What the Cumulative Incidence Measures
In competing-risks data, each person can experience at most one first-event cause. The probability of a particular cause by a specified time is therefore a central quantity: it describes the absolute probability of the outcome in the presence of the other possible first events. The cumulative incidence function (CIF) expresses that probability directly.
Let \(T\) be the time to the first event and \(J\) its cause. For a cause \(k\), the CIF at time \(t\) is the probability that the first event has occurred by \(t\) and was of cause \(k\). It is a marginal probability for the population being described, not a probability calculated only among people who remain event-free at \(t\).
The phrase “as the first event” matters. If a person first experiences a competing cause, they cannot later have cause \(k\) as their first event. Thus, \(F_k(t)\) includes the consequences of all competing causes in the population event process. It is not the probability that cause \(k\) would occur by \(t\) if other causes could somehow be removed.
Each \(F_k(t)\) is nondecreasing in \(t\): extending the time horizon cannot undo a cause-\(k\) first event that has already occurred. It lies between zero and one. When the listed causes exhaust the possible first events, their CIFs satisfy \(\sum_k F_k(t)=1-S(t)\). This says that the probability of any first event by \(t\) is divided among its mutually exclusive causes.
Why It Appears in the Subdistribution Hazard
The subdistribution hazard is defined for a specific cause \(k\), and the CIF for that same cause is built into its definition. In continuous time, when the cause-specific CIF has a density \(f_k(t)=dF_k(t)/dt\), the subdistribution hazard is $$ \widetilde{\lambda}_k(t)=\frac{f_k(t)}{1-F_k(t)}. $$ The denominator is the probability that cause \(k\) has not occurred as a first event by time \(t\). It is not, in general, the probability of still being event-free: people who experienced a competing first event have not experienced cause \(k\), so they remain part of the complement \(1-F_k(t)\).
This definition identifies the role of the CIF without making the hazard a probability. The CIF \(F_k(t)\) is a probability accumulated up to \(t\); \(f_k(t)\) describes its instantaneous rate of increase; and \(\widetilde{\lambda}_k(t)\) is that rate relative to \(1-F_k(t)\). Its units are inverse time, such as per year. A hazard value is not itself a chance of an event during a time interval.
For a differentiable CIF, the definition can also be written in small-interval form. For a small positive interval \(dt\), the probability of a cause-\(k\) first event during that interval is approximately \(f_k(t)dt\). Dividing this increment by the probability \(1-F_k(t)\) gives the limiting rate in the definition:
This is a rate approximation for a short interval, not an equality between the hazard and the interval probability. The approximation becomes useful when \(dt\) is small and the CIF is smooth over that interval. At event-time distributions with point masses or jumps, the derivative formula does not describe a jump by itself; the cumulative probability \(F_k(t)\) remains well-defined, while a jump-specific increment must be handled separately.
Worked Example: A CIF From Complete Follow-Up
Suppose a hypothetical cohort of 1,000 people is followed completely for two years. By two years, 120 people have a cause-A first event, 180 have a competing first event, and 700 have had no first event. Calculate the cause-A CIF at two years.
With complete follow-up, the observed proportion with a cause-A first event by two years estimates \(F_A(2)\). The calculation is $$ \widehat{F}_A(2)=\frac{120}{1000}=0.12. $$
Thus the estimated probability of a cause-A first event by two years is 0.12, or 12%. The competing-event count does not get added to the cause-A numerator: those 180 people experienced a different first-event cause. The counts also check the event-free proportion, \(700/1000=0.70\), and \(0.12+0.18=0.30=1-0.70\).
This simple calculation relies on the stated complete follow-up: every member of the cohort’s status by two years is known. If people were censored before two years, the raw fraction \(120/1000\) would not, by itself, generally estimate the two-year CIF; censoring must be accounted for.
Worked Example: Reading CIF Increments Over Time
In a second hypothetical cohort with complete follow-up, suppose the cumulative counts among 500 people are as follows. By year one, 25 have experienced cause A and 15 a competing cause. By year two, the cumulative counts are 55 for cause A and 35 for the competing cause. Calculate the empirical cause-A CIF at each time and its increase between years one and two.
At year one, the cause-A CIF estimate is $$ \widehat{F}_A(1)=\frac{25}{500}=0.050. $$ At year two, it is $$ \widehat{F}_A(2)=\frac{55}{500}=0.110. $$ The increase over the second year is \(0.110-0.050=0.060\), or 6 percentage points of the original cohort.
That increment corresponds to 30 additional cause-A events among the original 500 people: \((55-25)/500=30/500=0.060\). It is not the proportion of year-one event-free people who experience cause A in year two. At year one, 40 people have already had a first event of either cause, so the event-free group has 460 people; the second-year cause-A count divided by 460 would answer a different, conditional question.
The empirical CIF is nondecreasing here, as it must be: \(0.110\geq0.050\). The competing-cause counts help describe the event process, but they are not part of the cause-A cumulative count.
Worked Example: The CIF Complement in a Subdistribution Hazard
Consider two hypothetical populations at time \(t=2\) years. Suppose their smooth cause-A CIFs both have instantaneous slope \(f_A(2)=0.060\) per year. In population 1, \(F_A(2)=0.20\); in population 2, \(F_A(2)=0.40\). Calculate each subdistribution hazard at that time.
For population 1, \(1-F_A(2)=1-0.20=0.80\), so $$ \widetilde{\lambda}_{A,1}(2) =\frac{0.060\ \text{per year}}{0.80} =0.0750\ \text{per year}. $$ For population 2, \(1-F_A(2)=1-0.40=0.60\), so $$ \widetilde{\lambda}_{A,2}(2) =\frac{0.060\ \text{per year}}{0.60} =0.100\ \text{per year}. $$
The slopes are equal, but the subdistribution hazards differ because the CIF complements differ. These values are rates, not probabilities that a person will experience cause A in the next year. For a sufficiently short interval of length \(dt\), the corresponding first-order increment in the CIF is approximately \(f_A(2)dt\); the hazard instead scales that slope by \(1/[1-F_A(2)]\).
Estimation, Comparisons, and Limitations
With complete follow-up to a fixed time, the empirical fraction experiencing cause \(k\) by that time is a direct estimate of \(F_k(t)\). With right-censored observations, a person censored before \(t\) has an unknown event status after censoring, so this simple fraction is not generally adequate. The Aalen–Johansen estimator is a standard nonparametric estimator of the CIF in competing-risks data; its use requires appropriate handling of censoring, commonly justified by independent censoring in the population or within the groups used for analysis.
A useful contrast is the complement of a Kaplan–Meier estimate obtained by treating competing causes as censored. That calculation does not generally estimate the observed-world CIF for cause \(k\): it targets a hypothetical net-risk quantity that does not account for the competing events in the same way. For absolute probability of a particular first-event cause by a given time, use a CIF estimate appropriate to competing risks.
The subdistribution hazard and the CIF answer related but different questions. A CIF is directly interpretable as a probability by a stated time. A subdistribution hazard is a rate defined using the derivative of that CIF and its complement. In particular, a subdistribution hazard ratio is not a ratio of cumulative probabilities and should not be described as a relative probability of the event by a fixed time.
CIF estimates also depend on the population and follow-up context. Changing the distribution of competing events can change the observed probability of cause \(k\), even if the cause-\(k\) process is otherwise similar. A CIF is therefore an absolute-risk summary for the competing-risks process actually being considered, not a cause-specific biological rate isolated from other outcomes.
Common Mistakes and Checks
- Calling the CIF a conditional probability among event-free people. \(F_k(t)=P(T\leq t,J=k)\) is calculated in the original population. A conditional probability among people still event-free at \(t\) has a different denominator and answers a different question.
- Replacing \(1-F_k(t)\) with event-free survival \(S(t)\) in the subdistribution hazard. The defined denominator is the complement of the cause-\(k\) CIF. The two quantities generally differ when competing events occur.
- Calling a subdistribution hazard a probability. Its units are inverse time. A probability requires a specified time interval and is not given by the hazard value alone.
- Treating competing events as if they were ordinary censoring when estimating absolute risk. Doing so can yield a net-risk quantity rather than the CIF for the observed competing-risks setting.
- Ignoring data checks. Each estimated CIF should be nondecreasing and between zero and one. When causes are exhaustive, the sum of their CIF estimates should be consistent with the estimated probability of any first event by that time.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In a hypothetical cohort of 800 people, 64 experience cause B by year three. With complete follow-up, what is the empirical cause-B CIF at year three?
- At a time \(t\), suppose \(F_k(t)=0.25\) and \(f_k(t)=0.045\) per year. What is the subdistribution hazard at that time?
- Why is \(1-F_k(t)\) generally not interchangeable with event-free survival \(S(t)\)?
- A cause-specific CIF estimate decreases between two successive times. Is that compatible with the definition? Explain.
Solutions. (1) \(\widehat{F}_B(3)=64/800=0.080\), or 8% by year three. (2) \(\widetilde{\lambda}_k(t)=0.045/(1-0.25)=0.045/0.75=0.060\) per year. (3) \(1-F_k(t)\) includes everyone who has not had cause \(k\) as a first event, including people who have had a competing first event; \(S(t)\) includes only people who remain event-free. (4) No. A cumulative probability cannot decrease as the time horizon grows; such a pattern indicates a data, calculation, or plotting problem, subject to consistent time points and estimation procedures.