Generalized Additive Models

Title
Copyright
Dedication
Contents
Preface
1 Introduction
	1.1 What\'s in this book
	1.2 A word of caution
	1.3 How to read this book
	1.4 Data sets used in this book
	1.5 Notation
2 Smoothing
	2.1 What is a smoother?
	2.2 Scatterplot smoothing: definition
	2.3 Parametric regression
	2.4 Bin smoothers
	2.5 Running-mean and running-line smootht)rs
	2.6 Kernel smoothers
	2. 7 Running medians and enhancements
	2.8 Equivalent kernels
	2.9 Regression splines
		Cornp·utational aspects
	2.10 Cubic smoothing splines
		Computational aspects
	2.11 Locally-weighted running-line smoothers
	2.12 Smoothers for multiple predictors
	2.13 Bibliographic notes
	2.14 Further results and exercises 2
3 Smoothing in detail
	3.1 Introduction
	3.2 A formal model for scatterplot smoothing
	3.3 The bias-variance trade-off
	3.4 Automatic selection of smoothing parameters
		3.4.1 Crossvalidation
		3.4.2 The bias-variance trade-off for linear smoothers
		3.4.3 Cross-validation for linear smoothers
		3.4.4 The Gp statistic
		3.4.5 Variations on cross-validation and a comparison with Gp
		3.4.6 Discussion
	3.5 Degrees of freedom of a smoother
	3.6 A Bayesian model for smoothing
	3.7 Eigenanalysis of a smoother and spectral smoothing
	3.8 Variance of a smooth and confidence bands
		3.8.1 Pointwise standard-error bands
		3.8.2 Global confidence sets
	3.9 Approximate F-tests
	3.10 Asymptotic behaviour of smoothers
	3.11 Special topics
		3.11.1 Nonlinear smoothers
		3.11.2 Kriging
		3.11.3 Smoothing and penalized least-squares
		3.11.4 Weighted smoothing
		3.11.5 Tied predictor values
		3.11.6 Resistant smoothing
	3.12 Bibliographical notes
	3.13 Further results and exercises 3
4 Additive models
	4.1 Introduction
	4.2 Multiple regression and linear models
	4.3 Additive models
	4.4 Fitting additive models
	4.5 Generalized additive models: logistic regression
	4.6 Bibliographic notes
	4. 7 Further results and exercises 4
5 Some theory for additive models
	5.1 Introduction
	5.2 Estimating equations for additive models
		5.2.1 L2 fimction spaces
		5.2.2 Penalized least-squares
		5.2.3 Reproducing-kernel Hilbert-spnc1!s
	5.3 Solutions to the estimating equations
		5.3.1 Introduction
		5.3.2 Projection smoothers
		5.3.3 Scmi-parametric models
		5.3.4 Back fitting with two smoothers
		5.3.5 Existence and uniqueness: p-smoothers
		5.3.6 Convergence of backfitting: p-smoothers
		5.3.7 Summary of the main results of the section
	5.4 Special topics
		5.4.1 Weighted additive models
		5.4.2 A modified backfitting algorithm
		5.4.3 Explicit solutions to the estimating equations
		5.4.4 Standard errors
		5.4.5 Degree of freedom
		5.4.6 A Bayesian version of additive models
	5.5 Bibliographic notes
	5.6 Further results and exercises 5
6 Generalized additive models
	6.1 Introduction
	6.2 Fisher scoring for generalized linear models
	6.3 Local scoring for generalized additive models
	6.4 Illustrations
		6.4.1 Clotting times of blood
		6.4.2 Warm cardioplegia
	6.5 Derivation of the local-scoring procedure
		6.5.1 L2 function spaces
		6.5.2 Penalized likelihood
	6.6 Convergence of the local-scoring algorithm
	6. 7 Semi-parametric generalized linear models
	6.8 Inference
		6.8.1 Analysis of deviance
		6.8.2 Standard error bands
		6.8.3 Degrees of freedom
	6.9 Smoothing parameter selection
	6.10 Overinterpreting additive fits
	6.11 Missing predictor values
	6.12 Estimation of the link function
	6.13 Local-likelihood estimation
	6.14 Bibliographic notes
	6.15 Further results and exercises 6
7 Response transformation models
	7.1 Introduction
	7.2 The ACE algorithm
		7.2.1 Introduction
		7.2.2 ACE in L2 function spaces
		7.2.3 ACE and penalized least-squares
		7.2.4 Convergence of ACE with linear smoothers
		7.2.5 A close ancestor to ACE: canonical correlation
		7.2.6 Some anomalies of ACE
	7.3 Response transformations for regression
		7.3.1 lntrntlm:t·ion
		7.3.2 Generalizations of the Box-Cox procedure
		7.3.3 Comparison of genemlized Box-Cox and ACE
	7.4 Additivity and variance stabilization
		7.4.1 Some properties of AVAS
	7 .5 Further topics
		7.5.1 Prediction from a transformation model
		7.5.2 Methods for inference
	7.6 Bibliographical notes
	7.7 Further results and exercises 7
8 Extensions to other settings
	8.1 Introduction
	8.2 Matched case-control data
		8.2.1 Background
		8.2.2 Estimation
		8.2.3 Maxirnizfrig the conditional likelihood for the linear modle
		8.2.4 Spline estimation for a single function
		8.2.5 Algorithm for the additive model
		8.2.6 A simulated example
	8.3 The proportional-hazards model
		8.3.1 Background
		8.3.2 Estimation
		8.3.3 Further details of the computations
		8.3.4 An example
	8.4 The proportional-odds model
		8.4.1 Background
		8.4.2 Fitting the additiue model
		8.4.3 Illustration
	8.5 Seasonal decomposition of time series
		8.5.1 The STL procedure
		8.5.2 Eigen-analysis and bandwidth selection
	8.6 Bibliographic notes
	8. 7 Further results and exercises 8
9 Further topics
	9.1 Introduction
	9.2 Resistant fitting
		9.2.1 Resistant fitting of additive models
		9.2.2 Resistant fitting of generalized additive models
		9.2.3 Illustration
		9.2.4 Influence and resistance
	9.3 Parametric additive models
		9.3.1 Regression splines
		9.3.2 Simple knot-selection schemes for regression splines
		9.3.3 A simulated example
		9.3.4 Adaptive knot-selection strategies
		9.3.5 Discussion of regression splines
		9.3.6 Genemlizcd ridge regression — pseudo additive models
		9.3. 7 Illustration: diagnostics for additive models
	9.4 Model selection techniques
		9.4.1 Backward and forward stepwise selection techniques
		9.4.2 Adavtive regression splines: TURBO
		9.4.3 Adaptive backfitting: BRUTO
	9.5 Modelling interactions
		9.5.1 Simple interactions
		9.5.2 Interactions resulting in separate curves
		9.5.3 Hiemrchical models
		9.5.4 Examining residuals for interactions
		9.5.5 Illustration
		9.5.6 Regression trees
		9.5.7 Multivariate adaptive regression splines: MARS
	9.6 Bibliographic notes
	9.7 Further results and exercises 9
10 Case studies
	10.1 Introduction
	10.2 Kyphosis in laminectomy patients
	10.3 Atmospheric ozone concentration
APPENDIX A Data
	Kyphosis in Laminectomy patients
	Diabetes data
APPENDIX B Degrees of freedom
APPENDIX C Software
	GAIM
	S software
References
Author index
Subject index