Time-dynamic inference for non-Markov transition probabilities under independent right-censoring
In this article, weak convergence of the general non-Markov state transition probability estimator by Titman (2015) is established which, up to now, has not been verified yet for other general non-Markov estimators. A similar theorem is shown for the bootstrap, yielding resampling-based inference methods for statistical functionals. Formulas of the involved covariance functions are presented in detail. Particular applications include the conditional expected length of stay in a specific state, given occupation of another state in the past, as well as the construction of time-simultaneous confidence bands for the transition probabilities. The expected lengths of stay in the two-sample liver cirrhosis data-set by Andersen et al. (1993) are compared and confidence intervals for their difference are constructed. With borderline significance and in comparison to the placebo group, the treatment group has an elevated expected length of stay in the healthy state given an earlier disease state occupation. In contrast, the Aalen-Johansen estimator-based confidence interval, which relies on a Markov assumption, leads to a drastically different conclusion. Also, graphical illustrations of confidence bands for the transition probabilities demonstrate the biasedness of the Aalen-Johansen estimator in this data example. The reliability of these results is assessed in a simulation study.