Why the Conditioning Set Matters
A hazard is a rate defined by a conditional probability over a short time interval. The condition is essential: it specifies which people are in the denominator before the interval begins. For a cause-specific hazard, that condition is being free of every event up to that time. For a subdistribution hazard, the condition is different. Confusing the two can make a valid subdistribution hazard sound like an instantaneous event rate among people who are event-free.
Consider a study in which each person can experience one of several mutually exclusive first events. Choose one cause, \(k\), as the focal cause. A person whose first event is another cause has experienced a competing event. The examples below use a complete hypothetical cohort, so the distinction can be seen without additional complications from censoring.
The distinction can also be written in terms of a short interval. For a small \(\Delta t\), the event-free conditional probability of a focal event is approximately \(P(t<T\leq t+\Delta t,J=k\mid T>t)\), with the approximation to first order given by the cause-specific hazard times \(\Delta t\). The subdistribution hazard instead comes from the change in \(F_k(t)\) relative to \(1-F_k(t)\). That latter quantity includes people who have already had a competing event, as long as they have not previously had the focal event.
A Cohort Example: Two Different Conditions
Worked Example: Six Focal Events in a Short Interval
Suppose a hypothetical cohort contains 100 people. At time \(t=2\) years, 12 have already had the focal event, 28 have had a competing event, and 60 remain event-free. During the next \(0.1\) year, 6 of the 60 event-free people have the focal event, 3 have a competing event, and 51 have no event. No person can have more than one first event.
Event-free conditional calculation. The interval probability of a focal event among people still free of every event at year 2 is \(6/60=0.10\). Dividing by the interval length gives \(0.10/0.1=1.00\) per year as a short-interval rate approximation for the cause-specific hazard.
Subdistribution calculation. By year 2, the focal cumulative incidence is \(F_k(t)=12/100=0.12\). The focal incidence increases by \(\Delta F_k=6/100=0.06\) during the next interval. The fraction of the cohort not yet having had the focal event is \(1-F_k(t)=0.88\), or 88 people. The corresponding conditional increment is \(0.06/0.88=0.06818\). Dividing by \(0.1\) year gives a short-interval subdistribution hazard approximation of \(0.6818\) per year.
The calculations differ because they answer different conditional questions. The \(1.00\)-per-year approximation concerns the 60 people who are event-free at the interval's start. The \(0.6818\)-per-year approximation uses an increment in focal cumulative incidence relative to the 88 people who have not yet experienced the focal event. That set includes the 28 people who already had a competing event. It would therefore be incorrect to describe the subdistribution hazard as the instantaneous focal-event rate among people free of every event.
Where the Trap Enters the Probability Calculation
For a continuous event time, write \(f_k(t)\) for the rate of increase of \(F_k(t)\). Among people event-free at \(t\), a small-interval focal-event probability is approximately \(f_k(t)\Delta t/S(t)\). This gives the cause-specific hazard \(f_k(t)/S(t)\). By contrast, the subdistribution hazard is defined using \(1-F_k(t)\) in the denominator. The distinction is not a matter of changing the event of interest; it is a matter of changing the conditioning set.
The relationship follows directly by dividing the two expressions:
This identity is a useful check on the distinction. The event-free probability \(S(t)\) excludes everyone who has had any first event. The quantity \(1-F_k(t)\) excludes those who have already had the focal event, but it does not exclude people whose first event was a competing cause. Thus the two denominators generally describe different groups.
There is a precise conditional-probability statement behind the subdistribution expression. For a small interval after \(t\), the ratio \(\{F_k(t+\Delta t)-F_k(t)\}/\{1-F_k(t)\}\) is the conditional probability of a focal first event in that interval given that a focal first event has not occurred by \(t\). This condition is not the same as having had no event by \(t\). In particular, a person with a prior competing event satisfies “no focal event by \(t\),” even though that person is not event-free. A probability statement can therefore be mathematically valid while still being incorrectly described as conditioning on event-free survival.
Worked Example: Translating a Conditional Statement
In a second hypothetical cohort, at time \(t\), 20% have already had the focal event, 30% have had a competing event, and 50% are event-free. In the next \(0.05\) year, the focal cumulative incidence increases by 0.025.
The event-free conditional interval probability cannot be calculated from the incidence increment alone: its denominator is \(S(t)=0.50\), so it is \(0.025/0.50=0.050\), or 5.00%. The subdistribution conditional increment uses \(1-F_k(t)=1-0.20=0.80\), giving \(0.025/0.80=0.03125\), or 3.125%.
As short-interval rate approximations, these are \(0.050/0.05=1.00\) per year and \(0.03125/0.05=0.6250\) per year, respectively. The first is based on being event-free; the second is based on not having yet experienced the focal event. Calling both “the rate among people still event-free” would assign the same condition to two different calculations.
A Practical Check on Reported Interpretations
When a subdistribution hazard is reported, do not infer its conditioning set from the word “hazard” alone. Check how it was defined or modelled, then ask whether the interpretation being proposed conditions on no event of any type or only on no prior focal event. A useful first check is to write the numerator and denominator as probabilities or cohort counts. If the denominator is \(S(t)\), the calculation concerns the event-free set. If it is \(1-F_k(t)\), prior competing events have not been removed from the conditioning set.
Worked Example: Same Event-Free Count, Different Focal History
Compare two hypothetical cohorts of 100 people over a short interval of \(0.1\) year. In cohort A, 10 people have already had the focal event, 40 have had a competing event, and 50 are event-free. In cohort B, 30 have had the focal event, 20 have had a competing event, and 50 are event-free. In each cohort, 2 of the 50 event-free people have the focal event during the interval. Assume no other focal events occur in that interval.
In both cohorts, the event-free conditional interval probability is \(2/50=0.040\), corresponding to a short-interval rate approximation of \(0.040/0.1=0.400\) per year. The event-free group sizes and the numbers of focal events within them are identical.
For cohort A, the focal cumulative incidence rises by \(2/100=0.020\), and \(1-F_k(t)=0.90\). Its subdistribution conditional increment is \(0.020/0.90=0.02222\), giving an approximation of \(0.2222\) per year. For cohort B, the same incidence increment is divided by \(1-F_k(t)=0.70\): \(0.020/0.70=0.02857\), or \(0.2857\) per year. These results differ even though the event-free conditional calculation is the same. This example shows why an event-free interpretation cannot be recovered merely by looking at the focal-event count among people who are event-free.
Assumptions, Diagnostics, and Limits
The definitions and calculations here concern first events of mutually exclusive types. The expressions involving \(f_k(t)\) apply where the cumulative incidence has a density or rate of increase. The short-interval rate interpretations rely on taking \(\Delta t\) small; a calculation over a substantial interval is an interval probability or an average approximation, not automatically the instantaneous hazard at a single time. The count examples also assume complete knowledge of event status over their stated intervals, so their arithmetic does not require an adjustment for censoring.
In data analysis, the definitions do not by themselves guarantee that a particular estimator or model is appropriate. Incomplete follow-up, event-time recording, and the model specification must be handled by methods suited to the analysis. The narrow interpretive check remains the same: determine which event histories the definition includes in its conditioning set.
The main alternative for describing the instantaneous rate of a particular first event among people still event-free is the cause-specific hazard. The cumulative incidence, in turn, describes the probability of having experienced the focal event by a specified time. These quantities answer related but distinct questions. This tutorial's concern is the conditional-probability trap: a subdistribution hazard should not be relabelled as the event-free conditional rate.
- Check the event history in the condition. “No event yet” and “no focal event yet” are not interchangeable.
- Check the denominator. Event-free probability is \(S(t)\); the subdistribution expression uses \(1-F_k(t)\).
- Check the time scale. A conditional probability over an interval divided by its length is an approximation to a rate when the interval is short.
- Check the conclusion. Do not interpret a subdistribution hazard as an event-free rate simply because it concerns a particular event cause.
Check Your Understanding
Answer each question by naming the conditioning set, not just the event cause. Solutions follow the questions.
- At time \(t\), 15 of 100 people have had the focal event, 25 have had a competing event, and 60 are event-free. In a short interval, 3 people have the focal event. What event-free conditional interval probability is observed, and what is the subdistribution conditional increment?
- In words, why does a person with a prior competing event count in \(1-F_k(t)\) but not in \(S(t)\)?
- A colleague says, “The subdistribution hazard is the instantaneous rate of the focal event among people who have not yet had any event.” Identify the error and give the corrected interpretation.
- If \(F_k(t)=0.30\) and the focal cumulative incidence increases by \(0.014\) over a short interval, calculate the subdistribution conditional increment. What additional quantity is needed to calculate the event-free conditional increment?
- Two cohorts have the same event-free count and the same number of focal events during a short interval, but different numbers of prior focal events. Must their subdistribution conditional increments be the same? Explain.
Solutions. (1) The event-free interval probability is \(3/60=0.050\), or 5.00%. The subdistribution conditional increment is \(0.03/(1-0.15)=0.03/0.85=0.03529\), or 3.529%. (2) A prior competing event means the person has had an event, so the person is not in the event-free probability \(S(t)\). But the person has not had the focal event, so the person remains included in \(1-F_k(t)\). (3) The statement uses the event-free condition for a quantity whose denominator is based on not yet having the focal event. The corrected interpretation is that the subdistribution hazard is defined using the focal cumulative incidence and a condition that can include people with prior competing events; it is not the event-free instantaneous rate. (4) The subdistribution increment is \(0.014/(1-0.30)=0.014/0.70=0.020\), or 2.00%. To calculate the event-free conditional increment, one needs \(S(t)\), the probability of remaining free of every event. (5) No. The subdistribution increment divides the change in focal cumulative incidence by \(1-F_k(t)\), so different prior focal-event proportions can give different denominators even when the event-free counts and interval focal-event counts agree.