What a Subdistribution Rate Compares
A comparison of subdistribution hazards across populations concerns how the cumulative incidence of a particular event type accumulates over time. It does not compare the instantaneous rate of that event among people who are currently free of every event. The distinction is the denominator: a cause-specific hazard uses people still free of all events, while a subdistribution hazard uses a risk set that continues to include people who have already experienced a competing event.
This interpretation is useful when the question concerns a population’s experience of a particular event in the presence of competing events. It is not interchangeable with a comparison of instantaneous event rates in the currently event-free population, nor does a hazard ratio directly give a risk ratio.
The notation \(t-\) indicates the instant just before \(t\), which matters when events can be tied. For continuous event times, it is often written simply as \(t\). Intuitively, over a very short interval of length \(dt\), the chance of experiencing the focal event is approximately \(\widetilde{\lambda}_{k,g}(t)\,dt\), conditional on not having experienced that focal event earlier. That conditioning includes people with a prior competing event; it does not mean they are still event-free.
Why This Is Not an Event-Free Instantaneous Rate
Write \(S_g(t-)=P(T\geq t)\) for the probability of still being free of all event types just before \(t\). The cause-specific hazard for cause \(k\), when its density exists, is $$ \lambda_{k,g}(t)=\frac{f_{k,g}(t)}{S_g(t-)}. $$ Its denominator comprises people currently free of every event. By contrast, the subdistribution hazard divides the same focal-event density by \(1-F_{k,g}(t-)\). Because people who had a competing event have not had the focal event, they remain represented in this latter denominator.
The distinction can also be seen from the small-interval probabilities. The numerator \(f_{k,g}(t)\,dt\) describes focal events around \(t\), on the population probability scale. Dividing by \(S_g(t-)\) conditions on being event-free at \(t\); dividing by \(1-F_{k,g}(t-)\) conditions only on not yet having experienced cause \(k\). These conditions select different populations. Thus a cross-population subdistribution hazard comparison is a comparison of rates defined relative to the focal cause’s cumulative incidence process, not a comparison of current event-free patients’ instantaneous rates.
A Short Derivation and a Proportional-Hazards Result
For continuous event times, the probability of a focal event in a short interval is approximately \(f_{k,g}(t)\,dt\). Among the population fraction that has not yet experienced the focal event, that probability per unit time is therefore $$ \frac{P(t\leq T<t+dt,J=k)}{P(T\geq t\text{ or a competing event occurred before }t)}\cdot\frac{1}{dt} \ \approx\ \frac{f_{k,g}(t)\,dt}{1-F_{k,g}(t-)}\cdot\frac{1}{dt} = \widetilde{\lambda}_{k,g}(t). $$
This argument explains the denominator rather than making the subdistribution hazard a rate among currently event-free people. It also gives a useful relationship to cumulative incidence. Since \(F'_{k,g}(t)=f_{k,g}(t)\), $$ \frac{d}{dt}\{-\log[1-F_{k,g}(t)]\} = \frac{F'_{k,g}(t)}{1-F_{k,g}(t)} = \widetilde{\lambda}_{k,g}(t). $$
Consequently, the cumulative subdistribution hazard from time zero to \(t\), denoted \(\widetilde{\Lambda}_{k,g}(t)\), satisfies $$ \widetilde{\Lambda}_{k,g}(t)=-\log[1-F_{k,g}(t)], \qquad F_{k,g}(t)=1-\exp\{-\widetilde{\Lambda}_{k,g}(t)\}. $$ If a proportional subdistribution hazards model holds for two populations throughout the period of interest, so that \(\widetilde{\lambda}_{k,1}(t)=r\widetilde{\lambda}_{k,0}(t)\) with constant \(r\), integration gives \(\widetilde{\Lambda}_{k,1}(t)=r\widetilde{\Lambda}_{k,0}(t)\). It follows that $$ F_{k,1}(t)=1-\{1-F_{k,0}(t)\}^{r}. $$ This relationship helps interpret a constant subdistribution hazard ratio. It does not say the cumulative incidence ratio is \(r\), and it relies on proportional subdistribution hazards over the period considered.
Worked Example: Comparing Two Subdistribution Rates at One Time
Suppose two hypothetical populations are evaluated at \(t=2\) years for a focal event. At that time, population A has \(F_{k,A}(2)=0.30\), focal-event density \(f_{k,A}(2)=0.060\) per year, and event-free probability \(S_A(2)=0.50\). Population B has \(F_{k,B}(2)=0.10\), \(f_{k,B}(2)=0.030\) per year, and \(S_B(2)=0.75\). Treat these as population quantities, not estimates from a real study.
For population A, the probability of not yet having the focal event is \(1-0.30=0.70\). Its subdistribution hazard is $$ \widetilde{\lambda}_{k,A}(2)=\frac{0.060\text{ per year}}{0.70}=0.08571\text{ per year}. $$ For population B, the corresponding denominator is \(1-0.10=0.90\), so $$ \widetilde{\lambda}_{k,B}(2)=\frac{0.030\text{ per year}}{0.90}=0.03333\text{ per year}. $$ The subdistribution hazard ratio A to B at this time is \(0.08571/0.03333=2.571\). At \(t=2\), the focal-event density relative to the probability not yet having had that focal event is about 2.57 times as large in A as in B.
That is not the ratio of instantaneous rates among people currently free of all events. Those rates use \(S_g(2)\): for A, \(0.060/0.50=0.120\) per year; for B, \(0.030/0.75=0.0400\) per year. Their ratio is \(3.00\), not \(2.571\). The two comparisons differ because their denominators differ. Neither ratio alone establishes that population membership caused a difference.
Interpreting a Ratio Across Populations
A subdistribution hazard ratio greater than one means that, at the times and under the model being considered, the focal-event density is larger relative to the probability of not yet experiencing that focal event in the numerator population. A ratio below one means it is smaller by that same definition. The comparison is about rates on this particular cumulative-incidence-related scale; it does not directly state how many more people will experience the event, or how much faster event-free individuals experience it.
For a constant ratio to summarize the whole time course, the proportional subdistribution hazards assumption must be reasonable. If the ratio varies with time, one constant number can conceal important differences, including a change in the direction of comparison. Even when proportionality holds, the ratio is not a risk ratio: translating it into cumulative incidence requires the reference population’s cumulative incidence at the time of interest.
Worked Example: Translating a Constant Ratio Into Cumulative Incidence
Suppose a hypothetical comparison has a constant subdistribution hazard ratio of \(1.5\) for a focal event over follow-up. At a particular time, the reference population’s cumulative incidence is \(0.20\). Assume the proportionality condition holds from the start of follow-up through this time.
The proportional-hazards relationship gives $$ F_{k,1}(t)=1-\{1-F_{k,0}(t)\}^{1.5} =1-(1-0.20)^{1.5} =1-0.80^{1.5}. $$ Since \(0.80^{1.5}\approx0.7155\), the comparison population’s cumulative incidence is approximately \(1-0.7155=0.2845\), or \(28.45\%\), at that time.
The ratio of the two cumulative incidences is \(0.2845/0.20\approx1.423\), not \(1.5\). The hazard ratio therefore does not mean that the event probability is 50% higher. The calculation also depends on the reference cumulative incidence and on proportionality over follow-up.
Worked Example: Reporting a Ratio Below One
In a hypothetical analysis, the estimated subdistribution hazard ratio for a treatment group versus a reference group is \(0.72\). Suppose the model supports a constant ratio over follow-up and the reference group’s cumulative incidence at a specified time is \(0.25\). What does the ratio imply at that time under the model?
Using the proportional relationship, $$ F_{k,1}(t)=1-\{1-0.25\}^{0.72} =1-0.75^{0.72} \approx1-0.8129 =0.1871. $$ The model-implied cumulative incidence is approximately \(18.71\%\) in the treatment group, compared with \(25\%\) in the reference group, at the specified time. The ratio \(0.72\) describes a subdistribution hazard comparison, not a 28% reduction in the event probability. This is an association under the model; without a suitable design and assumptions, it is not evidence that treatment caused the difference.
Assumptions, Diagnostics, and Alternatives
The definitions above concern the underlying event-time distributions. Applying them to observed data requires correctly identifying the focal and competing event types and having event times measured on a common, meaningful time scale. When some participants are censored before an event is observed, estimation and model-based inference also require an appropriate censoring assumption, commonly that censoring is independent of the event process conditional on variables included in the analysis. A model-based constant ratio additionally requires proportional subdistribution hazards for the interval being summarized. These conditions should not be confused: a valid event-type definition does not guarantee valid handling of censoring, and neither guarantees proportionality.
Useful checks include comparing model-based fitted cumulative-incidence curves with estimated curves, examining whether the population contrast changes over time, and checking whether results depend strongly on how the comparison groups or adjustment variables are specified. A clearly changing contrast is a warning against reporting one constant ratio as a complete summary. If the estimated cumulative incidence or fitted curves behave implausibly, recheck the event coding, time origin, censoring information, and model fit before interpreting the ratio.
| Quantity | Denominator or scale | Question it addresses |
|---|---|---|
| Cause-specific hazard | People currently free of all events | Instantaneous rate of cause \(k\) among those still event-free |
| Subdistribution hazard | People not yet experiencing cause \(k\), including those with prior competing events | Rate defined relative to the cumulative incidence of cause \(k\) |
| Cumulative incidence | Whole population probability by time \(t\) | How many are expected to have experienced cause \(k\) by time \(t\)? |
Choose the quantity that matches the scientific question. For instantaneous rates among people currently free of every event, compare cause-specific hazards. For population probabilities of the focal event by a specified time, compare cumulative incidences. A subdistribution hazard comparison is useful when a model for the focal cumulative incidence is desired, but it should be reported with its risk-set meaning and, when applicable, the proportionality assumption. None of these comparisons alone establishes a causal effect.
Common Mistake
Wrong reasoning: “The subdistribution hazard ratio is 1.4, so among patients still free of all events the focal event is occurring 40% faster.” This assigns the wrong risk set to the subdistribution hazard.
Correction: A ratio of 1.4 compares focal-event densities relative to the probabilities of not yet experiencing the focal event, with people who previously experienced competing events still included in the subdistribution risk set. It is not a comparison restricted to people currently event-free, and it is not automatically a 40% increase in cumulative incidence. State the event, time scale, comparison groups, model assumption, and whether the reported contrast is a hazard ratio or a cumulative-incidence contrast.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- At a time \(t\), a population has \(F_k(t)=0.40\) and \(f_k(t)=0.030\) per year. Calculate its subdistribution hazard and state its units.
- Why can the subdistribution hazard denominator include people who have had a competing event?
- Under proportional subdistribution hazards with ratio \(r=0.5\), the reference cumulative incidence at time \(t\) is \(0.36\). Find the comparison population’s cumulative incidence under the model.
- A report calls a subdistribution hazard ratio of \(1.2\) “a 20% higher instantaneous rate among people still event-free.” Identify the error and give a corrected interpretation.
Solutions. 1. \(\widetilde{\lambda}_k(t)=0.030/(1-0.40)=0.030/0.60=0.0500\) per year. 2. The denominator is the probability of not yet experiencing the focal event, \(1-F_k(t-)\); a prior competing event is not the focal event, so it does not remove a person from that probability. 3. \(F_{k,1}(t)=1-(1-0.36)^{0.5}=1-\sqrt{0.64}=1-0.80=0.20\), or 20%, assuming the proportional relationship holds through \(t\). 4. The error is describing the subdistribution risk set as currently event-free. A corrected interpretation is that, under the model, the focal-event density relative to the probability of not yet having experienced that event is 1.2 times as large in the comparison group; this is not an event-free instantaneous-rate ratio or a 20% increase in cumulative incidence.