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A Primer on Linear Models


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Table of Contents

Examples of the General Linear Model
One-Sample Problem
Simple Linear Regression
Multiple Regression
First Discussion
The Two-Way Nested Model
Two-Way Crossed Model
Analysis of Covariance
The Linear Least Squares Problem
The Normal Equations
The Geometry of Least Squares
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
Multivariate Normal Distribution
Chi-Square and Related Distributions
Distribution of Quadratic Forms
Cochran's Theorem
Regression Models with Joint Normality
Statistical Inference
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
Further Topics in Testing
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
Variance Components: One Way
Variance Components: Two-Way Mixed ANOVA
Variance Components: General Case
The Split Plot
Predictions and BLUPs
The Multivariate Linear Model
The Multivariate Gauss-Markov Model
Inference under Normality Assumptions
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
A Summary, Notes, and Exercises appear at the end of most chapters.


"... 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

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