Linear Model Methodology
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Linear Model Methodology

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Given the importance of linear models in statistical theory and experimental research, a good understanding of their fundamental principles and theory is essential. Supported by a large number of examples, Linear Model Methodology provides a strong foundation in the theory of linear models and explores the latest developments in data analysis.





After presenting the historical evolution of certain methods and techniques used in linear models, the book reviews vector spaces and linear transformations and discusses the basic concepts and results of matrix algebra that are relevant to the study of linear models. Although mainly focused on classical linear models, the next several chapters also explore recent techniques for solving well-known problems that pertain to the distribution and independence of quadratic forms, the analysis of estimable linear functions and contrasts, and the general treatment of balanced random and mixed-effects models. The author then covers more contemporary topics in linear models, including the adequacy of Satterthwaite's approximation, unbalanced fixed- and mixed-effects models, heteroscedastic linear models, response surface models with random effects, and linear multiresponse models. The final chapter introduces generalized linear models, which represent an extension of classical linear models.





Linear models provide the groundwork for analysis of variance, regression analysis, response surface methodology, variance components analysis, and more, making it necessary to understand the theory behind linear modeling. Reflecting advances made in the last thirty years, this book offers a rigorous development of the theory underlying linear models.
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Product details

  • Hardback | 562 pages
  • 157.48 x 236.22 x 33.02mm | 929.86g
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 13 Illustrations, black and white
  • 1584884819
  • 9781584884811
  • 2,578,228

Table of contents

Linear Models: Some Historical Perspectives


The Invention of Least Squares


The Gauss-Markov Theorem


Estimability


Maximum Likelihood Estimation


Analysis of Variance (ANOVA)


Quadratic Forms and Craig's Theorem


The Role of Matrix Algebra


The Geometric Approach





Basic Elements of Linear Algebra


Introduction


Vector Spaces


Vector Subspaces


Bases and Dimensions of Vector Spaces


Linear Transformations





Basic Concepts in Matrix Algebra


Introduction and Notation


Some Particular Types of Matrices


Basic Matrix Operations


Partitioned Matrices


Determinants


The Rank of a Matrix


The Inverse of a Matrix


Eigenvalues and Eigenvectors


Idempotent and Orthogonal Matrices


Quadratic Forms


Decomposition Theorems


Some Matrix Inequalities


Function of Matrices


Matrix Differentiation





The Multivariate Normal Distribution


History of the Normal Distribution


The Univariate Normal Distribution


The Multivariate Normal Distribution


The Moment Generating Function


Conditional Distribution


The Singular Multivariate Normal Distribution


Related Distributions


Examples and Additional Results





Quadratic Forms in Normal Variables


The Moment Generating Function


Distribution of Quadratic Forms


Independence of Quadratic Forms


Independence of Linear and Quadratic Forms


Independence and Chi-Squaredness of Several Quadratic Forms


Computing the Distribution of Quadratic Forms


Appendix





Full-Rank Linear Models


Least-Squares Estimation


Properties of Ordinary Least-Squares Estimation


Generalized Least-Squares Estimation


Least-Squares Estimation under Linear Restrictions on ss


Maximum Likelihood Estimation


Inference Concerning ss


Examples and Applications


Less-Than-Full-Rank Linear Models


Parameter Estimation


Some Distributional Properties


Reparameterized Model


Estimable Linear Functions


Simultaneous Confidence Intervals on Estimable Linear Functions


Simultaneous Confidence Intervals on All Contrasts among the Means with Heterogeneous Group Variances


Further Results Concerning Contrasts and Estimable Linear Functions





Balanced Linear Models


Notation and Definitions


The General Balanced Linear Model


Properties of Balanced Models


Balanced Mixed Models


Complete and Sufficient Statistics


ANOVA Estimation of Variance Components


Confidence Intervals on Continuous Functions of the Variance Components


Confidence Intervals on Ratios of Variance Components





The Adequacy of Satterthwaite's Approximation


Satterthwaite's Approximation


Adequacy of Satterthwaite's Approximation


Measuring the Closeness of Satterthwaite's Approximation


Examples


Appendix





Unbalanced Fixed-Effects Models


The R-Notation


Two-Way Models without Interaction


Two-Way Models with Interaction


Higher-Order Models


A Numerical Example


The Method of Unweighted Means





Unbalanced Random and Mixed Models


Estimation of Variance Components


Estimation of Estimable Linear Functions


Inference Concerning the Random One-Way Model


Inference Concerning the Random Two-Way Model


Exact Tests for Random Higher-Order Models


Inference Concerning the Mixed Two-Way Model


Inference Concerning the Random Two-Fold Nested Model


Inference Concerning the Mixed Two-Fold Nested Model


Inference Concerning the General Mixed Linear Model


Appendix





Additional Topics in Linear Models


Heteroscedastic Linear Models


The Random One-Way Model with Heterogeneous Error Variances


A Mixed Two-Fold Nested Model with Heteroscedastic Random Effects


Response Surface Models


Response Surface Models with Random Block Effects


Linear Multiresponse Models





Generalized Linear Models


Introduction


The Exponential Family


Estimation of Parameters


Goodness of Fit


Hypothesis Testing


Confidence Intervals


Gamma-Distributed Response


Bibliography


Index


Exercises appear at the end of each chapter, except for Chapter 1.
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Review quote

This is a comprehensive and up-to-date textbook on the theory of linear models. ... Every chapter besides the first historical one contains many exercises ... . There is also a huge bibliography. The textbook represents an important source for all researchers and lectures in linear models.
-Hilmar Drygas, Zentralblatt MATH, 2012


The outstanding book, written by a prominent researcher and author, presents a wealth of materials on linear models in Chapters 1 though 12 and includes materials on generalized linear models in the last chapter. The material on linear models is an amazing collection of important topics that would benefit researchers, teachers, students, and practitioners and has added value to the book. Many illustrative examples are presented with SAS codes. The examples are practically important and thoughtfully chosen. Exercises at the end of Chapters 2-13 are excellent, and some are valuable resources for the researchers in this area. ... The author has to be commended for his success in executing this so elegantly.
-Subir Ghosh, Journal of Quality Technology, Vol. 44, January 2012


This book provides a very well-written and rigorous account of the theory of linear models. ... In sum, this is a carefully written and reliable book that reflects the experience of the author in teaching graduate level courses on linear models. I will certainly add it to the list of reference textbooks for the graduate one-quarter course on linear model theory taught at UC Santa Cruz.
-Raquel Prado, Journal of the American Statistical Association, September 2011, Vol. 106


This text is a possible choice for a second course in linear model theory.
-David J. Olive, Technometrics, May 2011


The material is well chosen and well organized, and includes many results that are not found in other textbooks. ... Throughout the book, the presentation is very clear and well organized, with a focus on mathematical developments. Most results are stated with proofs, some material is based on the author's own contributions to the field. Generally, many important special cases are treated in detail, which will make the book also highly useful as a reference. There are also many worked-out examples from different subject areas to illustrate the methods. Later chapters also include some instructions on how to use the methods in SAS. Furthermore, there are lots of exercises at the end of each chapter. ... The book is very accessible and encompassing ... the book will be an excellent choice both as a text and as a reference book.
-T. Mildenberger, Statistical Papers, April 2011


The material on which this book is based has been taught in a couple of courses at the University of Florida for about 20 years and the author's skills and experience in doing this are superbly represented in this fine text. ... there are numerous exercises that reinforce both the theoretical and the practical aspects of regression... This is an excellent, reliable, and comprehensive text.
-International Statistical Review (2010), 78


This book provides a thorough overview which is similar to other available texts but in a very different way. The choice of topics covered, their organization and presentation are the unique features that distinguish this book. ... This book is well structured as a textbook as well as a reference with every chapter explaining the definitions, principles and methods of the subject matter illustrated by data-based examples with the details on use of SAS software, wherever possible. ... the topics that are covered in Chapters 7-12 are not generally found in a single book. ... The book would make an excellent textbook for a course on linear models at masters and graduate levels. Moreover, some parts of the book can also be a part of a course on analysis of variance. Overall, the book is a valuable reference for those involved in research and teaching in this area.
-Journal of the Royal Statistical Society, Series A, 2010
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About Andre I. Khuri

Andre I. Khuri is a Professor Emeritus in the Department of Statistics at the University of Florida in Gainesville.
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