Reading Subdistribution Hazard Statements Carefully

What a Subdistribution Hazard Statement Says

A subdistribution hazard statement concerns how the cumulative incidence of a particular event cause changes over time. Its denominator is not simply the probability of remaining free of every event. That difference is essential: a person who has already experienced a competing event cannot later experience the focal first event, but is retained in the subdistribution risk set used to define this hazard.

Let \(T\) be the time to the first event, and let \(J\) indicate its cause. For a focal cause \(k\), the cause-specific cumulative incidence is \(F_k(t)=P(T\leq t,J=k)\). It is the probability of having experienced cause \(k\) by time \(t\), with competing causes treated as distinct outcomes, not as ordinary losses to follow-up.

The absolute-continuity condition matters: merely having a continuous event-time distribution does not guarantee that a density exists. The density-form definition and derivations here are for settings where the focal cumulative incidence is absolutely continuous. The subdistribution hazard has units of inverse time; it is a rate, not a probability.

In words, \(\lambda_k^*(t)\) is the instantaneous rate of increase in the focal cause’s cumulative incidence, relative to the probability that the focal event has not yet occurred. It does not describe the instantaneous rate among only people still free of every event. Nor does it give the probability that a person in the subdistribution risk set will experience cause \(k\) in the next moment.

From the Hazard to the Cumulative Incidence

The definition gives a direct way to translate a hazard statement into language about \(F_k\). Where the denominator is positive and the density exists,

Because \(F_k(0)=0\) when follow-up begins before any event, divide by \(1-F_k(t)\) and integrate from 0 to \(t\). This gives the cumulative subdistribution hazard \(\Lambda_k^*(t)\) and its relationship to the cumulative incidence:

The second equality follows because the derivative of \(-\log\{1-F_k(u)\}\) is \(F'_k(u)/\{1-F_k(u)\}\). Thus a larger subdistribution hazard, considered over time, corresponds to faster accumulation of the focal cumulative incidence, all else specified by the hazard path. A value of the hazard at one instant alone does not determine the cumulative incidence: the history of the hazard also matters.

Interpreting a Subdistribution Hazard Ratio

A common model assumption is proportional subdistribution hazards: for two groups, the treatment-group subdistribution hazard is a constant multiple of the reference-group hazard at every time. If the ratio is \(r\), then \(\lambda_1^*(t)=r\lambda_0^*(t)\), where the subscripts label groups. Under the absolute-continuity conditions above, integration gives \(\Lambda_1^*(t)=r\Lambda_0^*(t)\). Substituting \(\Lambda_g^*(t)=-\log\{1-F_g(t)\}\) yields:

This equation translates a proportional subdistribution hazard statement into a relationship between cumulative incidences at a specified time. The ratio \(r\) is not a risk ratio, and \(r-1\) is not the percentage change in cumulative incidence. The comparison depends on the reference group’s cumulative incidence at the time of interest.

The reverse direction is also useful. If \(r<1\), the model implies a lower cumulative incidence than the reference group at a time when \(F_0(t)>0\). The size of that difference still depends on \(F_0(t)\); the hazard ratio alone does not specify an absolute difference in percentage points.

Why the Denominator Is Not Event-Free Survival

Let \(S(t)=P(T>t)\) denote the probability of remaining free of every first event through time \(t\). In general, \(S(t)\) is not the subdistribution-hazard denominator. For two or more possible first-event causes, people who have had a competing event contribute to \(1-F_k(t)\), but not to \(S(t)\). For a focal cause \(k\), \(1-F_k(t)\) is the probability that cause \(k\) has not occurred; it is not the probability that no event has occurred.

The cause-specific hazard uses a different denominator: it describes the instantaneous rate of cause \(k\) among individuals still free of all event types just before time \(t\). By contrast, the subdistribution hazard is scaled by the probability of not yet having had the focal event. Both quantities involve cause \(k\), but they answer different questions and need not have the same numerical value. The earlier material on cause-specific hazards develops the event-free interpretation; here the key comparison is the denominator.

Assumptions, Diagnostics, and Limits

A careful interpretation begins by checking what the reported quantity actually is. A statement about a subdistribution hazard should identify the focal cause, the time scale, and—if it is a comparison—the groups and the model or summary used. The density-form interpretation requires an absolutely continuous focal cumulative incidence and a positive denominator. If \(F_k(t)\) reaches 1, the denominator is zero and this form is not defined there.

For an estimated hazard or model fitted to observed data, valid inference also depends on how censoring and the risk sets are handled. Independent censoring is a separate assumption: competing events are outcomes, not censoring, while losses to follow-up may require censoring assumptions or adjustment. A model-based hazard ratio is only a constant multiplier if proportional subdistribution hazards are adequately supported over the time range being summarized.

Useful warning signs include an interpretation that calls the denominator “people with no event,” a hazard ratio presented as a risk ratio, or an absolute risk claim made without giving a time point and baseline cumulative incidence. Also examine whether the proportionality assumption is plausible over time; a single ratio can conceal changing relative hazards. Even when the model fits, a subdistribution hazard is a particular rate-scale summary, not a direct probability and not automatically a causal effect.

Common Mistakes

• Wrong: “The subdistribution hazard is the rate among people who are still event-free.” Correction: That describes the event-free denominator used for a cause-specific hazard. The subdistribution denominator includes people who have had competing events, provided they have not had the focal event.
• Wrong: “A hazard ratio of 1.50 means 50% more people experience the event.” Correction: It is a ratio of subdistribution hazards. Under proportional subdistribution hazards, translate it using \(F_1(t)=1-\{1-F_0(t)\}^{1.50}\) at a specified time.
• Wrong: “The hazard is the chance of the event in the next month.” Correction: A hazard has units of inverse time and is not itself a probability. A probability over an interval requires an interval-specific calculation and assumptions about the hazard over that interval.
• Wrong: “A competing event is just censoring for the focal cause.” Correction: A competing event prevents the focal first event from occurring later. Treating it as ordinary censoring changes the interpretation and generally does not produce the focal cause’s cumulative incidence.