Nonparametric Models for Longitudinal Data: With Implementation in R, 1st Edition (Hardback) book cover

Nonparametric Models for Longitudinal Data

With Implementation in R, 1st Edition

By Colin O. Wu, Xin Tian

Chapman and Hall/CRC

552 pages | 72 B/W Illus.

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Description

Nonparametric Models for Longitudinal Data with Implementations in R presents a comprehensive summary of major advances in nonparametric models and smoothing methods with longitudinal data. It covers methods, theories, and applications that are particularly useful for biomedical studies in the era of big data and precision medicine. It also provides flexible tools to describe the temporal trends, covariate effects and correlation structures of repeated measurements in longitudinal data.

This book is intended for graduate students in statistics, data scientists and statisticians in biomedical sciences and public health. As experts in this area, the authors present extensive materials that are balanced between theoretical and practical topics. The statistical applications in real-life examples lead into meaningful interpretations and inferences.

Features:

  • Provides an overview of parametric and semiparametric methods
  • Shows smoothing methods for unstructured nonparametric models
  • Covers structured nonparametric models with time-varying coefficients
  • Discusses nonparametric shared-parameter and mixed-effects models
  • Presents nonparametric models for conditional distributions and functionals
  • Illustrates implementations using R software packages
  • Includes datasets and code in the authors’ website
  • Contains asymptotic results and theoretical derivations

Both authors are mathematical statisticians at the National Institutes of Health (NIH) and have published extensively in statistical and biomedical journals. Colin O. Wu earned his Ph.D. in statistics from the University of California, Berkeley (1990), and is also Adjunct Professor at the Georgetown University School of Medicine. He served as Associate Editor for Biometrics and Statistics in Medicine, and reviewer for National Science Foundation, NIH, and the U.S. Department of Veterans Affairs. Xin Tian earned her Ph.D. in statistics from Rutgers, the State University of New Jersey (2003). She has served on various NIH committees and collaborated extensively with clinical researchers.

Reviews

"This book will be a good reference book in the area of longitudinal data. It provides a self-contained treatment of structured nonparametric models for longitudinal data. Three commonly used method-kernel, polynomial splines, penalization/smoothing splines are all treated with enough depth. The book's coverage of time-varying coefficient models is comprehensive. Shared-parameter and mixed-effects models are very useful in longitudinal data analysis. Section V 'Nonparametric Models for Distribution' summarizes recent development on a new class of models. This section alone will make the book unique among all published books in longitudinal data analysis. Another unique feature of the book is the use of four actual longitudinal studies presented in Section 1.2. The book used data from these four studies when introducing every model/method. This approach motivates each method very well and also shows the usefulness of the method…The book will be an excellent addition to the literature."

~Jianhua Huang, Texas A&M University

Table of Contents

Preface

Author Bios

Author Bios

List of Figures

List of Tables

Introduction and Review

  1. Introduction
  2. 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

  3. Parametric and Semiparametric Methods
  4. 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

  5. Kernel and Local Polynomial Methods
  6. 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

  7. Basis Approximation Smoothing Methods
  8. 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

  9. Penalized Smoothing Spline Methods
  10. 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

  11. Smoothing with Time-Invariant Covariates
  12. 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

  13. The One-Step Local Smoothing Methods
  14. 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

  15. The Two-Step Local Smoothing Methods
  16. 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

  17. Global Smoothing Methods
  18. 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

  19. Models for Concomitant Interventions
  20. 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

  21. Nonparametric Mixed-Effects Models
  22. 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

  23. Unstructured Models for Distributions
  24. 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

  25. Time-Varying Transformation Models - I
  26. 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

  27. Time-Varying Transformation Models -
  28. 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

  29. 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

About the Authors

Both authors are mathematical statisticians at the National Institutes of Health (NIH) and have published extensively in statistical and biomedical journals. Colin O. Wu earned his Ph.D. in statistics from the University of California, Berkeley (1990), and is also Adjunct Professor at the Georgetown University School of Medicine. He served as Associate Editor for Biometrics and Statistics in Medicine, and reviewer for National Science Foundation, NIH, and the U.S. Department of Veterans Affairs. Xin Tian earned her Ph.D. in statistics from Rutgers, the State University of New Jersey (2003). She has served on various NIH committees and collaborated extensively with clinical researchers.

About the Series

Chapman & Hall/CRC Monographs on Statistics and Applied Probability

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Subject Categories

BISAC Subject Codes/Headings:
MAT029000
MATHEMATICS / Probability & Statistics / General