Multi-State Models for Interval-Censored Data with Transition Times from Gamma Processes

Abstract

Describing progression of a disease or the life history of an individual with multi-state models has been a topic of interest for many years. A challenge with these studies is that the data are often not continuously observed, i.e. the transition times are not recorded precisely and therefore interval-censored. The aim of this thesis is to introduce modeling of transition times as the threshold crossing times for Gamma processes in multi-state models for interval-censored data. To make this possible, we construct a suitable likelihood framework, where we set up a general likelihood for the three-state progressive model, the illness-death model, the four-state progressive model and a four-state illness-death model. The likelihood framework we create is general, meaning the transition times can be modeled by any parametric survival model. The fitting of our parametric models and the large-sample properties of the maximum likelihood estimates are also investigated using simulated data. Another central theme in this thesis is the Markov property. Multi-state models with interval-censored data often rely on the Markov property, and we therefore investigate the Markov property in our model framework. By calculating the transition probabilities, we prove that our model framework does not necessarily rely on the Markov property. For example, when we model the transition times as the threshold crossing times for Gamma processes, the Markov property does not hold. However, if the transition times are exponentially distributed, the Markov property is satisfied and we end up with a homogeneous Markov model. For application purposes, we consider a dataset on CAV (coronary allograft vasculopathy), a post-transplant complication. The disease progression of CAV is described with a four-state illness-death model. We model the transition times as the threshold crossing times for Gamma processes, and calculate the maximum likelihood estimates. In the end, we compare our results to homogeneous and inhomogeneous Markov models, both with and without covariates. Our findings indicate that the models with Gamma processes are preferred over both the homogeneous and inhomogeneous Markov models. This holds both with and without covariates.