What the Denominator Counts
For a chosen event cause \(k\), the denominator of the subdistribution hazard has a specific interpretation: it represents people who have not yet experienced cause \(k\). That group includes people who have experienced another, competing cause. It is therefore generally larger than the group who remain free of every event.
This distinction matters because competing events prevent a person from later experiencing cause \(k\) as the first event, but the subdistribution risk set continues to represent that person. The denominator is tied to the cumulative incidence of the focal cause, not to the probability of remaining entirely event-free.
The left limit \(F_k(t-)\) means that an event of cause \(k\) occurring exactly at \(t\) has not yet been counted when describing who is at risk immediately before \(t\). In a cohort of \(N\) people with complete event follow-up, the corresponding subdistribution risk-set size just before \(t\) is \(N\{1-F_k(t-)\}\). This is a population-style count: a person with a competing event remains represented in it, while a person who has already had cause \(k\) does not.
Why One Minus the Cumulative Incidence
The denominator follows from a set complement. By time \(t-\), a person is excluded from the subdistribution risk set exactly when that person has already experienced the focal cause. The complement of “has experienced cause \(k\) before \(t\)” is “has not experienced cause \(k\) before \(t\).” Competing events do not belong to the excluded set.
The two sets in the union are disjoint: one describes people with no event before \(t\), and the other describes people whose first event before \(t\) was a competing cause. Together they include everyone except those with a prior focal event. This is the essential justification; the denominator is not obtained by subtracting all event types from one.
For an interval \((u,v]\), the increase in the focal cumulative incidence is \(\Delta F_k=F_k(v)-F_k(u)\). In an uncensored cohort, the fraction of the subdistribution risk set at \(u\) that experiences cause \(k\) in this interval is
The denominator here is the proportion still represented for the focal cause at the start of the interval. This expression helps interpret the denominator, but it should not be mistaken for a formula that turns the continuous-time subdistribution hazard into a simple probability. The continuous-time hazard uses an instantaneous event rate; the interval expression summarizes a finite interval.
Worked Examples
Worked Example: Separating “No Focal Event” From “No Event”
Consider a hypothetical cohort of 100 people followed until a fixed time. By that time, 12 have experienced focal cause \(k\), 38 have experienced a competing cause, and 50 remain event-free. Find the fraction in the subdistribution risk set for cause \(k\), and compare it with the event-free fraction.
The cumulative incidence of cause \(k\) is \(F_k=12/100=0.12\). Thus, the fraction not yet experiencing cause \(k\) is
The corresponding subdistribution risk-set size is \(100(0.88)=88\) people. It consists of the 50 event-free people plus the 38 who already experienced a competing cause. By contrast, the event-free fraction is \(50/100=0.50\). Using 0.50 as the subdistribution denominator would exclude the 38 competing-event cases and would answer a different question.
Worked Example: Updating the Denominator Across Two Intervals
Suppose a hypothetical cohort of 100 people has complete follow-up. In the first interval, 10 experience cause \(k\), 20 experience a competing cause, and 70 remain event-free. In the second interval, 9 more experience cause \(k\), 14 experience a competing cause, and 47 remain event-free. Calculate the focal cumulative incidence and denominator at each interval boundary, then interpret the second-interval arithmetic.
At the end of the first interval, \(F_k=10/100=0.10\), so the fraction in the subdistribution risk set is \(1-0.10=0.90\), or 90 people. These 90 comprise 20 competing-event cases and 70 event-free people. The first-interval increase in cumulative incidence is 0.10; divided by the starting denominator of 1.00, it is \(0.10/1.00=0.10\).
At the end of the second interval, 19 people in total have experienced cause \(k\), so \(F_k=19/100=0.19\). The fraction not yet experiencing cause \(k\) is \(1-0.19=0.81\), or 81 people. Immediately before the second interval, the denominator was still 90: the 10 earlier focal events were excluded, while the 20 earlier competing events remained represented. The second-interval increase in cumulative incidence is \(9/100=0.09\), giving
Equivalently, 9 of the 90 people represented in the subdistribution risk set at the start of interval 2 experience cause \(k\) during that interval. The 14 competing events in that interval do not remove people from the denominator retroactively; they are competing outcomes during the interval, not earlier focal events.
Worked Example: A Competing Event Does Not Reduce This Denominator
In a hypothetical cohort of 80 people, 16 experience a competing cause before time \(t\), while no one has experienced focal cause \(k\). The other 64 remain event-free. What fraction of the cohort belongs to the subdistribution risk set for cause \(k\) just before \(t\)?
Here \(F_k(t-)=0/80=0\). Therefore,
All 80 people are represented for the focal cause: 64 are event-free and 16 have had a competing event. The result does not claim that all 80 remain physically able to experience cause \(k\) as a first event. It reflects the subdistribution risk-set convention: prior competing events remain represented. The count that remains physically event-free is only 64, which is not the denominator being interpreted here.
How This Denominator Differs From Other Risk Sets
The denominator can be understood most clearly by comparing what each group excludes. A cause-specific risk set removes anyone who has already had any first event. The subdistribution risk set for cause \(k\) removes only people who have already had cause \(k\). The event-free probability is therefore not interchangeable with \(1-F_k(t)\).
| Quantity at time \(t-\) | Who is excluded? | Interpretation |
|---|---|---|
| Event-free fraction \(P(T\geq t)\) | Anyone who has experienced any event | Fraction with no event before \(t\) |
| Cause-specific risk-set fraction | Anyone who has experienced any event | Fraction still event-free and at risk for a first event |
| Subdistribution risk-set fraction for cause \(k\), \(1-F_k(t-)\) | Anyone who has already experienced cause \(k\) | Fraction not yet experiencing the focal cause, including prior competing events |
This comparison is about the membership represented by each denominator; the cause-specific and subdistribution hazards answer different questions. In particular, substituting the event-free fraction into a subdistribution-hazard calculation changes the risk set and therefore changes the quantity being calculated.
Assumptions, Diagnostics, and Limitations
The interpretation assumes a well-defined first event time \(T\), a mutually exclusive cause label \(J\) for that first event, and a clearly specified focal cause \(k\). The cumulative incidence \(F_k(t)\) must refer to the same population and time origin as the risk set. If people can experience multiple events but the analysis is explicitly about the first event, later events do not alter the definition: only the first event type determines whether the person has experienced cause \(k\).
With complete follow-up in a finite cohort, the denominator can be checked directly: subtract the number with a prior focal event from the original cohort size. Do not instead subtract every event type. Also check that the event counts add up consistently with the cohort size and that each person’s first event is assigned only once.
With right censoring, the observed number still under follow-up is not generally equal to \(N\{1-F_k(t-)\}\). A censored person’s event status after censoring is unknown, and simply removing censored people from a raw count can make the observed denominator differ from the target population fraction. Estimation of the cumulative incidence must account for censoring under its relevant assumptions; the identity \(1-F_k(t-)\) describes the target population fraction, not a license to treat unobserved outcomes as known. Ties also require a consistent convention for whether the risk set is defined just before or just after the tied event time.
A useful diagnostic is to compare the proposed denominator with the event-free fraction. If many people have already had competing events, \(1-F_k(t-)\) should usually exceed the event-free fraction. If the two are equal despite competing events having occurred, check whether competing cases were mistakenly removed from the subdistribution risk set. Conversely, if people with prior focal events remain included, the denominator is too large.
Common Mistake
Wrong reasoning: “A competing event means the person can no longer experience the focal event, so that person must leave the denominator. Therefore the denominator is the fraction with no event of any type.”
Correction: That reasoning describes the event-free or cause-specific risk set, not the subdistribution risk set. For cause \(k\), the denominator is \(1-F_k(t-)\), which excludes prior cause-\(k\) events but retains people whose first event was a competing cause. It tracks the fraction that has not yet experienced the focal cause, not the fraction still free of all events.
Check Your Understanding
Answer each question, then compare with the solutions below.
- In a cohort of 200 people, 30 have experienced cause \(k\) and 70 have experienced competing causes by time \(t\). What is the subdistribution risk-set fraction and the corresponding count?
- Why is the event-free fraction not generally the denominator for the subdistribution hazard?
- At the start of an interval, \(F_k=0.20\). During the interval the focal cumulative incidence increases by 0.08. Calculate the interval increase divided by the starting subdistribution denominator.
- If no one has experienced cause \(k\) yet, but some people have experienced competing events, what is \(1-F_k(t-)\)? Explain.
- Why can the raw number still under observation fail to equal \(N\{1-F_k(t-)\}\) when there is censoring?
Solutions. (1) The fraction is \(1-30/200=0.85\), so the corresponding count is \(200(0.85)=170\). The 70 competing-event cases are included. (2) The event-free fraction removes people with any event, whereas the subdistribution denominator removes only people who have already experienced cause \(k\). (3) The starting denominator is \(1-0.20=0.80\); the ratio is \(0.08/0.80=0.10\). (4) It is 1, because no one has yet experienced the focal cause; competing events remain represented. (5) Censoring leaves later event status unknown, so the observed follow-up count need not equal the target population fraction who have not yet experienced cause \(k\).