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Multi-state models can be successfully used for describing complicated event history data, for example, describing stages in the disease progression of a patient. In these models one important goal is the estimation of the transition probabilities since they allow for long term prediction of the process. Traditionally these quantities have been estimated by the Aalen-Johansen estimator which is consistent if the process is Markovian. Recently, estimators have been proposed that outperform the Aalen-Johansen estimators in non-Markov situations. This paper considers a new proposal for the estimation of the transition probabilities in a multi-state system that is not necessarily Markovian. The proposed product-limit nonparametric estimator is defined in the form of a counting process, counting the number of transitions between states and the risk sets for leaving each state with an inverse probability of censoring weighted form. Advantages and limitations of the different methods and some practical recommendations are presented. We also introduce a graphical local test for the Markov assumption. Several simulation studies were conducted under different data scenarios. The proposed methods are illustrated with a real data set on colon cancer.

Estimation of the Transition Probabilities in Multi-state Survival Data: New Developments and Practical Recommendations