Nonparametric estimation of a Markov 'illness-death' process from interval-censored observations, with application to diabetes survival data

The nonparametric estimation of the cumulative transition intensity functions in a three-state time-nonhomogeneous Markov process with irreversible transitions, an 'illness-death' model, is considered when times of the intermediate transition, e.g. onset of a disease, are interval-censored. The times of 'death' are assumed to be known exactly or to be right-censored. In addition the observed process may be left-truncated. Data of this type arise when the process is sampled periodically. For example, when the patients are monitored through periodic examinations the observations on times of change in their disease status will be interval-censored. Under the sampling scheme considered here the Nelson-Aalen estimator (Aalen, 1978) for a cumulative transition intensity is not applicable. In the proposed method the maximum likelihood estimators of some of the transition intensities are derived from the estimators of the corresponding subdistribution functions. The maximum likelihood estimators are shown to have a self-consistency property. The self-consistency algorithm is developed for the computation of the estimators. This approach generalises the results from Turnbull (1976) and Frydman (1992). The methods are illustrated with diabetes survival data.