Semiparametric Estimation Methods for Survival and Biomarker Data

Abstract

In many clinical studies, the outcome of interest is an event time. In addition, longitudinal data on biomarkers may be collected, and such information may provide insight into underlying disease risk or severity. The goal in this dissertation is to develop models and estimation procedures that can incorporate survival and longitudinal data to provide patients and clinicians with knowledge of a subject’s disease risk, which may influence future treatment decisions.

Traditional survival analysis methods often place strong assumptions on the effect of covariates on a patient’s predicted risk. More flexible models have been proposed, but estimation of the cumulative baseline hazard function, which is of infinite dimension, proves difficult. In Chapter II, we consider two estimators of the cumulative baseline hazard: the nonparametric maximum likelihood estimator (NPMLE) and a Breslow-type estimator derived from a martingale estimating equation. The Breslow estimator relies solely on current event information, while the NPMLE depends on future data for risk predictions. We derive the asymptotic relative efficiency of the Breslow estimator in comparison to the NPMLE and demonstrate that while theoretically the Breslow estimator might not be fully efficient, in practice it is virtually identical to the NPMLE. The practical implication of this result is that the Breslow estimator may be used with minimal loss of efficiency while being conceptually and computationally more straightforward.

In Chapter III, we consider the role of internal or endogenous time-varying covariates in joint models of survival and biomarker data. The current belief, based largely on intuition, is that the future history of the endogenous marker should not be incorporated into the hazard function. To the contrary, we show that the hazard function conditional on an endogenous process is a construct incorporating missing data associated with the future unobserved trajectory of the process. In addition, in the presence of an endogenous covariate, the validity of the exponential relationship between the survival and the hazard function is questioned. In this chapter, we offer explicit theory and examples of such models and use it to derive a generalized hazard function that satisfies the exponential relationship.

In Chapter IV, we extend prior work on joint models of survival and biomarker data and utilize the framework developed in Chapter III. We consider a discretely observed marker process at measurement times that may be informative. We focus in particular on the case of marked survival, where the marker is measured at the event time for subjects who experience the event of interest. We assume that the marker process can be modeled as a Lévy process that is connected to a survival model through a time transformation. Considering the partially observed marker and an informative measurement time, we derive estimators of the marker parameters.