Multiple time scales in multi-state models

In multi-state models, it has been the tradition to model all transition intensities on one time scale, usually the time since entry into the study ('clock-forward' approach). The effect of time since an intermediate event has been accommodated either by changing the time scale to time since entry to the new state ('clock-back' approach) or by including the time at entry to the new state as a covariate. In this paper, we argue that the choice of time scale for the various transitions in a multi-state model should be dealt with as an empirical question, as also the question of whether a single time scale is sufficient. We illustrate that these questions are best addressed by using parametric models for the transition rates, as opposed to the traditional Cox-model-based approaches. Specific advantages are that dependence of failure rates on multiple time scales can be made explicit and described in informative graphical displays. Using a single common time scale for all transitions greatly facilitates computations of probabilities of being in a particular state at a given time, because the machinery from the theory of Markov chains can be applied. However, a realistic model for transition rates is preferable, especially when the focus is not on prediction of final outcomes from start but on the analysis of instantaneous risk or on dynamic prediction. We illustrate the various approaches using a data set from stem cell transplant in leukemia and provide supplementary online material in R.