# Discovering Statistics

A den for Learning

## Survival Analysis: Competing Risk Theory

### INTRODUCTION

A distinctive feature of survival data is the concept of censoring. And an implicit concept in the definition of censoring is that if the study had been prolonged (or if subjects had not dropped out), eventually the outcome of interest would have been observed to occur for all the subjects. Conventional statistical methods for the analysis of survival data make the important assumption of independent or non-informative censoring. This means that at a given point in time, subjects who remain under follow-up have the same future risk for the occurrence of the event as those subjects are no longer being followed (either because of censoring or study dropout), as if losses to follow-up were random and thus non-informative.

A competing risk is an event whose occurrence precludes, the occurrence of the primary event of interest. For instance, in a study in which the primary outcome was time to death due to a cardiovascular cause, a death due to a non-cardiovascular serves as a competing risk.

Conventional statistical methods for the analysis of survival data assume that competing risk are absent. Two competing risks are said to be independent if information about a subject’s risk of experiencing one type of event provides no information about the subject’s risk of experiencing the other type of event. The methods that will be described later on will involve impeding risks which are independent of one another and further also in which competing risks are not independent of one another.

In biomedical applications, the biology often suggests at least some dependence between competing risks, which in many cases may be quite strong. Accordingly, independent competing risks may be relatively rare in biomedical applications.

When analyzing survival data in which competing risks are present, analysts frequently censor subjects when a competing event occurs. Thus, when the outcome is time to death attributable to cardiovascular causes, an analyst may consider a subject as censored once that subject dies of noncardiovascular causes. However, censoring subjects at the time of death attributable to noncardiovascular causes may be problematic.

First, it may violate the assumption of noninformative censoring: it may be unreasonable to assume that subjects who died of noncardiovascular causes (and were thus treated as censored) can be represented by those subjects who remained alive and had not yet died of any cause.

Second, even when the competing events are independent, censoring subjects at the time of the occurrence of a competing event may lead to incorrect conclusions because the event probability being estimated is interpreted as occurring in a setting where the censoring (eg, the competing events) does not occur.

In the cardiovascular example described above, this corresponds to a setting where death from noncardiovascular causes is not a possibility. Although such probabilities may be of theoretical interest, they are of questionable relevance in many practical applications, and generally lead to overestimation of the cumulative incidence of an event in the presence of the competing events.