Dynamic Methods for the Prediction of Survival Outcomes using Longitudinal Biomarkers

Abstract

In medical research, predicting the probability of a time-to-event outcome is often of interest. Along with failure time data, we may longitudinally observe disease markers that can influence survival. These time-dependent covariates provide additional information that can improve the predictive capability of survival models. It is desirable to use a patient's changing marker information to produce updated survival predictions at future time points, which can in turn direct individualized care decisions. In this dissertation, we develop methods that incorporate time-dependent marker information collected during follow-up with the aim of dynamic prediction and inference.

In Chapter I, we compare two methods of dynamic prediction with a longitudinal binary marker, represented by an illness-death model. Joint modeling is a unified, principled approach that produces consistent predictions over time; however, it requires restrictive distributional assumptions and can involve computationally intensive estimation. Landmarking fits a Cox model at a sequence of prediction, or "landmark", times and is easily implemented, but does not produce a valid prediction function. We explore the theoretical justification and predictive capabilities of these methods, and propose extensions within the landmark framework to provide a better approximation to the true joint model.

In Chapter II, we present an approximate approach for obtaining dynamic predictions that combines the advantages of joint modeling and landmarking. We specify the marginal marker and failure time distributions conditional on surviving up to a prediction time, and use a Gaussian copula to link them over time with an association function. We use a single model for the time-to-event outcome from which the conditional survival is derived, achieving a greater level of consistency than landmarking. Estimation is conducted using a two-stage approach that reduces the computational burden associated with joint modeling.

In Chapter III, we introduce a model that incorporates the effects of a partially observed marker on failure time. We consider the marker to represent an underlying stochastic risk process that accumulates over time until a failure is experienced. We model this increasing risk as a Levy bridge process that has a multiplicative effect on the cumulative hazard. Using the mathematically tractable properties of the gamma process, we derive the marginal and conditional survival functions, and demonstrate estimation when the process is observed at the survival time. This approach can be extended to multiple measurement times, and applied to a variety of markers and disease settings where the correct marker distribution is not known or difficult to specify.