Semiparametric Theory and Missing Data (eBook)

Anastasios Tsiatis (Autor)

eBook Download: PDF

2007 | 2006
XVI, 388 Seiten
Springer New York (Verlag)
978-0-387-37345-4 (ISBN)

Infos

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.

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Missing data arise in almost all scientific disciplines. In many cases, the treatment of missing data in an analysis is carried out in a casual and ad-hoc manner, leading, in many cases, to invalid inference and erroneous conclusions. In the past 20 years or so, there has been a serious attempt to understand the underlying issues and difficulties that come about from missing data and their impact on subsequent analysis. There has been a great deal written on the theory developed for analyzing missing data for finite-dimensional parametric models. This includes an extensive literature on likelihood-based methods and multiple imputation. More recently, there has been increasing interest in semiparametric models which, roughly speaking, are models that include both a parametric and nonparametric component. Such models are popular because estimators in such models are more robust than in traditional parametric models. The theory of missing data applied to semiparametric models is scattered throughout the literature with no thorough comprehensive treatment of the subject.This book combines much of what is known in regard to 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 at a level that 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.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 15.1.2007
• Reihe/Serie: Springer Series in Statistics
• Reihe/Serie: Springer Series in Statistics
• Zusatzinfo: XVI, 388 p.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: average • Estimator • likelihood • Probability • semiparametric
• ISBN-10: 0-387-37345-4 / 0387373454
• ISBN-13: 978-0-387-37345-4 / 9780387373454