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Nonparametric Regression Methods for Longitudinal Data Analysis: Mixed-Effects Modeling ApproachesBy Hulin Wu and Jin-Ting Zang |
Preface
Contents
Datasets
Matlab Software
Order Book
Contents
1: Introduction
1.1 Motivating Longitudinal Data Examples
1.1.1 Progesterone Data
1.1.2 ACTG 388 Data
1.1.3 MACS Data
1.2 Mixed-Effects Modeling: from Parametric to Nonparametric
1.2.1 Parametric Mixed-Effects Models
1.2.2 Nonparametric Regression and Smoothing
1.2.3 Nonparametric Mixed-Effects Models
1.3 Scope of the Book
1.3.1 Building Blocks of the NPME Models
1.3.2 Fundamental Development of the NPMEModels
1.3.3 Further Extensions of the NPME Models
1.4 Implementation of Methodologies
1.5 Options for Reading This Book
1.6 Bibliographical Notes
2: Parametric Mixed-Effects Models
2.1 Introduction
2.2 Linear Mixed-Effects Model
2.2.1 Model Specification
2.2.2 Estimation of Fixed and Random-Effects
2.2.3 Bayesian Interpretation
2.2.4 Estimation of Variance Components
2.2.5 The EM-Algorithms
2.3 Nonlinear Mixed-Effects Model
2.3.1 Model Specification
2.3.2 Two-Stage Method
2.3.3 First-Order Linearization Method
2.3.4 Conditional First-Order Linearization Method
2.4 Generalized Mixed-Effects Model
2.4.1 Generalized Linear Mixed-Effects Model
2.4.2 Examples of GLME Model
2.4.3 Generalized Nonlinear Mixed-Effects Model
2.5 Summary and Bibliographical Notes
2.6 Appendix: Proofs
3 Nonparametric Regression Smoothers
3.1 Introduction
3.2 Local Polynomial Kernel Smoother
3.2.1 General Degree LPK Smoother
3.2.2 Local Constant and Linear Smoothers
3.2.3 Kernel Function
3.2.4 Bandwidth Selection
3.2.5 An Illustrative Example
3.3 Regression Splines 50
3.3.1 Truncated Power Basis
3.3.2 Regression Spline Smoother
3.3.3 Selection of Number and Location of Knots
3.3.4 General Basis-Based Smoother
3.4 Smoothing Splines
3.4.1 Cubic Smoothing Splines
3.4.2 General Degree Smoothing Splines
3.4.3 Connection between a Smoothing Spline and a LME Model
3.4.4 Connection between a Smoothing Spline and a State-Space Model
3.4.5 Choice of Smoothing Parameters
3.5 Penalized Splines
3.5.1 Penalized Spline Smoother
3.5.2 Connection between a Penalized Spline and a LME Model
3.5.3 Choice of the Knots and Smoothing Parameter Selection
3.5.4 Extension
3.6 Linear Smoother
3.7 Methods for Smoothing Parameter Selection
3.7.1 Goodness of Fit
3.7.2 Model Complexity
3.7.3 Cross-Validation
3.7.4 Generalized Cross-Validation
3.7.5 Generalized Maximum Likelihood
3.7.6 Akaike Information Criterion
3.7.7 Bayesian Information Criterion
3.8 Summary and Bibliographical Notes
4 Local Polynomial Methods
4.1 Introduction 71
4.2 Nonparametric Population Mean Model
4.2.1 Naive Local Polynomial Kernel Method
4.2.2 Local Polynomial Kernel GEE Method
4.2.3 Fan-Zhang’s Two-Step Method
4.3 Nonparametric Mixed-Effects Model
4.4 Local Polynomial Mixed-Effects Modeling
4.4.1 Local Polynomial Approximation
4.4.2 Local Likelihood Approach
4.4.3 Local Marginal Likelihood Estimation
4.4.4 Local Joint Likelihood Estimation
4.4.5 Component Estimation
4.4.6 A Special Case: Local ConstantMixed-Effects Model
4.5 Choosing Good Bandwidths
4.5.1 Leave-One-Subject-Out Cross-Validation
4.5.2 Leave-One-Point-Out Cross-Validation
4.5.3 Bandwidth Selection Strategies
4.6 LPME Backfitting Algorithm 90
4.7 Asymptotical Properties of the LPME Estimators
4.8 Finite Sample Properties of the LPME Estimators
4.8.1 Comparison of the LPME Estimators in Section 4.5.3
4.8.2 Comparison of Different Smoothing Methods
4.8.3 Comparisons of BCHB-Based versus Backfitting-Based LPME Estimators
4.9 Application to the Progesterone Data
4.10 Summary and Bibliographical Notes
4.11 Appendix: Proofs
4.11.1 Conditions
4.11.2 Proofs
5 Regression Spline Methods
5.1 Introduction
5.2 Naive Regression Splines
5.2.1 The NRS Smoother
5.2.2 Variability Band Construction
5.2.3 Choice of the Bases
5.2.4 Knot Locating Methods
5.2.5 Selection of the Number of Basis Functions
5.2.6 Example and Model Checking
5.2.7 Comparing GCV against SCV
5.3 Generalized Regression Splines
5.3.1 The GRS Smoother
5.3.2 Variability Band Construction
5.3.3 Selection of the Number of Basis Functions
5.3.4 Estimating the Covariance Structure
5.4 Mixed-Effects Regression Splines
5.4.1 Fits and Smoother Matrices
5.4.2 Variability Band Construction
5.4.3 No-Effect Test
5.4.4 Choice of the Bases
5.4.5 Choice of the Number of Basis Functions
5.4.6 Example and Model Checking
5.5 Comparing MERS against NRS
5.5.1 Comparison via the ACTG 388 Data
5.5.2 Comparison via Simulations
5.6 Summary and Bibliographical Notes
5.7 Appendix: Proofs
6 Smoothing Splines Methods
6.1 Introduction
6.2 Naive Smoothing Splines
6.2.1 The NSS Estimator
6.2.2 Cubic NSS Estimator
6.2.3 Cubic NSS Estimator for Panel Data
6.2.4 Variability Band Construction
6.2.5 Choice of the Smoothing Parameter
6.2.6 NSS Fit as BLUP of a LME Model
6.2.7 Model Checking
6.3 Generalized Smoothing Splines
6.3.1 Constructing a Cubic GSS Estimator
6.3.2 Variability Band Construction
6.3.3 Choice of the Smoothing Parameter
6.3.4 Covariance Matrix Estimation
6.3.5 GSS Fit as BLUP of a LME Model
6.4 Extended Smoothing Splines
6.4.1 Subject-Specific Curve Fitting
6.4.2 The ESS Estimators
6.4.3 ESS Fits as BLUPs of a LME Model
6.4.4 Reduction of the Number of Fixed-Effects Parameters
6.5 Mixed-Effects Smoothing Splines
6.5.1 The Cubic MESS Estimators
6.5.2 Bayesian Interpretation
6.5.3 Variance Components Estimation
6.5.4 Fits and Smoother Matrices
6.5.5 Variability Band Construction
6.5.6 Choice of the Smoothing Parameters
6.5.7 Application to the Conceptive Progesterone Data
6.6 General Degree Smoothing Splines
6.6.1 General Degree NSS
6.6.2 General Degree GSS
6.6.3 General Degree ESS
6.6.4 General Degree MESS
6.6.5 Choice of the Bases
6.7 Summary and Bibliographical Notes
6.8 Appendix: Proofs
7 Penalized Spline Methods
7.1 Introduction
7.2 Naive P-Splines
7.2.1 The NPS Smoother
7.2.2 NPS Fits and Smoother Matrix
7.2.3 Variability Band Construction
7.2.4 Degrees of Freedom
7.2.5 Smoothing Parameter Selection
7.2.6 Choice of the Number of Knots
7.2.7 NPS Fit as BLUP of a LME Model
7.3 Generalized P-Splines 203
7.3.1 Constructing the GPS Smoother
7.3.2 Degrees of Freedom
7.3.3 Variability Band Construction
7.3.4 Smoothing Parameter Selection
7.3.5 Choice of the Number of Knots
7.3.6 GPS Fit as BLUP of a LME Model
7.3.7 Estimating the Covariance Structure
7.4 Extended P-Splines
7.4.1 Subject-Specific Curve Fitting
7.4.2 Challenges for Computing the EPS Smoothers
7.4.3 EPS Fits as BLUPs of a LME Model
7.5 Mixed-Effects P-Splines
7.5.1 The MEPS Smoothers
7.5.2 Bayesian Interpretation
7.5.3 Variance Components Estimation
7.5.4 Fits and Smoother Matrices
7.5.5 Variability Band Construction
7.5.6 Choice of the Smoothing Parameters
7.5.7 Choosing the Numbers of Knots
7.6 Summary and Bibliographical Notes
7.7 Appendix: Proofs
8 Semiparametric Models
8.1 Introduction
8.2 Semiparametric Population Mean Model
8.2.1 Model Specification
8.2.2 Local Polynomial Method
8.2.3 Regression Spline Method
8.2.4 Penalized Spline Method
8.2.5 Smoothing Spline Method
8.2.6 Methods Involving No Smoothing
8.2.7 MACS Data
8.3 Semiparametric Mixed-Effects Model
8.3.1 Model Specification
8.3.2 Local Polynomial Method
8.3.3 Regression Spline Method
8.3.4 Penalized Spline Method
8.3.5 Smoothing Spline Method
8.3.6 ACTG 388 Data Revisited
8.3.7 MACS Data Revisted
8.4 Semiparametric Nonlinear Mixed-Effects Model
8.4.1 Model Specification
8.4.2 Wu and Zhang’s Approach
8.4.3 Ke and Wang’s Approach
8.4.4 Generalizations of Ke and Wang’s Approach
8.5 Summary and Bibliographical Notes
9 Time-Varying Coefficient Models
9.1 Introduction
9.2 Time-Varying Coefficient NPM Model
9.2.1 Local Polynomial Kernel Method
9.2.2 Regression Spline Method
9.2.3 Penalized Spline Method
9.2.4 Smoothing Spline Method
9.2.5 Smoothing Parameter Selection
9.2.6 Backfitting Algorithm
9.2.7 Two-Step Method
9.2.8 TVC-NPM Models with Time-Independent Covariates
9.2.9 MACS Data
9.2.10 Progesterone Data
9.3 Time-Varying Coefficient SPM Model
9.4 Time-Varying Coefficient NPME Model
9.4.1 Local Polynomial Method
9.4.2 Regression Spline Method
9.4.3 Penalized Spline Method
9.4.4 Smoothing Spline Method
9.4.5 Backfitting Algorithms
9.4.6 MACS Data Revisted
9.4.7 Progesterone Data Revisted
9.5 Time-Varying Coefficient SPME Model
9.5.1 Backfitting Algorithm
9.5.2 Regression Spline Method
9.6 Summary and Bibliographical Notes
10 Discrete Longitudinal Data
10.1 Introduction
10.2 Generalized NPM Model
10.3 Generalized SPM Model
10.4 Generalized NPME Model
10.4.1 Penalized Local Polynomial Estimation
10.4.2 Bandwidth Selection
10.4.3 Implementation
10.4.4 Asymptotic Theory
10.4.5 Application to an AIDS Clinical Study
10.5 Generalized TVC-NPME Model
10.6 Generalized SAME Model
10.7 Summary and Bibliographical Notes
10.8 Appendix: Proofs