Multi-states models play an important role in describing the evolution of a process of interest. Multi-states models naturally generalize the survival models, and deal not only with a final event (generally the death of patients in the medical field), but with different relevant events. A multi-state model is a stochastic process, namely (X(t), t ≥ 0), which at a given time t occupies one state {i} from a discrete set E = {1, 2, . . . , K}. It is characterized by its transition intensities from one state {i} to another {j} at time t conditional to its history F t − , namely λ ij (t|F t − ). Some hypotheses may be done on this transition structure, which leads us to the specific class of Markov and semi-Markov (SM) models. A non-homogenous Markov (NHM) process is defined by dependent functions of the chronological time, λ ij (t). We talk about a homogenous Markov (HM) model when the transition intensities are constant with time, λ ij . Markov models have been extensively used in medicine ([KEI 89], [AAL 97], [JOL 98], [COM 02], [MAT 06]) and have delivered interesting results.However, another time scale reveals to be important in the medical field: the duration of time in the previous state. Indeed, generally, the longer patients stay in critical disease states, the more severe their evolution is. In this context, the SM model is well adapted. Let T N (t − ) be the time of the previous transition just before time t, and let Chapter written by Eve MATHIEU-DUPAS, Claudine GRAS-AYGON and Jean-Pierre DAURÈS.
Mathematical Methods in Survival Analysis, Reliability and Quality of Life Edited by Catherine Huber, Nikolaos Limnios, Mounir Mesbah & Mikhail NikulinCopyright 0 2008, ISTE Ltd.
Mathematical Methods in Survival Analysis, Reliability and Quality of Life
Methods
Model description and notationLet us consider E = {1, . . . , K} a discrete state space and (Ω, , P ) a probability space. We define the following random variables [JAN 01], ∀n ≥ 0:where J n represents the state at the n-th transition and S n represents the chronological time of the n-th transition. Let N (t) be the counting process (N (t), t ≥ 0) associated
