When risks compete, a curve can help predict which event is more likely to occur

How long will a patient live after treatment? This is one of the standard questions addressed in survival analysis. In practice, however, the question is often more complex, and mere knowledge of the duration may not be enough: we also need to understand which risk will bring it to an end.

In survival analysis, the word “survival” has a broader meaning: it is the time until one of the events under consideration occurs, not necessarily death. After cancer treatment, for example, one may study the time until the cancer returns, a serious treatment-related complication develops, or the patient dies before either of these events occurs. Each event identifies the risk that has materialized. Once the first event has been observed, the others can no longer be observed as the first outcome. In statistics, this situation is known as competing risks.

Competing risks arise well beyond medicine. In engineering, a machine can break down because different components fail. In actuarial science, an insurance contract may terminate for different reasons. Although the settings differ, the underlying statistical problem is the same.

A new paper we published with Claudio Del Sole in the Journal of the American Statistical Association addresses this problem. Claudio Del Sole earned a PhD in Statistics and Computer Science at Bocconi University and is currently a research fellow in the Department of Economics, Management and Statistics at the University of Milano-Bicocca. The paper is part of his doctoral thesis.

Not only when but also which event

Methods for competing risks can estimate the probability that no event has been observed yet, as well as the incidence functions associated with each type of event. We also address a different question: given the data already available, if we know that an event will occur at a given time, which type of event is most likely?

Imagine a transplant patient facing two possible, mutually exclusive events: for example, the onset of a complication or a relapse. Under classical statistical models, it is possible to evaluate the probability of each event over time. Our approach, however, naturally leads to a different question: if we know which events have occurred in the past in similar cases and the time at which an event-free period ends for a new patient, which of the competing events is most likely to have ended it?

This is where the paper’s main innovation comes in: the principled estimation of prediction curves. For each possible event time, the prediction curve gives the probability that a future event will be of a particular type, taking into account what has previously been observed. It therefore provides more than a single percentage: it shows how the probabilities of the different events change with the time at which the event occurs.

For example, a certain complication may be the most likely outcome immediately after treatment. A few weeks later, another complication may become more likely. Simply saying that the first complication is generally more likely than the second would omit an essential piece of information: the relative probabilities of the two events can change over time, in light of the data observed in the past. The prediction curve makes this change visible. 

When different risks share information

We have used a Bayesian nonparametric approach. The proposed model places few constraints on the shape of the cause-specific hazard rates, allowing the patterns that emerge from the data to play a greater role. At each point in time, these rates describe the propensity for each type of event to occur among individuals who have not yet experienced any of the events. However, the different causes are not analysed in isolation. The model links them through a hierarchical structure, so that information learned about one risk can also contribute, in part, to understanding the others.

This structure enables what is known as borrowing of information: information is shared among the rates associated with the different risks. This is particularly useful when some events are much less frequent than others and, considered on their own, would provide little data on which to base estimation and prediction. At the same time, the model does not force the risks to be homogeneous; each retains its own specific features. In our simulations, the hierarchical model produces consistently more accurate estimates than a version in which the causes are treated independently.

When medical outcomes compete

We illustrate our method using data from the European Blood and Marrow Transplant Group (EBMT), concerning 400 patients with acute myeloid leukaemia who underwent stem-cell transplantation. The main event of interest is the onset of Graft-versus-Host Disease (GvHD), a complication in which immune cells from the donor attack the recipient’s tissues. Death or relapse without prior GvHD are the competing events.

The main interest of the application lies in showing how the model simultaneously estimates the survival curve, the incidence functions associated with the different events and, above all, the prediction curves. Together, these provide an integrated view of how long patients remain event-free, how the different events accumulate over time, and which event is most likely to occur at any given time. The method also accommodates right-censored observations, namely patients for whom none of these events was observed before the end of follow-up. A second clinical illustration concerns patients with stage I melanoma, where the competing events are death from melanoma and death from other causes.

From medicine to other fields

Although the paper’s applications focus on medical problems, the methodology has much broader potential. In engineering, for example, if a machine breaks down after several years of operation, which component is most likely responsible? In actuarial science, how does the likelihood of different reasons for leaving an insured population depend on the time elapsed? Similar questions arise in economics, where a period of unemployment may end because someone finds a job or leaves the labour force. In all these settings, prediction curves could help reveal how the relative probabilities of different outcomes depend on when an event occurs.

We have introduced a coherent probabilistic approach that combines the estimation in competing risks models with interpretable predictions about which event will occur, given its timing. A prediction curve cannot say with certainty what will happen to one individual. It offers a different kind of information: it uses what has already been observed to show how the relative weight of the different risks changes over time.