One Cause, Two Denominators
A hazard for a particular event cause is a rate, but the population used to define that rate matters. In competing-risks data, a person who has already had a competing event cannot subsequently experience the focal event as a first event. The cause-specific hazard excludes that person from its risk set. The subdistribution hazard, by contrast, retains that person in its denominator. These are not two calculations of the same conditional rate: they answer different questions.
The comparison below concerns a single focal cause \(k\) among mutually exclusive first-event causes. Let \(T\) be time to the first event and \(J\) its cause, with \(J=k\) denoting the focal event. Write \(F_k(t)=P(T\leq t,J=k)\) for the cumulative incidence of cause \(k\), and \(S(t)=P(T\geq t)\) for the probability of remaining free of every event through time \(t\). We use continuous event times to give the hazards in density form; this avoids complications from tied event times.
In the subdistribution definition, a person with a prior competing event is included in the conditioning group, but cannot contribute a later cause-\(k\) first event. This is a population-level conditional definition: the denominator \(1-F_k(t)\) is the probability of not having experienced cause \(k\) by \(t\), not the probability of being event-free. Both hazards have units of inverse time. Neither is itself a probability.
Risk Sets and Conditional Interpretations
The cause-specific risk set answers: among people who have had no first event yet, at what instantaneous rate does cause \(k\) occur? A competing event removes a person from this risk set because that person is no longer event-free and cannot have cause \(k\) as their first event.
The subdistribution risk set answers a different conditional question: among people who have not had cause \(k\) as a first event by time \(t\), at what rate does cause \(k\) occur? It retains people with prior competing events, as well as people still event-free. People who have already had the focal cause are excluded. The retained competing-event group changes the denominator, even though its members cannot contribute new focal events.
| Feature | Cause-specific hazard for cause \(k\) | Subdistribution hazard for cause \(k\) |
|---|---|---|
| Denominator | \(S(t)\): event-free of all causes at \(t\) | \(1-F_k(t)\): not yet experienced cause \(k\) |
| Prior competing event | Person is outside the risk set | Person remains in the risk set |
| Prior focal event | Person is outside the risk set | Person is outside the risk set |
| Conditional interpretation | Instantaneous cause-\(k\) rate among those still event-free | Instantaneous cause-\(k\) rate relative to everyone who has not yet had cause \(k\) |
Worked Example: Comparing Denominators at One Time
Consider a hypothetical cohort of 1,000 people at a specified time \(t\). Suppose 600 remain event-free, 200 have already had a competing event, and 200 have already had the focal event. At \(t\), suppose the focal-event density is \(f_k(t)=0.06\) per year. These values are hypothetical and describe the population at this time.
The event-free proportion is \(S(t)=600/1000=0.60\). Therefore the cause-specific hazard is $$ \lambda_k(t)=\frac{f_k(t)}{S(t)} =\frac{0.06}{0.60}=0.10\ \text{per year}. $$
The proportion that has not yet had the focal event is \(1-F_k(t)=(600+200)/1000=0.80\). It includes both the event-free group and the 200 people with prior competing events. Thus $$ \widetilde{\lambda}_k(t)=\frac{f_k(t)}{1-F_k(t)} =\frac{0.06}{0.80}=0.075\ \text{per year}. $$
The two values differ because the denominator for the subdistribution hazard is larger. The cause-specific rate is the instantaneous focal-event rate among the 600 people still event-free; the subdistribution rate uses the larger group of 800 people who have not had the focal event. The 200 with a prior competing event are included in that second denominator, not treated as people who can later have the focal first event.
Why the Hazards Are Related
For a continuous first-event time, a cause-\(k\) event in a very short interval after \(t\) must come from someone who is event-free at \(t\). Consequently the cause-\(k\) event density factors into the probability of being event-free and the cause-specific hazard. The subdistribution hazard uses that same event density but divides by a different probability.
Among people event-free at \(t\), the chance of a cause-\(k\) event in a short interval of width \(\Delta t\) is approximately \(\lambda_k(t)\Delta t\).
The proportion event-free at \(t\) is \(S(t)\). Thus the probability of a cause-\(k\) event in that interval is approximately \(S(t)\lambda_k(t)\Delta t\), so \(f_k(t)=S(t)\lambda_k(t)\).
By definition, \(\widetilde{\lambda}_k(t)=f_k(t)/(1-F_k(t))\). Substitution gives the relationship below.
This identity makes the source of the difference explicit. The numerator \(f_k(t)\) is shared; the denominators are not. Since \(1-F_k(t)=S(t)+\sum_{j\ne k}F_j(t)\) when the competing causes exhaust the first-event types, prior competing events add probability to the subdistribution denominator. The identity is a population relationship, not a claim that the two hazards have the same conditional meaning.
Worked Example: Calculating a Subdistribution Hazard From a Cause-Specific Hazard
Suppose, hypothetically, that at time \(t=3\) years, \(S(t)=0.50\), \(F_k(t)=0.25\), and the cause-specific hazard for cause \(k\) is \(0.12\) per year. Assume the cause-specific hazard and the population probabilities refer to the same time and event definition. First obtain the cause-\(k\) density: $$ f_k(t)=S(t)\lambda_k(t) =(0.50)(0.12)=0.060\ \text{per year}. $$
The subdistribution denominator is \(1-F_k(t)=1-0.25=0.75\). Therefore $$ \widetilde{\lambda}_k(t) =\frac{0.060}{0.75}=0.080\ \text{per year}. $$
The subdistribution hazard is \(0.080\) per year at year 3, whereas the cause-specific hazard is \(0.12\) per year. The former is scaled by \(S(t)/(1-F_k(t))=0.50/0.75=2/3\). These rates describe different conditioning groups; the numerical comparison does not mean that an individual’s underlying focal-event rate changed when the denominator changed.
Reading Risk-Set Counts Carefully
The definitions also clarify the conceptual risk sets used when describing observed data. In a short interval, the cause-specific denominator is the people still event-free and under observation just before that interval. The subdistribution denominator is conceptually the people who have not had the focal cause, including those with prior competing events. In an observed cohort, censoring can remove people from what is observed; it is not itself a competing event. Estimation therefore needs to account appropriately for censoring rather than treating censored people as permanently retained or as having experienced a competing cause.
The following count-based calculation treats the interval as short enough that a rate can be approximated by events divided by risk-set exposure. The figures are hypothetical.
Worked Example: Short-Interval Rates From Risk-Set Exposure
At the start of a hypothetical 0.1-year interval, a cohort has 600 people still event-free, 200 people who previously had a competing event, and 200 who previously had the focal event. Suppose 6 focal events occur during the interval among the people still event-free, and assume for this calculation that everyone in the two relevant groups is observed throughout the interval.
For the cause-specific calculation, the event-free group contributes \(600(0.1)=60\) person-years. The estimated short-interval rate is $$ \frac{6}{60}=0.10\ \text{per year}. $$
For the subdistribution risk set, the 200 people with prior competing events are also included, giving \(800(0.1)=80\) person-years in this simple rate approximation. The corresponding rate is $$ \frac{6}{80}=0.075\ \text{per year}. $$
The same 6 focal events appear in both numerators, but the denominators differ by 20 person-years. The calculation illustrates the risk-set distinction; for non-negligible intervals, actual event probabilities and risk-set changes within the interval matter, so this approximation should not be treated as an exact continuous-time hazard.
Choosing and Checking an Interpretation
The choice depends on the question. The cause-specific hazard describes the instantaneous rate among people who are currently eligible to experience any first event. The subdistribution hazard is constructed with a risk set tied to the cumulative incidence of the focal cause: it compares focal-event occurrence with the group that has not yet had that cause. Thus it is directly linked to \(F_k(t)\) through \(f_k(t)=\widetilde{\lambda}_k(t)\{1-F_k(t)\}\), while the cause-specific hazard is linked to the event-free population through \(f_k(t)=\lambda_k(t)S(t)\).
These distinctions are useful when checking an analysis or explaining a reported rate. Ask which people are in the denominator, whether prior competing events are included, and whether the stated interpretation matches that conditioning group. A hazard ratio from one framework should not be described as a hazard ratio from the other merely because both concern the same event cause.
- Denominator check: if the risk set includes people with a prior competing event, it is not the ordinary cause-specific risk set.
- Focal-event check: people who have already had the focal first event must not remain in the subdistribution denominator \(1-F_k(t)\).
- Units check: a hazard is a rate, such as per year, not a percentage probability over an unspecified interval.
- Censoring check: censoring is distinct from a competing event; observed risk sets require appropriate handling of censoring.
A common mistake is to say that the subdistribution hazard is the rate of cause \(k\) among people still event-free, then retain people with prior competing events in its denominator. Those two statements conflict: event-free conditioning defines the cause-specific hazard. The correction is to state the subdistribution conditioning group explicitly—people who have not had cause \(k\)—and to explain that it includes prior competing events even though those individuals cannot subsequently experience cause \(k\) as a first event.
Check Your Understanding
Answer each question using the definitions above. Brief solutions follow the questions.
- At time \(t\), what does the denominator \(S(t)\) represent, and who is excluded from that risk set?
- Why does \(1-F_k(t)\) include people with a prior competing event?
- Suppose \(S(t)=0.40\), \(F_k(t)=0.20\), and \(\lambda_k(t)=0.15\) per year. Calculate \(f_k(t)\) and \(\widetilde{\lambda}_k(t)\).
- A report describes a rate as the instantaneous cause-\(k\) rate among people who remain free of all events. Which hazard definition matches that interpretation?
- Why is it incorrect to interpret a hazard of \(0.08\) per year as an 8% probability of the event in every year?
Solutions. (1) \(S(t)\) is the probability of being free of all event causes at \(t\); people with any prior event are excluded. (2) \(F_k(t)\) counts only people who have had cause \(k\), so its complement includes everyone who has not had cause \(k\), including people with another first event. (3) \(f_k(t)=S(t)\lambda_k(t)=(0.40)(0.15)=0.060\) per year, and \(\widetilde{\lambda}_k(t)=0.060/(1-0.20)=0.075\) per year. (4) The cause-specific hazard. (5) A hazard is an instantaneous rate with units of inverse time; a probability over a specified interval depends on the rate over that interval and the evolving event-free or cause-free probability.