Markov multi-state models for survival analysis with recurrent events
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Markov models are a major class within the system of multi-state models for the analysis of lifetime or event-time data. Applications abound, including the estimation of lifetime of ultra-cold neutrons, the bias correction of the apparent magnitude distribution of the stars in a certain area of the sky, and the survival analysis of clinical trials. This thesis addresses some of the problems arising in the analysis of right-censored lifetime data. Clinical trials are used as examples to investigate these problems. A Markov model that takes a patient's disease development into account for the analysis of right-censored data was first constructed by Fix and Neyman (1951). The Fix-Neyman (F-N) model is a homogeneous Markov process with two transient and two absorbing states that describes a patient's status over a period of time during a cancer clinical trial. This thesis extends the F-N model by assuming the transition rates (hazard rates) to be both state and time dependent. Recurrent transitions between relapse and recovery are allowed in the extended model. By relaxing the condition of time-independent hazard rates, the extension increases the applicability of the Markov models. The extended models are used to compute the model survival functions, cumulative hazard functions that take into consideration of right censored observations as it has been done in the celebrated Kaplan-Meier estimator. Using the Fix-Neyman procedure and the Kolmogorov forward equations, closed-form solutions are obtained for certain irreversible 4-state extended models while numerical solutions are obtained for the model with recurrent events. The 4-state model is motivated by an Aplastic Anemia data set. The computational method works for general irreversible and reversible models with no restriction on the number of states. Simulations of right-censored Markov processes are performed by using a sequence of competing risks models. Simulated data are used for checking the performance of nonparametric estimators for various sample sizes. In addition, applying Aalen's (1978) results, estimators are shown have asymptotic normal distributions. A brief review of some of the literature relevant to this thesis is provided. References are readily available from a vast literature on the survival analysis including many text books. General Markov process models for survival analysis are described, e.g., in Andersen, Borgan, Gill and Keiding (1993).