Tolusso and Cook have a
paper in Communications in Statistics - Theory and Methods, based on chapter 4 of the first author's
PhD thesis. They propose a method of robust estimation for the state occupancy probabilities in progressive multi-state models when the transition times are interval censored. The method is in the spirit of the
Pepe estimator being based on the differences between the marginal survival distributions of state entry or exit times. These marginal survival distributions can be estimated either through the NPMLE using self-consistency algorithms, through weakly parametric piecewise-constant hazard assumptions, or via local-likelihood.
The method is applied to a three-state illness death model, where the absorbing state is death and times of entry into the absorbing state are known exactly. Prevalence in state 1 is estimated by the interval-censored survival estimate of exit from state 1, prevalence in state 3 is estimated through the Kaplan-Meier estimate of overall survival and prevalence in state 2 is based on the difference between these functions.
In this case the method can be thought of as a less computationally intensive alternative to using
Frydman and Szarek's NPMLE, with the added advantage that it is not necessary to make the Markov assumption.
An unrecognized problem with the method in the case of exactly known death times is that, for the healthy state survival function, the upper boundary of the censoring interval is not independent of the process. If a patient dies then they will be censored in some interval
where
is the time of death. However, if they died from state 1, their exit time from state 1 was
. Thus the sojourn time in state 1 will tend to be underestimated and consequentially state 2 occupation will be overestimated. The extent of bias will depend on the chance of death from state 1 and the severity of interval censoring.
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