A Primer on Linear Models

A Primer on Linear Modelspresents 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 bookenables complete comprehension of the material by taking a general, unifying approach to the theory, fundamentals, and exact results of linear models.

Preface	p. xiii
Examples of the General Linear Model	p. 1
Introduction	p. 1
One-Sample Problem	p. 1
Simple Linear Regression	p. 2
Multiple Regression	p. 2
One-Way ANOVA	p. 3
First Discussion	p. 4
Two-Way Nested Model	p. 5
Two-Way Crossed Model	p. 6
Analysis of Covariance	p. 8
Autoregression	p. 8
Discussion	p. 9
Summary	p. 10
Notes	p. 10
Exercises	p. 11
The Linear Least Squares Problem	p. 13
The Normal Equations	p. 13
The Geometry of Least Squares	p. 15
Reparameterization	p. 23
Gram-Schmidt Orthonormalization	p. 27
Summary of Important Results	p. 30
Notes	p. 30
Exercises	p. 31
Estimability and Least Squares Estimators	p. 37
Assumptions for the Linear Mean Model	p. 37
Confounding, Identifiability, and Estimability	p. 37
Estimability and Least Squares Estimators	p. 38
First Example: One-Way ANOVA	p. 41
Second Example: Two-Way Crossed without Interaction	p. 45
Two-Way Crossed with Interaction	p. 48
Reparameterization Revisited	p. 50
Imposing Conditions for a Unique Solution to the Normal Equations	p. 56
Constrained Parameter Space	p. 60
Summary	p. 64
Exercises	p. 64
Gauss-Markov Model	p. 71
Model Assumptions	p. 71
The Gauss-Markov Theorem	p. 73
Variance Estimation	p. 75
Implications of Model Selection	p. 76
Underfitting or Misspecification	p. 77
Overfitting and Multicollinearity	p. 79
The Aitken Model and Generalized Least Squares	p. 82
Estimability	p. 82
Linear Estimator	p. 83
Generalized Least Squares Estimators	p. 83
Estimation of ¿2	p. 83
Application: Aggregation Bias	p. 87
Best Estimation in a Constrained Parameter Space	p. 88
Summary	p. 90
Notes	p. 90
Exercises	p. 91
Addendum: Variance of Variance Estimator	p. 95
Distributional Theory	p. 99
Introduction	p. 99
Multivariate Normal Distribution	p. 99
Chi-Square and Related Distributions	p. 103
Distribution of Quadratic Forms	p. 110
Cochran's Theorem	p. 113
Regression Models with Joint Normality	p. 115
Summary	p. 118
Notes	p. 118
Exercises	p. 119
Statistical Inference	p. 125
Introduction	p. 125
Results from Statistical Theory	p. 125
Testing the General Linear Hypothesis	p. 128
The Likelihood Ratio Test and Change in SSE	p. 136
First Principles Test and LRT	p. 139
Confidence Intervals and Multiple Comparisons	p. 141
Identifiability	p. 147
Summary	p. 150
Notes	p. 151
Exercises	p. 151
Further Topics in Testing	p. 157
Introduction	p. 157
Reparameterization	p. 157
Applying Cochran's Theorem for Sequential SS	p. 160
Orthogonal Polynomials and Contrasts	p. 169
Pure Error and the Lack of Fit Test	p. 173
Heresy: Testing Nontestable Hypotheses	p. 175
Summary	p. 177
Exercises	p. 177
Variance Components and Mixed Models	p. 181
Introduction	p. 181
Variance Components: One Way	p. 181
Variance Components: Two-Way Mixed ANOVA	p. 186
Variance Components: General Case	p. 189
Maximum Likelihood	p. 190
Restricted Maximum Likelihood (REML)	p. 192
The ANOVA Approach	p. 193
The Split Plot	p. 194
Predictions and BLUPs	p. 199
Summary	p. 202
Notes	p. 202
Exercises	p. 203
The Multivariate Linear Model	p. 207
Introduction	p. 207
The Multivariate Gauss-Markov Model	p. 207
Inference under Normality Assumptions	p. 211
Testing	p. 216
First Principles Again	p. 217
Likelihood Ratio Test and Wilks' Lambda	p. 220
Other Test Statistics	p. 223
Power of Tests	p. 224
Repeated Measures	p. 228
Confidence Intervals	p. 231
Summary	p. 233
Notes	p. 233
Exercises	p. 234
Review of Linear Algebra	p. 237
Notation and Fundamentals	p. 237
Rank, Column Space, and Nullspace	p. 239
Some Useful Results	p. 244
Solving Equations and Generalized Inverses	p. 245
Projections and Idempotent Matrices	p. 251
Trace, Determinants, and Eigenproblems	p. 254
Definiteness and Factorizations	p. 257
Notes	p. 260
Exercises	p. 261
Lagrange Multipliers	p. 269
Main Results	p. 269
Notes	p. 271
Exercises	p. 272
Bibliography	p. 273
Index	p. 277
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A Primer on Linear Models

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