Methods and Applications of Linear Models

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 1.5 Concluding Comments, 21 Exercises, 21 2 Regression on Functions of One Variable 23 2.1 The Simple Linear Regression Model, 23 2.2 Parameter Estimation, 25 2.3 Properties of the Estimators and Test Statistics, 34 2.4 The Analysis of Simple Linear Regression Models, 39 2.5 Examining the Data and the Model, 50 2.6 Polynomial Regression Models, 63 Exercises, 72 3 Transforming the Data 81 3.1 The Need for Transformations, 81 3.2 Weighted Least Squares, 82 3.3 Variance Stabilizing Transformations, 85 3.4 Transformations to Achieve a Linear Model, 86 3.5 Analysis of the Transformed Model, 92 Exercises, 95 4 Regression on Functions of Several Variables 99 4.1 The Multiple Linear Regression Model, 99 4.2 Preliminary Data Analysis, 100 4.3 Analysis of the Multiple Linear Regression Model, 103 4.4 Partial Correlation and Added-Variable Plots, 113 4.5 Variable Selection, 119 4.6 Model Specification, 130 Exercises, 137 5 Collinearity in Multiple Linear Regression 142 5.1 The Collinearity Problem, 142 5.2 An Example with Collinearity, 150 5.3 Collinearity Diagnostics, 156 5.4 Remedial Solutions: Biased Estimators, 166 Exercises, 178 6 Influential Observations in Multiple Linear Regression 182 6.1 The Influential Data Problem, 182 6.2 The Hat Matrix, 183 6.3 The Effects of Deleting Observations, 188 6.4 Numerical Measures of Influence, 192 6.5 The Dilemma Data, 197 6.6 Plots for Identifying Unusual Cases, 201 6.7 Robust/Resistant Methods in Regression Analysis, 209 Exercises, 213 7 Polynomial Models and Qualitative Predictors 216 7.1 Polynomial Models, 216 7.2 The Analysis of Response Surfaces, 220 7.3 Models with Qualitative Predictors, 225 Exercises, 247 8 Additional Topics 254 8.1 Nonlinear Regression Models, 254 8.2 Nonparametric Model-Fitting Methods, 260 8.3 Generalized Linear Models, 265 8.4 Random Input Variables, 274 8.5 Errors in the Inputs, 276 8.6 Calibration, 277 Exercises, 278 PART II THE ANALYSIS OF VARIANCE 283 9 Classification Models I: Introduction 285 9.1 Background Information, 285 9.2 The One-Way Classification Model, 286 9.3 The Two-Way Classification Model: Balanced Data, 304 9.4 The Two-Way Classification Model: Unbalanced Data, 322 9.5 The Two-Way Classification Model: No Interaction, 334 9.6 Concluding Comments, 347 Exercises, 347 10 The Mathematical Theory of Linear Models 359 10.1 The Distribution of Linear and Quadratic Forms, 359 10.2 Estimation and Inference for Linear Models, 368 10.3 Tests of Linear Hypotheses on ß, 380 10.4 Confidence Regions and Intervals, 392 Exercises, 395 11 Classification Models II: Multiple Crossed and Nested Factors 405 11.1 The Three-Factor Cross-Classified Model, 406 11.2 A General Structure for Balanced, Factorial Models, 412 11.3 The Twofold Nested Model, 417 11.4 A General Structure for Balanced, Nested Models, 426 11.5 A Three-Factor, Nested-Factorial Model, 429 11.6 A General Structure for Balanced, Nested-Factorial Models, 434 Exercises, 438 12 Mixed Models I: The AOV Method with Balanced Data 443 12.1 Introduction, 443 12.2 Examples of the Analysis of Mixed Models, 444 12.3 The General Analysis for Balanced, Mixed Models, 464 12.4 Additional Examples, 479 12.5 Alternative Developments of Mixed Models, 487 Exercises, 493 13 Mixed Models II: The AVE Method with Balanced Data 499 13.1 Introduction, 499 13.2 The Two-Way Cross-Classification Model, 500 13.3 The Three-Factor, Cross-Classification Model, 511 13.4 Nested Models, 515 13.5 Nested-Factorial Models, 518 13.6 A General Description of the AVE Table, 524 13.7 Additional Examples, 531 13.8 The Computational Procedure for the AVE Method, 537 Exercises, 537 14 Mixed Models III: Unbalanced Data 543 14.1 Introduction, 543 14.2 Parameter Estimation: Likelihood Methods, 545 14.3 ML and REML Estimates with Balanced Data, 554 14.4 The EM Algorithm for REML Estimation, 558 14.5 Diagnostic Analysis with the EM Algorithm, 572 14.6 Models with Covariates, 581 14.7 Summary, 585 Exercises, 585 15 Simultaneous Inference: Tests and Confidence Intervals 591 15.1 Simultaneous Tests, 591 15.2 Simultaneous Confidence Intervals, 610 Exercises, 612 Appendix A Mathematics 615 A.I Matrix Algebra, 615 A.I.1 Notation, 615 A.I.2 The Rank of a Matrix, 616 A.I.3 The Trace of a Matrix, 617 A.I.4 Eigenvalues and Eigenvectors, 617 A.I.5 Quadratic Forms and Definite Matrices, 618 A.I.6 Special Matrices, 619 A.I.7 The Diagonalization of Matrices, 620 A.I.8 Kronecker Products of Matrices, 620 A.I.9 Factorization of Matrices, 621 A.I.10 Matrix Inversion, 622 A.I.11 The Solution of Linear Equations, 624 A.I.12 Generalized Inverses, 627 A.I.13 Cauchy-Schwartz Inequalities, 630 A.II Optimization, 630 A.II.1 The Differentiation of Matrices and Determinants, 630 A.II.2 The Differentiation of a Function with Respect to a Vector, 631 A.II.3 The Optimization of a Function, 632 Appendix B Statistics 634 B.I Distributions, 634 B.I.1 The Normal Distribution, 634 B.I.2 The chi2-Distribution, 637 B.I.3 The t-Distribution, 638 B.I.4 The F-distribution, 639 B.II The Distribution of Quadratic Forms, 639 B.III Estimation, 642 B.III.1 Maximum Likelihood Estimation, 642 B.III.2 Constrained Maximum Likelihood Estimation, 642 B.III.3 Complete, Sufficient Statistics, 643 B.IV Tests of Hypotheses and Confidence Regions, 643 B.IV.1 Tests of Hypotheses, 643 B.IV.2 Confidence Intervals and Regions, 644 Appendix C Data Tables 645 C.I Downloading Data Files from FTP Server, 645 C.II Listing of Data Set Files, 645 Appendix D Statistical Tables 660 References 669 Index 677