2005最新出版

Preface xiii

1 Scatterplots and Regression 1

1.1 Scatterplots, 1

1.2 Mean Functions, 9

1.3 Variance Functions, 11

1.4 Summary Graph, 11

1.5 Tools for Looking at Scatterplots, 12

1.5.1 Size, 13

1.5.2 Transformations, 14

1.5.3 Smoothers for the Mean Function, 14

1.6 Scatterplot Matrices, 15
Problems, 17

2 Simple Linear Regression 19

2.1 Ordinary Least Squares Estimation, 21

2.2 Least Squares Criterion, 23

2.3 Estimating σ ²,25

2.4 Properties of Least Squares Estimates, 26

2.5 Estimated Variances, 27

2.6 Comparing Models: The Analysis of Variance, 28

2.6.1 The F -Test for Regression, 30

2.6.2 Interpreting p-values, 31

2.6.3 Power of Tests, 31

2.7 The Coefficient of Determination, R²,31

2.8 Confidence Intervals and Tests, 32

2.8.1 The Intercept, 32

2.8.2 Slope, 33

2.8.3 Prediction, 34

2.8.4 Fitted Values, 35

2.9 The Residuals, 36
Problems, 38

3 Multiple Regression 47

3.1 Adding a Term to a Simple Linear Regression Model, 47

3.1.1 Explaining Variability, 49

3.1.2 Added-Variable Plots, 49

3.2 The Multiple Linear Regression Model, 50

3.3 Terms and Predictors, 51

3.4 Ordinary Least Squares, 54

3.4.1 Data and Matrix Notation, 54

3.4.2 Variance-Covariance Matrix of e,56

3.4.3 Ordinary Least Squares Estimators, 56

3.4.4 Properties of the Estimates, 57

3.4.5 Simple Regression in Matrix Terms, 58

3.5 The Analysis of Variance, 61

3.5.1 The Coefficient of Determination, 62

3.5.2 Hypotheses Concerning One of the Terms, 62

3.5.3 Relationship to the t-Statistic, 63

3.5.4 t-Tests and Added-Variable Plots, 63

3.5.5 Other Tests of Hypotheses, 64

3.5.6 Sequential Analysis of Variance Tables, 64

3.6 Predictions and Fitted Values, 65
Problems, 65

4 Drawing Conclusions 69

4.1 Understanding Parameter Estimates, 69

4.1.1 Rate of Change, 69

4.1.2 Signs of Estimates, 70

4.1.3 Interpretation Depends on Other Terms in the Mean Function, 70

4.1.4 Rank Deficient and Over-Parameterized Mean
Functions, 73

4.1.5 Tests, 74

4.1.6 Dropping Terms, 74

4.1.7 Logarithms, 76

4.2 Experimentation Versus Observation, 77 4.3 Sampling from a Normal Population, 80

4.4 More on R²,81

4.4.1 Simple Linear Regression and R²,83

4.4.2 Multiple Linear Regression, 84

4.4.3 Regression through the Origin, 84

4.5 Missing Data, 84

4.5.1 Missing at Random, 84

4.5.2 Alternatives, 85

4.6 Computationally Intensive Methods, 87

4.6.1 Regression Inference without Normality, 87

4.6.2 Nonlinear Functions of Parameters, 89

4.6.3 Predictors Measured with Error, 90
Problems, 92

5 Weights, Lack of Fit, and More 96

5.1 Weighted Least Squares, 96

5.1.1 Applications of Weighted Least Squares, 98

5.1.2 Additional Comments, 99

5.2 Testing for Lack of Fit, Variance Known, 100

5.3 Testing for Lack of Fit, Variance Unknown, 102

5.4 General F Testing, 105

5.4.1 Non-null Distributions, 107

5.4.2 Additional Comments, 108

5.5 Joint Confidence Regions, 108
Problems, 110

6 Polynomials and Factors 115

6.1 Polynomial Regression, 115

6.1.1 Polynomials with Several Predictors, 117

6.1.2 Using the Delta Method to Estimate a Minimum or a Maximum, 120

6.1.3 Fractional Polynomials, 122

6.2 Factors, 122

6.2.1 No Other Predictors, 123

6.2.2 Adding a Predictor: Comparing Regression Lines, 126

6.2.3 Additional Comments, 129

6.3 Many Factors, 130

6.4 Partial One-Dimensional Mean Functions, 131

6.5 Random Coefficient Models, 134
Problems, 137

7 Transformations 147

7.1 Transformations and Scatterplots, 147

7.1.1 Power Transformations, 148

7.1.2 Transforming Only the Predictor Variable, 150

7.1.3 Transforming the Response Only, 152

7.1.4 The Box and Cox Method, 153

7.2 Transformations and Scatterplot Matrices, 153

7.2.1 The 1D Estimation Result and Linearly Related Predictors, 156

7.2.2 Automatic Choice of Transformation of Predictors, 157

7.3 Transforming the Response, 159

7.4 Transformations of Nonpositive Variables, 160
Problems, 161

8 Regression Diagnostics: Residuals 167

8.1 The Residuals, 167

8.1.1 Difference Between eˆand e, 168

8.1.2 The Hat Matrix, 169

8.1.3 Residuals and the Hat Matrix with Weights, 170

8.1.4 The Residuals When the Model Is Correct, 171

8.1.5 The Residuals When the Model Is Not Correct, 171

8.1.6 Fuel Consumption Data, 173

8.2 Testing for Curvature, 176

8.3 Nonconstant Variance, 177

8.3.1 Variance Stabilizing Transformations, 179

8.3.2 A Diagnostic for Nonconstant Variance, 180

8.3.3 Additional Comments, 185

8.4 Graphs for Model Assessment, 185

8.4.1 Checking Mean Functions, 186

8.4.2 Checking Variance Functions, 189
Problems, 191

9 Outliers and Influence 194

9.1 Outliers, 194

9.1.1 An Outlier Test, 194

9.1.2 Weighted Least Squares, 196

9.1.3 Significance Levels for the Outlier Test, 196

9.1.4 Additional Comments, 197

9.2 Influence of Cases, 198

9.2.1 Cook’s Distance, 198 9.2.2 Magnitude of Di , 199

9.2.3 Computing Di , 200

9.2.4 Other Measures of Influence, 203

9.3 Normality Assumption, 204
Problems, 206

10 Variable Selection 211

10.1 The Active Terms, 211

10.1.1 Collinearity, 214

10.1.2 Collinearity and Variances, 216

10.2 Variable Selection, 217

10.2.1 Information Criteria, 217

10.2.2 Computationally Intensive Criteria, 220

10.2.3 Using Subject-Matter Knowledge, 220

10.3 Computational Methods, 221

10.3.1 Subset Selection Overstates Significance, 225

10.4 Windmills, 226

10.4.1 Six Mean Functions, 226

10.4.2 A Computationally Intensive Approach, 228
Problems, 230

11 Nonlinear Regression 233

11.1 Estimation for Nonlinear Mean Functions, 234

11.2 Inference Assuming Large Samples, 237

11.3 Bootstrap Inference, 244

11.4 References, 248
Problems, 248

12 Logistic Regression 251

12.1 Binomial Regression, 253

12.1.1 Mean Functions for Binomial Regression, 254

12.2 Fitting Logistic Regression, 255

12.2.1 One-Predictor Example, 255

12.2.2 Many Terms, 256

12.2.3 Deviance, 260

12.2.4 Goodness-of-Fit Tests, 261

12.3 Binomial Random Variables, 263

12.3.1 Maximum Likelihood Estimation, 263

12.3.2 The Log-Likelihood for Logistic Regression, 264

12.4 Generalized Linear Models, 265
Problems, 266

Appendix 270

A.1 Web Site, 270

A.2 Means and Variances of Random Variables, 270

A.2.1 E Notation, 270

A.2.2 Var Notation, 271

A.2.3 Cov Notation, 271

A.2.4 Conditional Moments, 272

A.3 Least Squares for Simple Regression, 273

A.4 Means and Variances of Least Squares Estimates, 273

A.5 Estimating E(Y |X) Using a Smoother, 275

A.6 A Brief Introduction to Matrices and Vectors, 278

A.6.1 Addition and Subtraction, 279

A.6.2 Multiplication by a Scalar, 280

A.6.3 Matrix Multiplication, 280

A.6.4 Transpose of a Matrix, 281

A.6.5 Inverse of a Matrix, 281

A.6.6 Orthogonality, 282

A.6.7 Linear Dependence and Rank of a Matrix, 283

A.7 Random Vectors, 283

A.8 Least Squares Using Matrices, 284

A.8.1 Properties of Estimates, 285

A.8.2 The Residual Sum of Squares, 285

A.8.3 Estimate of Variance, 286

A.9 The QR Factorization, 286

A.10 Maximum Likelihood Estimates, 287

A.11 The Box-Cox Method for Transformations, 289

A.11.1 Univariate Case, 289

A.11.2 Multivariate Case, 290

A.12 Case Deletion in Linear Regression, 291

References 293

Author Index 301

Subject Index 305

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