Linear Models: The Theory and Application of Analysis of VarianceISBN: 9780470025666
272 pages
August 2008

Linear Models explores the theory of linear models and the dynamic relationships that these models have with Analysis of Variance (ANOVA), experimental design, and random and mixedmodel effects. This oneofakind book emphasizes an approach that clearly explains the distribution theory of linear models and experimental design starting from basic mathematical concepts in linear algebra.
The author begins with a presentation of the classic fixedeffects linear model and goes on to illustrate eight common linear models, along with the value of their use in statistics. From this foundation, subsequent chapters introduce concepts pertaining to the linear model, starting with vector space theory and the theory of leastsquares estimation. An outline of the Helmert matrix is also presented, along with a thorough explanation of how the ANOVA is created in both typical twoway and higher layout designs, ultimately revealing the distribution theory. Other important topics covered include:

Vector space theory

The theory of least squares estimation

GaussMarkov theorem

Kronecker products

Diagnostic and robust methods for linear models

Likelihood approaches to estimation
A discussion of Bayesian theory is also included for purposes of comparison and contrast, and numerous illustrative exercises assist the reader with uncovering the nature of the models, using both classic and new data sets. Requiring only a working knowledge of basic probability and statistical inference, Linear Models is a valuable book for courses on linear models at the upperundergraduate and graduate levels. It is also an excellent reference for practitioners who use linear models to conduct research in the fields of econometrics, psychology, sociology, biology, and agriculture.
Acknowledgments.
Notation.
1. Introduction.
1.1 The Linear Model and Examples.
1.2 What Are the Objectives?.
1.3 Problems.
2. Projection Matrices and Vector Space Theory.
2.1 Basis of a Vector Space.
2.2 Range and Kernel.
2.3 Projections.
2.3.1 Linear Model Application.
2.4 Sums and Differences of Orthogonal Projections.
2.5 Problems.
3. Least Squares Theory.
3.1 The Normal Equations.
3.2 The GaussMarkov Theorem.
3.3 The Distribution of S_{Ω}.
3.4 Some Simple Significance Tests.
3.5 Prediction Intervals.
3.6 Problems.
4. Distribution Theory.
4.1 Motivation.
4.2 NonCentral X^{2} and F Distributions.
4.2.1 NonCentral FDistribution.
4.2.2 Applications to Linear Models.
4.2.3 Some Simple Extensions.
4.3 Problems.
5. Helmert Matrices and Orthogonal Relationships.
5.1 Transformations to Independent Normally Distributed Random Variables.
5.2 The Kronecker Product.
5.3 Orthogonal Components in TwoWay ANOVA: One Observation Per Cell.
5.4 Orthogonal Components in TwoWay ANOVA with Replications.
5.5 The GaussMarkov Theorem Revisited.
5.6 Orthogonal Components for Interaction.
5.6.1 Testing for Interaction: One Observation Per Cell.
5.6.2 Example Calculation of Tukey’s One's Degree of Freedom Statistic.
5.7 Problems.
6. Further Discussion of ANOVA.
6.1 The Different Representations of Orthogonal Components.
6.2 On the Lack of Orthogonality.
6.3 The Relationship Algebra.
6.4 The Triple Classification.
6.5 Latin Squares.
6.6 2^{k} Factorial Designs.
6.6.1 Yates’ Algorithm.
6.7 The Function of Randomization.
6.8 Brief View of Multiple Comparison Techniques.
6.9 Problems.
7. Residual Analysis: Diagnostics and Robustness.
7.1 Design Diagnostics.
7.1.1 Standardized and Studentized Residuals.
7.1.2 Combining Design and Residual Effects on Fit  DFITS.
7.1.3 The CookDStatistic.
7.2 Robust Approaches.
7.2.1 Adaptive Trimmed Likelihood Algorithm.
7.3 Problems.
8. Models That Include Variance Components.
8.1 The OneWay Random Effects Model.
8.2 The Mixed TwoWay Model.
8.3 A Split Plot Design.
8.3.1 A Traditional Model.
8.4 Problems.
9. Likelihood Approaches.
9.1 Maximum Likelihood Estimation.
9.2 REML.
9.3 Discussion of Hierarchical Statistical Models.
9.3.1 Hierarchy for the Mixed Model (Assuming Normality).
9.4 Problems.
10. Uncorrelated Residuals Formed from the Linear Model.
10.1 Best Linear Unbiased Error Estimates.
10.2 The Best Linear Unbiased ScalarCovarianceMatrix Approach.
10.3 Explicit Solution.
10.4 Recursive Residuals.
10.4.1 Recursive Residuals and their Properties.
10.5 Uncorrelated Residuals.
10.5.1 The Main Results.
10.5.2 Final Remarks.
10.6 Problems.
11. Further inferential questions relating to ANOVA.
References.
Index.