Choosing A Hazard Interpretation For A Clinical Question

Start With the Clinical Question

A hazard is an instantaneous rate, not a probability over a fixed period. But in competing-risks settings, the phrase “the hazard of event $k$” can refer to two different rates, built from different risk sets. Choosing between them is not a matter of which definition is generally better. It depends on the clinical question: are we asking about the rate of event $k$ among people who are still event-free, or about a rate tied to the cumulative incidence of event $k$ in the original population?

For example, a clinician might ask how quickly a complication occurs among patients who are alive and have not yet had any first event. A health-services researcher might instead ask how a treatment is associated with the population probability of hospitalization before death. The first question points to a cause-specific hazard; the second points to a subdistribution hazard. Neither answer alone gives the probability of an event by a stated time.

Definition: Let $T$ be time to the first event and $J$ its cause, with causes $1,\ldots,K$. For a focal cause $k$, the cause-specific hazard at time $t$ is the instantaneous rate of cause $k$ among people who have not yet experienced any event: $$ \lambda_k(t)=\lim_{\Delta t\to 0^+} \frac{P(t\leq T<t+\Delta t,\ J=k\mid T\geq t)}{\Delta t}. $$ The subdistribution hazard for cause $k$ is the instantaneous rate defined among people who have not yet experienced cause $k$, including in its constructed risk set those who previously experienced a competing cause. In continuous time it is $$ \lambda_k^*(t)=\lim_{\Delta t\to 0^+} \frac{P(t\leq T<t+\Delta t,\ J=k\mid \text{no cause-}k\text{ event before }t)}{\Delta t} =\frac{f_k(t)}{1-F_k(t)}, $$ where $F_k(t)=P(T\leq t,J=k)$ is the cumulative incidence for cause $k$, and $f_k(t)$ is its density where that density exists.

The subdistribution risk set is a mathematical construction. A person who has already experienced a competing event cannot subsequently have the first event of cause $k$, but is retained in this risk set so that the hazard is linked directly to $F_k(t)$. By contrast, a person who has had any first event leaves the cause-specific risk set. These definitions therefore answer different questions, even when both are called hazards for the same cause.

Assumptions and a Short Derivation

The formulas above describe a first-event outcome with a clearly defined time origin and mutually exclusive cause labels: each observed first event has one cause. The density form for the subdistribution hazard assumes a continuous-time formulation with a density for cause $k$; with discrete event times or substantial ties, the corresponding increments and conventions must be stated instead. Censoring is not a competing cause. Any estimation from censored data also requires an appropriate independent-censoring assumption, possibly conditional on measured covariates. The definitions themselves do not make censoring independent.

The relationship between the subdistribution hazard and cumulative incidence follows directly from a short-interval conditional probability. Among people who have not yet had cause $k$, the probability of cause $k$ in a short interval is the increase in $F_k$ divided by the proportion who have not yet had cause $k$. Dividing by the interval length and taking the limit gives:

\lambda_k^*(t) =\lim_{\Delta t\to 0^+} \frac{F_k(t+\Delta t)-F_k(t)} {\Delta t\{1-F_k(t)\}} =\frac{f_k(t)}{1-F_k(t)}.

This derivation clarifies the interpretation: the denominator is $1-F_k(t)$, not the proportion still free of every event. For the cause-specific hazard, the denominator instead corresponds to remaining event-free. In continuous time, if $S(t)=P(T\geq t)$, then $f_k(t)=S(t)\lambda_k(t)$. Thus the cause-specific hazard describes an event rate within the event-free group, while the subdistribution hazard incorporates both that group and the population’s history of competing events through $F_k(t)$.

Match the Estimand to the Question

Use the cause-specific hazard when the target is the current event rate among people still at risk of any first event. This is a natural choice for questions about the rate at which a cause occurs in the event-free state, or how that rate differs between groups. A person with a competing first event is no longer at risk of another first event, so that person does not remain in this risk set.

Use the subdistribution hazard when the target concerns the cumulative incidence of a particular cause and the question is framed in terms of a rate associated with that cumulative-incidence process. The retained competing-event history matters to this definition. A subdistribution hazard ratio can summarize an association with the cumulative incidence under a suitable model, but it is not itself a risk ratio or a difference in event probabilities.

Worked Example: An Instantaneous Rate Among Event-Free Patients

Hypothetical scenario. A study asks whether a new treatment is associated with the instantaneous rate of a first kidney complication among patients who have not yet had kidney failure or died. The question explicitly concerns people who remain event-free.

Suppose, for illustration, that during a short interval there are 400 person-years at risk in each treatment group. There are 12 kidney complications in the treated group and 8 in the comparison group; no participant contributes time after a first event. Treating the interval event rate as a rough rate estimate gives:

\widehat{\lambda}_{k,\mathrm{treated}} =\frac{12\text{ complications}}{400\text{ person-years}} =0.0300\text{ per person-year},

\widehat{\lambda}_{k,\mathrm{comparison}} =\frac{8\text{ complications}}{400\text{ person-years}} =0.0200\text{ per person-year}, \qquad \frac{0.0300}{0.0200}=1.50.

The matching interpretation is that the estimated rate in the treated group is 1.50 times the comparison-group rate over this interval among patients still event-free. These simple rates are not, by themselves, an adjusted hazard model or evidence of a treatment effect. The chosen interpretation follows from the question’s event-free risk set; deaths and other first events remove patients from that set.

Worked Example: A Rate Linked to Cumulative Incidence

Hypothetical scenario. Investigators ask how a program is associated with the cumulative incidence of hospitalization before death by one year. At a particular time, suppose $F_k(t)=0.12$, and the increase in hospitalization cumulative incidence over the next quarter-year is 0.015. The corresponding short-interval subdistribution hazard approximation is:

\widehat{\lambda}_k^*(t) \approx \frac{0.015}{(1-0.12)(0.25\text{ years})} = \frac{0.015}{0.88\times 0.25} =0.06818\text{ per year}.

The denominator $0.88$ is the proportion that has not yet had hospitalization by time $t$. It includes people who have already died without a prior hospitalization. The result is a subdistribution-rate approximation tied to the hospitalization cumulative incidence, not a probability of hospitalization during the quarter-year. The clinical wording about the cumulative-incidence process supports this interpretation; a question about the rate among patients still alive and event-free would instead call for a cause-specific hazard.

Why the Two Rates Can Differ

A competing event changes who remains event-free, and it also affects the cumulative incidence of the focal cause. This can make the numerical values of the two hazards quite different. The following calculation uses hypothetical population proportions at a time $t$, followed by a short interval of 0.1 year. Suppose 60% remain free of every first event, 10% have already had the focal cause, and 30% have already had a competing event. Suppose a further 0.006 of the original population experiences the focal cause during the interval.

Worked Example: The Same Event Increment, Two Denominators

The cause-specific rate approximation conditions on the 0.60 who are still event-free. The 0.006 population increment is therefore a conditional increment of $0.006/0.60=0.0100$ among that group:

\widehat{\lambda}_k(t) \approx \frac{0.006/0.60}{0.1\text{ years}} =0.100\text{ per year}.

For the subdistribution rate, the proportion without the focal cause is $1-0.10=0.90$. This includes the 0.30 who have had the competing event. The calculation is:

\widehat{\lambda}_k^*(t) \approx \frac{0.006/0.90}{0.1\text{ years}} =0.06667\text{ per year}.

The same population increment produces different rates because the denominators encode different questions. The first rate is among those still event-free; the second is among those not yet having the focal event, including people with a prior competing event. Neither calculation says that an individual’s probability of the focal event in the interval is 10.0% or 6.67%; those are rates per year under a short-interval approximation.

This distinction also matters when comparing groups. A group can have a different cause-specific hazard because its event rate among currently event-free patients differs. Its subdistribution hazard can differ because the focal cause’s cumulative incidence differs, which can reflect both focal-event rates and the occurrence of competing events. Therefore, two groups’ cause-specific hazard ratio and subdistribution hazard ratio need not agree, and neither should be translated automatically into the other.

Practical Checks, Alternatives, and Limits

Before selecting a hazard, state the event, the time origin, the time scale, and the population represented by the risk set. Then check that the proposed denominator matches the question. If the question says “among patients still alive and free of any event,” a subdistribution hazard is not the matching definition. If it asks about a cause’s cumulative-incidence pattern while accounting for competing events, a cause-specific hazard alone does not directly describe that population probability.

Model assumptions matter once a hazard ratio is used as a summary. For a proportional-hazards interpretation, the ratio of the modeled hazards is assumed constant over time, conditional on any covariates in the model. If that assumption is implausible, one constant ratio may conceal time-varying associations. The hazard definition can still be meaningful, but a single proportional-hazards coefficient may not summarize it adequately. Censoring assumptions must also be appropriate for the analysis; a competing event is not ordinary censoring simply because it prevents observation of a later focal first event.

Neither hazard is automatically causal. A group comparison can reflect differences in prognosis, treatment selection, or other factors, and a hazard ratio is not an individual-level probability statement. The choice of hazard also does not replace reporting event probabilities at clinically relevant times when those probabilities are the decision target. Cause-specific hazards are often suited to questions about rates in the event-free state; cumulative-incidence-based summaries are suited to questions about how often a particular first event occurs in the presence of competing outcomes. These are complementary estimands, not interchangeable methods for the same quantity.

Diagnostic question: Would a person with a competing first event still belong in the denominator described by the clinical question? If yes, a subdistribution interpretation may fit; if no, the cause-specific interpretation may fit.
Time-scale check: A rate per day and a rate per year differ by units, even with the same underlying process. State the time unit and use a consistent time origin.
Failure mode: A fitted hazard ratio is reported as “the treatment reduces the one-year risk by 20%.” A hazard ratio is not a risk reduction; calculate or report the relevant cumulative-incidence contrast for that claim.
Failure mode: Deaths are treated as if they were independent loss to follow-up when the target is a first hospitalization before death. That changes the question and can misrepresent the event probability of interest.

Key takeaway: Choose the hazard by naming its risk set. Cause-specific hazard means an instantaneous rate among people free of every first event. Subdistribution hazard means a rate linked to the focal cause’s cumulative incidence, with competing-event cases retained in its constructed risk set.

Check Your Understanding

For each item, identify the hazard interpretation that matches the question. Then compare your reasoning with the solutions.

A clinician wants the instantaneous rate of stroke among patients who are still alive and have had no first event. Which hazard is appropriate?
A report asks how a care pathway is associated with the cumulative incidence of discharge before death. Which hazard is the closer match, and who remains in its constructed risk set?
At time $t$, a cause has cumulative incidence 0.20. Its cumulative incidence increases by 0.008 over 0.2 year. Approximate its subdistribution hazard over that interval.
Why can a cause-specific hazard ratio and a subdistribution hazard ratio differ between the same two groups?

Solutions. (1) The cause-specific hazard, because its risk set consists of patients free of every first event. (2) The subdistribution hazard, because the question concerns a cause’s cumulative-incidence process; the constructed risk set includes those without the focal event, including people with a prior competing event. (3) Using the short-interval approximation, $0.008/\{(1-0.20)(0.2\text{ years})\}=0.008/0.16=0.0500$ per year. This is a rate, not a 5% probability. (4) They use different denominators and target different quantities: the cause-specific hazard is conditional on remaining event-free, whereas the subdistribution hazard is tied to the focal cause’s cumulative incidence and retains competing-event cases in its risk set.