Who Is In The Cause-Specific Risk Set

Why the Risk Set Needs a Precise Rule

A cause-specific hazard concerns first events of a particular cause. Its risk set at time $t$ must therefore represent subjects who could still experience a first event at that time, given what has happened up to just before $t$. In competing-risks data, a subject who has already had a competing event is not merely missing an outcome: that first event has ended the subject’s eligibility to experience any other cause as the first event.

The distinction matters in practice. Counting every enrolled subject in the denominator at every time would include people who had an earlier event or whose observation had already ended. The resulting set would not describe those still under observation and event-free just before $t$.

Definition: Let $T_i$ be subject $i$’s time to the first event, $J_i$ its cause, $C_i$ the censoring time, and $L_i$ the entry time into observation. The observed follow-up time is $X_i=\min(T_i,C_i)$. Subject $i$ is in the cause-specific risk set immediately before time $t$ if the subject has entered observation and is observed through that time: $$ Y_i(t)=I(L_i<t,\ X_i\ge t), \qquad R(t)=\{i:Y_i(t)=1\}. $$ The convention here includes a subject whose first event occurs at $t$: the subject was event-free immediately before that event. Tied censoring and event times require a stated convention; here, an event recorded at $t$ is treated as observed at $t$.

With no delayed entry, everyone enters at time zero, so the entry condition is automatic for $t>0$. For a subject whose event occurs before $t$, $X_i<t$, and the subject is out of the risk set. This applies whether the earlier event was the target cause or a competing cause. A subject censored before $t$ is also out: their event status after censoring is unknown, so they cannot be counted as under observation at $t$.

The risk set is defined by status just before $t$, not by whether a subject will eventually experience the target cause. Someone who has not yet had any event remains eligible even if a competing event later occurs. Conversely, once any first event occurs, the subject cannot remain eligible for a different cause as a first event.

Why Prior Competing Events Exclude a Subject

For each cause, the event of interest is a first event. If subject $i$ has a competing event at time $u<t$, then $T_i=u$ and $J_i$ records that competing cause. The first-event time has already occurred. There is no later first event of the target cause for this subject, so the subject cannot contribute a target-cause event at $t$.

This is a consequence of how the outcome is defined, not an assumption that a competing event makes a later event biologically impossible. A person might experience a later event of another type in some broader clinical history, but it would not be the first event in the competing-risks outcome being analyzed. The risk set follows that outcome definition.

T_i<t \ \Longrightarrow\ X_i<t \ \Longrightarrow\ Y_i(t)=0.

The same risk set applies when considering each cause-specific hazard: immediately before the first event, the cause has not yet been determined by an observed event. After a first event of any cause, the subject leaves the event-free state. Thus, the set is not formed by keeping only subjects who will eventually have the target cause.

Worked Example: Membership at a Target-Cause Event Time

Consider eight hypothetical subjects followed in years. The target cause is cause A; cause B is a competing cause. Follow-up is complete through each listed event or censoring time.

Subject	First event or censoring time	Status
A	3	Cause A
B	4	Cause B
C	6	Cause A
D	8	Censored
E	2	Censored
F	5	Cause A
G	7	Cause B
H	9	Censored

Find the risk set immediately before $t=5$. Subjects A and B had first events before 5, so both are excluded; B is excluded specifically because the competing event has already ended the first-event follow-up. Subject E was censored at 2, before 5, so E is also excluded. Subjects C, D, F, G, and H had no event or censoring before 5 and are in the set. In particular, F is included because the cause A event occurs at 5, not before 5.

Therefore, $$ R(5)=\{C,D,F,G,H\}, \qquad |R(5)|=5. $$ The count is five subjects at risk immediately before the event time. This calculation describes membership; it does not by itself estimate an event probability.

Deriving the Membership Rule from First-Event Data

The risk-set rule can be justified directly from the event definition. Write $N_{ik}(t)$ for the indicator that subject $i$ has had a first event of cause $k$ by time $t$. A cause-$k$ event at time $t$ can occur only if the subject has not had any first event before $t$, has entered observation, and is still observed at $t$. A subject with a prior competing event fails the first of these conditions.

Check entry.
The subject must have entered observation before $t$, so $L_i<t$.

Check prior event history.
If a first event occurred before $t$, then $T_i<t$, regardless of its cause, and the subject is not event-free immediately before $t$.

Check observation.
If censoring occurred before $t$, the subject’s status at $t$ is unknown and the subject is not observed in the risk set.

Include those meeting all checks.
Equivalently, retain subjects with $L_i<t$ and $X_i\ge t$. An event at $t$ is included because the subject was event-free immediately beforehand.

This rule is about membership, not about whether censoring is statistically independent of event times. Membership can be determined from the recorded data either way. However, analyses that use the observed risk set to estimate population quantities generally need assumptions about censoring; the particular assumptions depend on the analysis and are not part of the definition of $R(t)$.

Worked Example: A Competing Event Before the Time of Interest

Suppose five hypothetical subjects have the following observed histories, with times in months: P has cause B at month 2; Q has no event and is censored at month 10; R has cause A at month 6; S has cause B at month 8; and U has no event and is censored at month 4. Determine the cause-A risk set immediately before month 5.

P is excluded because the competing cause B occurred at month 2, before the time of interest. U is excluded because censoring occurred at month 4, before month 5. Q and R remain observed and event-free through just before month 5; R’s later event at month 6 does not remove R earlier. S also remains event-free and observed at month 5, since the cause B event occurs at month 8.

Thus, $$ R(5)=\{Q,R,S\}, \qquad |R(5)|=5-1-1=3. $$ The two subtractions are for P’s earlier competing event and U’s earlier censoring. Q, R, and S are counted because each is under observation and has had no first event before month 5.

The fact that S later has a competing event does not affect S’s earlier membership. Risk-set membership at $t$ uses information about follow-up through $t$, not future outcomes.

Entry, Censoring, and Tied Times

Delayed entry occurs when a subject becomes observable only after the time origin. Such a subject cannot belong to a risk set before entering. For example, if a subject enters at month 3, that subject is not included at month 3 under the $L_i<t$ convention, but is eligible just after entry if still event-free and observed. This prevents counting time during which the subject was not under observation.

At an event time $t$, a subject whose event occurs exactly at $t$ belongs to the risk set immediately before $t$. A subject whose event occurred at $t-\epsilon$, for any positive $\epsilon$, does not. Censoring also removes a subject after follow-up ends. If an event and censoring time are recorded as tied, software and analyses may use different conventions; the data definition and analysis must specify which status takes priority. Do not silently switch conventions within an analysis.

Worked Example: Delayed Entry and Censoring

Four hypothetical subjects are observed on a time scale measured in days. A enters at day 0 and has no event before censoring at day 9. B enters at day 3 and has cause B at day 7. C enters at day 1 and has cause A at day 5. D enters at day 0 and is censored at day 4. Determine the risk set immediately before day 5.

A is included: A entered before day 5 and remains observed beyond it. B is included: B entered at day 3, before day 5, and the cause B event is later, at day 7. C is included because the cause A event is at day 5 and so occurs after the instant just before day 5. D is excluded because censoring at day 4 occurred before day 5. Therefore, $$ R(5)=\{A,B,C\}, \qquad |R(5)|=4-1=3. $$ The delayed-entry check matters for B: B would not have belonged to a risk set before entering at day 3. The event-time convention matters for C: C is included immediately before the event at day 5.

Common Errors, Diagnostics, and Alternatives

When checking a risk-set calculation, sort each subject’s entry time, first-event time, and censoring time on the same time scale. Then ask whether the subject was under observation and event-free immediately before the time being considered. This simple timeline check catches most membership errors.

Incorrect approach	Why it fails	Correction
Keep a subject with a prior competing event in the cause-A risk set.	The subject’s first event has already occurred, so a later cause-A event cannot be a first event.	Remove the subject from every first-event cause-specific risk set after any first event.
Count everyone originally enrolled at every time.	Some subjects have had an event, been censored, or not yet entered.	Recalculate membership at each time from entry and observed follow-up.
Exclude a subject whose target event occurs exactly at $t$.	The risk set is defined just before $t$; that subject is event-free up to that instant.	Include the subject at $t$ under the stated event-time convention.
Exclude a subject because of a competing event that occurs after $t$.	This uses future information to define earlier membership.	Use only event and censoring history through $t$.

A different denominator may be appropriate for a different estimand or model, but it must not be substituted without checking its membership rule. In particular, the cause-specific risk set is the set still event-free and observed immediately before $t$; it is not simply the original cohort size or the number who will eventually experience the target cause. The cause-specific risk set is useful precisely because it matches the first-event question. Its limitations follow from observation: censoring can hide later outcomes, and risk-set membership alone does not correct for informative loss to follow-up.

Key takeaway: At time $t$, the cause-specific risk set contains subjects who entered observation and remain observed without any first event just before $t$. A prior competing event removes a subject because it is already the subject’s first event, not because the subject is merely unlikely to have the target cause.

Check Your Understanding

For each question, give a risk-set conclusion and the reason for it.

A subject has a competing event at time 4. Is the subject in the cause-A risk set immediately before time 6?
A subject has a cause-A event at time 6. Is the subject in the risk set immediately before time 6 under the convention used here?
A subject enters at time 2 and is event-free until censoring at time 5. Is the subject in the risk set immediately before time 5? What about immediately before time 6?
At time 7, a subject is observed and event-free; the subject has a competing event at time 9. Does the future event change membership at time 7?

Solutions. (1) No. The competing event at 4 is a prior first event, so the subject left the event-free state before time 6. (2) Yes. The event at 6 occurs after the instant immediately before 6, so the subject is included at that time. (3) Yes, immediately before 5, because censoring is at 5 and the subject remains observed up to that time; no, immediately before 6, because censoring at 5 ended observation before then. (4) No. The subject is in the risk set at 7 because membership uses history through 7; the event at 9 does not alter earlier membership.