Generalized Linear Models

A valuable overview of the most important ideas and results in statistical modeling Written by a highly-experienced author, Foundations of Linear and Generalized Linear Models is a clear and comprehensive guide to the key concepts and results of linearstatistical models. The book presents a broad, in-depth overview of the most commonly usedstatistical models by discussing the theory underlying the models, R software applications,and examples with crafted models to elucidate key ideas and promote practical modelbuilding. The book begins by illustrating the fundamentals of linear models, such as how the model-fitting projects the data onto a model vector subspace and how orthogonal decompositions of the data yield information about the effects of explanatory variables. Subsequently, the book covers the most popular generalized linear models, which include binomial and multinomial logistic regression for categorical data, and Poisson and negative binomial loglinear models for count data. Focusing on the theoretical underpinnings of these models, Foundations ofLinear and Generalized Linear Models also features: An introduction to quasi-likelihood methods that require weaker distributional assumptions, such as generalized estimating equation methods An overview of linear mixed models and generalized linear mixed models with random effects for clustered correlated data, Bayesian modeling, and extensions to handle problematic cases such as high dimensional problems Numerous examples that use R software for all text data analyses More than 400 exercises for readers to practice and extend the theory, methods, and data analysis A supplementary website with datasets for the examples and exercises An invaluable textbook for upper-undergraduate and graduate-level students in statistics and biostatistics courses, Foundations of Linear and Generalized Linear Models is also an excellent reference for practicing statisticians and biostatisticians, as well as anyone who is interested in learning about the most important statistical models for analyzing data.

Provides an easy-to-understand guide to statistical linear models and its uses in data analysis This book defines a broad spectrum of statistical linear models that is useful in the analysis of data. Considerable rewriting was done to make the book more reader friendly than the first edition. Linear Models, Second Edition is written in such a way as to be self-contained for a person with a background in basic statistics, calculus and linear algebra. The text includes numerous applied illustrations, numerical examples, and exercises, now augmented with computer outputs in SAS and R. Also new to this edition is: • A greatly improved internal design and format • A short introductory chapter to ease understanding of the order in which topics are taken up • Discussion of additional topics including multiple comparisons and shrinkage estimators • Enhanced discussions of generalized inverses, the MINQUE, Bayes and Maximum Likelihood estimators for estimating variance components Furthermore, in this edition, the second author adds many pedagogical elements throughout the book. These include numbered examples, end-of-example and end-of-proof symbols, selected hints and solutions to exercises available on the book’s website, and references to “big data” in everyday life. Featuring a thorough update, Linear Models, Second Edition includes: • A new internal format, additional instructional pedagogy, selected hints and solutions to exercises, and several more real-life applications • Many examples using SAS and R with timely data sets • Over 400 examples and exercises throughout the book to reinforce understanding Linear Models, Second Edition is a textbook and a reference for upper-level undergraduate and beginning graduate-level courses on linear models, statisticians, engineers, and scientists who use multiple regression or analysis of variance in their work. SHAYLE R. SEARLE, PhD, was Professor Emeritus of Biometry at Cornell University. He was the author of the first edition of Linear Models, Linear Models for Unbalanced Data, and Generalized, Linear, and Mixed Models (with Charles E. McCulloch), all from Wiley. The first edition of Linear Models appears in the Wiley Classics Library. MARVIN H. J. GRUBER, PhD, is Professor Emeritus at Rochester Institute of Technology, School of Mathematical Sciences. Dr. Gruber has written a number of papers and has given numerous presentations at professional meetings during his tenure as a professor at RIT. His fields of interest include regression estimators and the improvement of their efficiency using shrinkage estimators. He has written and published two books on this topic. Another of his books, Matrix Algebra for Linear Models, also published by Wiley, provides good preparation for studying Linear Models. He is a member of the American Mathematical Society, the Institute of Mathematical Statistics and the American Statistical Association.

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.

This 1971 classic on linear models is once again available–as a Wiley Classics Library Edition. It features material that can be understood by any statistician who understands matrix algebra and basic statistical methods.

Since Lord Rayleigh introduced the idea of viscous damping in his classic work «The Theory of Sound» in 1877, it has become standard practice to use this approach in dynamics, covering a wide range of applications from aerospace to civil engineering. However, in the majority of practical cases this approach is adopted more for mathematical convenience than for modeling the physics of vibration damping. Over the past decade, extensive research has been undertaken on more general «non-viscous» damping models and vibration of non-viscously damped systems. This book, along with a related book Structural Dynamic Analysis with Generalized Damping Models: Identification, is the first comprehensive study to cover vibration problems with general non-viscous damping. The author draws on his considerable research experience to produce a text covering: dynamics of viscously damped systems; non-viscously damped single- and multi-degree of freedom systems; linear systems with non-local and non-viscous damping; reduced computational methods for damped systems; and finally a method for dealing with general asymmetric systems. The book is written from a vibration theory standpoint, with numerous worked examples which are relevant across a wide range of mechanical, aerospace and structural engineering applications. Contents 1. Introduction to Damping Models and Analysis Methods. 2. Dynamics of Undamped and Viscously Damped Systems. 3. Non-Viscously Damped Single-Degree-of-Freedom Systems. 4. Non-viscously Damped Multiple-Degree-of-Freedom Systems. 5. Linear Systems with General Non-Viscous Damping. 6. Reduced Computational Methods for Damped Systems

Linear Statistical Models Developed and refined over a period of twenty years, the material in this book offers an especially lucid presentation of linear statistical models. These models lead to what is usually called «multiple regression» or «analysis of variance» methodology, which, in turn, opens up a wide range of applications to the physical, biological, and social sciences, as well as to business, agriculture, and engineering. Unlike similar books on this topic, Linear Statistical Models emphasizes the geometry of vector spaces because of the intuitive insights this approach brings to an understanding of the theory. While the focus is on theory, examples of applications, using the SAS and S-Plus packages, are included. Prerequisites include some familiarity with linear algebra, and probability and statistics at the postcalculus level. Major topics covered include: * Methods of study of random vectors, including the multivariate normal, chi-square, t and F distributions, central and noncentral * The linear model and the basic theory of regression analysis and the analysis of variance * Multiple regression methods, including transformations, analysis of residuals, and asymptotic theory for regression analysis. Separate sections are devoted to robust methods and to the bootstrap. * Simultaneous confidence intervals: Bonferroni, Scheffe, Tukey, and Bechhofer * Analysis of variance, with two- and three-way analysis of variance * Random component models, nested designs, and balanced incomplete block designs * Analysis of frequency data through log-linear models, with emphasis on vector space viewpoint. This chapter alone is sufficient for a course on the analysis of frequency data.

The essential introduction to the theory and application of linear models—now in a valuable new edition Since most advanced statistical tools are generalizations of the linear model, it is neces-sary to first master the linear model in order to move forward to more advanced concepts. The linear model remains the main tool of the applied statistician and is central to the training of any statistician regardless of whether the focus is applied or theoretical. This completely revised and updated new edition successfully develops the basic theory of linear models for regression, analysis of variance, analysis of covariance, and linear mixed models. Recent advances in the methodology related to linear mixed models, generalized linear models, and the Bayesian linear model are also addressed. Linear Models in Statistics, Second Edition includes full coverage of advanced topics, such as mixed and generalized linear models, Bayesian linear models, two-way models with empty cells, geometry of least squares, vector-matrix calculus, simultaneous inference, and logistic and nonlinear regression. Algebraic, geometrical, frequentist, and Bayesian approaches to both the inference of linear models and the analysis of variance are also illustrated. Through the expansion of relevant material and the inclusion of the latest technological developments in the field, this book provides readers with the theoretical foundation to correctly interpret computer software output as well as effectively use, customize, and understand linear models. This modern Second Edition features: New chapters on Bayesian linear models as well as random and mixed linear models Expanded discussion of two-way models with empty cells Additional sections on the geometry of least squares Updated coverage of simultaneous inference The book is complemented with easy-to-read proofs, real data sets, and an extensive bibliography. A thorough review of the requisite matrix algebra has been addedfor transitional purposes, and numerous theoretical and applied problems have been incorporated with selected answers provided at the end of the book. A related Web site includes additional data sets and SAS® code for all numerical examples. Linear Model in Statistics, Second Edition is a must-have book for courses in statistics, biostatistics, and mathematics at the upper-undergraduate and graduate levels. It is also an invaluable reference for researchers who need to gain a better understanding of regression and analysis of variance.

Since Lord Rayleigh introduced the idea of viscous damping in his classic work «The Theory of Sound» in 1877, it has become standard practice to use this approach in dynamics, covering a wide range of applications from aerospace to civil engineering. However, in the majority of practical cases this approach is adopted more for mathematical convenience than for modeling the physics of vibration damping. Over the past decade, extensive research has been undertaken on more general “non-viscous” damping models and vibration of non-viscously damped systems. This book, along with a related book Structural Dynamic Analysis with Generalized Damping Models: Analysis, is the first comprehensive study to cover vibration problems with general non-viscous damping. The author draws on his considerable research experience to produce a text covering: parametric senistivity of damped systems; identification of viscous damping; identification of non-viscous damping; and some tools for the quanitification of damping. The book is written from a vibration theory standpoint, with numerous worked examples which are relevant across a wide range of mechanical, aerospace and structural engineering applications. Contents 1. Parametric Sensitivity of Damped Systems. 2. Identification of Viscous Damping. 3. Identification of Non-viscous Damping. 4. Quantification of Damping. About the Authors Sondipon Adhikari is Chair Professor of Aerospace Engineering at Swansea University, Wales. His wide-ranging and multi-disciplinary research interests include uncertainty quantification in computational mechanics, bio- and nanomechanics, dynamics of complex systems, inverse problems for linear and nonlinear dynamics, and renewable energy. He is a technical reviewer of 97 international journals, 18 conferences and 13 funding bodies.He has written over 180 refereed journal papers, 120 refereed conference papers and has authored or co-authored 15 book chapters.

A self-contained introduction to matrix analysis theory and applications in the field of statistics Comprehensive in scope, Matrix Algebra for Linear Models offers a succinct summary of matrix theory and its related applications to statistics, especially linear models. The book provides a unified presentation of the mathematical properties and statistical applications of matrices in order to define and manipulate data. Written for theoretical and applied statisticians, the book utilizes multiple numerical examples to illustrate key ideas, methods, and techniques crucial to understanding matrix algebra’s application in linear models. Matrix Algebra for Linear Models expertly balances concepts and methods allowing for a side-by-side presentation of matrix theory and its linear model applications. Including concise summaries on each topic, the book also features: Methods of deriving results from the properties of eigenvalues and the singular value decomposition Solutions to matrix optimization problems for obtaining more efficient biased estimators for parameters in linear regression models A section on the generalized singular value decomposition Multiple chapter exercises with selected answers to enhance understanding of the presented material Matrix Algebra for Linear Models is an ideal textbook for advanced undergraduate and graduate-level courses on statistics, matrices, and linear algebra. The book is also an excellent reference for statisticians, engineers, economists, and readers interested in the linear statistical model.

An advanced discussion of linear models with mixed or random effects. In recent years a breakthrough has occurred in our ability to draw inferences from exact and optimum tests of variance component models, generating much research activity that relies on linear models with mixed and random effects. This volume covers the most important research of the past decade as well as the latest developments in hypothesis testing. It compiles all currently available results in the area of exact and optimum tests for variance component models and offers the only comprehensive treatment for these models at an advanced level. Statistical Tests for Mixed Linear Models: Combines analysis and testing in one self-contained volume. Describes analysis of variance (ANOVA) procedures in balanced and unbalanced data situations. Examines methods for determining the effect of imbalance on data analysis. Explains exact and optimum tests and methods for their derivation. Summarizes test procedures for multivariate mixed and random models. Enables novice readers to skip the derivations and discussions on optimum tests. Offers plentiful examples and exercises, many of which are numerical in flavor. Provides solutions to selected exercises. Statistical Tests for Mixed Linear Models is an accessible reference for researchers in analysis of variance, experimental design, variance component analysis, and linear mixed models. It is also an important text for graduate students interested in mixed models.