Wu / Tian Nonparametric Models for Longitudinal Data

With Implementation in R
1. Auflage 2018
ISBN: 978-0-429-93908-2
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 582 Seiten

Reihe: Chapman & Hall/CRC Monographs on Statistics and Applied Probability

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

Autoren/Hrsg.

Wu, Colin O.

Tian, Xin

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik

Inhaltsverzeichnis

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

Über Autor(innen)

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.