State-split approach for an extra time scale in multi-state models

Abstract

In illness-death multi-state models, time from intermediate state is often an extra time scale of interest. This extra time scale can be taken as a time-dependent covariate or as a secondary time scale into the model [1]. Both options will result in a non-Markov model. The loss of the Markov property implies that the standard probability theories such as Aalen-Johansen estimator [2] cannot be used to compute the transition probabilities. In this paper, we introduce the ‘state-split’ approach. By using ‘time to intermediate state’ as a categorical indicator, we split the intermediate state into sub-states. In a simple case, the intermediate state can be split into two sub-states ‘early entry to the intermediate state’ and ‘late entry to the intermediate state’. We illustrate the machinery of ‘state-split’ approach including when to split, where to split and when to stop with splitting. When a resulting model is obtained through this mechanism, we compare this model to the model from Iacobelli et al. [1]. As the criterion of comparison, empirical Brier score is used [3]. The result shows that, for the CML (chronic myeloid leukemia) dataset, the ‘state-split’ model delivers similar prediction performance as the model of Iacobelli et al.