A Primer on Linear Models presents a unified, thorough, and rigorous development of the theory behind the statistical methodology of regression and analysis of variance (ANOVA). It seamlessly incorporates these concepts using non-full-rank design matrices and emphasizes the exact, finite sample theory supporting common statistical methods.
With coverage steadily progressing in complexity, the text first provides examples of the general linear model, including multiple regression models, one-way ANOVA, mixed-effects models, and time series models. It then introduces the basic algebra and geometry of the linear least squares problem, before delving into estimability and the Gauss–Markov model. After presenting the statistical tools of hypothesis tests and confidence intervals, the author analyzes mixed models, such as two-way mixed ANOVA, and the multivariate linear model. The appendices review linear algebra fundamentals and results as well as Lagrange multipliers.
This book enables complete comprehension of the material by taking a general, unifying approach to the theory, fundamentals, and exact results of linear models.
Examples of the General Linear Model
Introduction
One-Sample Problem
Simple Linear Regression
Multiple Regression
One-Way ANOVA
First Discussion
The Two-Way Nested Model
Two-Way Crossed Model
Analysis of Covariance
Autoregression
Discussion
The Linear Least Squares Problem
The Normal Equations
The Geometry of Least Squares
Reparameterization
Gram–Schmidt Orthonormalization
Estimability and Least Squares Estimators
Assumptions for the Linear Mean Model
Confounding, Identifiability, and Estimability
Estimability and Least Squares Estimators
First Example: One-Way ANOVA
Second Example: Two-Way Crossed without Interaction
Two-Way Crossed with Interaction
Reparameterization Revisited
Imposing Conditions for a Unique Solution to the Normal Equations
Constrained Parameter Space
Gauss–Markov Model
Model Assumptions
The Gauss–Markov Theorem
Variance Estimation
Implications of Model Selection
The Aitken Model and Generalized Least Squares
Application: Aggregation Bias
Best Estimation in a Constrained Parameter Space
Addendum: Variance of Variance Estimator
Distributional Theory
Introduction
Multivariate Normal Distribution
Chi-Square and Related Distributions
Distribution of Quadratic Forms
Cochran’s Theorem
Regression Models with Joint Normality
Statistical Inference
Introduction
Results from Statistical Theory
Testing the General Linear Hypothesis
The Likelihood Ratio Test and Change in SSE
First Principles Test and LRT
Confidence Intervals and Multiple Comparisons
Identifiability
Further Topics in Testing
Introduction
Reparameterization
Applying Cochran’s Theorem for Sequential SS
Orthogonal Polynomials and Contrasts
Pure Error and the Lack-of-Fit Test
Heresy: Testing Nontestable Hypotheses
Variance Components and Mixed Models
Introduction
Variance Components: One Way
Variance Components: Two-Way Mixed ANOVA
Variance Components: General Case
The Split Plot
Predictions and BLUPs
The Multivariate Linear Model
Introduction
The Multivariate Gauss–Markov Model
Inference under Normality Assumptions
Testing
Repeated Measures
Confidence Intervals
Appendix A: Review of Linear Algebra
Notation and Fundamentals
Rank, Column Space, and Nullspace
Some Useful Results
Solving Equations and Generalized Inverses
Projections and Idempotent Matrices
Trace, Determinants, and Eigenproblems
Definiteness and Factorizations
Appendix B: Lagrange Multipliers
Main Results
Bibliography
A Summary, Notes, and Exercises appear at the end of most chapters.
Biography
John F. Monahan
"… I found the book very helpful. … the result is very nice, very readable, and in particular I like the idea of avoiding leaps in the development and proofs, or referring to other sources for the details of the proofs. This is a useful well-written instructive book."
—International Statistical Review"This work provides a brief, and also complete, foundation for the theory of basic linear models . . . can be used for graduate courses on linear models."
– Nicoleta Breaz, Zentralblatt Math". . . well written . . . would serve well as the textbook for an introductory course in linear models, or as references for researchers who would like to review the theory of linear models."
– Justine Shults, Department of Biostatistics, University of Pennsylvania School of Medicine, Journal of Biopharmaceutical Statistics
