Regression models for interval-censored semi-competing risks data with missing intermediate transition status

We propose a multi-state model for analyzing semi-competing risks data with interval-censored or missing intermediate events. This model is an extension of the 'illness-death model', which composes three states, such as 'healthy', 'diseased', and 'dead'. The state of 'diseased' can be considered as an intermediate event. Two more states are added into the illness-death model to describe missing events caused by a loss of follow-up before the end of the study. One of them is a state of `LTF', representing a lost-to-follow-up, and the other is an unobservable state that represents the intermediate event experienced after LTF occurred. Given covariates, we employ the Cox proportional hazards model with a normal frailty and construct a full likelihood to estimate transition intensities between states in the multi-state model. Marginalization of the full likelihood is completed using the adaptive Gaussian quadrature, and the optimal solution of the regression parameters is achieved through the iterative Newton-Raphson algorithm. Simulation studies are carried out to investigate the finite-sample performance of the proposed estimation procedure in terms of the empirical coverage probability of the true regression parameter. Our proposed method is also illustrated with the dataset adapted from Helmer et al. (2001).