Why the Risk Set Ends at a Competing Event
A cause-specific Cox model asks about the rate of one event type among people who are still event-free. A competing event changes who belongs to that group: after a person experiences a competing event, that person is no longer able to experience the target event as a first event. Removing that person from the target-cause risk set is therefore part of defining the estimand, not an assertion that the competing event is unrelated to the target event.
This distinction matters because the same data coding can support different questions. In a target-cause Cox fit, a competing event is commonly coded as a non-target endpoint so that the person leaves the risk set at that time. That fit estimates the target cause-specific hazard. It does not, by itself, estimate the target event's cumulative incidence or the risk that would have occurred in a hypothetical world where competing events were eliminated.
For a Cox model, a common specification is \(\lambda_k(t\mid Z)=\lambda_{0k}(t)\exp(\beta^\mathsf{T}Z)\), where \(\lambda_{0k}(t)\) is an unspecified baseline hazard and \(\beta\) is the target-cause coefficient vector. The hazard ratio for a one-unit covariate contrast is conditional on being event-free at time \(t\); it is not a ratio of cumulative risks.
What the Risk Set Represents
Suppose the observed follow-up time is the earlier of the event time and the censoring time. Immediately before time \(t\), the cause-specific risk set consists of people who are still under observation and have not yet experienced any event. A target event at \(t\) is an event in this risk set. A competing event at time \(u\) removes its subject after \(u\), just as a target event does. A person lost to follow-up is also removed, but for a different reason: their event status is no longer observed.
That distinction is substantive. A competing event is observed and tells us the person has left the event-free state. Loss to follow-up is incomplete observation. To use the observed risk set for inference about the target cause-specific hazard, ordinary right-censoring must be independent of the event process, at least conditional on the modeled covariates and relevant observed history. The exact assumption depends on the study design and analysis, but informative loss to follow-up cannot be repaired just by coding the last contact time as censoring.
The cause-specific analysis does not require a competing event to be independent of the target event in some hypothetical latent-time sense. The estimand concerns the target event rate among people who are actually event-free at each time. Dependence between causes, shared risk factors, or a competing event that is more common in one covariate group does not change that definition. It can, however, affect who remains in the risk set and may make the target cause-specific hazards differ over time or between groups.
Worked Example: A Cause-Specific Partial-Likelihood Contribution
Consider hypothetical follow-up with a target event \(k\) and a competing event. At time 4 years, subject A experiences the target event. Immediately before time 4, A has covariate \(x=1\), while B and E have \(x=0\). Subject C, with \(x=1\), experienced a competing event at year 2. Subject D, with \(x=0\), was lost to follow-up at year 3. E remains event-free through year 4 and experiences the target event later, at year 5. Assume all five subjects entered follow-up before year 2.
The risk set just before the target event at year 4 is \(\{A,B,E\}\). C has left the event-free state because of the competing event; D is no longer observed after loss to follow-up. Under the cause-specific Cox model with coefficient \(\beta\), the partial-likelihood contribution for A's event is $$ \frac{\exp(\beta x_A)} {\sum_{j\in R(4)}\exp(\beta x_j)}. $$
For illustration, set \(\beta=\log(2)\), so that \(\exp(\beta)=2\). Substitution gives $$ \frac{\exp(\log(2)\cdot 1)} {\exp(\log(2)\cdot 1)+\exp(\log(2)\cdot 0)+\exp(\log(2)\cdot 0)} =\frac{2}{2+1+1}=0.500. $$
This is the conditional contribution for which A, rather than another member of the year-4 risk set, has the target event. C is not in the denominator because C could no longer have a first target event after year 2. D is not in it because D's outcome is unobserved after year 3. The reasons for exit differ, even though both are absent from this risk set.
Why This Gives Inference for the Target Hazard
The event-count formulation makes the risk-set logic explicit. Let \(Y_i(t)\) indicate that subject \(i\) is observed and event-free immediately before \(t\), and let \(N_{ik}(t)\) count subject \(i\)'s target-cause events by \(t\). Given the history just before \(t\), the probability of a target event in the next short interval is approximately the cause-specific hazard times the interval length for a person with \(Y_i(t)=1\). Thus, to first order in \(dt\),
If a competing event occurs, \(Y_i(t)\) changes from 1 to 0 after that event. The subject contributes exposure to the target hazard only while event-free; there is no target-event exposure afterward. This is why target events are compared with the covariate profiles of people in the actual event-free risk set. Under a Cox proportional-hazards model and suitable independent-censoring conditions, the resulting partial likelihood provides inference for \(\beta\) in the target cause-specific hazard. The partial likelihood eliminates the unspecified baseline hazard by conditioning on the risk set at each target-event time.
The crucial point is that this argument uses the observed process: which people remain event-free and at risk at each time. It does not posit that each person has an unobserved target-event time that would still occur after a competing event. In particular, no independence assumption between such latent cause-specific times is needed to define or estimate the cause-specific hazard.
Worked Example: A Cause-Specific Rate With Competing Exits
Suppose a hypothetical cohort contributes 360 person-years while event-free and under observation. During that time, 18 target events and 27 competing events occur. For a simple illustration, assume the target cause-specific hazard is constant over the period and that censoring is noninformative. Under a constant hazard, the target-event count has exposure equal to the event-free person-time, so the rate estimate is the target-event count divided by that exposure: $$ \widehat{\lambda}_k=\frac{18\ \text{target events}}{360\ \text{person-years}} =0.0500\ \text{per person-year}. $$
The 27 competing events are not added to the target-event numerator. Each subject contributes person-time up to their competing event, then leaves the target-cause risk set. The estimate describes the target event rate among people who remain event-free; it is not the probability of a target event within one year. Under a changing hazard, the rate need not be constant, and a Cox model can represent covariate effects without imposing a constant baseline rate.
Why “Censored” Does Not Mean “Ordinary Loss to Follow-Up”
Software may accept a target-event indicator that labels both competing events and loss to follow-up as “not the target event.” That is a computational convention for forming a target-specific event process. It should not erase the different meanings of the two exits. Competing events are observed transitions out of the event-free state; ordinary censoring is an end to observation while the person may still be event-free.
For estimating a cause-specific hazard, both exits end observed membership in the risk set. For estimating absolute risk, however, the competing events must also affect the probability of remaining event-free. A cause-specific Cox model by itself does not turn a competing event into a harmless censoring event and then yield the actual target-event probability.
Worked Example: Cause-Specific Hazard Versus Cumulative Incidence
Consider a hypothetical cohort of 10 people with complete observation at three event times. At time 1, two have the target event, leaving eight event-free. At time 2, four of those eight have a competing event, leaving four event-free. At time 3, two of the remaining four have the target event.
The observed target cumulative incidence by time 3 is the proportion of the original cohort that has had the target event: \(2/10+2/10=4/10=0.400\). The second increment is \(2/10\), not \(2/4\), because the four people at risk at time 3 are only 40% of the original cohort.
If the four competing events at time 2 are instead treated as ordinary censoring in a Kaplan–Meier calculation for the target event, the target-event-free estimate is \((1-2/10)(1-2/4)=0.8(0.5)=0.400\). Its complement is \(0.600\). This is a net-risk-style quantity under a hypothetical elimination of competing events, not the observed target cumulative incidence of 0.400. The gap arises because the competing events remove 40% of the original cohort before time 3. The cause-specific hazard remains a valid rate estimand, but its Kaplan–Meier complement is not the actual target-event probability when competing risks are present.
Assumptions, Diagnostics, and Alternatives
A cause-specific Cox analysis relies on several conditions. Subjects should be independent, the target-cause hazard model should be adequately specified, and the proportional-hazards assumption should be reasonable for the covariates whose effects are summarized by constant hazard ratios. Censoring should be independent of the event process conditional on the variables and history used in the analysis. There must also be adequate at-risk support: if nearly everyone with a covariate profile has already experienced a competing event or been censored, its target hazard is weakly informed by later follow-up.
Practical diagnostics should therefore include checking risk-set sizes and target-event counts over time, especially within important covariate groups. A sharp decline in the risk set can make late estimates unstable. Compare censoring patterns across covariate groups and consider whether loss to follow-up could depend on unmeasured prognosis. Assess whether a constant hazard-ratio summary is plausible; changing covariate effects or sparse target events can undermine a simple Cox fit. A well-behaved numerical fit does not establish independent censoring or proportional hazards.
The main alternatives answer different questions. Cumulative-incidence methods target the probability of experiencing a particular cause by a time horizon while accounting for competing events. Fine-Gray regression models a subdistribution hazard associated with that cumulative-incidence question. By contrast, cause-specific Cox regression models the instantaneous target-event rate among those still event-free. The choice is driven by the scientific estimand, not by which model is easier to fit.
| Approach | Risk-set interpretation | Primary quantity |
|---|---|---|
| Cause-specific Cox model | People still observed and event-free | Target cause-specific hazard and its covariate associations |
| Cumulative-incidence analysis | Accounts for event-free survival and competing events | Observed probability of the target event by a time |
| Fine-Gray regression | Uses a subdistribution-hazard construction for the target cause | Association with the target cumulative incidence |
| Kaplan–Meier complement with competing events censored | Treats competing events as if removed by censoring | A net-risk-style quantity, not generally observed cumulative incidence |
Common Mistake
Wrong reasoning: “The competing event is coded as censored, so the analysis assumes it is independent of the target event just like loss to follow-up.” That confuses a risk-set rule with an assumption about missing observation. A competing event is known to have occurred and makes a first target event impossible afterward; it must end membership in the event-free risk set. Independent censoring is a separate condition needed for unobserved follow-up to support inference. The cause-specific hazard is conditional on actually remaining event-free, and does not require independence of hypothetical latent event times. Nor does a cause-specific Cox fit, without additional calculations, estimate the target cumulative incidence.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- What population is conditioned on in the definition of a cause-specific hazard?
- A subject experiences a competing event at year 3. Should that subject be in the target-cause risk set just before year 4? Why?
- In a target-specific Cox fit, does valid cause-specific hazard inference require independence between latent times to different causes?
- In the worked cohort of 10 people, why is the observed target cumulative incidence by time 3 equal to 0.400 rather than 0.600?
- Name one condition concerning loss to follow-up and one diagnostic related to risk-set support.
Solutions. (1) It is conditional on having had no event of any type before the time being considered, as well as on the stated covariates. (2) No. The competing event ended the subject's event-free status at year 3, so the subject cannot experience a first target event after that time. (3) No. The target cause-specific hazard is defined from the observed event-free process; independent censoring is needed for loss to follow-up, but latent-time independence across causes is not required. (4) Four of the original 10 people experienced the target event, so the observed cumulative incidence is \(4/10=0.400\). The value 0.600 is the complement of the target-only Kaplan–Meier estimate when competing events are treated as censoring, and is not the observed cumulative incidence. (5) Censoring should be independent of the event process conditional on modeled variables and relevant history; inspect risk-set sizes and target-event counts over time, particularly within covariate groups.