Statistical inference for state occupation and transition probabilities in non-Markov multi-state models subject to both random left-truncation and right-censoring

The Aalen-Johansen estimator generalizes the Kaplan-Meier estimator for independently left-truncated and right-censored survival data to estimate the transition probability matrix of a time-inhomogeneous Markov model with finite state space. Such multi-state models have a wide range of applications for modelling complex courses of a disease over the course of time, but the Markov assumption may often be in doubt. If censoring is entirely unrelated to the multi-state data, it has been suggested that the Aalen-Johansen estimator, standardized by the initial empirical distribution of the multi-state model, still consistently estimates the state occupation probabilities. Recently, this approach has been extended to transition probabilities using landmarking, which is, inter alia, useful for dynamic prediction. However, there have been recent concerns about the mathematical arguments leading to the former result. These findings are complemented in three ways. Firstly, a rigorous proof of consistency of the Aalen-Johansen estimator for state occupation probabilities, on which also correctness of the landmarking approach hinges, is presented correcting and simplifying the earlier result. Secondly, delayed study entry is a common phenomenon in observational studies, and the earlier results are extended to multi-state data also subject to left-truncation. Thirdly, the rigorous proof is suggestive of wild bootstrap resampling. Studying wild bootstrap is motivated by the fact that it is desirable to have a technique that works for models where left-truncation and right-censoring need not be entirely random, then requiring a Markov assumption, and that may still perform reasonably with non-Markov models subject to random left-truncation and right-censoring. The developments for left-truncation are motivated by a prospective observational study on the occurrence and the impact of a multi-resistant infectious organism in patients undergoing surgery. Both the real data example and simulation studies are presented. The case for wild bootstrapping is illustrated for event-driven trials, where the data are censored once a prespecified number of events have been observed.