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Competing Risks and Multistate Models with R

verfasst von: Jan Beyersmann, Arthur Allignol, Martin Schumacher

Verlag: Springer New York

Buchreihe : Use R!

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book covers competing risks and multistate models, sometimes summarized as event history analysis. These models generalize the analysis of time to a single event (survival analysis) to analysing the timing of distinct terminal events (competing risks) and possible intermediate events (multistate models). Both R and multistate methods are promoted with a focus on nonparametric methods.

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Inhaltsverzeichnis

Frontmatter

Data examples and some mathematical background

Frontmatter

1. Data examples

Abstract

In this book, we use both real and simulated data. One idea behind using simulated data is to illustrate that competing risks and multistate data can be conveniently approached from an algorithmic perspective. The data simulations are explained in their respective places in the book. In this section, we briefly introduce the real data examples. All of them are publicly available as part of the R packages used in this book.

Jan Beyersmann, Martin Schumacher, Arthur Allignol

2. An informal introduction to hazard-based analyses

Abstract

This chapter explains in a non-technical manner why methods for analysing standard survival data — one endpoint, observation of which is subject to right-censoring — transfer to more complex models, namely competing risks and multistate models, this book’s topic.

Jan Beyersmann, Martin Schumacher, Arthur Allignol

Competing risks

Frontmatter

3. Multistate modelling of competing risks

Abstract

Competing risks models analyse the time until some first event and the event type that occurs at that time. In contrast, standard survival analysis considers the time until some first event only. Examples include disease-free survival and length of hospital stay. Disease-free survival is observed either at the time the disease in question is diagnosed or at the time of death without prior disease. Length of hospital stay ends with either discharge alive or hospital death. In the latter example, survival models consider length of stay with a combined endpoint discharge alive/hospital death. Competing risks also model the endpoint type. Competing risks do not model subsequent events such as death after hospital discharge. To do this, more complex multistate models are needed, which is the topic of the multistate part of this book.

Jan Beyersmann, Martin Schumacher, Arthur Allignol

4. Nonparametric estimation

Abstract

We introduce the key nonparametric estimators, the Nelson-Aalen estimator of the cumulative cause-specific hazards and the Aalen-Johansen estimator of the cumulative incidence functions, in Section 4.1. We analyse the simulated data of Section 3.2 in Section 4.2; this section also introduces in detail the functionality offered by the R packages mvna, etm, cmprsk, and survival for nonparametric estimation in a competing risks model. We analyse a real data example in Section 4.3; this section emphasizes interpretation of competing risks analyses. The usual nonparametric estimators for standard single endpoint survival analysis, i.e., the Nelson-Aalen estimator of the cumulative all-cause hazard and the Kaplan-Meier estimator of the survival function, are included in the present account.

Jan Beyersmann, Martin Schumacher, Arthur Allignol

5. Proportional hazards models

Abstract

This chapter discusses the most widely used regression models in competing risks. Following an introduction in Section 5.1, Section 5.2 discusses proportional cause-specific hazards models, and Section 5.3 discusses the proportional subdistribution hazards model. The cause-specific hazards are as defined in Chapter 3. The subdistribution hazard is a different hazard notion, namely the hazard ‘attached’ to the cumulative incidence function of interest as explained below. Both modelling approaches have their relative merits, and both approaches make the proportional hazards assumption solely for interpretational and technical convenience. It is not uncommon that both approaches are employed in one data analysis, although one model assumption usually precludes the other. A justification for employing both models sideby- side is provided in Section 5.4. Goodness-of-fit methods are described in Section 5.5, and Section 5.6 gives a brief overview of regression models that go beyond the familiar proportional hazards assumption together with their availability in R.

Jan Beyersmann, Martin Schumacher, Arthur Allignol

6. Nonparametric hypothesis testing

Abstract

The log-rank test is arguably the most widely used test in survival analysis. In this brief Chapter, we explain the idea of the log-rank test and how it translates to competing risks. The key issue is that the log-rank test compares hazards and may consequently be used to compare cause-specific hazards, too. As we have seen earlier, differences between cause-specific hazards do not translate into differences of the cumulative event probabilities in a straightforward manner. Therefore, cumulative incidence functions are often compared by a log-rank-type test for the subdistribution hazard rather than for the cause-specific hazards. As we show below, we may settle for a brief Chapter, because these tests have already been computed as a byproduct of the Cox-type models in Chapter 5.

Jan Beyersmann, Martin Schumacher, Arthur Allignol

7. Further topics in competing risks

Abstract

So far, our treatment of competing risks has been restricted to two competing event states only. An exception was the analysis of drug-exposed pregnancies in Sections 4.4 and 5.2.2, where we handled three competing event states without much ado. This effortlessness shows that otherwise focusing on two competing events has been for ease of presentation only. The aim of the present section is to briefly demonstrate that everything that has been said before for two competing risks easily generalizes to J competing risks, where J is some finite number as in Section 2.2.3.

Jan Beyersmann, Martin Schumacher, Arthur Allignol

Multistate models

Frontmatter

8. Multistate models and their connection to competing risks

Abstract

Except for Section 2.2.4, this book has so far focused on competing risks, which model time until first event and type of that first event. We now turn to more complex multistate models, which, e.g., would also allow us to study the occurrence of subsequent events. Such investigations are often of subject matter interest. E.g., in medical applications they will allow for a more detailed study of the course of a patient’s disease. In the ONKO-KISS example of bloodstream infection in stem-cell transplanted patients studied in Section 5.2.2, such a model would allow us also to study mortality after infectious complication. In the SIR3 example of Sections 4.3, 5.2.2, 5.3.3 and Chapter 6, we investigated the impact of pneumonia admission diagnosis on intensive care unit mortality. A more complex multistate model would allow the further study of subsequent events which happen during unit stay such as ventilation switched on or off, catheter usage, or occurrence of hospitalacquired infections.

Jan Beyersmann, Martin Schumacher, Arthur Allignol

9. Nonparametric estimation

Abstract

We consider n individuals under study with individual multistate processes \((x-^{i}{t}) _{t\geq O,} X^{i} _{t} \epsilon {0,1,2,...,J} , i = 1,2,...n\) We assume that the n processes are, conditional on the initial states X(i) 0 , independent replicates of a multistate process as in Section 8.1. Observation of the individual multistate data is subject to a right-censoring time Ci and possibly also to a left-truncation time Li. We assume that right-censoring and left-truncation are independent as explained in Section 2.2.2.

Jan Beyersmann, Martin Schumacher, Arthur Allignol

10. Proportional transition hazards models

Abstract

As with competing risks, the most widely used regression model for multistate data assumes a proportional hazards form for the transition hazards of the multistate model. We re-emphasize that the proportional hazards assumption is made for interpretational and technical convenience. As in Chapter 9, we consider n individuals under study with individual multistate data subject to independent right-censoring and/or left-truncation. This entails that rightcensoring and left-truncation may depend on covariates included in the model. The n multistate processes are assumed to be conditionally independent given the baseline covariate values and given the initial states.

Jan Beyersmann, Martin Schumacher, Arthur Allignol

11. Time-dependent covariates and multistate models

Abstract

Our treatment of regression models for cause-specific or transition hazards in Chapters 5 and 10 has so far been restricted to baseline covariates Z only, those covariates whose value is measured or known at time origin. It is a remarkable strength of regression models for hazards that they can also incorporate covariates Z(t) whose value may change with time. Although this extension is a good deal technically straightforward, new interpretational challenges arise. Some of these may conveniently be addressed from a multistate perspective, which is what we do in this chapter.

Jan Beyersmann, Martin Schumacher, Arthur Allignol

12. Further topics in multistate modelling

Abstract

In Section 5.2 on proportional cause-specific hazards models, we illustrated that one typically has to investigate the effect of a covariate on all transition hazards. We also outlined that some covariates may have a common effect on some of the cause-specific hazards, but this is rarely used in practical applications. In more complex multistate models, however, sample size restrictions may motivate more parsimonious models. In practice, this is typically achieved by analysing an extended data frame as in Section 5.2.2 (see the data frame xl) with one row for each individual and each transition. Such an extended data frame for multistate data has been discussed in Section 10.2. As illustrated in Section 5.2.2, transition-specific covariates are used which allow single covariates to have a common effect on some transition hazards and different effects on other hazards. One may also impose that a covariate has no effect on a certain hazard by setting the corresponding entry of the transition-specific covariate to zero.

Jan Beyersmann, Martin Schumacher, Arthur Allignol

Backmatter

Titel: Competing Risks and Multistate Models with R
verfasst von: Jan Beyersmann
Arthur Allignol
Martin Schumacher
Copyright-Jahr: 2012
Verlag: Springer New York
Electronic ISBN: 978-1-4614-2035-4
Print ISBN: 978-1-4614-2034-7
DOI: https://doi.org/10.1007/978-1-4614-2035-4

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