Phase-type methods for semi-Markov models

Andrew Titman, Lancaster University

Multi-state models are widely used in event history analysis, particularly to model the progression of chronic diseases through a set of discrete disease states. Often it is only possible to observe a patient's disease state at clinic visits which may be unequally spaced and subject specific, leading to panel data. Due to computational difficulties, most analyses for such data assume a Markov model. However, if may often be more preferable to allow the transition intensities to depend on the time spent in the current state implying a semi-Markov model. The likelihood for general semi-Markov models is somewhat intractable. However much of the computationally issues can be avoided by restricting the semi-Markov model to have a continuous-time hidden Markov representation such that the sojourn distributions are phase-type.

Two main parametric approaches can be taken, either allowing the model to be directly parametrised in terms of phase-type distributions or else using the phase-type distributions to approximate Weibull or Gamma sojourn distributions. Extensions to non-parametric methods using penalised likelihood will also be briefly discussed. The methods are illustrated on data relating to the progression of bronchiolitis obliterans syndrome in post-lung-transplantation patients.