Brenton R. Clarke 
Linear Models 
The Theory and Application of Analysis of Variance

An insightful approach to the analysis of variance in the study of
linear models

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 mixed-model
effects. This one-of-a-kind 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
fixed-effects 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 least-squares estimation. An outline of the
Helmert matrix is also presented, along with a thorough explanation
of how the ANOVA is created in both typical two-way and higher
layout designs, ultimately revealing the distribution theory. Other
important topics covered include:

* Vector space theory

* The theory of least squares estimation

* Gauss-Markov 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
upper-undergraduate 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.

Inhoudsopgave

Preface.

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 Gauss-Markov Theorem.

3.3 The Distribution of SOmega.

3.4 Some Simple Significance Tests.

3.5 Prediction Intervals.

3.6 Problems.

4. Distribution Theory.

4.1 Motivation.

4.2 Non-Central X² and F Distributions.

4.2.1 Non-Central F-Distribution.

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 Two-Way ANOVA: One Observation Per
Cell.

5.4 Orthogonal Components in Two-Way ANOVA with
Replications.

5.5 The Gauss-Markov 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 Cook-D-Statistic.

7.2 Robust Approaches.

7.2.1 Adaptive Trimmed Likelihood Algorithm.

7.3 Problems.

8. Models That Include Variance Components.

8.1 The One-Way Random Effects Model.

8.2 The Mixed Two-Way 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 Scalar-Covariance-Matrix
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.

Over de auteur

Brenton R. Clarke, Ph D, is Senior Lecturer in Mathematics and Statistics at Murdoch University, Australia. A former president of the Western Australian Branch of the Statistical Society of Australia, Dr. Clarke has published numerous journal articles in his areas of research interest, which include linear models, robust statistics, and time series analysis.