Linear Models: The Theory and Application of Analysis of Variance (Wiley Series in Probability and Statistics)

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. Linear Models: The Theory and Application of Analysis of Variance CONTENTS 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 SW 3.4 Some Simple Significance Tests 3.5 Prediction Intervals 3.6 Problems 4 Distribution Theory 4.1 Motivation 4.2 Noncentral X 2 and F Distributions 4.2.1 Noncentral 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-Degree-of- Freedom Test Statistic 5.7 Problems 6 Further Discussion of ANOVA 6.1 The Various Representations of Orthogonal Components 6.2 On the Lack of Orthogonality 6.3 Relationship Algebra 6.4 Triple Classification 6.5 Latin Squares 6.6 2k Factorial Design 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 Cook's 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 Estimatest 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 propertiest†† 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

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.

Find a serious approach to the study of linear models and their applications through discussion of analysis of variance (ANOVA) techniques in Linear Models: The Theory and Application of Analysis of Variance . This book represents an equal emphasis on orthogonal representations that delve from the history of the subject and on the vector-matrix approach that has surfaced in recent years. Over 400 exercises varying in degrees of difficulty, orientation and application with some solutions reinforce applications to linear regression .. This graduate-level book succinctly describes the role of analysis of variance (ANOVA) in the study of linear models. Applications to linear regression (both simple and multiple) and to experimental design are included in order to help explain the theory behind and the analysis of ANOVA techniques