A New Parametric Accelerated Failure Time Model for Semi‐Competing Risks Data

In cohort studies, the occurrence of a specific disease is often of main interest. For proper modeling, death as a competing risk should be accounted for. Especially, the "semi-competing" character of the data has to be acknowledged: Disease occurrence (the non-terminal event) can be observed before death (the terminal event), but not vice versa. Hence, the terminal event might censor the non-terminal event but remains observable if the non-terminal event occurs first. The underlying setting can be described by an illness-death model incorporating transitions between the three states "healthy," "diseased," and "dead." This article aims to introduce a new method employing accelerated failure time (AFT) models for each of the three transitions between the states of the illness-death model. The major advantage of this approach is its intuitive and straightforward interpretation based on the survival instead of the hazard function, facilitating communication of results. We propose a parametric model by assuming Weibull distributions for the ages in the different states, modeling (1) the age at disease onset, (2) the age at death for patients without diagnosis, and (3) the age at death after diagnosis. We add random effects to adjust for intra-individual correlations, yielding a trivariate model. Model parameters are estimated by the maximum likelihood principle. The likelihood function incorporates left truncation for both terminal and non-terminal events, reflecting the delayed entry into the study cohort. As in cohort studies the diagnosis is usually made at intermittent follow-up visits and the exact age at disease onset is only known to lie in the interval between the last two visits, we finally consider interval censoring for the disease occurrence. The model is illustrated using data from the PAQUID study, focusing on dementia as a non-terminal event. Our approach leads to plausible results, confirming the findings by other existing methods. The overall performance is assessed by a simulation study, yielding promising results concerning the accuracy and numerical robustness of our model.