Modeling and maximum likelihood based inference of interval‐censored data with unknown upper limits and time‐dependent covariates

Due to the nature of study design or other reasons, the upper limits of the interval‐censored data with multiple visits are unknown. A naïve approach is to treat the last observed time as the exact event time, which may induce biased estimators of the model parameters. In this paper, we first develop a Cox model with time‐dependent covariates for the event time and a proportional hazards model with frailty for the gap time. We then construct the upper limits using the latent gap times to resolve the issue of interval‐censored event time data with unknown upper limits. A data‐augmentation technique and a Monte Carlo EM (MCEM) algorithm are developed to facilitate computation. Theoretical properties of the computational algorithm are also investigated. Additionally, new model comparison criteria are developed to assess the fit of the gap time data as well as the fit of the event time data conditional on the gap time data. Our proposed method compares favorably with competing methods in both simulation study and real data analysis.