Review and implementation of cure models based on first hitting times for Wiener processes

Abstract

The development of models and methods for cure rate estimation has recently burgeoned into an important subfield of survival analysis. Much of the literature focuses on the standard mixture model. Recently, process-based models have been suggested. We focus on several models based on first passage times for Wiener processes. Whitmore and others have studied these models in a variety of contexts. Lee and Whitmore (Stat Sci 21(4):501-513, 2006) give a comprehensive review of a variety of first hitting time models and briefly discuss their potential as cure rate models. In this paper, we study the Wiener process with negative drift as a possible cure rate model but the resulting defective inverse Gaussian model is found to provide a poor fit in some cases. Several possible modifications are then suggested, which improve the defective inverse Gaussian. These modifications include: the inverse Gaussian cure rate mixture model; a mixture of two inverse Gaussian models; incorporation of heterogeneity in the drift parameter; and the addition of a second absorbing barrier to the Wiener process, representing an immunity threshold. This class of process-based models is a useful alternative to the standard model and provides an improved fit compared to the standard model when applied to many of the datasets that we have studied. Implementation of this class of models is facilitated using expectation-maximization (EM) algorithms and variants thereof, including the gradient EM algorithm. Parameter estimates for each of these EM algorithms are given and the proposed models are applied to both real and simulated data, where they perform well.

Two new defective distributions based on the Marshall–Olkin extension

“…A deeper study was made by Tweedie (1945). It is also where the name inverse Gaussian arose (Balka et al 2009). …”

“…However, here we use an alternative methodology based on defective models, a concept introduced by Balka et al (2009). In these models it is possible to estimate a cure rate with the use of a naturally improper distribution.…”

Research on two dimensional Wiener stochastic degradation model based on the wear model

“…Nicolai [9] et al proposed an application to coatings on steel structures by comparing models for measurable deterioration with Wiener processes. Balka [10] et al reviewed and implemented the cure models based on first hitting times of Wiener processes. Si [11] et al estimated remaining useful life based on a nonlinear diffusion degradation process with monitoring degraded signal.…”

A new class of defective models based on the Marshall–Olkin family of distributions for cure rate modeling

“…The use of these defective distributions became more appealing after the works of [5] and [6], although some previous papers have used the same idea. In [41], the term "defective" was used to refer to the inverse Gaussian distribution that allows one of its parameters to be negative.…”

“…More details were studied in [40] and [41]. Defective versions were investigated in [5] and [6], with classical and Bayesian approaches.…”