Nonparametric Regression for Assessing Time-Varying Effects in Survival Analysis Open Access

Soh, Jae Eui (Summer 2019)

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Most regression models in survival analysis tacitly assume constant effects of covariates on event times. However, this assumption may not always be realistic in practice. In a clinical study for AIDS patients, for example, a treatment might take time to reach its full efficacy rather than right after randomization; meanwhile, the treatment effect might also erode over time as drug resistance develops, e.g., Eshleman et al. (2001). In this dissertation, we present three projects to develop regression models that accommodate time-varying effects of covariates; two projects are for the analysis of recurrent events, and the third one is for the analysis of univariate survival data. 

In the first project, we propose a varying-coefficient model for the mean frequency of recurrent events. We develop an estimation procedure that fully exploits observed data, and a resampling-based inference procedure. Consistency and weak convergence of the proposed estimator are established. Simulation studies demonstrate utility of the estimator with practical sample sizes. Two real data analyses are presented for illustration of the proposed method.

Most models for recurrent events consider the study-time scale, but gap times between recurrent events are of natural interest in many applications. The second project is concerned with the gap-time scale of recurrent events, and we propose a marginal varying-coefficient model for the cumulative hazard function of the gap time. Estimation and inference procedures are developed. We establish consistency and weak convergence of the proposed estimator, and Monte Carlo simulations demonstrate utility of the proposed estimator. An analysis of the bladder tumor trial data is presented for illustration.

In the third project, we propose a semiparametric survival regression model for the analysis of univariate survival data. With a mixture of time-varying and constant effects of covariates, the proposed model generalizes the proportional hazards model of Cox (1972), while being a sub-model of the temporal survival regression of Peng and Huang (2007). We develop an iterative estimation procedure and an inference procedure. Extensive simulations are conducted to assess finite-sample behaviors of the proposed estimator. The proposed method is illustrated by an analysis of the Veterans' Administration lung cancer trial data.

Table of Contents

1.    Introduction and Background

1.1.  Time-To-Event Data

1.1.1.    Survival Time Example: the VA Lung Cancer Trial Data

1.1.2.    Recurrence Time Example: the Bladder Tumor Trial Data

1.2.  Nonparametric Estimators for the One-Sample Problem

1.2.1.    Counting Processes and Nonparametric Estimators

1.3.  Existing Regression Models in Survival Analysis

1.3.1.    Constant-Effects Regression Models for Univariate Survival Time

1.3.2.    Varying-Coefficient Models

1.4.  Existing Models in Recurrent Events

1.4.1.    Intra-Individual Correlation and Marginal Models

1.4.2.    Varying-Coefficient Models

1.5.  Overview

2.    Dynamic Regression with Recurrent Events

2.1.  Model

2.2.  Estimation and Inference

2.2.1.    Point Estimation

2.2.2.    Large Sample Properties

2.2.3.    Interval Estimation

2.2.4.    Average Effect and Test for Varying Effect

2.3.  Simulation Studies

2.3.1.    Simulation 1: Single Covariate with Constant Effect

2.3.2.    Simulation 2: Single Covariate with Time-Varying Effect

2.3.3.    Simulation 3: Two Covariates with Constant and Time-Varying Effects

2.4.  Real Data Analysis

2.4.1.    Analysis of the Bladder Tumor Trial Data

2.4.2.    Analysis of the DISC Trial Data

3.    A Varying-Coefficient Model for Gap Times Between Recurrent Events

3.1.  Model

3.2.  Estimation and Inference

3.2.1.    Point Estimation

3.2.2.    Large Sample Properties

3.2.3.    Interval Estimation

3.2.4.    Average Effect and Test for Varying Effect

3.3.  Simulation Studies

3.3.1.    Simulation 1: Single Covariate with Constant Effect

3.3.2.    Simulation 2: Single Covariate with Time-Varying Effect

3.3.3.    Simulation 3: Two Covariates with Constant and Time-Varying Effects

3.4.  Analysis of the Bladder Tumor Trial Data

4.    Semiparametric Survival Regression with a Mixture of Time-Varying and Constant Effects

4.1.  Model

4.2.  Estimation and Inference

4.2.1.    Estimation Procedure

4.2.2.    Interval Estimation

4.3.  Monte Carlo Simulations under the Mixture Effect Model

4.4.  Efficiency-Loss Study when the Cos Model Holds

4.5.  Analysis of the VA Lung Cancer Trial Data

5.    Summary and Future Work

5.1.  Summary

5.2.  Future Work

Appendix A.    Proofs of Consistency and Weak Convergence in Chapter 2

Appendix B.    Proofs of Consistency and Weak Convergence in Chapter 3


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