Why Retention Needs an Explanation
A person who has experienced a competing event is no longer event-free. In the observed first-event process, that person cannot subsequently experience the focal cause as their first event. It may therefore seem contradictory that the person remains represented in the subdistribution risk set for that focal cause.
The key is that this set is a mathematical construction for the focal cause’s cumulative incidence, not a list of people who are currently able to experience any first event. It keeps track of everyone who has not yet had the focal event. That group includes people whose first event was a different cause. The earlier material on the subdistribution hazard and subdistribution risk set introduces these quantities; here, the population accounting explains why the retained members belong in the construction.
This definition concerns the first observed event. Once a competing event has occurred, the person is not restored to the event-free group, nor is a later event reclassified as their first event. Retention means only that the person continues to count in the constructed set because the focal event has not occurred.
The Population Partition Behind Retention
Assume that each person begins event-free, experiences at most one first event, and has complete follow-up through the time being considered. The possible statuses just before \(t\) partition the population into three groups: event-free, already experienced the focal cause, or already experienced a competing cause. These groups are mutually exclusive and exhaustive.
Here \(F_j(t-)=P(T<t,J=j)\) is the cumulative incidence for cause \(j\) strictly before \(t\). The event-free group is represented by \(P(T\geq t)\); \(F_k(t-)\) is the group already removed by the focal event; and the remaining terms are people already experiencing competing causes. Rearranging the partition gives
The left side is the population proportion in the modified risk set: still event-free people plus people with prior competing events. The right side is the proportion that has not yet experienced the focal event. This accounting identity is the reason for retention. If people with prior competing events were deleted, the remaining set would instead represent only the event-free group, which is a different set.
For a cohort of \(n\) people with complete follow-up, the same argument applies to counts. If \(n_k(t-)\) have already had the focal event, then \(n-n_k(t-)\) have not. That complement consists of the event-free people and those with a prior competing event. Therefore the modified-set size is \(n-n_k(t-)\), not merely the number still event-free.
Worked Examples
Worked Example: Classifying People Just Before a Focal Event Time
Consider a hypothetical cohort of eight people. Cause A is focal; cause B is competing. Their observed first-event times are: person 1, A at time 2; person 2, B at time 3; person 3, B at time 4; person 4, A at time 5; person 5, event-free through time 5; person 6, B at time 7; person 7, event-free through time 5; and person 8, event-free through time 5. Who is represented in the modified risk set just before time 5?
Person 1 is excluded because A occurred before time 5. Persons 2 and 3 are retained because they had B, not A. Person 4 is included just before time 5 because their A event occurs at time 5. Persons 5, 7, and 8 are included because they remain event-free through time 5. Person 6 is included because their competing event occurs later. Thus the modified set contains seven people: persons 2 through 8.
The count agrees with the complement calculation. One person has already experienced A, so \(8-1=7\) people have not yet experienced A. Of these seven, two are not event-free: they had the competing event before time 5. Their inclusion is exactly what makes the set count all people who have not yet had A.
Worked Example: The Same Complement in Cohort Counts
In a hypothetical cohort of 100 people with complete follow-up, by time \(t\), 10 have had focal cause A, 18 have had competing events, and 72 remain event-free. Compute the modified risk-set size just after time \(t\), taking the event counts as occurring before the next interval.
The three groups sum to the cohort size: \(10+18+72=100\). The modified set excludes the 10 people who already had A but retains the 18 with competing events and the 72 who are event-free:
The complement calculation gives the same result: \(100-10=90\). As a proportion, the set is \(90/100=0.90\), while the cumulative incidence of A through \(t\) is \(10/100=0.10\). Thus the set’s proportion is \(1-0.10=0.90\). Deleting the 18 people with prior competing events would leave only 72, which is the event-free count, not the modified-set count.
Worked Example: Why the Denominator Changes When Prior Competing Events Are Deleted
Suppose a hypothetical cohort has 200 people. Before the next interval, 30 have experienced focal cause A, 50 have experienced competing events, and 120 are event-free. During the interval, 8 people experience A as their first event. Work out the focal cumulative incidence before and after the interval, and check the risk-set accounting.
Before the interval, the cumulative incidence is \(30/200=0.15\). The modified risk set includes the 50 prior competing events and 120 event-free people, for \(170\) people, or \(170/200=0.85\) of the cohort. The interval adds 8 focal events, so the new cumulative incidence is \((30+8)/200=0.19\), an increase of \(8/200=0.04\).
Relative to the modified-set proportion, the increase is \(0.04/0.85=0.04706\). Multiplying back by the proportion in that set recovers the population increase: \(0.85\times0.04706=0.04000\), apart from rounding. The 50 people with prior competing events contribute to the modified-set count even though the 8 observed focal events must come from people who had not already experienced a first event. The count is an accounting device for the focal cumulative incidence, not a claim that all 170 people remain physically susceptible to a first event.
What the Risk Set Does—and Does Not—Claim
The cause-specific risk set answers a different question: who is event-free and therefore still at risk of experiencing a first event of any cause? A prior competing event removes a person from that set. By contrast, the modified subdistribution risk set is arranged around whether the focal event has occurred. A prior competing event removes the person from the event-free group but not from the group that has yet to experience the focal event.
| Status just before \(t\) | Cause-specific risk set | Modified subdistribution risk set for cause \(k\) |
|---|---|---|
| Event-free | Included | Included |
| Prior event of cause \(k\) | Excluded | Excluded |
| Prior competing event | Excluded | Retained |
This distinction matters whenever the target is the cumulative incidence of a particular cause. The event-free set shrinks after every first event; the modified set shrinks when the focal event occurs. The two sets therefore track different populations. Confusing them can lead to the wrong denominator and a mismatch between the risk-set construction and the focal cumulative incidence it is intended to represent.
Assumptions, Diagnostics, and Limits
The simple partition and count examples rely on a clearly defined first-event outcome: causes are mutually exclusive labels for the first event, and a person is not counted as having two different first-event causes. Complete follow-up is assumed so that someone whose status is unknown is not mistakenly classified as event-free or as having a competing event. The identity is about status through a specified time; it does not require competing causes to be independent.
In an analysis with censoring, censored people cannot simply be assigned to one of the three observed-status groups after their censoring time. Appropriate methods must account for censoring; the unadjusted count identity above is not, by itself, a censoring correction. Similarly, if the outcome is not a first event, or if the event categories are not mutually exclusive, the three-group partition must be reconsidered before applying this reasoning.
A useful diagnostic is to check the bookkeeping at each time: the modified-set count should equal the cohort count minus the number of prior focal events, under complete follow-up and the stated first-event setup. It should also equal the event-free count plus the count with prior competing events. If those totals disagree, check event timing conventions, cause labels, and whether censored observations have been handled as though their later status were known.
Common Mistake
Wrong reasoning: “A competing event has happened, so that person is no longer at risk; remove them from every risk set.” This is correct for the event-free cause-specific risk set, but it does not define the modified set for a focal subdistribution. Correction: Ask what the set is constructed to count. For the focal subdistribution, it represents people who have not yet experienced the focal event. A prior competing event ends event-free follow-up, but it does not make the focal event part of that person’s observed first-event history, so the person remains represented in the complement of the focal cumulative incidence.
Check Your Understanding
Answer each question using the first-event setup and complete follow-up unless stated otherwise. Brief solutions follow the questions.
- A cohort has 12 people: 3 have had the focal cause, 4 have had a competing event, and 5 remain event-free. What is the modified risk-set count?
- Why does a person with a prior competing event not belong to the event-free cause-specific risk set?
- State the population partition that explains the modified risk-set proportion.
- What additional issue arises if some people are censored before their event status at time \(t\) is known?
Solutions. (1) The count is \(4+5=9\), equivalently \(12-3=9\). (2) The first event has already occurred, so the person is no longer event-free and cannot experience another first event. (3) Event-free, prior focal event, and prior competing event are disjoint groups whose probabilities sum to 1; the modified set combines the first and third groups, giving \(1-F_k(t-)\). (4) Their status after censoring is unknown, so they cannot be treated as known event-free or known to have had a competing event; censoring must be handled appropriately.