Methods and Applications of Linear Models: Regression and the Analysis of Variance

Praise for the Second Edition "An essential desktop reference book . . . it should definitely be on your bookshelf." —Technometrics A thoroughly updated book, Methods and Applications of Linear Models: Regression and the Analysis of Variance, Third Edition features innovative approaches to understanding and working with models and theory of linear regression. The Third Edition provides readers with the necessary theoretical concepts, which are presented using intuitive ideas rather than complicated proofs, to describe the inference that is appropriate for the methods being discussed. The book presents a unique discussion that combines coverage of mathematical theory of linear models with analysis of variance models, providing readers with a comprehensive understanding of both the theoretical and technical aspects of linear models. With a new focus on fixed effects models, Methods and Applications of Linear Models: Regression and the Analysis of Variance, Third Edition also features: Newly added topics including least squares, the cell means model, and graphical inspection of data in the AVE method Frequent conceptual and numerical examples for clarifying the statistical analyses and demonstrating potential pitfalls Graphics and computations developed using JMP® software to accompany the concepts and techniques presented Numerous exercises presented to test readers and deepen their understanding of the material An ideal book for courses on linear models and linear regression at the undergraduate and graduate levels, the Third Edition of Methods and Applications of Linear Models: Regression and the Analysis of Variance is also a valuable reference for applied statisticians and researchers who utilize linear model methodology.

Preface to the Third Edition xvii Preface to the Second Edition xix Preface to the First Edition xxi Part I Regression 1 1 Introduction to Linear Models 3 1.1 Background Information 3 1.2 Mathematical and Statistical Models 5 1.3 Definition of the Linear Model 8 1.4 Examples of Regression Models 13 Exercises 22 2 Regression on Functions of One Variable 25 2.1 The Simple Linear Regression Model 25 2.2 Parameter Estimation 27 2.3 Properties of the Estimators and Test Statistics 36 2.4 Analysis of Simple Linear Regression Models 41 2.5 Examining the Data and the Model 52 2.6 Polynomial Regression Models 69 Exercises 78 3 Transforming the Data 87 3.1 The Need for Transformations 87 3.2 Weighted Least Squares 88 3.3 Variance Stabilizing Transformations 91 3.4 Transformations to Achieve a Linear Model 92 3.5 Analysis of the Transformed Model 98 Exercises 101 4 Regression on Functions of Several Variables 105 4.1 The Multiple Linear Regression Model 105 4.2 Preliminary Data Analysis 106 4.3 Analysis of the Multiple Linear Regression Model 109 4.4 Partial Correlation and Added–Variable Plots 120 4.5 Variable Selection 128 4.6 Model Specification 139 Exercises 146 5 Collinearity in Multiple Linear Regression 151 5.1 Collinearity Problem 151 5.2 An Example With Collinearity 160 5.3 Collinearity Diagnostics 166 5.4 Remedial Solutions: Biased Estimators 177 Exercises 188 6 Influential Observations in Multiple Linear Regression 193 6.1 Influential Data Problem 193 6.2 The Hat Matrix 194 6.3 The Effects of Deleting Observations 199 6.4 Numerical Measures of Influence 203 6.5 Dilemma Data 207 6.6 Plots for Identifying Unusual Cases 213 6.7 Robust/Resistant Methods in Regression Analysis 221 Exercises 225 7 Polynomial Models and Qualitative Predictors 229 7.1 Polynomial Models 229 7.2 The Analysis of Response Surfaces 234 7.3 Models with Qualitative Predictors 238 Exercises 263 8 Additional Topics 271 8.1 Non–Linear Regression Models 271 8.2 Non–Parametric Model–Fitting Methods 277 8.3 Generalized Linear Models 282 8.4 Random Input Variables 290 8.5 Errors in the Inputs 293 8.6 Calibration 294 Exercises 295 Part II Analysis of Variance 299 9 CLASSIFICATION MODELS I: INTRODUCTION 301 9.1 Background Information 301 9.2 The One–Way Classification Model 302 9.3 The Two–Way Classification Model: Balanced Data 320 9.4 The Two–Way Classification Model: Unbalanced Data 337 9.5 The Two–Way Classification Model: No Interaction 349 Exercises 362 10 The Mathematical Theory of Linear Models 375 10.1 The Distribution of Linear and Quadratic Forms 375 10.2 Estimation and Inference for Linear Models 383 10.3 Test of Linear Hypotheses on β 395 10.4 Confidence Regions and Intervals 407 Exercises 410 11 Classification Models II: Multiple Crossed and Nested Factors 419 11.1 The Three–Factor Cross–Classified Model 420 11.2 A General Structure for Balanced, Factorial Models 427 11.3 The Two–Fold Nested Model 432 11.4 A General Structure for Balanced, Nested Models 441 11.5 A Three–Factor, Nested Factorial Model 443 11.6 A General Structure for Balanced, Nested–Factorial Mod4e4ls8 Exercises 452 12 Mixed Models I: The AOV Method with Unbalanced Data 457 12.1 Introduction 457 12.2 Examples of the Analysis of Mixed Models 458 12.3 The General Analysis for Balanced, Mixed Models 478 12.4 Additional Examples 492 12.5 Alternative Developments for Mixed Models 500 Exercises 506 13 Mixed Models II: The AVE Method with Balanced Data 511 13.1 Introduction 511 13.2 The Two–Way Cross–Classification Model 512 13.3 The Three–Factor, Cross–Classification Model 524 13.4 Nested Models 529 13.5 Nested Factorial Models 532 13.6 A General Description of the AVE Table 537 13.7 Additional Examples 545 13.8 The Computational Procedure for the AVE Method 551 Exercises 552 14 Mixed Models III: Unbalanced Data 557 14.1 Introduction 557 14.2 Parameter Estimation: Likelihood Methods 559 14.3 ML and REML Estimates with Balanced Data 567 14.4 EM Algorithm for REML Estimation 572 14.5 Diagnostic Analysis with the EM Algorithm 584 14.6 Models with Covariates 593 Exercises 597 15 Simultaneous Inference: Tests and Confidence Intervals 603 15.1 Simultaneous Tests 603 15.2 Simultaneous Confidence Intervals 623 Exercises 625 Appendix A: Mathematics 627 A.I Matrix Algebra 627 A.II Optimization 642 Appendix B: Statistics 647 B.I Distributions 647 B.II The Distribution of Quadratic Forms 651 B.III Estimation 654 B.IV Tests of Hypotheses and Confidence Intervals 656 Appendix C: Data Tables 659 References 675 Index 685

RONALD R. HOCKING, PhD, is Professor Emeritus in the Department of Statistics and Founder of the Ronald R. Hocking Lecture Series at Texas A&M University. A Fellow of the American Statistical Association, Dr. Hocking is the recipient of numerous honors in the statistical community including the Shewell Award, the Youden Award, the Wilcoxon Award, the Snedecor Award, and the Owen Award.