Threshold Regression for Survival Analysis: Modeling Event Times by a Stochastic Process Reaching a Boundary

Abstract

Many researchers have investigated first hitting times as models for survival data. First hitting times arise naturally in many types of stochastic processes, ranging from Wiener processes to Markov chains. In a survival context, the state of the underlying process represents the strength of an item or the health of an individual. The item fails or the individual experiences a clinical endpoint when the process reaches an adverse threshold state for the first time. The time scale can be calendar time or some other operational measure of degradation or disease progression. In many applications, the process is latent (i.e., unobservable). Threshold regression refers to first-hitting-time models with regression structures that accommodate covariate data. The parameters of the process, threshold state and time scale may depend on the covariates. This paper reviews aspects of this topic and discusses fruitful avenues for future research.

Reliability Analysis of the High-speed Train Bearing Based on Wiener Process

“…In addition, as a result of uncertain random factors in the testing environment, there is a rising trend in the curve in Figure 1. In light of the above features of the bearing relative temperature, we used the Wiener process for modeling [6][7][8][9][10][11]. This mathematical model can be expressed as follows:…”

TheRPackagethregto Implement Threshold Regression Models

“…For example, one can have gamma processes, curvilinear boundaries, Markov chains with absorbing states, and many more. Lee and Whitmore (2006) present a wide-ranging review of the general methodology and types of applications.…”

A First Passage Time Model for Long-Term Survivors with Competing Risks

“…The idea of studying survival time data based on the notion of an underlying stochastic process leading to 'failure' has been studied recently by Aalen and co-workers (see Aalen and Gjessing, 2001, for example), and earlier by Whitmore in a series of papers (see Lee and Whitmore, 2006, and the references therein for an excellent review). A stimulating discussion of this 'process' point of view is given in the recent text by Aalen, Borgan, and Gjessing (2008).…”

“…While this latter text is canonical in developing the counting process paradigm, the emphasis has been to a large extent on the classical Cox proportional hazards model. Lee and Whitmore (2006) consider a family of stochastic models for the underlying process leading to the failure of interest. Here, we study only one of these, namely a Wiener process X(t), such that when X(t) reaches a barrier at c, say, the failure of interest occurs.…”

“…If one or more competing risks exist, considering a multidimensional Wiener process to model the various risks simultaneously is possible. Such an approach is discussed, for example, by Lee and Whitmore (2006). In the two illustrative examples discussed in Section 3, death due to breast cancer and recurrence of melanoma are the respective events of interest, and the competing risk is 'cure' or death from other causes.…”