Tsiatis Semiparametric Theory and Missing Data

1. Auflage 2007
ISBN: 978-0-387-37345-4
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 388 Seiten

Reihe: Springer Series in Statistics

This book summarizes current knowledge regarding the theory of estimation for semiparametric models with missing data, in an organized and comprehensive manner. It starts with the study of semiparametric methods when there are no missing data. The description of the theory of estimation for semiparametric models is both rigorous and intuitive, relying on geometric ideas to reinforce the intuition and understanding of the theory. These methods are then applied to problems with missing, censored, and coarsened data with the goal of deriving estimators that are as robust and efficient as possible.

Autoren/Hrsg.

Tsiatis, Anastasios

Inhaltsverzeichnis

1;Preface;7
2;Contents;10
3;1 Introduction to Semiparametric Models;16
3.1;1.1 What Is an Infinite-Dimensional Space?;17
3.2;1.2 Examples of Semiparametric Models;18
3.3;1.3 Semiparametric Estimators;23
4;2 Hilbert Space for Random Vectors;25
4.1;2.1 The Space of Mean-Zero q-dimensional Random Functions;25
4.2;2.2 Hilbert Space;27
4.3;2.3 Linear Subspace of a Hilbert Space and the Projection Theorem;28
4.4;2.4 Some Simple Examples of the Application of the Projection Theorem;29
4.5;2.5 Exercises for Chapter 2;33
5;3 The Geometry of Influence Functions;34
5.1;3.1 Super-Efficiency;37
5.2;3.2 m-Estimators (Quick Review);42
5.3;3.3 Geometry of Influence Functions for Parametric Models;51
5.4;3.4 Efficient Influence Function;55
5.5;3.5 Review of Notation for Parametric Models;62
5.6;3.6 Exercises for Chapter 3;63
6;4 Semiparametric Models;65
6.1;4.1 GEE Estimators for the Restricted Moment Model;66
6.2;4.2 Parametric Submodels;71
6.3;4.3 Influence Functions for Semiparametric RAL Estimators;73
6.4;4.4 Semiparametric Nuisance Tangent Space;75
6.5;4.5 Semiparametric Restricted Moment Model;85
6.6;4.6 Adaptive Semiparametric Estimators for the Restricted Moment Model;105
6.7;4.7 Exercises for Chapter 4;110
7;5 Other Examples of Semiparametric Models;112
7.1;5.1 Location-Shift Regression Model;112
7.2;5.2 Proportional Hazards Regression Model with Censored Data;124
7.3;5.3 Estimating the Mean in a Nonparametric Model;136
7.4;5.4 Estimating Treatment Difference in a Randomized Pretest- Posttest Study or with Covariate Adjustment;137
7.5;5.5 Remarks about Auxiliary Variables;144
7.6;5.6 Exercises for Chapter 5;146
8;6 Models and Methods for Missing Data;148
8.1;6.1 Introduction;148
8.2;6.2 Likelihood Methods;154
8.3;6.3 Imputation;155
8.4;6.4 Inverse Probability Weighted Complete- Case Estimator;157
8.5;6.5 Double Robust Estimator;158
8.6;6.6 Exercises for Chapter 6;161
9;7 Missing and Coarsening at Random for Semiparametric Models;162
9.1;7.1 Missing and Coarsened Data;162
9.2;7.2 The Density and Likelihood of Coarsened Data;167
9.3;7.3 The Geometry of Semiparametric Coarsened- Data Models;174
9.4;7.4 Example: Restricted Moment Model with Missing Data by Design;185
9.5;7.5 Recap and Review of Notation;192
9.6;7.6 Exercises for Chapter 7;194
10;8 The Nuisance Tangent Space and Its Orthogonal Complement;196
10.1;8.1 Models for Coarsening and Missingness;196
10.2;8.2 Estimating the Parameters in the Coarsening Model;199
10.3;8.3 The Nuisance Tangent Space when Coarsening Probabilities Are Modeled;201
10.4;8.4 The Space Orthogonal to the Nuisance Tangent Space;203
10.5;8.5 Observed-Data Influence Functions;204
10.6;8.6 Recap and Review of Notation;206
10.7;8.7 Exercises for Chapter 8;207
11;9 Augmented Inverse Probability Weighted Complete- Case Estimators;209
11.1;9.1 Deriving Semiparametric Estimators for ß;209
11.2;9.2 Additional Results Regarding Monotone Coarsening;217
11.3;9.3 Censoring and Its Relationship to Monotone Coarsening;223
11.4;9.4 Recap and Review of Notation;228
11.5;9.5 Exercises for Chapter 9;230
12;10 Improving Efficiency and Double Robustness with Coarsened Data;231
12.1;10.1 Optimal Observed-Data Influence Function Associated with Full- Data Influence Function;231
12.2;10.2 Improving Efficiency with Two Levels of Missingness;235
12.3;10.3 Improving Efficiency with Monotone Coarsening;249
12.4;10.4 Remarks Regarding Right Censoring;264
12.5;10.5 Improving Efficiency when Coarsening Is Nonmonotone;265
12.6;10.6 Recap and Review of Notation;277
12.7;10.7 Exercises for Chapter 10;280
13;11 Locally Efficient Estimators for Coarsened- Data Semiparametric Models;283
13.1;11.1 The Observed-Data Efficient Score;287
13.2;11.2 Strategy for Obtaining Improved Estimators;295
13.3;11.3 Concluding Thoughts;301
13.4;11.4 Recap and Review of Notation;302
13.5;11.5 Exercises for Chapter 11;303
14;12 Approximate Methods for Gaining Efficiency;304
14.1;12.1 Restricted Class of AIPWCC Estimators;304
14.2;12.2 Optimal Restricted (Class 1) Estimators;309
14.3;12.3 Example of an Optimal Restricted ( Class 1) Estimator;318
14.4;12.4 Optimal Restricted (Class 2) Estimators;322
14.5;12.5 Recap and Review of Notation;330
14.6;12.6 Exercises for Chapter 12;331
15;13 Double-Robust Estimator of the Average Causal Treatment Effect;332
15.1;13.1 Point Exposure Studies;332
15.2;13.2 Randomization and Causality;335
15.3;13.3 Observational Studies;336
15.4;13.4 Estimating the Average Causal Treatment Effect;337
15.5;13.5 Coarsened-Data Semiparametric Estimators;338
15.6;13.6 Exercises for Chapter 13;346
16;14 Multiple Imputation: A Frequentist Perspective;347
16.1;14.1 Full- Versus Observed-Data Information Matrix;350
16.2;14.2 Multiple Imputation;352
16.3;14.3 Asymptotic Properties of the Multiple- Imputation Estimator;354
16.4;14.4 Asymptotic Distribution of the Multiple- Imputation Estimator;362
16.5;14.5 Estimating the Asymptotic Variance;370
16.6;14.6 Proper Imputation;374
16.7;14.7 Surrogate Marker Problem Revisited;379
17;References;383
18;Index;389