Unified Methods for Censored Longitudinal Data and Causality

Transcription

1 Mark J. van der Laan James M. Robins Unified Methods for Censored Longitudinal Data and Causality Springer

2 Preface v Notation 1 1 Introduction Motivation, Bibliographic History, and an Overview of the book Tour through the General Estimation Problem Estimation in a high-dimensional full data model The curse of dimensionality in the full data model Coarsening at random The curse of dimensionality revisited The observed data model General method for construction of locally efficient estimators Comparison with maximum likelihood estimation Example: Causal Effect of Air Pollution on Short-Term Asthma Response Estimating Functions Orthogonal complement of a nuisance tangent space Review of efficiency theory Estimating functions Orthogonal complement of a nuisance tangent space in an observed data model 64

3 viii Contents Basic useful results to compute projections Robustness of Estimating Functions Robustness of estimating functions against misspecification of linear convex nuisance parameters Double robustness of observed data estimating functions Understanding double robustness for a general semiparametric model Doubly robust estimation in censored data models Using Cross-Validation to Select Nuisance Parameter Models A semiparametric model selection criterian Forward/backward selection of a nuisance parameter model based on cross-validation with respect to the parameter of interest Data analysis example: Estimating the causal relationship between boiled water use and diarrhea in HIV-positive men 99 2 General Methodology The General Model and Overview Full Data Estimating Functions Orthogonal complement of the nuisance tangent space in the multivariate generalized linear regression model (MGLM) Orthogonal complement of the nuisance tangent space in the multiplicative intensity model Linking the orthogonal complement of the nuisance tangent space to estimating functions Ill 2.3 Mapping into Observed Data Estimating Functions Initial mappings and reparametrizing the full data estimating functions Initial mapping indexed by censoring and protected nuisance parameter Extending a mapping for a restricted censoring model to a complete censoring model Inverse weighting a mapping developed for a restricted censoring model Beating a given RAL estimator Orthogonalizing an initial mapping w.r.t. G: Double robustness Ignoring information on the censoring mechanism improves efficiency Optimal Mapping into Observed Data Estimating Functions 137

4 ix The corresponding estimating equation Discussion of ingredients of a one-step estimator Guaranteed Improvement Relative to an Initial Estimating Function Construction of Confidence Intervals Asymptotics of the One-Step Estimator Asymptotics assuming consistent estimation of the censoring mechanism Proof of Theorem Asymptotics assuming that either the censoring mechanism or the full data distribution is estimated consistently Proof of Theorem The Optimal Index Finding the optimal estimating function among a given class of estimating functions Estimation of the Optimal Index Reparametrizing the representations of the optimal full data function Estimation of the optimal full data structure estimating function Locally Efficient Estimation with Score-Operator Representation 170 Monotone Censored Data Data Structure and Model Cause-specific censoring Examples Right-censored data on a survival time Right-censored data on quality-adjusted survival time Right-censored data on a survival time with reporting delay Univariately right-censored multivariate failure time data Inverse Probability Censoring Weighted (IPCW) Estimators Identifiability condition Estimation of a marginal multiplicative intensity model Extension to proportional rate models Projecting on the tangent space of the Cox proportional hazards model of the censoring mechanism Optimal Mapping into Estimating Functions 195

5 3.5 Estimation of Q Regression approach: Assuming that the censoring mechanism is correctly specified Maximum likelihood estimation according to a multiplicative intensity model: Doubly robust Maximum likelihood estimation for discrete models: Doubly robust Regression approach: Doubly robust Estimation of the Optimal Index The multivariate generalized regression model The multivariate generalized regression model when covariates are always observed Multivariate failure time regression model Simulation and data analysis for the nonparametric full data model Rigorous Analysis of a Bivariate Survival Estimate Proof of Theorem Prediction of Survival ; General methodology Prediction of survival with Regression Trees Cross-Sectional Data and Right-Censored Data Combined Model and General Data Structure Cause Specific Monitoring Schemes Overview The Optimal Mapping into Observed Data Estimating Functions Identifiability condition Estimation of a parameter on which we have current status data Estimation of a parameter on which we have rightcensored data Estimation of a joint-distribution parameter on which we have current status data and rightcensored data Estimation of the Optimal Index in the MGLM Example: Current Status Data with Time-Dependent Covariates Regression with current status data Previous work and comparison with our results An initial estimator The locally efficient one-step estimator Implementation issues Construction of confidence intervals 255

6 xi A doubly robust estimator Data-adaptive selection of the location parameter Simulations Example 1: No unmodeled covariate Example 2: Unmodeled covariate Data Analysis: California Partners' Study Example: Current Status Data on a Process Until Death 262 Multivariate Right-Censored Multivariate Data General Data Structure Modeling the censoring mechanism Overview Mapping into Observed Data Estimating Functions The initial mapping into observed estimating data functions Generalized Dabrowska estimator of the survival function in the nonparametric full data model Simulation study of the generalized Dabrowka estimator The proposed mapping into observed data estimating functions Choosing the full data estimating function in MGLM Bivariate Right-Censored Failure Time Data Introduction Locally efficient estimation with bivariate rightcensored data Implementation of the locally efficient estimator Inversion of the information operator Asymptotic performance and confidence intervals Asymptotics Simulation methods and results for the nonparametric full data model Data analysis: Twin age at appendectomy Unified Approach for Causal Inference and Censored Data General Model and Method of Estimation Causal Inference with Marginal Structural Models Closed Form Formula for the Inverse of the Nonparametric Information Operator in Causal Inference Models Double Robustness in Point Treatment MSM Marginal Structural Model with Right-Censoring

7 xii Contents Doubly robust estimators in marginal structural models with right-censoring Data Analysis: SPARCS A simulation for estimators of a treatment-specific survival function Structural Nested Model with Right-Censoring The orthogonal complement of a nuisance tangent space in a structural nested model without censoring A class of estimating functions for the marginal structural nested model Analyzing dynamic treatment regimes Simulation for dynamic regimes in point treatment studies Right-Censoring with Missingness Interval Censored Data Interval censoring and right-censoring combined. 368 References 371 Author index 388 Subject index 394 Example index 397