Title: Semiparametric Inference of Censored Data with Time-dependent Covariates
Description: This thesis develops two semiparametric methods for censored survival data when the covariates involved are time-dependent. Respectively in the two parts of this thesis, we introduce an interquantile regression model and a censored quantile regression model that account for the commonly observed time-dependent covariates in survival analysis. The proposed quantile-based techniques offer a greater model flexibility comparing to the Cox proportional hazards model and the accelerated failure time model. The first half of this thesis introduces a censored interquantile regression model with time-dependent covariates. Conventionally, censored quantile regression stipulates a specific, pointwise conditional quantile of the survival time given covariates. Despite its model flexibility and straightforward interpretation, the pointwise formulation oftentimes yields rather unstable estimates across neighbouring quantile levels with large variances. In view of this phenomenon, we propose a new class of censored interquantile regression models with time-dependent covariates that can capture the relationship between the failure time and the covariate processes of a target population that falls within a specific quantile bracket.

The pooling of information within a homogeneous neighbourhood facilitates more efficient estimates hence more consistent conclusion on statistical significances of the variables concerned. This new formulation can also be regarded as a generalization of the accelerated failure time model for survival data in the sense that it relaxes the assumption of global homogeneity for the error at all quantile levels. By introducing a class of weighted rank-based estimation procedure, our framework allows a quantile-based inference on the covariate effect with a less restrictive set of assumptions. Numerical studies demonstrate that the proposed estimator outperforms existing alternatives under various settings in terms of smaller empirical bias and standard deviation. A perturbation-based resampling method is also developed to reconcile the asymptotic distribution of the parameter estimates. Finally, consistency and weak convergence of the proposed estimator are established via empirical process theory.

In the second half of this thesis, we propose a class of censored quantile regression models for right censored failure time data with time-dependent covariates that only requires a standard conditionally independent censorship. Upon a quantile based transformation, a system of functional estimating equations for the quantile parameters is derived based on the martingale construction. While time-dependent covariates naturally arise in time to event analysis, the few existing literature requires either an independent censoring mechanism or a fully observed covariate process even after the event has occured. The proposed formulation extends the existing censored quantile regression model so that only the covariate history up to the observed event time is required as in the Cox proportional hazards model for time-dependent covariates. A recursive algorithm is developed to evaluate the estimator numerically. Asymptotic properties including uniform consistency and weak convergence of the proposed estimator as a process of the quantile level is established.

Monte Carlo simulations and numerical studies on the clinical trial data of the AIDS Clinical Trials Group is presented to illustrate the numerical performance of the proposed estimator.