Brenton R. Clarke

Linear Models (eBook, PDF)

The Theory and Application of Analysis of Variance

• Format: PDF

Produktbeschreibung

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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Produktdetails

• Verlag: John Wiley & Sons
• Seitenzahl: 272
• Erscheinungstermin: 27. August 2008
• Englisch
• ISBN-13: 9780470377970
• Artikelnr.: 37291729

Autorenporträt

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.

Inhaltsangabe

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 S(c). 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.

Rezensionen

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