Clarke Linear Models

The Theory and Application of Analysis of Variance
1. Auflage 2008
ISBN: 978-0-470-37797-0
Verlag: John Wiley & Sons
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 272 Seiten, E-Book

Reihe: Wiley Series in Probability and Statistics

An insightful approach to the analysis of variance in the study oflinear models
Linear Models explores the theory of linear models and thedynamic relationships that these models have with Analysis ofVariance (ANOVA), experimental design, and random and mixed-modeleffects. This one-of-a-kind book emphasizes an approach thatclearly explains the distribution theory of linear models andexperimental design starting from basic mathematical concepts inlinear algebra.
The author begins with a presentation of the classicfixed-effects linear model and goes on to illustrate eight commonlinear models, along with the value of their use in statistics.From this foundation, subsequent chapters introduce conceptspertaining to the linear model, starting with vector space theoryand the theory of least-squares estimation. An outline of theHelmert matrix is also presented, along with a thorough explanationof how the ANOVA is created in both typical two-way and higherlayout designs, ultimately revealing the distribution theory. Otherimportant 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 ofcomparison and contrast, and numerous illustrative exercises assistthe reader with uncovering the nature of the models, using bothclassic and new data sets. Requiring only a working knowledge ofbasic probability and statistical inference, Linear Models is avaluable book for courses on linear models at theupper-undergraduate and graduate levels. It is also an excellentreference for practitioners who use linear models to conductresearch in the fields of econometrics, psychology, sociology,biology, and agriculture.

Autoren/Hrsg.

Clarke, Brenton R.

Inhaltsverzeichnis

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 RandomVariables.
5.2 The Kronecker Product.
5.3 Orthogonal Components in Two-Way ANOVA: One Observation PerCell.
5.4 Orthogonal Components in Two-Way ANOVA withReplications.
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 ofFreedom 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 LinearModel.
10.1 Best Linear Unbiased Error Estimates.
10.2 The Best Linear Unbiased Scalar-Covariance-MatrixApproach.
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.

Über Autor(innen)

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.