Why the Fine–Gray Risk Set Is Different
A cause-specific risk set contains people who have not yet experienced any event. The Fine–Gray risk set answers a different question: who has not yet experienced the event of interest? Those sets differ because a person who experiences a competing event is known not to experience the event of interest as their first event, but remains part of the population used to define its cumulative incidence.
That distinction is the reason for the Fine–Gray pseudo risk set. It is not a claim that someone with a competing event remains biologically event-free. Rather, it is a bookkeeping device that makes the risk-set denominator match the subdistribution hazard. Censoring is different: if follow-up ends before any event, the person’s later event status is unknown, so that person cannot simply be treated as permanently free of the event of interest.
The comparison is made immediately before \(t\), which is why the cumulative incidence denominator uses \(F_j(t-)\), the probability of a cause-\(j\) event strictly before \(t\). A person who has already had cause \(j\) is removed; a person who has had another cause is retained. At an event time with tied observations, the risk set is defined just before that time, so people having any event at \(t\) are still at risk immediately before \(t\).
Deriving the Pseudo Risk Set
The subdistribution hazard for cause \(j\) is built from increments of \(F_j\) divided by the proportion that has not yet had a cause-\(j\) event. In a continuous-time formulation, its hazard rate is \(f_j(t)/[1-F_j(t)]\), where \(f_j(t)\) is the cause-\(j\) cumulative-incidence density. The denominator has a direct probability interpretation:
The two events on the right are disjoint. The first means no event has happened before \(t\); the second means an event has happened before \(t\), but its cause was not \(j\). Their union is exactly the event “cause \(j\) has not occurred before \(t\).” Thus the pseudo risk set is the sample counterpart of the population represented by \(1-F_j(t-)\).
Because the two indicators cannot both equal one, their sum is an indicator. It equals one for someone still free of all events at \(t-\), and also for someone who previously had a competing event. It equals zero for someone who previously had the event of interest. This identity derives the membership rule; it is not an additional assumption about how events occur.
For a small time interval after \(t\), the numerator of the subdistribution hazard is the probability of a cause-\(j\) event in that interval, while the denominator is the probability of being in this pseudo risk set. In continuous time, dividing the cause-\(j\) incidence density by that denominator gives the subdistribution hazard rate. This explains why the risk set must retain prior competing events: removing them would replace \(1-F_j(t-)\) with the probability of remaining free of every event, which is a different denominator.
Worked Example: Membership at a Target-Cause Event Time
Suppose a hypothetical study follows eight people for two event types. Let A be the target event and B the competing event. The observed records are:
| Person | Observed time | Record |
|---|---|---|
| 1 | 2 | A event |
| 2 | 3 | B event |
| 3 | 4 | B event |
| 4 | 5 | Censored, no event observed |
| 5 | 7 | A event |
| 6 | 9 | B event |
| 7 | 9 | Censored, no event observed |
| 8 | 10 | Censored at administrative study end |
Find the observed pseudo risk set for A immediately before time \(t=6\). Persons 2 and 3 had B events before 6, so both remain represented. Persons 5, 6, 7, and 8 have observed follow-up through at least 6 without an A event before 6, so they are also represented. Person 1 had A before 6 and is excluded. Person 4 was censored at 5 without an event; their later status is unknown, so they are not directly represented after 5.
Therefore the observed set is \(\{2,3,5,6,7,8\}\), with six members. It is not the ordinary event-free set: that set would remove persons 2 and 3 after their B events. It is also not all eight people: person 1 had already experienced A, and person 4’s status after censoring is unobserved.
Censoring Changes What Can Be Represented
Write \(C_i\) for the censoring time, \(Y_i=\min(T_i,C_i)\) for the observed follow-up time, and \(\Delta_i=I(T_i\le C_i)\) for whether an event was observed. When a competing event is observed before censoring, its cause is known. That person can be retained in the pseudo risk set at later times, even if follow-up ends afterward: the first-event history already establishes that cause \(j\) did not occur first.
By contrast, a person censored before any event has an unknown future event time and cause. For an observed-data risk set, this person is included only up to the censoring time. A concise membership rule immediately before \(t\) is:
The first term includes people observed through \(t-\), including those who have a cause-\(j\) event exactly at \(t\). The second retains people with a previously observed competing event. A prior observed cause-\(j\) event satisfies neither term; a censoring time before \(t\) without an event also satisfies neither. This is the directly observable representation, not by itself a guarantee that censoring has no effect on estimation.
To infer the population subdistribution hazard when censoring occurs, Fine–Gray estimation must account for the fact that censored people are missing from later observed risk sets. The usual justification requires censoring to be independent of the event process, either unconditionally or conditional on the covariates used in the analysis, and requires a positive probability of remaining uncensored over the times being analyzed. Under these conditions, censoring adjustment can recover the relevant population contributions. Without adequate censoring adjustment, simply dropping censored people can distort the pseudo risk-set representation. The specific weighted likelihood construction is a separate estimation step.
Worked Example: Competing Event Versus Censoring
Consider two hypothetical people and a target cause A. Person X has a B event at time 2 and is censored at time 4. Person Z has no observed event and is censored at time 2.5. Consider membership just before time \(t=5\).
For X, the B event was observed before censoring. The event history is known, so X remains in the underlying A pseudo risk set after time 2, including at time 5. Censoring at 4 does not erase the established fact that B occurred first.
For Z, no event was observed before censoring. At time 5, Z might have had A, B, or no event later; the data do not say which. Z therefore cannot be directly classified as a member of the observed pseudo risk set at 5. A method that treats Z as permanently free of A would invent an event history; a method that treats Z as having had A would invent one as well.
This is why “competing events stay, censored people leave” is a statement about what is known from observed histories. It does not mean censored people are irrelevant to the target population: under independent censoring, their missing later contributions are handled through censoring adjustment rather than by assigning them a definite event status.
Relation to the Cause-Specific Risk Set
For the cause-specific hazard of A, the risk set at \(t-\) contains only people who have not experienced any event before \(t\). A person with B at time 2 leaves that risk set at time 2. For the Fine–Gray subdistribution hazard of A, the same person remains represented after time 2. The methods differ in the quantity their denominator represents, not merely in a coding convention.
| History before \(t\) | Cause-specific risk set for A | Fine–Gray pseudo risk set for A |
|---|---|---|
| No event before \(t\) | Included | Included |
| Cause A before \(t\) | Excluded | Excluded |
| Competing cause before \(t\) | Excluded | Retained |
| Censored before \(t\), no event observed | Not directly included after censoring | Not directly included after censoring; censoring adjustment is needed |
The cause-specific risk set supports a hazard conditional on being free of all event types. The Fine–Gray pseudo risk set supports a subdistribution hazard whose denominator is \(1-F_j(t-)\). Neither is universally preferable: each is tied to a different estimand. The earlier material on cause-specific risk sets develops the former interpretation; the key distinction here is the retained competing-event history.
Assumptions, Diagnostics, and Common Errors
The set derivation itself follows from the definition of \(F_j\). Its application to observed data depends on additional conditions and careful event coding:
- Competing events are mutually exclusive first events. The event type \(J_i\) must identify which first event occurred. If multiple event types can occur sequentially and the target is not defined as a first event, this risk-set construction does not automatically apply.
- Event and censoring times are recorded consistently. The analysis must distinguish a cause-\(j\) event, a competing event, and censoring. A competing event miscoded as censoring loses a known history; censoring miscoded as a competing event falsely establishes one.
- Censoring assumptions support the adjustment. If censoring depends on future event risk after conditioning on the modeled information, an adjustment based on independent censoring may not recover the population risk set. If the probability of remaining uncensored approaches zero, later risk-set contributions cannot be reliably recovered.
- The risk-set construction does not establish proportionality. It explains the denominator for a subdistribution hazard. Whether a proportional subdistribution-hazards model is suitable is a distinct modeling question.
A practical diagnostic is to inspect records around each event time: a previously observed competing event should remain represented in the pseudo history, while a cause-\(j\) event should not. Separately inspect censoring times and the amount of follow-up available at later times. A sudden loss of follow-up or sparse support can make late-time estimates unstable even if the event coding is correct.
Common mistake: “A person with a competing event is no longer at risk of anything, so they must leave every risk set.” The first clause may describe their subsequent event process, but the conclusion is wrong for the Fine–Gray denominator. The subdistribution risk set is defined by absence of the target event, not by absence of all events; a competing event therefore leaves the person represented. The opposite mistake is to retain someone censored before any event as though their later status were known. That person must leave the directly observed risk set, with censoring addressed through an appropriate adjustment.
Check Your Understanding
Answer each question, then compare with the solutions below.
- For target cause A, classify a person with a B event at time 4 for the risk set immediately before time 7.
- In one sentence, derive the probability represented by the pseudo risk set from \(F_A(t-)\).
- A person is censored at time 3 without an event. Is that person directly represented in the observed pseudo risk set immediately before time 6? Why?
- State one difference between the cause-specific risk set and the Fine–Gray pseudo risk set when a competing event occurs before \(t\).
Solutions. (1) Include the person: the observed B event establishes a competing event before 7, so A has not occurred first. (2) \(1-F_A(t-)=P(T\ge t)+P(T<t,J\ne A)\), the probability of no A event before \(t\). (3) No. Follow-up ended before 6 without an observed event, so later cause status is unknown; censoring adjustment, not an assumed event history, is needed. (4) The cause-specific set removes the person after the competing event, whereas the Fine–Gray pseudo risk set retains them because the target event has not occurred.