Semi-Markov models with phase-type sojourn distributions

Titman and Sharples have a new paper in Biometrics. This concerns the fitting of semi-Markov models to panel observed data. They propose to fit models where the sojourn time in each state has a phase-type sojourn distribution - i.e. corresponds to the time to absorption of some time homogeneous Markov model. The advantage of this specification is that, unlike general semi-Markov models, the likelihood remains analytically tractable, falling within a hidden Markov model framework. This also makes the extension to models where the observations are subject to misclassification error straightforward, at least theoretically. A two-phase Coxian phase-type distribution is proposed for the sojourn time, allowing increasing, decreasing or constant hazards with respect to time since entry into the state.

While the phase-type framework makes computation of the likelihood more straightforward, model fitting is still potentially problematic due to possible problems of parameter estimability. Also since certain parameters of the phase-type model are unidentifiable under a Markov model meaning an (approximate) modified likelihood ratio test is required to test the Markov assumption.