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E-raamat: Nonparametric Models for Longitudinal Data: With Implementation in R

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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

Arvustused

"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

". . . , this book provides a comprehensive review of the structure of longitudinal data, as well as several applicable nonparametric models. It should prove useful for those working with biomedical data, including real world evidence. The focus is on theorems and proofs; references demonstrate the depth of the research. Real life examples are provided in the form of data descriptions, as well as R code and output. These examples will allow the reader to employ the models and gain expertise in the interpretation of the R output." ~Journal of Biopharmaceutical Statistics, Darcy Hille, Merck Research Laboratories

"The authors are to be commended for such a thorough and well written book, which would certainly be of interest to anyone involved in analysing complex longitudinal data or with an interest in nonparametric approaches." ~David Hughes, ISCB Newsletter

"This book gives a good summary of major advances in unstructured nonparametric models, time-varying models (smoothing models), shared-parameter and mixed-effects models and nonparametric models for distributions. It covers methods, theories and applications, presents R codes for programming which is useful for graduate students in statistics, data scientists and statisticians in biomedical sciences and public health." ~ Rózsa Horváth-Bokor (Budakalász), zbMath

List of Figures xvii
List of Tables xxiii
Preface xxv
About the Authors xxix
I Introduction and Review 1(64)
1 Introduction
3(30)
1.1 Scientific Objectives of Longitudinal Studies
3(2)
1.2 Data Structures and Examples
5(6)
1.2.1 Structures of Longitudinal Data
5(1)
1.2.2 Examples of Longitudinal Studies
6(4)
1.2.3 Objectives of Longitudinal Analysis
10(1)
1.3 Conditional-Mean Based Regression Models
11(6)
1.3.1 Parametric Models
12(1)
1.3.2 Semiparametric Models
12(1)
1.3.3 Unstructured Nonparametric Models
13(1)
1.3.4 Structured Nonparametric Models
14(3)
1.4 Conditional-Distribution Based Models
17(6)
1.4.1 Conditional Distribution Functions and Functionals
17(3)
1.4.2 Parametric Distribution Models
20(1)
1.4.3 Semiparametric Distribution Models
20(1)
1.4.4 Unstructured Nonparametric Distribution Models
21(1)
1.4.5 Structured Nonparametric Distribution Models
22(1)
1.5 Review of Smoothing Methods
23(7)
1.5.1 Local Smoothing Methods
24(3)
1.5.2 Global Smoothing Methods
27(3)
1.6 Introduction to R
30(1)
1.7 Organization of the Book
31(2)
2 Parametric and Semiparametric Methods
33(32)
2.1 Linear Marginal and Mixed-Effects Models
33(7)
2.1.1 Marginal Linear Models
34(1)
2.1.2 The Linear Mixed-Effects Models
35(1)
2.1.3 Conditional Maximum Likelihood Estimation
36(1)
2.1.4 Maximum Likelihood Estimation
37(1)
2.1.5 Restricted Maximum Likelihood Estimation
38(1)
2.1.6 Likelihood-Based Inferences
39(1)
2.2 Nonlinear Marginal and Mixed-Effects Models
40(3)
2.2.1 Model Formulation and Interpretation
40(2)
2.2.2 Likelihood-Based Estimation and Inferences
42(1)
2.2.3 Estimation of Subject-Specific Parameters
43(1)
2.3 Semiparametric Partially Linear Models
43(10)
2.3.1 Marginal Partially Linear Models
44(1)
2.3.2 Mixed-Effects Partially Linear Models
45(1)
2.3.3 Iterative Estimation Procedure
46(2)
2.3.4 Profile Kernel Estimators
48(3)
2.3.5 Semiparametric Estimation by Splines
51(2)
2.4 R Implementation
53(10)
2.4.1 The BMACS CD4 Data
53(5)
2.4.2 The ENRICHD BDI Data
58(5)
2.5 Remarks and Literature Notes
63(2)
II Unstructured Nonparametric Models 65(82)
3 Kernel and Local Polynomial Methods
67(30)
3.1 Least Squares Kernel Estimators
67(2)
3.2 Least Squares Local Polynomial Estimators
69(1)
3.3 Cross-Validation Bandwidths
70(3)
3.3.1 The Leave-One-Subject-Out Cross-Validation
70(1)
3.3.2 A Computation Procedure for Kernel Estimators
71(1)
3.3.3 Heuristic Justification of Cross-Validation
72(1)
3.4 Bootstrap Pointwise Confidence Intervals
73(4)
3.4.1 Resampling-Subject Bootstrap Samples
73(1)
3.4.2 Two Bootstrap Confidence Intervals
74(1)
3.4.3 Simultaneous Confidence Bands
75(2)
3.5 R Implementation
77(6)
3.5.1 The HSCT Data
77(2)
3.5.2 The BMACS CD4 Data
79(4)
3.6 Asymptotic Properties of Kernel Estimators
83(12)
3.6.1 Mean Squared Errors
84(1)
3.6.2 Assumptions for Asymptotic Derivations
85(1)
3.6.3 Asymptotic Risk Representations
86(7)
3.6.4 Useful Special Cases
93(2)
3.7 Remarks and Literature Notes
95(2)
4 Basis Approximation Smoothing Methods
97(26)
4.1 Estimation Method
97(4)
4.1.1 Basis Approximations and Least Squares
97(2)
4.1.2 Selecting Smoothing Parameters
99(2)
4.2 Bootstrap Inference Procedures
101(5)
4.2.1 Pointwise Confidence Intervals
101(1)
4.2.2 Simultaneous Confidence Bands
102(1)
4.2.3 Hypothesis Testing
103(3)
4.3 R Implementation
106(4)
4.3.1 The HSCT Data
106(2)
4.3.2 The BMACS CD4 Data
108(2)
4.4 Asymptotic Properties
110(10)
4.4.1 Conditional Biases and Variances
110(1)
4.4.2 Consistency of Basis Approximation Estimators
111(5)
4.4.3 Consistency of B-Spline Estimators
116(1)
4.4.4 Convergence Rates
117(1)
4.4.5 Consistency of Goodness-of-Fit Test
118(2)
4.5 Remarks and Literature Notes
120(3)
5 Penalized Smoothing Spline Methods
123(24)
5.1 Estimation Procedures
123(3)
5.1.1 Penalized Least Squares Criteria
123(1)
5.1.2 Penalized Smoothing Spline Estimator
124(1)
5.1.3 Cross-Validation Smoothing Parameters
125(1)
5.1.4 Bootstrap Pointwise Confidence Intervals
125(1)
5.2 R Implementation
126(4)
5.2.1 The HSCT Data
126(3)
5.2.2 The NGHS BMI Data
129(1)
5.3 Asymptotic Properties
130(16)
5.3.1 Assumptions and Equivalent Kernel Function
130(2)
5.3.2 Asymptotic Distributions, Risk and Inferences
132(4)
5.3.3 Green's Function for Uniform Density
136(3)
5.3.4 Theoretical Derivations
139(7)
5.4 Remarks and Literature Notes
146(1)
III Time-Varying Coefficient Models 147(154)
6 Smoothing with Time-Invariant Covariates
149(44)
6.1 Data Structure and Model Formulation
149(3)
6.1.1 Data Structure
149(1)
6.1.2 The Time-Varying Coefficient Model
150(1)
6.1.3 A Useful Component-wise Representation
151(1)
6.2 Component-wise Kernel Estimators
152(5)
6.2.1 Construction of Estimators through Least Squares
152(2)
6.2.2 Cross-Validation Bandwidth Choices
154(3)
6.3 Component-wise Penalized Smoothing Splines
157(4)
6.3.1 Estimators by Component-wise Roughness Penalty
157(2)
6.3.2 Estimators by Combined Roughness Penalty
159(1)
6.3.3 Cross-Validation Smoothing Parameters
159(2)
6.4 Bootstrap Confidence Intervals
161(1)
6.5 R Implementation
162(5)
6.5.1 The BMACS CD4 Data
162(3)
6.5.2 A Simulation Study
165(2)
6.6 Asymptotic Properties for Kernel Estimators
167(13)
6.6.1 Mean Squared Errors
167(2)
6.6.2 Asymptotic Assumptions
169(1)
6.6.3 Asymptotic Risk Representations
170(3)
6.6.4 Remarks and Implications
173(1)
6.6.5 Useful Special Cases
174(2)
6.6.6 Theoretical Derivations
176(4)
6.7 Asymptotic Properties for Smoothing Splines
180(11)
6.7.1 Assumptions and Equivalent Kernel Functions
180(2)
6.7.2 Asymptotic Distributions and Mean Squared Errors
182(3)
6.7.3 Theoretical Derivations
185(6)
6.8 Remarks and Literature Notes
191(2)
7 The One-Step Local Smoothing Methods
193(48)
7.1 Data Structure and Model Interpretations
193(6)
7.1.1 Data Structure
193(1)
7.1.2 Model Formulation
194(1)
7.1.3 Model Interpretations
195(1)
7.1.4 Remarks on Estimation Methods
196(3)
7.2 Smoothing Based on Local Least Squares Criteria
199(8)
7.2.1 General Formulation
199(1)
7.2.2 Least Squares Kernel Estimators
200(1)
7.2.3 Least Squares Local Linear Estimators
200(2)
7.2.4 Smoothing with Centered Covariates
202(4)
7.2.5 Cross-Validation Bandwidth Choice
206(1)
7.3 Pointwise and Simultaneous Confidence Bands
207(3)
7.3.1 Pointwise Confidence Intervals by Bootstrap
207(2)
7.3.2 Simultaneous Confidence Bands
209(1)
7.4 R Implementation
210(6)
7.4.1 The NGHS BP Data
210(4)
7.4.2 The BMACS CD4 Data
214(2)
7.5 Asymptotic Properties for Kernel Estimators
216(23)
7.5.1 Asymptotic Assumptions
216(1)
7.5.2 Mean Squared Errors
217(3)
7.5.3 Asymptotic Risk Representations
220(5)
7.5.4 Asymptotic Distributions
225(7)
7.5.5 Asymptotic Pointwise Confidence Intervals
232(7)
7.6 Remarks and Literature Notes
239(2)
8 The Two-Step Local Smoothing Methods
241(18)
8.1 Overview and Justifications
241(3)
8.2 Raw Estimators
244(3)
8.2.1 General Expression and Properties
244(2)
8.2.2 Component Expressions and Properties
246(1)
8.2.3 Variance and Covariance Estimators
246(1)
8.3 Refining the Raw Estimates by Smoothing
247(4)
8.3.1 Rationales for Refining by Smoothing
248(1)
8.3.2 The Smoothing Estimation Step
248(3)
8.3.3 Bandwidth Choices
251(1)
8.4 Pointwise and Simultaneous Confidence Bands
251(3)
8.4.1 Pointwise Confidence Intervals by Bootstrap
251(2)
8.4.2 Simultaneous Confidence Bands
253(1)
8.5 R Implementation
254(2)
8.5.1 The NGHS BP Data
254(2)
8.6 Remark on the Asymptotic Properties
256(1)
8.7 Remarks and Literature Notes
256(3)
9 Global Smoothing Methods
259(42)
9.1 Basis Approximation Model and Interpretations
259(2)
9.1.1 Data Structure and Model Formulation
259(1)
9.1.2 Basis Approximation
260(1)
9.1.3 Remarks on Estimation Methods
260(1)
9.2 Estimation Method
261(13)
9.2.1 Approximate Least Squares
261(2)
9.2.2 Remarks on Basis and Weight Choices
263(1)
9.2.3 Least Squares B-Spline Estimators
264(2)
9.2.4 Cross-Validation Smoothing Parameters
266(3)
9.2.5 Conditional Biases and Variances
269(2)
9.2.6 Estimation of Variance and Covariance Structures
271(3)
9.3 Resampling-Subject Bootstrap Inferences
274(8)
9.3.1 Pointwise Confidence Intervals
274(2)
9.3.2 Simultaneous Confidence Bands
276(3)
9.3.3 Hypothesis Testing for Constant Coefficients
279(3)
9.4 R Implementation with the NGHS BP Data
282(5)
9.4.1 Estimation by B-Splines
282(3)
9.4.2 Testing Constant Coefficients
285(2)
9.5 Asymptotic Properties
287(12)
9.5.1 Integrated Squared Errors
287(2)
9.5.2 Asymptotic Assumptions
289(1)
9.5.3 Convergence Rates for Integrated Squared Errors
289(3)
9.5.4 Theoretical Derivations
292(4)
9.5.5 Consistent Hypothesis Tests
296(3)
9.6 Remarks and Literature Notes
299(2)
IV Shared-Parameter and Mixed-Effects Models 301(102)
10 Models for Concomitant Interventions
303(60)
10.1 Concomitant Interventions
303(6)
10.1.1 Motivation for Outcome-Adaptive Covariate
303(2)
10.1.2 Two Modeling Approaches
305(1)
10.1.3 Data Structure with a Single Intervention
306(3)
10.2 Naive Mixed-Effects Change-Point Models
309(9)
10.2.1 Justifications for Change-Point Models
310(1)
10.2.2 Model Formulation and Interpretation
311(2)
10.2.3 Biases of Naive Mixed-Effects Models
313(5)
10.3 General Structure for Shared Parameters
318(2)
10.4 The Varying-Coefficient Mixed-Effects Models
320(9)
10.4.1 Model Formulation and Interpretation
320(2)
10.4.2 Special Cases of Conditional-Mean Effects
322(1)
10.4.3 Likelihood-Based Estimation
323(3)
10.4.4 Least Squares Estimation
326(1)
10.4.5 Estimation of the Covariances
327(2)
10.5 The Shared-Parameter Change-Point Models
329(12)
10.5.1 Model Formulation and Justifications
330(1)
10.5.2 The Linear Shared-Parameter Change-Point Model
331(1)
10.5.3 The Additive Shared-Parameter Change-Point Model
332(1)
10.5.4 Likelihood-Based Estimation
333(3)
10.5.5 Gaussian Shared-Parameter Change-Point Models
336(4)
10.5.6 A Two-Stage Estimation Procedure
340(1)
10.6 Confidence Intervals for Parameter Estimators
341(2)
10.6.1 Asymptotic Confidence Intervals
341(1)
10.6.2 Bootstrap Confidence Intervals
342(1)
10.7 R Implementation to the ENRICHD Data
343(8)
10.7.1 The Varying-Coefficient Mixed-Effects Models
343(4)
10.7.2 Shared-Parameter Change-Point Models
347(4)
10.8 Consistency
351(9)
10.8.1 The Varying-Coefficient Mixed-Effects Models
351(4)
10.8.2 Maximum Likelihood Estimators
355(1)
10.8.3 The Additive Shared-Parameter Models
355(5)
10.9 Remarks and Literature Notes
360(3)
11 Nonparametric Mixed-Effects Models
363(40)
11.1 Objectives of Nonparametric Mixed-Effects Models
363(1)
11.2 Data Structure and Model Formulation
364(14)
11.2.1 Data Structure
364(1)
11.2.2 Mixed-Effects Models without Covariates
365(3)
11.2.3 Mixed-Effects Models with a Single Covariate
368(6)
11.2.4 Extensions to Multiple Covariates
374(4)
11.3 Estimation and Prediction without Covariates
378(6)
11.3.1 Estimation with Known Covariance Matrix
379(1)
11.3.2 Estimation with Unknown Covariance Matrix
380(1)
11.3.3 Individual Trajectories
381(1)
11.3.4 Cross-Validation Smoothing Parameters
382(2)
11.4 Functional Principal Components Analysis
384(6)
11.4.1 The Reduced Rank Model
385(1)
11.4.2 Estimation of Eigenfunctions and Eigenvalues
386(3)
11.4.3 Model Selection of Reduced Ranks
389(1)
11.5 Estimation and Prediction with Covariates
390(3)
11.5.1 Models without Covariate Measurement Error
390(1)
11.5.2 Models with Covariate Measurement Error
391(2)
11.6 R Implementation
393(8)
11.6.1 The BMACS CD4 Data
393(5)
11.6.2 The NGHS BP Data
398(3)
11.7 Remarks and Literature Notes
401(2)
V Nonparametric Models for Distributions 403(126)
12 Unstructured Models for Distributions
405(38)
12.1 Objectives and General Setup
405(4)
12.1.1 Objectives
405(1)
12.1.2 Applications
406(2)
12.1.3 Estimation of Conditional Distributions
408(1)
12.1.4 Rank-Tracking Probability
408(1)
12.2 Data Structure and Conditional Distributions
409(7)
12.2.1 Data Structure
409(2)
12.2.2 Conditional Distribution Functions
411(1)
12.2.3 Conditional Quantiles
411(1)
12.2.4 Rank-Tracking Probabilities
412(2)
12.2.5 Rank-Tracking Probability Ratios
414(1)
12.2.6 Continuous and Time-Varying Covariates
415(1)
12.3 Estimation Methods
416(13)
12.3.1 Conditional Distribution Functions
416(2)
12.3.2 Conditional Cumulative Distribution Functions
418(2)
12.3.3 Conditional Quantiles and Functionals
420(1)
12.3.4 Rank-Tracking Probabilities
421(2)
12.3.5 Cross-Validation Bandwidth Choices
423(4)
12.3.6 Bootstrap Pointwise Confidence Intervals
427(2)
12.4 R Implementation
429(3)
12.4.1 The NGHS BMI Data
429(3)
12.5 Asymptotic Properties
432(9)
12.5.1 Asymptotic Assumptions
433(1)
12.5.2 Asymptotic Mean Squared Errors
434(4)
12.5.3 Theoretical Derivations
438(3)
12.6 Remarks and Literature Notes
441(2)
13 Time-Varying Transformation Models - I
443(28)
13.1 Overview and Motivation
443(1)
13.2 Data Structure and Model Formulation
444(2)
13.2.1 Data Structure
444(1)
13.2.2 The Time-Varying Transformation Models
445(1)
13.3 Two-Step Estimation Method
446(11)
13.3.1 Raw Estimates of Coefficients
446(2)
13.3.2 Bias, Variance and Covariance of Raw Estimates
448(3)
13.3.3 Smoothing Estimators
451(2)
13.3.4 Bandwidth Choices
453(3)
13.3.5 Bootstrap Confidence Intervals
456(1)
13.4 R Implementation
457(3)
13.4.1 The NGHS Data
457(3)
13.5 Asymptotic Properties
460(9)
13.5.1 Conditional Mean Squared Errors
461(1)
13.5.2 Asymptotic Assumptions
461(1)
13.5.3 Asymptotic Risk Expressions
462(2)
13.5.4 Theoretical Derivations
464(5)
13.6 Remarks and Literature Notes
469(2)
14 Time-Varying Transformation- Models - II
471(40)
14.1 Overview and Motivation
471(2)
14.2 Data Structure and Distribution Functionals
473(6)
14.2.1 Data Structure
473(1)
14.2.2 Conditional Distribution Functions
473(1)
14.2.3 Conditional Quantiles
474(1)
14.2.4 Rank-Tracking Probabilities
475(1)
14.2.5 Rank-Tracking Probability Ratios
476(1)
14.2.6 The Time-Varying Transformation Models
477(2)
14.3 Two-Step Estimation and Prediction Methods
479(9)
14.3.1 Raw Estimators of Distribution Functions
479(1)
14.3.2 Smoothing Estimators for Conditional CDFs
480(3)
14.3.3 Smoothing Estimators for Quantiles
483(1)
14.3.4 Estimation of Rank-Tracking Probabilities
483(2)
14.3.5 Estimation of Rank-Tracking Probability Ratios
485(1)
14.3.6 Bandwidth Choices
485(3)
14.4 R Implementation
488(4)
14.4.1 Conditional CDF for the. NGHS SBP Data
488(2)
14.4.2 RTP and RTPR for the NGHS SBP Data
490(2)
14.5 Asymptotic Properties
492(18)
14.5.1 Asymptotic Assumptions
492(1)
14.5.2 Raw Baseline and Distribution Function Estimators
493(4)
14.5.3 Local Polynomial Smoothing Estimators
497(6)
14.5.4 Theoretical Derivations
503(7)
14.6 Remarks and Literature Notes
510(1)
15 Tracking with Mixed-Effects Models
511(18)
15.1 Data Structure and Models
511(3)
15.1.1 Data Structure
511(1)
15.1.2 The Nonparametric Mixed-Effects Models
512(1)
15.1.3 Conditional Distributions and Tracking Indices
512(2)
15.2 Prediction and Estimation Methods
514(9)
15.2.1 B-spline Prediction of Trajectories
514(3)
15.2.2 Estimation with Predicted Outcome Trajectories
517(4)
15.2.3 Estimation Based on Split Samples
521(1)
15.2.4 Bootstrap Pointwise Confidence Intervals
522(1)
15.3 R Implementation with the NGHS Data
523(3)
15.3.1 Rank-Tracking for BMI
523(3)
15.3.2 Rank-Tracking for SBP
526(1)
15.4 Remarks and Literature Notes
526(3)
Bibliography 529(12)
Index 541
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.
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