New Statistical Methods for Complex Survival Analysis Problems Open Access
Wei, Bo (Summer 2021)
Abstract
In biomedical studies, the analysis of time-to-event data may encounter various complex problems. One such scenario is that the observation of recurrent events can be terminated by a dependent event. Another example is that treatment choice is not random, possibly outcome-dependent, and therefore standard approaches comparing treated group versus untreated group generally do not lead to valid estimates for the causal treatment effect of interest. In this dissertation, we develop new statistical methods to handle these complications in survival analysis.
In the first project, we propose two sensible adaptations of the generalized accelerated recurrence time (GART) model (Sun et al., 2016) to handle the recurrent events terminated by a dependent event. Our modeling strategies align with the rationale underlying the use of the survivors' rate function or the adjusted rate function to account for the presence of the dependent terminal event. We establish the asymptotic properties of the new estimators. Simulation studies demonstrate good finite-sample performance of the proposed methods. An application to a dataset from the Cystic Fibrosis Foundation Patient Registry (CFFPR) illustrates the practical utility of the new methods.
In the second project, we propose a new IV framework with randomly censored outcomes where the causal treatment effect is quantified as complier quantile causal effect (CQCE). Employing the special characteristic of IV and adapting the principle of conditional score, we uncover a simple weighting scheme that can be incorporated into the standard censored quantile regression procedure to estimate CQCE. We develop robust nonparametric estimation of the derived weights in the first stage, which permits stable implementation of the second stage estimation based on existing software. We establish rigorous asymptotic properties for the proposed estimator, and confirm its validity and satisfactory finite-sample performance via extensive simulations. The proposed method is applied to a dataset from the Center for International Blood and Marrow Transplant Research (CIBMTR) to evaluate the causal effect of rituximab in diffuse large B-cell lymphoma (DLBCL) patients.
In the third project, we study the IV estimation of the population quantile causal effect (PQCE) with the randomly censored data. Employing the rank similarity assumption, we propose an estimating equation based on the observed quantities. We develop a simple and easily implemented two-step estimation procedure to solve the non-monotonous estimating equation, and propose a sample-based inference approach to avoid computation burden in resampling-based approaches. We rigorously justify the asymptotic properties for the proposed estimator. Extensive simulations have been conducted to confirm its validity and satisfactory finite-sample performance. An application to a dataset from the Center for International Blood and Marrow Transplant Research (CIBMTR) demonstrates the practical utility of the proposed method.
Table of Contents
1 Introduction 1
1.1 Background 2
1.2 Literature Review 5
1.2.1 Existing regression methods for recurrent events data subject to a dependent terminal event 5
1.2.2 Existing work on IV methods in time-to-event data 7
1.2.2.1 Existing work on IV methods for estimating complier causal effect in time-to-event data 8
1.2.2.2 Existing work on IV methods for estimating population causal effect in time-to-event data 9
1.3 Outline 12
2 Generalized Accelerated Recurrence Time Model in the Presence of a Dependent Terminal Event 14
2.1 Notation and Data Scenario 15
2.2 A Review of the GART model 15
2.3 The Proposed Models and Inference Procedure 17
2.3.1 An extension of the GART model based on survivors' rate function 17
2.3.2 Extension of the GART model based on the adjusted rate function 19
2.3.3 Asymptotic properties 21
2.3.4 Inference 22
2.4 Numerical Studies 23
2.4.1 Monte-Carlo simulations 23
2.4.2 An application to a dataset from the Cystic Fibrosis Foundation
Patient Registry 28
2.5 Remarks 32
2.6 Appendix 34
2.6.1 Appendix A: Justification of the Counting Process Formulations of Model (4) and Model (6) 34
2.6.2 Appendix B: Proofs of Theorem 2.1 and Theorem 2.2 36
3 Estimation of Complier Causal Quantile Effects with a Binary Instrumental Variable and Censored Data 42
3.1 Potential Outcomes Framework and Assumptions 43
3.2 The Proposed Method 44
3.2.1 A causal censored quantile regression model 44
3.2.2 Estimation procedure with randomly censored data 45
3.2.3 Asymptotic properties 49
3.2.4 Inference 53
3.3 Numerical Studies 54
3.3.1 Monte-Carlo simulations 54
3.3.2 An application to a dataset from the Center for International Blood and Marrow Transplant Research 59
3.4 Remarks 62
3.5 Appendix 63
3.5.1 Appendix C: Propositions and their proofs 63
3.5.2 Appendix D: Proofs of theorems 3.1 and Theorems 3.2 66
3.5.2.1 Technical lemmas and their proofs 66
3.5.2.2 Proof of theorems 75
4 Estimation of Population Causal Quantile Effects with Instrumental Variables and Censored Data 90
4.1 Potential Outcomes Framework and Assumptions 91
4.2 The Proposed Model 92
4.2.1 Censored population quantile causal effect (CPQCE) model 92
4.2.2 Estimation procedure with randomly censored Data 93
4.2.3 Asymptotic properties 95
4.3 Inference 98
4.4 Numerical Studies 100
4.4.1 Monte-Carlo simulations 100
4.4.2 Application to bone marrow transplant Dataset 102
4.5 Remarks 109
4.6 Appendix 111
4.6.1 Appendix E: Propositions 4.1 and 4.2 and their proofs 111
4.6.1.1 Proposition 4.1 and its proof 111
4.6.1.2 Proposition 4.2 and its proof 112
4.6.2 Appendix F: Proofs of Theorems 4.1 and 4.2 114
4.6.2.1 Proof of Theorem 4.1 114
4.6.2.2 Lemma 4.1 and its proof 119
4.6.2.3 Proof of Theorem 4.2 119
4.6.3 Appendix G: Justification of the Proposed Covariance Estimator 121
5 Summary and Future Work 123
5.1 Summary 124
5.2 Future Work 125
Bibliography 127
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