Why the Risk Set Changes the Meaning of a Hazard
In competing-risks data, a person can experience one of several mutually exclusive event types, and the first event prevents observation of the others. A biomarker may be associated with the rate of one particular event among people who are still event-free. Alternatively, a question may concern how that biomarker relates to the probability of that event occurring by a given time, in the presence of competing events. These are different targets. The distinction between cause-specific and subdistribution hazards makes the target explicit.
The earlier material on cause-specific hazards establishes the event-free risk set: at time \(t\), it consists of people who have experienced no event before \(t\). The subdistribution hazard uses a different, modified risk set. Understanding this difference is essential in joint models, because the event-process likelihood is naturally written in terms of event hazards among those still event-free, whereas cumulative incidence also depends on what happens under every competing cause.
For continuous event times, write \(f_k(t)\) for the density of an event of cause \(k\) at time \(t\), so \(F_k(t)=\int_0^t f_k(u)\,du\). The subdistribution hazard is \(\alpha_k(t)=f_k(t)/\{1-F_k(t)\}\). Its denominator is the probability of not yet having experienced the event of interest by \(t\), not the probability of remaining free of all event types. In the associated modified risk set, people who have already experienced a competing event are retained; people who have already experienced cause \(k\) are not. In practice, censoring before \(t\) must also be handled, often with appropriate weighting.
Deriving the Relationship
The two hazards use different denominators, but they describe the same underlying cause-specific event density and cumulative incidence. Let \(S(t)=P(T\geq t)\) be the probability of remaining event-free through time \(t\). By the definition of the cause-specific hazard, the density of a cause-\(k\) event is \(f_k(t)=S(t)\lambda_k(t)\). Therefore,
This gives a direct explanation of why a cause-specific hazard alone does not determine the cumulative incidence: \(F_k(t)\) also depends on \(S(t)\), which is affected by hazards for all event causes. Conversely, \(\alpha_k(t)\) is defined using the density and cumulative incidence for cause \(k\), and does not use the ordinary event-free risk set as its denominator.
The relationship can also be seen from the modified risk set. Its probability of not yet having had cause \(k\) is \(1-F_k(t)\), since it includes people who remain event-free and people whose first event was a competing cause. Over a short interval, the probability of a cause-\(k\) event is approximately \(f_k(t)\Delta t\). Dividing by the modified risk-set probability and then by \(\Delta t\) gives \(\alpha_k(t)=f_k(t)/\{1-F_k(t)\}\). This is a definition-based derivation; it does not make the modified risk set equivalent to the ordinary set of people still biologically capable of experiencing any event.
Worked Example: Tracking the Two Risk Sets
Consider four hypothetical participants followed for a time-to-first-event outcome with causes 1 and 2. Participant A has a cause-1 event at time 2; B has a cause-2 event at time 1; C is event-free through time 2; and D is censored at time 1.5. Compare the risk sets just before time 2, setting aside any censoring weights.
Cause-specific risk set for cause 1. Just before time 2, A has not yet had an event, and C remains event-free. B is excluded because the competing event already occurred, and D is no longer observed. Thus A and C are in the ordinary event-free risk set; A contributes the cause-1 event at time 2.
Modified risk set for cause 1. B is retained despite having had cause 2, because B has not had cause 1. A and C are also in the modified set just before time 2, while D is still censored and cannot be included without adjustment. The modified set is designed for the cause-1 cumulative incidence, not to represent people still at risk of every event.
The cause-2 event removes B from the cause-specific risk set immediately, but not from the cause-1 subdistribution risk set. Treating these sets as interchangeable changes the meaning of the fitted hazard.
Worked Example: Calculating Cause-Specific and Subdistribution Hazards
Suppose a hypothetical population has constant cause-specific hazards \(\lambda_1(t)=0.10\) per year and \(\lambda_2(t)=0.05\) per year, with no other event causes. Assume continuous event times and no censoring for this calculation. The event-free survival probability is \(S(t)=\exp\{-0.15t\}\), because the total hazard is \(0.10+0.05=0.15\) per year.
The cause-1 event density is \(f_1(t)=S(t)\lambda_1(t)=0.10\exp(-0.15t)\). At \(t=2\) years, \(S(2)=\exp(-0.30)=0.7408182\), so \(f_1(2)=0.10(0.7408182)=0.0740818\) per year.
Integrating that density gives the cause-1 cumulative incidence by 2 years:
The cause-specific hazard at 2 years is \(0.10\) per year, by assumption. The subdistribution hazard is instead
Thus the cause-specific hazard is \(0.10\) per year, while the subdistribution hazard is approximately \(0.08956\) per year at 2 years. The first is an event rate among those still event-free. The second is based on the modified risk set and the cause-1 cumulative incidence. Neither number is the probability of a cause-1 event by 2 years; that probability is \(F_1(2)\approx0.1728\).
Why Cause-Specific Hazards Arise Naturally in a Joint Likelihood
A shared-random-effects joint model links longitudinal measurements to event processes through latent subject-specific effects. Let \(b_i\) denote subject \(i\)'s random effects, and suppose the cause-specific hazards conditional on those effects are \(\lambda_{ik}(t\mid b_i)\). Conditional on \(b_i\), the probability of remaining free of all event types to time \(t\) is
If the observed event is of cause \(k\) at time \(t\), its conditional density is \(S_i(t\mid b_i)\lambda_{ik}(t\mid b_i)\). If the subject is censored at time \(c\) without an event, the event-process contribution is \(S_i(c\mid b_i)\), assuming the censoring mechanism is handled as noninformative conditional on the model covariates and random effects. These forms follow from the competing event process: survival requires avoiding every cause, and an observed event contributes the hazard for its recorded cause.
The longitudinal measurements contribute their conditional model given \(b_i\). The shared-random-effects construction combines that longitudinal contribution with the conditional event contribution, then integrates over the distribution of \(b_i\). This structure makes cause-specific hazards the natural components of the event-process likelihood. A subdistribution hazard can also be modeled, but it is not the ordinary event density factor \(S_i(t\mid b_i)\lambda_{ik}(t\mid b_i)\); it is a transformation involving the cause-specific density and cumulative incidence.
Worked Example: A Conditional Event Contribution With Shared Effects
For illustration, suppose a hypothetical subject has a scalar random effect \(b_i=0.6\), and the two conditional cause-specific hazards are constant in time: \(\lambda_{i1}(t\mid b_i)=0.08\exp(0.5b_i)\) per year and \(\lambda_{i2}(t\mid b_i)=0.04\exp(-0.2b_i)\) per year. The calculation is conditional on this value of the random effect; it is not yet integrated over the population distribution of \(b_i\).
Substituting \(b_i=0.6\), the cause-1 hazard is \(0.08\exp(0.30)\approx0.1080\) per year, and the cause-2 hazard is \(0.04\exp(-0.12)\approx0.03548\) per year. Using the unrounded hazard values, their sum is approximately \(0.1434655\) per year. The conditional event-free survival at 2 years is therefore \(S_i(2\mid b_i)=\exp\{-2(0.1434655)\}\approx0.75056\).
If this subject has a cause-1 event at time 2, the conditional event contribution is
This product uses the cause-1 hazard and survival from every cause. The shared random effect connects the marker trajectory to both conditional hazards in this example. A population-level likelihood would integrate the longitudinal and event contributions over the distribution of \(b_i\); substituting one fixed \(b_i\) value illustrates the conditional factor, not that integration.
Choosing and Interpreting a Formulation
| Feature | Cause-specific hazard | Subdistribution hazard |
|---|---|---|
| Denominator or risk set | People still free of every event | People without the event of interest, retaining those with competing events |
| Direct interpretation | Instantaneous rate of a particular cause among event-free people | Rate defined through the cause-specific density and cumulative incidence |
| Connection to cumulative incidence | Requires combining the cause-specific hazards for all causes to obtain event-free survival | Directly uses \(F_k(t)\) through \(1-F_k(t)\) |
| Role in a standard shared-effects event likelihood | Provides the cause-specific event density and all-cause event-free survival | Can be modeled, but is not the ordinary cause-specific event density factor |
A cause-specific hazard ratio compares event rates among individuals who are currently event-free, conditional on the model specification. A subdistribution hazard ratio compares the modeled subdistribution hazards; it is not a risk ratio, and it is not automatically a causal effect. In particular, neither ratio alone is a probability contrast. The cumulative incidence is a probability of experiencing a particular cause by a specified time, and it depends on the competing causes as well as on the cause of interest.
In a joint model, the chosen formulation should match the scientific question and the probability model. If the goal is to describe cause-specific event rates conditional on the longitudinal process and shared effects, cause-specific hazards fit the event-process structure directly. If the target is the relationship with the cumulative incidence of a particular event, subdistribution hazards provide a formulation tied to that incidence. Modeling one does not make the other irrelevant: cause-specific hazards for all causes imply cumulative incidences, while a subdistribution model focuses on one cause's incidence-related quantity.
Diagnostics, Limitations, and a Common Pitfall
Check first that event types are mutually exclusive first events in the analysis, that event times and censoring are recorded consistently, and that each fitted model matches its stated risk set. For a cause-specific model, inspect whether the conditional hazard form and covariate effects are plausible over time, and whether the event-free risk set has adequate support. For a subdistribution model, verify that the fitted cumulative incidence is coherent, lies between 0 and 1, and is nondecreasing. In either approach, sparse causes, heavy censoring, and few events can make estimates unstable. In joint models, weak information about the random-effects distribution or association parameters can add further uncertainty.
A subdistribution risk set retains people with competing events by construction; this does not mean those people remain physically capable of having the event of interest as their first event. Nor does a cause-specific analysis require pretending that competing events did not occur: they remove people from the event-free risk set because a later first event is no longer possible. The two formulations answer different questions, so a difference between their coefficients is not itself evidence that one model is wrong.
- Common mistake: “A cause-2 event is a competing event, so it should be treated as ordinary independent censoring in every analysis.” Correction: For a cause-specific hazard, a cause-2 event ends membership in the event-free risk set. For a cause-1 subdistribution hazard, the modified risk set retains that person. These choices reflect different estimands.
- Common mistake: “A subdistribution hazard ratio is the fold-change in the probability of the event.” Correction: It is a ratio of subdistribution hazards. The event probability by time \(t\) is a cumulative incidence and is not generally obtained by interpreting that ratio as a risk ratio.
The main limitation is interpretational and structural: a hazard ratio is tied to its risk set and model, and a subdistribution hazard does not describe an event rate among only those still free of all events. In a shared-random-effects model, be explicit about whether hazards and associations are conditional on the latent effects; averaging over those effects is a separate step and can change the interpretation.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- At time \(t\), who is included in the cause-specific risk set for cause \(k\), and who is additionally retained in the modified subdistribution risk set?
- In the numerical example, what is the cause-1 cumulative incidence by 2 years, and what are the cause-specific and subdistribution hazards at that time?
- Why does a cause-\(k\) event density in a shared-random-effects competing-risks likelihood involve survival from all event causes?
- Does a subdistribution hazard ratio directly equal a ratio of cumulative-incidence probabilities? Explain.
Solutions. (1) The cause-specific set contains people still free of every event; the modified set also retains people whose first event was a competing cause, while excluding those with cause \(k\). (2) \(F_1(2)=0.1727879\), the cause-specific hazard is \(0.10\) per year, and the subdistribution hazard is approximately \(0.08956\) per year. (3) Remaining event-free requires avoiding every cause, so \(S_i(t\mid b_i)\) contains the sum of all conditional cause-specific hazards; the observed cause then contributes its own hazard. (4) No. It is a ratio of subdistribution hazards, not generally a ratio of cumulative-incidence probabilities.