Introduction to Linear Regression Analysis

PREFACE xiii
1. INTRODUCTION 1
1.1 Regression and Model Building 1
1.2 Data Collection 5
1.3 Uses of Regression 9
1.4 Role of the Computer 10
2. SIMPLE LINEAR REGRESSION 12
2.1 Simple Linear Regression Model 12
2.2 Least-Squares Estimation of the Parameters 13
2.3 Hypothesis Testing on the Slope and Intercept 22
2.4 Interval Estimation in Simple Linear Regression 29
2.5 Prediction of New Observations 33
2.6 Coeffi cient of Determination 35
2.7 A Service Industry Application of Regression 37
2.8 Using SAS and R for Simple Linear Regression 39
2.9 Some Considerations in the Use of Regression 42
2.10 Regression Through the Origin 45
2.11 Estimation by Maximum Likelihood 51
2.12 Case Where the Regressor x is Random 52
3. MULTIPLE LINEAR REGRESSION 67
3.1 Multiple Regression Models 67
3.2 Estimation of the Model Parameters 70
3.3 Hypothesis Testing in Multiple Linear Regression 84
3.4 Confidence Intervals in Multiple Regression 97
3.5 Prediction of New Observations 104
3.6 A Multiple Regression Model for the Patient Satisfaction Data 104
3.7 Using SAS and R for Basic Multiple Linear Regression 106
3.8 Hidden Extrapolation in Multiple Regression 107
3.9 Standardized Regression Coeffi cients 111
3.10 Multicollinearity 117
3.11 Why Do Regression Coeffi cients Have the Wrong Sign? 119
4. MODEL ADEQUACY CHECKING 129
4.1 Introduction 129
4.2 Residual Analysis 130
4.3 PRESS Statistic 151
4.4 Detection and Treatment of Outliers 152
4.5 Lack of Fit of the Regression Model 156
5. TRANSFORMATIONS AND WEIGHTING TO CORRECT MODEL INADEQUACIES 171
5.1 Introduction 171
5.2 Variance-Stabilizing Transformations 172
5.3 Transformations to Linearize the Model 176
5.4 Analytical Methods for Selecting a Transformation 182
5.5 Generalized and Weighted Least Squares 188
5.6 Regression Models with Random Effect 194
6. DIAGNOSTICS FOR LEVERAGE AND INFLUENCE 211
6.1 Importance of Detecting Infl uential Observations 211
6.2 Leverage 212
6.3 Measures of Infl uence: Cook's D 215
6.4 Measures of Infl uence: DFFITS and DFBETAS 217
6.5 A Measure of Model Performance 219
6.6 Detecting Groups of Infl uential Observations 220
6.7 Treatment of Infl uential Observations 220
7. POLYNOMIAL REGRESSION MODELS 223
7.1 Introduction 223
7.2 Polynomial Models in One Variable 223
7.3 Nonparametric Regression 236
7.4 Polynomial Models in Two or More Variables 242
7.5 Orthogonal Polynomials 248
8. INDICATOR VARIABLES 260
8.1 General Concept of Indicator Variables 260
8.2 Comments on the Use of Indicator Variables 273
8.3 Regression Approach to Analysis of Variance 275
9. MULTICOLLINEARITY 285
9.1 Introduction 285
9.2 Sources of Multicollinearity 286
9.3 Effects of Multicollinearity 288
9.4 Multicollinearity Diagnostics 292
9.5 Methods for Dealing with Multicollinearity 303
9.6 Using SAS to Perform Ridge and Principal-Component Regression 321
10. VARIABLE SELECTION AND MODEL BUILDING 327
10.1 Introduction 327
10.2 Computational Techniques for Variable Selection 338
10.3 Strategy for Variable Selection and Model Building 351
10.4 Case Study: Gorman and Toman Asphalt Data Using SAS 354
11. VALIDATION OF REGRESSION MODELS 372
11.1 Introduction 372
11.2 Validation Techniques 373
11.3 Data from Planned Experiments 385
12. INTRODUCTION TO NONLINEAR REGRESSION 389
12.1 Linear and Nonlinear Regression Models 389
12.2 Origins of Nonlinear Models 391
12.3 Nonlinear Least Squares 395
12.4 Transformation to a Linear Model 397
12.5 Parameter Estimation in a Nonlinear System 400
12.6 Statistical Inference in Nonlinear Regression 409
12.7 Examples of Nonlinear Regression Models 411
12.8 Using SAS and R 412
13. GENERALIZED LINEAR MODELS 421
13.1 Introduction 421
13.2 Logistic Regression Models 422
13.3 Poisson Regression 444
13.4 The Generalized Linear Model 450
14. REGRESSION ANALYSIS OF TIME SERIES DATA 474
14.1 Introduction to Regression Models for Time Series Data 474
14.2 Detecting Autocorrelation: The Durbin-Watson Test 475
14.3 Estimating the Parameters in Time Series Regression Models 480
15. OTHER TOPICS IN THE USE OF REGRESSION ANALYSIS 500
15.1 Robust Regression 500
15.2 Effect of Measurement Errors in the Regressors 511
15.3 Inverse Estimation--The Calibration Problem 513
15.4 Bootstrapping in Regression 517
15.5 Classifi cation and Regression Trees (CART) 524
15.6 Neural Networks 526
15.7 Designed Experiments for Regression 529
APPENDIX A. STATISTICAL TABLES 541
APPENDIX B. DATA SETS FOR EXERCISES 553
APPENDIX C. SUPPLEMENTAL TECHNICAL MATERIAL 574
C.1 Background on Basic Test Statistics 574
C.2 Background from the Theory of Linear Models 577
C.3 Important Results on SSR and SSRes 581
C.4 Gauss-Markov Theorem, Var(epsilon) = sigma2I 587
C.5 Computational Aspects of Multiple Regression 589
C.6 Result on the Inverse of a Matrix 590
C.7 Development of the PRESS Statistic 591
C.8 Development of S2 (i) 593
C.9 Outlier Test Based on R-Student 594
C.10 Independence of Residuals and Fitted Values 596
C.11 Gauss-Markov Theorem, Var(epsilon) = V 597
C.12 Bias in MSRes When the Model Is Underspecifi ed 599
C.13 Computation of Infl uence Diagnostics 600
C.14 Generalized Linear Models 601
APPENDIX D. INTRODUCTION TO SAS 613
D.1 Basic Data Entry 614
D.2 Creating Permanent SAS Data Sets 618
D.3 Importing Data from an EXCEL File 619
D.4 Output Command 620
D.5 Log File 620
D.6 Adding Variables to an Existing SAS Data Set 622
APPENDIX E. INTRODUCTION TO R TO PERFORM LINEAR REGRESSION ANALYSIS 623
E.1 Basic Background on R 623
E.2 Basic Data Entry 624
E.3 Brief Comments on Other Functionality in R 626
E.4 R Commander 627
REFERENCES 628
INDEX 642