# Survival Analysis: Competing Risk Theory

## Statistical Methods for the Analysis of Survival Data in the Presence of Competing Risks

Estimating Crude Incidence: Suppose, the baseline time in the cohort is well defined and that T denotes the time from baseline time until the occurrence of the event of interest. In the absence of competing risks, the survival function,

S(t) = Pr[T>t]

describes the distribution of event times. One minus the survival function (ie, the complement of the survival function),

F(t)= 1 -S(t) = Pr[T\leq t]

describes the incidence of the event over the duration of follow-up. Two key properties of the survival function being

S(0) = 0 \\
\textit{ i.e. at the beginning of the study, the event}\\ \textit{has not yet occurred for any subjects}, \\
----------------\\
\underset{t \rightarrow \infty}{lim } S(t) = 0 \\
\textit{ i.e. eventually the event of interest occurs for all subjects}


In the absence of competing risks, F(t) may be a good measure when describing the incidence of occurrence of event(deaths). However, in the presence of competing risks, the deaths due to competing risks are treated as censored observations(most of the times, loss during to follow up) , as a result an absurd assumption is taken into consideration that the event (death) occurs ultimately due to the primary cause. This results to a heavy bias in estimation.

To do away with this demerit, the Cumulative Incidence Function which allows for estimation of the incidence of the occurrence of an event while taking competing risk into account, is a much better measure. The cumulative incidence function for the kth cause is defined as:

CIF_k(t) = Pr[T\leq t , D=k] \\
; \textit{where D is a random variable denoting} \\ \textit{the type of event that occured.}

A key point is that, in the competing risks setting, only 1 event type can occur, such that the occurrence of 1 event precludes the subsequent occurrence of other event types. The function CIFk(t) denotes the probability of experiencing the kth event before time t and before the occurrence of a different type of event. The CIF has the desirable property that the sum of the CIF estimates of the incidence of each of the individual outcomes will equal the CIF estimates of the incidence of the composite outcome consisting of all of the competing events. Unlike the survival function in the absence of competing risks, CIFk(t) will not necessarily approach unity as time becomes large, because of the occurrence of competing events that preclude the occurrence of events of type k.