Regression analysis of the illness-death model with a shared frailty when all transition times are interval censored

In biomedical or clinical studies, semi-competing risks data are often encountered in which one type of event may censor an other event, but not vice versa. An illness-death model is proposed to analyze these semi-competing risks data in the presence of interval censoring on both intermediate and terminal events. The Cox proportional hazards model is employed with a frailty effect to incorporate a dependent structure between non-fatal and fatal events and individual-specific variations as well. Weight allocations on sub-intervals of censored intervals are used to construct the modified likelihood functions. Marginalization of the full likelihood is accomplished using adaptive importance sampling, and the optimal solution of the regression parameters is achieved through the iterative quasi-Newton algorithm. The proposed methodology is illustrated with several simulation studies and real data.