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2021 | Book

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Survival Analysis

Proportional and Non-Proportional Hazards Regression

Author: John O'Quigley

Publisher: Springer International Publishing

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book provides an extensive coverage of the methodology of survival analysis, ranging from introductory level material to deeper more advanced topics. The framework is that of proportional and non-proportional hazards models; a structure that is broad enough to enable the recovery of a large number of established results as well as to open the way to many new developments. The emphasis is on concepts and guiding principles, logical and graphical. Formal proofs of theorems, propositions and lemmas are gathered together at the end of each chapter separate from the main presentation.

The intended audience includes academic statisticians, biostatisticians, epidemiologists and also researchers in these fields whose focus may be more on the applications than on the theory. The text could provide the basis for a two semester course on survival analysis and, with this goal in mind, each chapter includes a section with a range of exercises as a teaching aid for instructors.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
The chapters of this book correspond to a collection of broad themes. Taken together, these themes make up a sizable component of what can be considered to be the topic of modern survival analysis. A feature that we maintain throughout the text is to summarize the salient features of each chapter in a brief section headed “Chapter summary”. Section 1.1 is the first such summary. Following these summaries comes a section headed “Context and motivation”, the purpose of which is to motivate our interest as well as to recall any key notions we may need as an aid in pursuing the topic of the chapter.
John O’Quigley
Chapter 2. Survival analysis methodology
Abstract
We recall some elementary definitions concerning probability distributions, putting an emphasis toward one minus the usual cumulative distribution function, i.e., the survival function.
John O’Quigley
Chapter 3. Survival without covariates
Abstract
The marginal survival function is of central interest even when dealing with covariates. We need to find good estimates of this function and we use those estimates in several different contexts. Good estimates only become possible under certain assumptions on the censoring mechanism. Attention is paid to the exponential and piecewise exponential models, both of which are particularly transparent.
John O’Quigley
Chapter 4. Proportional hazards modelsRegression models
Abstract
We consider several models that describe survival in the presence of observable covariates, these covariates measuring subject heterogeneity. The most general situation can be described by a model with a parameter of high, possibly unbounded, dimension. We refer to this as the general or non-proportional hazards model since dependence is expressed via a parameter, \(\beta (t),\) that is not constrained or restricted. Proportional hazards models have the same form but constrain \(\beta (t)\) to be a constant. We write the constant as \(\beta ,\) sometimes \(\beta _0\), since it does not change with time. When the covariate itself is constant, the dependence structure corresponds to the Cox regression model. We describe this model, its connection to the well-known log-rank test, and its use in many applications. We recall the founding paper of Cox ((Cox, 1972)) and the many discussions that surrounded that paper. Some of the historical backgrounds that lay behind Cox’s proposal is also recalled in order to for the new reader to quickly appreciate that, brilliant though Professor Cox’s insights were, they leant on more than just his imagination. They did not emerge from a vacuum. Some discussion on how the model should be used in practice is given.
John O’Quigley
Chapter 5. Proportional hazards models in epidemiology
Abstract
The basic questions of epidemiology  are reconsidered in this chapter from the standpoint of a survival model. We rework the calculations of relative risk, where the time factor is now age, and we see how our survival models can be used to control for the effects of age. Series of \(2 \times 2\) tables, familiar to epidemiologists, can be structured within the regression model setting. The well-known Mantel-Haenszel test arises as a model-based score test. Logistic regression, conditional logistic regression as well as stratified regression are all considered. These various models, simple proportional hazards model, stratified models, and time-dependent models can all be exploited in order to better evaluate risk factors, how they interrelate, and how they relate to disease incidence in various situations.
John O’Quigley
Chapter 6. Non-proportional hazards models
Abstract
The most general model, described in Chapter 4 covers a very broad spread of possibilities and, in this chapter, we consider some special cases.
John O’Quigley
Chapter 7. Model-based estimating equations
Abstract
The regression effect process, described in Chapter 9, shapes our main approach to inference. At its heart are differences between observations and their model-based expectations. The flavor is very much that of linear estimating equations (Appendix D.1). Before we study this process, we consider here an approach to inference that makes a more direct appeal to estimating equations. The two chapters are closely related and complement one another. This chapter leans less heavily on stochastic processes and links in a natural and direct way to the large body of theory available for estimating equations. Focusing attention on the expectation operator, leaning upon different population models and different working assumptions, makes several important results transparent. For example, it is readily seen that the so-called partial likelihood estimator is not consistent for average effect, \(E\{\beta (T) \},\) under independent censoring and non-constant \(\beta (t).\) One example we show, under heavy censoring, indicates the commonly used partial likelihood estimate to converge to a value greater than 4 times its true value. Linear estimating equations provide a way to investigate statistical behavior of estimates for small samples. Several examples are considered.
John O’Quigley
Chapter 8. Survival given covariate information
Abstract
We begin by considering the probability that one subject with particular covariates will have a greater survival time than another subject with different covariates, i.e., \(\text{ Pr }\,(T_i>T_j|Z_i,Z_j).\) Note that this also provides a Kendall \(\tau \)-type measure of predictive strength (Gönen and Heller, 2005) and, although not explored in this work, provides a potential alternative to the \(R^2\) that we recommend. Confidence intervals are simple to construct and maintain the same coverage properties as those for \(\beta \). Using the main results of Chapter 7 we obtain a simple expression for survival probability given a particular covariate configuration, i.e., \(S(t|Z\in H)\) where H is some given covariate subspace.
John O’Quigley
Chapter 9. Regression effect process
Abstract
In this chapter we describe the regression effect process. This can be established in different ways and provides all of the essential information that we need in order to gain an impression of departures from some null structure, the most common null structure corresponding to an absence of regression effect. Departures in specific directions enable us to make inferences on model assumptions and can suggest, of themselves, richer more plausible models. The regression effect process, in its basic form, is much like a scatterplot for linear regression telling us, before any formal statistical analysis, whether the dependent variable really does seem to depend on the explanatory variable as well as the nature, linear or more complex, of that relationship. Our setting is semi-parametric and the information on the time variable is summarized by its rank within the time observations.
John O’Quigley
Chapter 10. Model construction guided by regression effect process
Abstract
The basic ideas of linear regression run through this chapter is structured as follows: In Section 10.3, we present a literature review of graphical methods and goodness of fit tests for proportional hazards models. In Section 10.6, we consider the \(R^2\) coefficient for non-proportional hazards models, which is the measure of predictive ability used from here on.
John O’Quigley
Chapter 11. Hypothesis tests based on regression effect processHypothesis tests
Abstract
We revisit the standard log-rank test and several modifications of it that come under the heading of weighted log-rank tests. Taken together these provide us with an extensive array of tools for the hypothesis testing problem. Importantly, all of these tests can be readily derived from within the proportional and non-proportional hazards framework. Given our focus on the regression effect process, it is equally important to note that these tests can be based on established properties of this process under various assumptions. These properties allow us to cover a very broad range of situations. With many different tests, including goodness-of-fit tests, coming under the same heading, it makes it particularly straightforward to carry out comparative studies on the relative merits of different choices. Furthermore, we underline the intuitive value of the regression effect process since it provides us with a clear visual impression of the possible presence of effects as well as the nature of any such effects. In conjunction with formal testing, the investigator has a powerful tool to study dependencies and co-dependencies in survival data.
John O’Quigley
Backmatter
Metadata
Title
Survival Analysis
Author
John O'Quigley
Copyright Year
2021
Electronic ISBN
978-3-030-33439-0
Print ISBN
978-3-030-33438-3
DOI
https://doi.org/10.1007/978-3-030-33439-0

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