Primer on Linear Models

ISBN-10: 1420062018

ISBN-13: 9781420062014

Edition: 2008

Authors: John F. Monahan

List price: $59.95
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Description:

A Primer on Linear Modelspresents a concise yet complete foundation for understanding basic linear models. Designed for a one-semester graduate course, this textbook begins with a practical discussion of basic algebraic and geometric concepts as they apply to the linear model. The books two distinguishing features are the constant use of non-full-rank design matrices to seamlessly incorporate regression, analysis of variance (ANOVA), and various mixed models and attention to the exact, finite sample theory supporting common statistical methods. A Primer on Linear Modelsprovides a brief, yet complete foundation for the understanding of basic linear models. It focuses on the theory behind the methodology students learn in other courses. As a result, the style follows the route of theorem/proof/corollary/example/exercise with discussions of the motivations for the assumptions from theory and practice. The coverage progresses steadily in both difficulty and sophistication ofthe proofs, discussion, and exercises, paralleling the growth of students as they master the subject. Avoiding leaps in the development and proofs, or referring to other sources for the details of the proofs, the author demonstrates the complete development of the material. This gives students learning the material for the first time enough information to understand complex concepts with confidence. Pedagocial Features Proofs and discussion from both algebraic and geometric viewpoints Exercises at the end of each chapter escalate in degrees of difficulty Multipart exercises taken from past quizzes and final exams 196 equations Summaries, discussions, notes, and exercises at the end of each chapter
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Book details

List price: $59.95
Copyright year: 2008
Publisher: CRC Press LLC
Publication date: 3/31/2008
Binding: Paperback
Pages: 304
Size: 5.75" wide x 9.25" long x 0.55" tall
Weight: 0.946
Language: English

John F. Monahan is a Professor of Statistics at North Carolina State University where he joined the faculty in 1978 and has been a professor since 1990. His research has appeared in numerous computational as well as statistical journals. He is also the author of A Primer on Linear Models (2008).

Preface
Examples of the General Linear Model
Introduction
One-Sample Problem
Simple Linear Regression
Multiple Regression
One-Way ANOVA
First Discussion
Two-Way Nested Model
Two-Way Crossed Model
Analysis of Covariance
Autoregression
Discussion
Summary
Notes
Exercises
The Linear Least Squares Problem
The Normal Equations
The Geometry of Least Squares
Reparameterization
Gram-Schmidt Orthonormalization
Summary of Important Results
Notes
Exercises
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
Summary
Exercises
Gauss-Markov Model
Model Assumptions
The Gauss-Markov Theorem
Variance Estimation
Implications of Model Selection
Underfitting or Misspecification
Overfitting and Multicollinearity
The Aitken Model and Generalized Least Squares
Estimability
Linear Estimator
Generalized Least Squares Estimators
Estimation of �2
Application: Aggregation Bias
Best Estimation in a Constrained Parameter Space
Summary
Notes
Exercises
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
Summary
Notes
Exercises
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
Summary
Notes
Exercises
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
Summary
Exercises
Variance Components and Mixed Models
Introduction
Variance Components: One Way
Variance Components: Two-Way Mixed ANOVA
Variance Components: General Case
Maximum Likelihood
Restricted Maximum Likelihood (REML)
The ANOVA Approach
The Split Plot
Predictions and BLUPs
Summary
Notes
Exercises
The Multivariate Linear Model
Introduction
The Multivariate Gauss-Markov Model
Inference under Normality Assumptions
Testing
First Principles Again
Likelihood Ratio Test and Wilks' Lambda
Other Test Statistics
Power of Tests
Repeated Measures
Confidence Intervals
Summary
Notes
Exercises
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
Notes
Exercises
Lagrange Multipliers
Main Results
Notes
Exercises
Bibliography
Index
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