Contents
Preface page xiii
Guide to Notation xv
1 Introduction 1
1.1 Assessing the Carcinogenicity of Phenolphthalein 3
1.2 Salinity and Fishing in North Carolina 4
1.3 Management of a Retirement Fund 5
1.4 Biomonitoring of Airborne Mercury 7
1.5 Term Structure of Interest Rates 7
1.6 Air Pollution and Mortality in Milan: The Harvesting Effect 11
2 Parametric Regression 15
2.1 Introduction 15
2.2 Linear Regression Models 15
2.3 Regression Diagnostics 20
2.4 Inference 28
2.5 Parametric Additive Models 36
2.6 Model Selection 44
2.7 Polynomial Regression Models 46
2.8 Nonlinear Regression 48
2.9 Transformations in Regression 51
2.10 Bibliographic Notes 55
2.11 Summary of Formulas 55
3 Scatterplot Smoothing 57
3.1 Introduction 57
3.2 Preliminary Ideas 58
3.3 Practical Implementation 62
3.4 Automatic Knot Selection 64
3.5 Penalized Spline Regression 65
3.6 Quadratic Spline Bases 67
3.7 Other Spline Models and Bases 69
3.8 Other Penalties 74
3.9 General Definition of a Penalized Spline 75
3.10 Linear Smoothers 76
3.11 Error of a Smoother 76
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viii Contents
3.12 Rank of a Smoother 78
3.13 Degrees of Freedom of a Smoother 80
3.14 Residual Degrees of Freedom 82
3.15 Other Approaches to Scatterplot Smoothing 84
3.16 Choosing a Scatterplot Smoother 87
3.17 Bibliographical Notes 88
3.18 Summary of Formulas 89
4 Mixed Models 91
4.1 Introduction 91
4.2 Mixed Models 91
4.3 Prediction 95
4.4 The Linear Mixed Model (LMM) 98
4.5 Estimation and Prediction in LMM 98
4.6 Estimated BLUP (EBLUP) 101
4.7 Standard Error Estimation 102
4.8 Hypothesis Testing 104
4.9 Penalized Splines as BLUPs 108
4.10 Bibliographical Notes 110
4.11 Summary of Formulas 110
5 Automatic Scatterplot Smoothing 112
5.1 Introduction 112
5.2 The Likelihood Approach 113
5.3 The Model Selection Approach 114
5.4 Caveats of Automatic Parameter Selection 120
5.5 Choosing the Knots and Basis Functions 123
5.6 Automatic Selection of the Number of Knots 127
5.7 Bibliographical Notes 131
5.8 Summary of Formulas 131
6 Inference 133
6.1 Introduction 133
6.2 Variability Bands 133
6.3 Confidence and Prediction Intervals 135
6.4 Inference for Penalized Splines 137
6.5 Simultaneous Confidence Bands 142
6.6 Testing the Adequacy of Parametric Models 145
6.7 Testing for No Effect 149
6.8 Inference Using First Derivatives 151
6.9 Testing for Existence of a Feature 156
6.10 Bibliographical Notes 158
6.11 Summary of Formulas 159
7 Simple Semiparametric Models 161
7.1 Introduction 161
7.2 Beyond Scatterplot Smoothing 161
Contents ix
7.3 Semiparametric Binary Offset Model 162
7.4 Additivity and Interactions 164
7.5 General Parametric Component 164
7.6 Inference 167
7.7 Bibliographical Notes 168
8 Additive Models 170
8.1 Introduction 170
8.2 Fitting an Additive Model 171
8.3 Degrees of Freedom 174
8.4 Smoothing Parameter Selection 176
8.5 Hypothesis Testing 181
8.6 Model Selection 183
8.7 Bibliographical Notes 185
9 Semiparametric Mixed Models 186
9.1 Introduction 186
9.2 Additive Mixed Models 186
9.3 Subject-Specific Curves 191
9.4 Bibliographical Notes 192
10 Generalized Parametric Regression 194
10.1 Introduction 194
10.2 Binary Response Data 194
10.3 Logistic Regression 195
10.4 Other Generalized Linear Models 197
10.5 Iteratively Reweighted Least Squares 200
10.6 Hat Matrix, Degrees of Freedom, and Standard Errors 201
10.7 Overdispersion and Variance Functions: Pseudolikelihood 201
10.8 Generalized Linear Mixed Models 203
10.9 Deviance 209
10.10 Technical Details 210
10.11 Bibliographical Notes 213
11 Generalized Additive Models 214
11.1 Introduction 214
11.2 Generalized Scatterplot Smoothing 215
11.3 Generalized Additive Mixed Models 217
11.4 Degrees-of-Freedom Approximations 219
11.5 Automatic Smoothing Parameter Selection 220
11.6 Hypothesis Testing 220
11.7 Model Selection 221
11.8 Density Estimation 221
11.9 Bibliographical Notes 222
12 Interaction Models 223
12.1 Introduction 223
12.2 Binary-by-Continuous Interaction Models 224
x Contents
12.3 Factor-by-Curve Interactions in Additive Models 226
12.4 Varying Coefficient Models 234
12.5 Continuous-by-Continuous Interactions 235
12.6 Bibliographical Notes 237
13 Bivariate Smoothing 238
13.1 Introduction 238
13.2 Choice of Bivariate Basis Functions 240
13.3 Kriging 242
13.4 General Radial Smoothing 248
13.5 Default Automatic Bivariate Smoother 256
13.6 Geoadditive Models 258
13.7 Additive Plus Interaction Models 259
13.8 Generalized Bivariate Smoothing 259
13.9 Appendix: Equivalence of BLUP using ZR and ZP 259
13.10 Bibliographical Notes 260
14 Variance Function Estimation 261
14.1 Introduction 261
14.2 Formulation 263
14.3 Application to the LIDAR Data 264
14.4 Quasilikelihood and Variance Functions 266
14.5 Bibliographical Notes 267
15 Measurement Error 268
15.1 Introduction 268
15.2 Formulation 269
15.3 The Expectation Maximization (EM) Algorithm 270
15.4 Simulated Example Revisited 273
15.5 Sensitivity Analysis Example 273
15.6 Bibliographical Notes 275
16 Bayesian Semiparametric Regression 276
16.1 Introduction 276
16.2 General Framework 277
16.3 Scatterplot Smoothing 279
16.4 Linear Mixed Models 285
16.5 Generalized Linear Mixed Models 288
16.6 Rao–Blackwellization 291
16.7 Bibliographical Notes 292
17 Spatially Adaptive Smoothing 293
17.1 Introduction 293
17.2 A Local Penalty Method 294
17.3 Completely Automatic Algorithm 295
17.4 Bayesian Inference 296
Contents xi
17.5 Simulations 298
17.6 LIDAR Example 304
17.7 Additive Models 305
17.8 Bibliographical Notes 307
18 Analyses 308
18.1 Cancer Rates on Cape Cod 308
18.2 Assessing the Carcinogenicity of Phenolphthalein 308
18.3 Salinity and Fishing in North Carolina 308
18.4 Management of a Retirement Fund 313
18.5 Biomonitoring of Airborne Mercury 314
18.6 Term Structure of Interest Rates 315
18.7 Air Pollution and Mortality in Milan: The Harvesting Effect 319
19 Epilogue 320
19.1 Introduction 320
19.2 Minimalist Statistics 320
19.3 Some Omitted Topics 321
19.4 Future Research 325
A Technical Complements 326
A.1 Introduction 326
A.2 Matrix Definitions and Results 326
A.3 Linear Algebra 331
A.4 Probability Definitions and Results 333
A.5 Maximum Likelihood Estimation 335
A.6 Bibliographical Notes 335
B Computational Issues 336
B.1 Fast Computation of Penalized Spline Smooths 336
B.2 Computation of Covariance Matrix Estimators 351
B.3 Software 353
Bibliography 361
Author Index 375
Notation Index 380
Example Index 381
Subject Index 382