Multi-State Survival Models for Interval-Censored Data

Ardo van den Hout

• Publisher: Chapman and Hall/CRC
• Year: 2016
• ISBN:
  9781315356730
• Edition: 1
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Multi-State Survival Models for Interval-Censored Data introduces methods to describe stochastic processes that consist of transitions between states over time. It is targeted at researchers in medical statistics, epidemiology, demography, and social statistics. One of the applications in the book is a three-state process for dementia and survival in the older population. This process is described by an illness-death model with a dementia-free state, a dementia state, and a dead state. Statistical modelling of a multi-state process can investigate potential associations between the risk of moving to the next state and variables such as age, gender, or education. A model can also be used to predict the multi-state process.

The methods are for longitudinal data subject to interval censoring. Depending on the definition of a state, it is possible that the time of the transition into a state is not observed exactly. However, when longitudinal data are available the transition time may be known to lie in the time interval defined by two successive observations. Such an interval-censored observation scheme can be taken into account in the statistical inference.

Multi-state modelling is an elegant combination of statistical inference and the theory of stochastic processes. Multi-State Survival Models for Interval-Censored Data shows that the statistical modelling is versatile and allows for a wide range of applications.

Chapter 1

Introduction

1.1 Multi-state survival models

Multi-state models are routinely used in research where change of status over time is of interest. In epidemiology and medical statistics, for example, the models are used to describe health-related processes over time, where status is defined by a disease or a condition. In social statistics and in demography, the models are used to study processes such as region of residence, work history, or marital status. A multi-state model that includes a dead state is called a multi-state survival model.

The specification and estimation of a multi-state survival model depend partly on the study design which generated the longitudinal data that are under investigation. An important distinction is whether or not exact times are observed for transitions between the states. This book considers mainly study designs where death times are known (or right-censored) and where times of transitions between the living states are interval-censored. Many applications in epidemiology and medical statistics have this property as it is often hard to measure the exact time of onset of a disease or condition. Examples are dementia, cognitive decline, disability in old age, and infectious diseases.

Closely related to the interval censoring is the choice between discrete-time and continuous-time models. A discrete-time model assumes a stepwise transition process, where the fixed time between successive steps is not part of the model. There are applications where this model is appropriate, and other applications for which this model is a good approximation of the process of interest. Continuous-time models, the topic of this book, allow changes of state at any time and will be more realistic in many situations. This type of model is also more flexible with respect to the study design for the observation times.

A continuous-time multi-state model can be seen as an extension of the standard survival model. The latter can be defined as a two-state model where a one-off change of status is the event of interest. The archetype example in medical statistics is the transition from the state of being alive to the state of being dead. Often there will be additional information in the data on other stochastic events that may be associated with the risk of a transition. An example of this is the onset of dementia in a study of survival in the older population. In such a case, the onset of dementia can be taken into account in the survival model by including it as binary time-dependent covariate for the risk of dying. The multi-state model approach would in this case consist of defining three states: a dementia-free state, a dementia state, and a dead state.

Assuming that both models are parametric, there are some clear advantages of the three-state model over the two-state model. Firstly, because the onset of dementia is part of the model, the three-state model can be used for prediction. Even though the two-state model can be used to study the effect of dementia on survival, this model cannot be used for prediction without additional modelling of the stochastic process which underlies the onset of dementia. Secondly, the onset of dementia is a latent process—observation will always be interval-censored. Dealing with this interval censoring is not straightforward when the onset is included in the two-state model as a time-dependent covariate. The multi-state models in this book, however, are explicitly defined for interval-censored transitions between living states.

Even in applications where survival is not of immediate interest, multistate survival model can still be very useful. Especially in epidemiological and medical research, a longitudinal outcome variable of interest may be correlated with survival. If this is the case, the risk of dying cannot be ignored in the data analysis. Examples of such longitudinal outcomes are blood pressure, cognitive function in the older population, and biomarkers for cardiovascular disease. One option is to specify a model for the longitudinal process of interest and combine this model with a standard survival model. This is called a joint model, and it is often specified with random effects which are shared by the two constituent models. But if the longitudinal outcome can be adequately discretised by a set of living states, then a multi-state survival model can be defined by adding a dead state. This provides an elegant alternative to a joint model: the multi-state model can be defined as an overall fixed-effects model for the process of interest as well as for survival.

A multi-state process is called a Markov chain if all information about the future is contained in the present state. If, for example, time spent in the present state affects the risk of a transition to the next state, then the process is not a Markov chain. Although many processes in real life are not Marko-vian, a statistical model based upon a Markov chain may still provide a good approximation of the process of interest. Most of the multi-state models in this book are not Markovian in the strict sense. By linking age and values of covariates to the risk of a transition, information about the future is contained not only in the present state, but also in current age and additional background characteristics.

This first chapter introduces the basic concepts that are used in multi-state survival models, discusses the relevant type of data, and illustrates the scope of the statistical modelling by an example. Details of parameter estimation and statistical inference are postponed to later chapters. At the end of this chapter an overview will be given of the methods and the examples in the rest of the book.

1.2 Basic concepts

The standard two-state survival model is defined by distinguishing a living state and dead state. The two main features of the standard survival model are that there is one event of interest (the transition from alive to dead) and that the timing of this event may be right-censored, in which case it is known that the event has not happened yet.

As an example, say patients are followed up for a year after a risky medical operation and the time scale is months since the operation. When the event is defined by death, the information per patient is either time of death or the time at which death is right-censored. The latter is typically the end of the follow-up, in this example 12 months. Alternatively, the patient drops out of the study during the follow-up and the censored time is the last time the patient was seen alive. The statistical modelling of survival in the presence of right censoring is often undertaken by assuming a model for the hazard. The hazard is the instantaneous risk of the event. As a quantity it is an unbounded positive value and should be distinguished from a probability, which is a value between zero and one. In the example, the probability of the event is linked to a specified time interval, for example, the probability of dying within a year, whereas the hazard is defined for a moment in time specified in months.

In a multi-state survival model there is more than one event of interest. An example with two states is the functioning of a machine with working (state 1) versus being repaired (state 2). The events are the transition from state 1 to state 2 and the transition from state 2 to...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• Acknowledgments
• 1 Introduction
• 2 Modelling Survival Data
• 3 Progressive Three-State Survival Model
• 4 General Multi-State Survival Model
• 5 Frailty Models
• 6 Bayesian Inference for Multi-State Survival Models
• 7 Residual State-Specific Life Expectancy
• 8 Further Topics
• A Matrix P(t) When Matrix Q Is Constant
• B Scoring for the Progressive Three-State Model
• C Some Code for the R and BUGS Software
• Bibliography
• Index