Statistical Analysis of Illness–Death Processes and Semicompeting Risks Data


  • Jinfeng Xu,

    Corresponding author
    1. Department of Statistics and Applied Probability, Risk Management Institute, National University of Singapore, Singapore 117546, Singapore
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  • John D. Kalbfleisch,

    1. Department of Biostatistics, University of Michigan, 1420 Washington Heights, Ann Arbor, Michigan 48109-2029, U.S.A.
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  • Beechoo Tai

    1. Department of Epidemiology and Public Health, Yong Loo Lin School of Medicine, National University of Singapore, Singapore 117597, Singapore
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Summary In many instances, a subject can experience both a nonterminal and terminal event where the terminal event (e.g., death) censors the nonterminal event (e.g., relapse) but not vice versa. Typically, the two events are correlated. This situation has been termed semicompeting risks (e.g., Fine, Jiang, and Chappell, 2001, Biometrika88, 907–939; Wang, 2003, Journal of the Royal Statistical Society, Series B65, 257–273), and analysis has been based on a joint survival function of two event times over the positive quadrant but with observation restricted to the upper wedge. Implicitly, this approach entertains the idea of latent failure times and leads to discussion of a marginal distribution of the nonterminal event that is not grounded in reality. We argue that, similar to models for competing risks, latent failure times should generally be avoided in modeling such data. We note that semicompeting risks have more classically been described as an illness–death model and this formulation avoids any reference to latent times. We consider an illness–death model with shared frailty, which in its most restrictive form is identical to the semicompeting risks model that has been proposed and analyzed, but that allows for many generalizations and the simple incorporation of covariates. Nonparametric maximum likelihood estimation is used for inference and resulting estimates for the correlation parameter are compared with other proposed approaches. Asymptotic properties, simulations studies, and application to a randomized clinical trial in nasopharyngeal cancer evaluate and illustrate the methods. A simple and fast algorithm is developed for its numerical implementation.