First Hitting Time Regression Models

Lifetime Data Analysis Based on Underlying Stochastic Processes

Chrysseis Caroni

• Publisher: Wiley-ISTE
• Year: 2017
• Print ISBN: 9781848218895
• eBook ISBN: 9781119437222
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

This book aims to promote regression methods for analyzing lifetime (or time-to-event) data that are based on a representation of the underlying process, and are therefore likely to offer greater scientific insight compared to purely empirical methods.

In contrast to the rich statistical literature, the regression methods actually employed in lifetime data analysis are limited, particularly in the biomedical field where D. R. Cox's famous semi-parametric proportional hazards model predominates. Practitioners should become familiar with more flexible models. The first hitting time regression models (or threshold regression) presented here represent observed events as the outcome of an underlying stochastic process. One example is death occurring when the patient's health status falls to zero, but the idea has wide applicability – in biology, engineering, banking and finance, and elsewhere. The central topic is the model based on an underlying Wiener process, leading to lifetimes following the inverse Gaussian distribution. Introducing time-varying covariates and many other extensions are considered. Various applications are presented in detail.

1
Introduction to Lifetime Data and Regression Models

1.1. Basics

The principal aim of this book is to present a particular approach to the regression analysis of lifetime data in detail, and to discuss its features and advantages in comparison to older-established approaches. This objective will first involve undertaking a general review of the various regression methods that can be found in the literature.

This opening chapter provides a brief review of the basic features of lifetime data and its modeling. First of all, what is meant by lifetime data? A great deal of statisticians’ activity goes into studying whether or not an event occurs and how various factors influence its occurrence. If we move on from the simple yes/no fact of occurrence to also examining how long it takes until the event occurs, we enter the realm of “time to event” or “lifetime” data. Basic examples include how long a machine operates until it breaks down (the event is the breakdown) and how long a patient lives after undergoing heart transplantation (the event is death). These examples show two major areas of application. One is in engineering and technology (where the subject is usually known as reliability modeling) and the other is in biomedical sciences (known as survival analysis). However, other areas of application include all those in which statistics is used - in other words, in virtually every science. As the two examples of a machine’s breakdown and a patient’s survival suggest, applications of lifetime data analysis can have immense practical importance. Well-known textbooks with wide coverage of lifetime data analysis include Lawless [LAW 03], Collett [COL 14], Kalbfleisch and Prentice [KAL 02] and Klein and Moeschberger [KLE 03]. A brief review was given by Hougaard [HOU 99]. Other papers reviewing the analysis of lifetime data include Kiefer [KIE 88] in economics and Chung et al. [CHU 91] in criminology.

In mathematical terms, lifetime is denoted by T. No two machines are identical or operate under identical conditions; no two people are quite alike. Consequently, we treat T as a random variable, which follows some distribution in the relevant population of machines or people (in general, units). We note that T must be non-negative. Furthermore, in this book, we will follow the vast majority of the literature in treating the time scale as continuous. Consequently, we suppose that T ∼ f(t), t > 0 for some probability density function (pdf) f(.), and hence that

. The functions that present particular interest are the following:

1. – the survival function
   
   
2. – the hazard function h(t) = f(t)/S(t).

The former is P (T > t), the probability of survival for at least time t - the probability that the machine is still operating, or that the patient is still alive, after this time. In engineering and technological applications, this probability is called reliability and the notation R(t) is usually used instead of S(t). The term hazard function is replaced by failure rate. Other terms in use for the same function include force of mortality (in demography) and intensity.

The survival function or reliability P (T > t) is a quantity of basic scientific and practical importance. For example, in medical settings, a patient’s prognosis might be expressed as his or her five-year survival probability, and in manufacturing, reliability is obviously related to how long a guarantee period can be offered for a product. The hazard function can be interpreted as the instantaneous rate of failure at time t, given that the unit has survived that long, and hence the term failure rate. However, it is important to remember that the hazard refers to failure conditionally on survival to that time (the unconditional failure rate is of course given by the pdf of the lifetime distribution). More precisely, the hazard ...

Table of contents

• Cover
• Table of Contents
• Title
• Copyright
• Preface
• 1 Introduction to Lifetime Data and Regression Models
• 2 First Hitting Time Regression Models
• 3 Model Fitting and Diagnostics
• 4 Extensions to Inverse Gaussian First Hitting Time Regression Models
• 5 Relationship of First Hitting Time Models to Proportional Hazards and Accelerated Failure Time Models
• 6 Applications
• Bibliography
• Index
• End User License Agreement