Piecewise Linear Multiple Change Point Model: An Application to Survival Analysis

Abstract

The hazard function is a frequently used function for modeling and evaluating the time related events .In the Cox-proportional hazard model this hazard function assumes to be constant. Earlier many statisticians investigated this assumption and found that this function is not constant over time and it has some change point in survival time of patient. Many authors Matthews and Farewell (1982), Pons(2002), Gijbels and Gurler (2003) and Zhao et al.(2009) investigated about one change point in hazard function through piecewise constant hazard model. Moreover, Goodman et al. (2006, 2007), Qian and Zhang (2014) investigated about multiple change point in hazard function through piecewise linear multiple change point model by Nelder-mead simplex method. Rois et al. (2017) investigated about multiple change point in hazard function through piecewise linear multiple change point model by Cross entropy algorithm. In our paper we calculate the likelihood function for piecewise linear multiple change point model considering three change point. Since this likelihood function is non-linear so we use non-linear minimization method to estimate the parameters of the model. Later we make comparison among the parameters estimated by Non-linear method with those estimated by Nelder-mead simplex method and Cross entropy algorithm in a simulation study. That study shows that the standard error of the estimates estimated by Non-linear minimization method is lowest among the three method. A Wald type test shows that the estimated parameters are significant. So, change point estimation in hazard function by non-linear minimization method can be considered as an improvement in change point analysis.