Bootstrapping stochastic systems in survival analysis

Abstract

The main focus of this paper is the first passage time distribution in a semi-Markov processes from an initial state to an absorbing state. The single bootstrap is implemented using a saddlepoint approximation to determine estimates for the survival and hazard functions of first passage. The double bootstrap is also implemented by resampling saddlepoint inversions and provides BCa confidence bands for these functions. Confidence intervals for the mean and variance of first passage times are easily computed. A characterization of the asymptotic hazard rate for survival times is presented and leads to an indirect method for constructing its bootstrap confidence intervals. The lifetime of a patient in the presence of independent left- and right-censoring is considered from a systems theoretic point-of-view. The systems development introduces semi-Markov flowgraphs to represent the transitions of the patient into and out of censoring states up until the time of death. The empirical flowgraph. determined from the lifetime data subject to censoring, provides an estimate of the population version and is itself a semi-Markov flowgraph. Solving the empirical flowgraph has the effect of removing the censoring risk with both Kaplan-Meier and Turnbull survival estimators resulting. The fact that these estimators are produced highlights the strength of this particular point of view. Now consider passage through a semi-Markov system in the presence of independent right-censoring. Interest lies in estimating the first passage time distribution with the censoring risk removed. Data is observed from a censored-system, producing an empirical flowgraph. This is an estimate of the population flowgraph and is also a semi-Markov system. The main idea is to amend the empirical flowgraph so that the censoring risk is removed. Since the system is free of censoring, previous saddlepoint and bootstrap methods can be used to compute confidence intervals for the survival and hazard functions of first passage. The methods are illustrated by providing a bootstrap/saddlepoint solution to the classical Fix and Neyman (1951) survival model.