The Rank of a Matrix

The Inverse of a Matrix

Eigenvalues and Eigenvectors

Idempotent and Orthogonal Matrices

Quadratic Forms

Decomposition Theorems

Some Matrix Inequalities

Function of Matrices

Matrix Differentiation

The Multivariate Normal Distribution

History of the Normal Distribution

The Univariate Normal Distribution

The Multivariate Normal Distribution

The Moment Generating Function

Conditional Distribution

The Singular Multivariate Normal Distribution

Related Distributions

Examples and Additional Results

Quadratic Forms in Normal Variables

The Moment Generating Function

Distribution of Quadratic Forms

Independence of Quadratic Forms

Independence of Linear and Quadratic Forms

Independence and Chi-Squaredness of Several Quadratic Forms

Computing the Distribution of Quadratic Forms

Appendix

Full-Rank Linear Models

Least-Squares Estimation

Properties of Ordinary Least-Squares Estimation

Generalized Least-Squares Estimation

Least-Squares Estimation under Linear Restrictions on β

Maximum Likelihood Estimation

Inference Concerning β

Examples and Applications

Less-Than-Full-Rank Linear Models

Parameter Estimation

Some Distributional Properties

Reparameterized Model

Estimable Linear Functions

Simultaneous Confidence Intervals on Estimable Linear Functions

Simultaneous Confidence Intervals on All Contrasts among the Means with Heterogeneous Group Variances

Further Results Concerning Contrasts and Estimable Linear Functions

Balanced Linear Models

Notation and Definitions

The General Balanced Linear Model

Properties of Balanced Models

Balanced Mixed Models

Complete and Sufficient Statistics

ANOVA Estimation of Variance Components

Confidence Intervals on Continuous Functions of the Variance Components

Confidence Intervals on Ratios of Variance Components

The Adequacy of Satterthwaite’s Approximation

Satterthwaite’s Approximation

Adequacy of Satterthwaite’s Approximation

Measuring the Closeness of Satterthwaite’s Approximation

Examples

Appendix

Unbalanced Fixed-Effects Models

The R-Notation

Two-Way Models without Interaction

Two-Way Models with Interaction

Higher-Order Models

A Numerical Example

The Method of Unweighted Means

Unbalanced Random and Mixed Models

Estimation of Variance Components

Estimation of Estimable Linear Functions

Inference Concerning the Random One-Way Model

Inference Concerning the Random Two-Way Model

Exact Tests for Random Higher-Order Models

Inference Concerning the Mixed Two-Way Model

Inference Concerning the Random Two-Fold Nested Model

Inference Concerning the Mixed Two-Fold Nested Model

Inference Concerning the General Mixed Linear Model

Appendix

Additional Topics in Linear Models

Heteroscedastic Linear Models

The Random One-Way Model with Heterogeneous Error Variances

A Mixed Two-Fold Nested Model with Heteroscedastic Random Effects

Response Surface Models

Response Surface Models with Random Block Effects

Linear Multiresponse Models

Generalized Linear Models

Introduction

The Exponential Family

Estimation of Parameters

Goodness of Fit

Hypothesis Testing

Confidence Intervals

Gamma-Distributed Response

Bibliography

Index

Exercises appear at the end of each chapter, except for Chapter 1.