A unified likelihood approach for progressive multistate frailty models under double censoring: application to retinopathy of prematurity

In multistate survival studies, transition times may exhibit within-subject dependence and be simultaneously subject to left and right censoring. Existing approaches typically address these features separately. The joint analysis of progressive multistate transitions, shared frailty dependence, and double censoring within a unified likelihood framework has received limited attention. This study proposes a parametric progressive multistate survival model that simultaneously accommodates shared frailty and double censoring. Transition intensities are modeled using Weibull distributions. Parameters are estimated via maximum likelihood with log-stabilized numerical integration over the gamma frailty distribution to ensure computational stability. Simulations were conducted under varying sample sizes (100, 200, and 600), frailty variances (0.5, 1, and 1.5), and proportions of left censoring (20%, 40%, and 60%), with performance assessed using bias, root mean square error (RMSE), and coverage probability of 95% confidence intervals. Under the simulation settings examined, the proposed model showed reduced bias and improved estimation accuracy and confidence-interval coverage compared with models that ignore frailty and/or left censoring, with gains becoming more pronounced under higher frailty variance and higher left-censoring proportions. The proposed framework provides a practical approach for progressive multistate data with correlated transition times and double censoring. The method is illustrated using data on retinal vascular development in premature infants with retinopathy of prematurity, demonstrating its utility for correlated multistate processes in biomedical studies.