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.

Table of Contents:
Preface ix Chapter 1 Introduction to Lifetime Data and Regression Models 1 1.1 Basics 1 1.2 The classic lifetime distribution: the Weibull distribution 5 1.3 Regression models for lifetimes 9 1.4 Proportional hazards models 10 1.5 Checking the proportional hazards assumption 13 1.6 Accelerated failure time models 17 1.7 Checking the accelerated failure time assumption 20 1.8 Proportional odds models 22 1.9 Proportional mean residual life models 25 1.10 Proportional reversed hazard rate models 26 1.11 The accelerated hazards model 27 1.12 The additive hazards model 29 1.13 PH, AFT and PO distributions 30 1.14 Cox’s semi-parametric PH regression model 33 1.15 PH versus AFT 35 1.16 Residuals 39 1.17 Cured fraction or long-term survivors 43 1.18 Frailty 45 1.19 Models for discrete lifetime data 47 1.20 Conclusions 52 Chapter 2 First Hitting Time Regression Models 55 2.1 Introduction 55 2.2 First hitting time models 58 2.3 First hitting time regression models based on an underlying Wiener process 60 2.4 Long-term survivors 63 2.5 FHT versus PH 65 2.6 Randomized drift in the Wiener process 69 2.7 First hitting time regression models based on an underlying Ornstein-Uhlenbeck process 71 2.8 The Birnbaum-Saunders distribution 74 2.9 Gamma processes 75 2.10 The inverse Gaussian process 77 2.11 Degradation and markers 77 Chapter 3 Model Fitting and Diagnostics 81 3.1 Introduction 81 3.2 Fitting the FHT regression model by maximum likelihood 82 3.3 The stthreg package 84 3.4 The threg package 86 3.5 The invGauss package 86 3.6 Fitting FHT regressions using the EM algorithm 87 3.7 Bayesian methods 88 3.8 Checking model fit 89 3.9 Issues in fitting inverse Gaussian FHT regression models 90 3.9.1 Possible collinearity? 90 3.9.2 Fitting inverse Gaussian FHT regression: a simulation study 92 3.9.3 Fitting the wrong model 95 3.10 Influence diagnostics for an inverse Gaussian FHT regression model 97 3.11 Variable selection 99 Chapter 4 Extensions to Inverse Gaussian First Hitting Time Regression Models 103 4.1 Introduction 103 4.2 Time-varying covariates 103 4.3 Recurrent events 106 4.4 Individual random effects 107 4.5 First hitting time regression model for recurrent events with random effects 110 4.6 Multiple outcomes 116 4.7 Extensions of the basic FHT model in a study of low birth weights: a mixture model and a competing risks model 119 4.7.1 Mixture model 120 4.7.2 Competing risks model 121 4.7.3 Comparative results 122 4.8 Semi-parametric modeling of covariate effects 123 4.9 Semi-parametric model for data with a cured fraction 125 4.10 Semi-parametric time-varying coefficients 126 4.11 Bivariate Wiener processes for markers and outcome 128 Chapter 5 Relationship of First Hitting Time Models to Proportional Hazards and Accelerated Failure Time Models 131 5.1 Introduction 131 5.2 FHT and PH models: direct comparisons by case studies 131 5.2.1 Case study 1: mortality after cardiac surgery 131 5.2.2 Case study 2: lung cancer in a cohort of nurses 134 5.3 FHT and PH models: theoretical connections 134 5.3.1 Varying the time scale 135 5.3.2 Varying the boundary 137 5.3.3 Estimation 137 5.4 FHT and AFT models: theoretical connections 138 Chapter 6 Applications 141 6.1 Introduction 141 6.2 Lung cancer risk in railroad workers 143 6.3 Lung cancer risk in railroad workers: a case-control study 144 6.4 Occupational exposure to asbestos 147 6.5 Return to work after limb injury 147 6.6 An FHT mixture model for a randomized clinical trial with switching 148 6.7 Recurrent exacerbations in COPD 150 6.7.1 COPD in lung cancer 153 6.8 Normalcy and discrepancy indexes for birth weight and gestational age 153 6.9 Hip fractures 155 6.10 Annual risk of death in cystic fibrosis 158 6.11 Disease resistance in cows 159 6.12 Balka, Desmond and McNicholas: an application of their cure rate models 161 6.13 Progression of cervical dilation 163 Bibliography 165 Index 181