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Linear Models: The Theory and Application of Analysis of Variance

ISBN: 978-0-470-02566-6
272 pages
August 2008
Linear Models: The Theory and Application of Analysis of Variance (0470025662) cover image
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.

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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 SΩ.

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 2k 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.

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Brenton R. Clarke, PhD, 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.
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"Solved examples throughout bring clarity to otherwise difficult, complex concepts. Highly recommended." (CHOICE, March 2009)
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