Non-parametric estimation in multi-state survival models: An update to msaj

Background: Multi-state survival models are a useful tool when disease pathways are complex and there are multiple events of interest. The multistate package in Stata can provide a range of predictions from parametric multi-state models via the predictms command. However, non-parametric estimates produced by the accompanying msaj command were limited. The aim of this work was to update msaj to provide a comprehensive set of non-parametric estimates. Methods: Two useful metrics in a multi-state model are transition probabilities and expected length of stay. Transition probabilities from a Markov model can be estimated non-parametrically using the empirical Aalen–Johansen estimator (analogous to the Kaplan–Meier estimator in standard survival). Expected length of stay can be estimated by integrating the transition probabilities. In this setting, this involves a summation of rectangles, as the Aalen–Johansen estimator is a step function. Updates to msaj: Previously, only transition probabilities from state 1 at time 0 could be obtained using msaj, along with corresponding confidence intervals. Following the update, the starting state, entry time and exit time can be specified. Estimates can now also be produced for bidirectional models and expected length of stay can be obtained. Illustrative example: A non-parametric analysis was performed on hospital epidemiology data, which demonstrated how msaj can be implemented. Three parametric multi-state models were also fitted to illustrate how non-parametric estimates can be used as a reference to informally compare models. Transition probabilities and expected length of stay were estimated from state 1 at time 0 and from state 2 at time 3 (relevant metrics for this dataset). Conclusion: The updated msaj provides a comprehensive set of non-parametric predictions, allowing for analyses with no assumptions made on transition rates and providing a reference for parametric models. Extensions could include fixed horizon predictions and confidence intervals for expected length of stay.