Bayesian Nonparametric Estimation of Hazard Rate in Survival Analysis Using Gibbs Sampler

This text describes the way of estimating the hazard rate of survival data based on techniques which were introduced by Arjas, E., Gasbarra, D. (1994). The point is the assumption of piecewise constant hazard rate with varying number and size of intervals. The estimation is slighty shifted to hierarchical model and changing the dimension of the model via adding of new jumps or deleting the existing ones was introduced in wider scale. To provide the estimation MCMC algorithms were used. The method is demonstrated on simulated data. Introduction Many of standard approaches in dealing with various statistical problems are restricted to fixed dimension of the model which could be sometimes rather limitative. The example of such type of statistical problems could be the multiple change-point problem as well as variable selection in regression or choice between models in Bayesian context. Models which are able to switch between parametric spaces with different dimensions are likely based on Bayesian inference, where the dimensionality of the parameter vector may be open, though it is adherent to specifying the priors. Computational difficulties for such type of models could be overcome using MCMC algorithms. In next pages providing of such type of modeling is demonstrated. The method approximates the hazard rate in survival data using piecewise constant functions with a random number and locations of jump times. Survival Analysis: Basic Definitions and Formulas Let T be a continuous non-negative random variable representing e.g. the survival times of individuals in some population. Let f(t) denote the probability density function of T and let F (t) be the distribution function of T . Then the probability of an individual’s surviving till time t is given by survival function S(t), t ∈ [0,∞): S(t) = 1− F (t) = P (T > t) The hazard function, h(t), also called the hazard rate, is the instantaneous rate of failure at time t defined by h(t) = lim ∆t→0 1 ∆t P (t < T ≤ t+∆t|T > t) and after multiplying with ∆t could be interpreted as an approximate probability of failure in (t, t+∆t]. From the definition of the hazard rate we may easily see that h(t) = lim ∆t→0 1 ∆t P (t < T ≤ t+∆t, T > t) P (T > t) = f(t) S(t) = − d dt logS(t) and after integrating both sides and exponentiating we get