Modeling survival data by Coxian phase-type models
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Survival analysis is for analyzing time to event data. The event is normally considered as death or failure in the literature. It has been used to analyze credit risk, predict medical survival and reliability in engineering. Survival models can be viewed as ordinary regression models. But the presence of censored or truncated data separate it different from other analyses. Survival and hazard function is the central idea in survival analysis. Survival data is usually continuous, always positive and may contains incomplete observations. Censoring and truncation are two main features often seen in some survival studies. Modelling survival data can be challenging due to the asymmetrically distributed, censoring and truncation. Even though the asymmetry can be handle by transformation, but it is more satisfying to use suitable models to fit the original survival data. As phase-type distribution is dense on all positive-valued distribution, it is a good fit for survival data. In this thesis, Coxian phase-type distribution is focused. Coxian phase-type distribution is a special case of general phase-type distribution. It has fewer parameters, therefore, it is easier to estimate. This thesis gives a great detail of how phase-type distribution can be used in survival analysis and the special cases, Coxian phase-type distribution as well as the corresponding competing risks model. Simulated data set with or without censoring are used to examine the way of estimation. Here right-censored data is analyzed. In addition identifiability problem for Coxian phase-type model is taken care of. New parameters are introduced to avoid identifiability problem. Numerical optimization is used to maximize likelihood function in order to estimate the parameters with more than one transient state. L-BFGS-B method is used to estimate parameters in R. At the end, the new parameters applied to simulated data and pneumonia on admission data. Survival and hazard function were used to examine the estimation.