Nonparametric Models for Longitudinal Data: With Implementation in R

This book covers the recent advancement of statistical methods for the analysis of longitudinal data. Real datasets from four large NIH-supported longitudinal clinical trials and epidemiological studies illustrate the practical applications of the statistical methods. This book focuses on the nonparametric approaches, which have gained tremendous popularity in biomedical studies. These approaches have the flexibility to answer many scientific questions that cannot be properly addressed by the existing parametric approaches, such as the linear and nonlinear mixed effects models.

Table of Contents:
Preface Author Bios Author Bios List of Figures List of Tables Introduction and Review Introduction Scientific Objectives of Longitudinal Studies Data Structures and Examples Structures of Longitudinal Data Examples of Longitudinal Studies Objectives of Longitudinal Analysis Conditional-Mean Based Regression Models Parametric Models Semiparametric Models Unstructured Nonparametric Models Structured Nonparametric Models Conditional-Distribution Based Models Conditional Distribution Functions and Functionals Parametric Distribution Models Semiparametric Distribution Models Unstructured Nonparametric Distribution Models Structured Nonparametric Distribution Models Review of Smoothing Methods Local Smoothing Methods Global Smoothing Methods Introduction to R Organization of the Book Parametric and Semiparametric Methods Linear Marginal and Mixed-Effects Models Marginal Linear Models The Linear Mixed-Effects Models Conditional Maximum Likelihood Estimation Maximum Likelihood Estimation Restricted Maximum Likelihood Estimation Likelihood based Inferences Nonlinear Marginal and Mixed-Effects Models Model Formulation and Interpretation Likelihood-based Estimation and Inferences Estimation of Subject-Specific Parameters Semiparametric Partially Linear Models Marginal Partially Linear Models Mixed-Effects Partially Linear Models Iterative Estimation Procedure Profile Kernel Estimators Semiparametric Estimation by Splines R Implementation The BMACS CD Data The ENRICHD BDI Data Remarks and Literature Notes Unstructured Nonparametric Models Kernel and Local Polynomial Methods Least-Squares Kernel Estimators Least-Squares Local Polynomial Estimators Cross-Validation Bandwidths The Leave-One-Subject-Out Cross-Validation A Computation Procedure for Kernel Estimators Heuristic Justification of Cross-Validation Bootstrap Pointwise Confidence Intervals Resampling-Subject Bootstrap Samples Two Bootstrap Confidence Intervals Simultaneous Confidence Bands R Implementation The HSCT Data The BMACS CD Data Asymptotic Properties of Kernel Estimators Mean Squared Errors Assumptions for Asymptotic Derivations Asymptotic Risk Representations Useful Special Cases Remarks and Literature Notes Basis Approximation Smoothing Methods Estimation Method Basis Approximations and Least Squares Selecting Smoothing Parameters Bootstrap Inference Procedures Pointwise Confidence Intervals Simultaneous Confidence Bands Hypothesis Testing R Implementation The HSCT Data The BMACS CD Data Asymptotic Properties Conditional Biases and Variances Consistency of Basis Approximation Estimators Consistency of B-Spline Estimators Convergence Rates Consistency of Goodness-of-Fit Test Remarks and Literature Notes Penalized Smoothing Spline Methods Estimation Procedures Penalized Least Squares Criteria Penalized Smoothing Spline Estimator Cross-Validation Smoothing Parameters Bootstrap Pointwise Confidence Intervals R Implementation The HSCT Data The NGHS BMI Data Asymptotic Properties Assumptions and Equivalent Kernel Function Asymptotic Distributions, Risk and Inferences Green’s Function for Uniform Density Theoretical Derivations Remarks and Literature Notes Time-Varying Coefficient Models Smoothing with Time-Invariant Covariates Data Structure and Model Formulation Data Structure The Time-Varying Coefficient Model A Useful Componentwise Representation Componentwise Kernel Estimators Construction of Estimators through Least Squares Cross-Validation Bandwidth Choices Componentwise Penalized Smoothing Splines Estimators by Componentwise Roughness Penalty Estimators by Combined Roughness Penalty Cross-Validation Smoothing Parameters Bootstrap Confidence Intervals R Implementation The BMACS CD Data A Simulation Study Asymptotic Properties for Kernel Estimators Mean Squared Errors Asymptotic Assumptions Asymptotic Risk Representations Remarks and Implications Useful Special Cases Theoretical Derivations Asymptotic Properties for Smoothing Splines Assumptions and Equivalent Kernel Functions Asymptotic Distributions and Mean Squared Errors Theoretical Derivations Remarks and Literature Notes The One-Step Local Smoothing Methods Data Structure and Model Interpretations Data Structure Model Formulation Model Interpretations Remarks on Estimation Methods Smoothing Based on Local Least Squares Criteria General Formulation Least Squares Kernel Estimators Least Squares Local Linear Estimators Smoothing with Centered Covariates Cross-Validation Bandwidth Choice Pointwise and Simultaneous Confidence Bands Pointwise Confidence Intervals by Bootstrap Simultaneous Confidence Bands R Implementation The NGHS BP Data The BMACS CD Data Asymptotic Properties for Kernel Estimators Asymptotic Assumptions Mean Squared Errors Asymptotic Risk Representations Asymptotic Distributions Asymptotic Pointwise Confidence Intervals Remarks and Literature Notes The Two-Step Local Smoothing Methods Overview and Justifications Raw Estimators General Expression and Properties Component Expressions and Properties Variance and Covariance Estimators Refining the Raw Estimates by Smoothing Rationales for Refining by Smoothing The Smoothing Estimation Step Bandwidth Choices Pointwise and Simultaneous Confidence Bands Pointwise Confidence Intervals by Bootstrap Simultaneous Confidence Bands R Implementation The NGHS BP Data Remark on the Asymptotic Properties Remarks and Literature Notes Global Smoothing Methods Basis Approximation Model and Interpretations Data Structure and Model Formulation Basis Approximation Remarks on Estimation Methods Estimation Method Approximate Least Squares Remarks on Basis and Weight Choices Least Squares B-Spline Estimators Cross-Validation Smoothing Parameters Conditional Biases and Variances Estimation of Variance and Covariance Structures Resampling-Subject Bootstrap Inferences Pointwise Confidence Intervals Simultaneous Confidence Bands Hypothesis Testing for Constant Coefficients R Implementation with the NGHS BP Data Estimation by B-Splines Testing Constant Coefficients Asymptotic Properties Integrated Squared Errors Asymptotic Assumptions Convergence Rates for Integrated Squared Errors Theoretical Derivations Consistent Hypothesis Tests Remarks and Literature Notes Shared-Parameter and Mixed-Effects Models Models for Concomitant Interventions Concomitant Interventions Motivation for Outcome-Adaptive Covariate Two Modeling Approaches Data Structure with a Single Intervention Naive Mixed-Effects Change-Point Models Justifications for Chang-Point Models Model Formulation and Interpretation Biases of Naive Mixed-Effects Models General Structure for Shared-Parameters The Varying-Coefficient Mixed-Effects Models Model Formulation and Interpretation Special Cases of Conditional Mean Effects Likelihood-Based Estimation Least Squares Estimation Estimation of the Covariances The Shared-Parameter Change-Point Models Model Formulation and Justifications The Linear Shared-Parameter Change-Point Model The Additive Shared-Parameter Change-Point Model Likelihood-Based Estimation Gaussian Shared-Parameter Change-Point Models A Two-Stage Estimation Procedure Confidence Intervals for Parameter Estimators Asymptotic Confidence Intervals Bootstrap Confidence Intervals R Implementation to the ENRICHD Data Varying-Coefficient Mixed-Effects Models Shared-Parameter Change-Point Models Asymptotic Consistency The Varying-Coefficient Mixed-Effects Models Maximum Likelihood Estimators The Additive Shared-Parameter Models Remarks and Literature Notes Nonparametric Mixed-Effects Models Objectives of Nonparametric Mixed-Effects Models Data Structure and Model Formulation Data Structure Mixed-Effects Models without Covariates Mixed-Effects Models with a Single Covariate Extensions to Multiple Covariates Estimation and Prediction without Covariates Estimation with Known Covariance Matrix Estimation with Unknown Covariance Matrix Individual Trajectories Cross-Validation Smoothing Parameters Functional Principal Components Analysis The Reduced Rank Model Estimation of Eigenfunctions and Eigenvalues Model Selection of Reduced Ranks Estimation and Prediction with Covariates Models without Covariate Measurement Error Models with Covariate Measurement Error R Implementation The BMACS CD Data The NGHS BP Data Remarks and Literature Notes Nonparametric Models for Distributions Unstructured Models for Distributions Objectives and General Setup Objectives Applications Estimation of Conditional Distributions Rank-Tracking Probability Data Structure and Conditional Distributions Data Structure Conditional Distribution Functions Conditional Quantiles Rank-Tracking Probabilities Rank-Tracking Probability Ratios Continuous and Time-Varying Covariates Estimation Methods Conditional Distribution Functions Conditional Cumulative Distribution Functions Conditional Quantiles and Functionals Rank-Tracking Probabilities Cross-Validation Bandwidth Choices Bootstrap Pointwise Confidence Intervals R Implementation The NGHS BMI Data Asymptotic Properties Asymptotic Assumptions Asymptotic Mean Squared Errors Theoretical Derivations Remarks and Literature Notes Time-Varying Transformation Models - I Overview and Motivation Data Structure and Model Formulation Data Structure The Time-Varying Transformation Models Two-Step Estimation Method Raw Estimates of Coefficients Bias, Variance and Covariance of Raw Estimates Smoothing Estimators Bandwidth Choices Bootstrap Confidence Intervals Implementation and Numerical Results The NGHS Data Asymptotic Properties Conditional Mean Squared Errors Asymptotic Assumptions Asymptotic Risk Expressions Theoretical Derivations Remarks and Literature Notes Time-Varying Transformation Models - Overview and Motivation Data Structure and Distribution Functionals Data Structure Conditional Distribution Functions Conditional Quantiles Rank-Tracking Probabilities Rank-Tracking Probability Ratios The Time-Varying Transformation Models Two-Step Estimation and Prediction Methods Raw Estimators of Distribution Functions Smoothing Estimators for Conditional CDFs Smoothing Estimators for Quantiles Estimation of Rank-Tracking Probabilities Estimation of Rank-Tracking Probability Ratios Bandwidth Choices R Implementation Conditional CDF for the NGHS SBP Data RTP and RTPR for the NGHS SBP Data Asymptotic Properties Asymptotic Assumptions Raw Baseline and Distribution Function Estimators Local Polynomial Smoothing Estimators Theoretical Derivations Remarks and Literature Notes Tracking with Mixed-Effects Models Data Structure and Models Data Structure The Nonparametric Mixed-Effects Models Conditional Distributions and Tracking Indices Prediction and Estimation Methods B-spline Prediction of Trajectories Estimation with Predicted Outcome Trajectories Estimation based on Split Samples Bootstrap Pointwise Confidence Intervals R Implementation with the NGHS Data Rank-Tracking for BMI Rank-Tracking for SBP Remarks and Literature Notes Bibliography Index