One Clinical Question, Two Hazard Definitions
Suppose a clinical team follows patients after treatment and records the first occurrence of hospitalization or death. If the question is about the rate of first hospitalization, death before hospitalization matters: it prevents a later hospitalization from being the first event. Yet it can enter the definition of a hospitalization hazard in different ways. The cause-specific hazard describes the instantaneous hospitalization rate among patients still free of either first event. The subdistribution hazard uses a different risk set that continues to include patients whose first event was a competing event such as death.
The distinction is not a choice between correct and incorrect definitions. Each answers a different question. This worked example keeps the endpoint fixed—hospitalization as the event of interest—and makes explicit which patients contribute to each interpretation.
Both hazards have units of inverse time, such as per month. They are rates, not probabilities. For a sufficiently short interval \(\Delta t\), a hazard multiplied by \(\Delta t\) approximates a conditional event probability, subject to the usual small-interval approximation. Neither hazard alone is the probability of hospitalization by a given time.
Defining First Events in the Hospital Example
For this example, the study endpoint is the first of hospitalization or death after enrollment. These are alternative first-event types: a participant can have only one of them as the first event. If someone is hospitalized first and dies later, that later death does not change the first-event label. If someone dies before hospitalization, the first event is death, even if hospitalization would otherwise have occurred later. The event definition must be chosen in advance and applied consistently.
Worked Example: Classifying First Events
Consider these hypothetical follow-up histories, with time measured in months after enrollment. Classify each history by its first event.
| Patient | Hypothetical history | First-event classification |
|---|---|---|
| A | Hospitalized at month 2; dies at month 7 | Hospitalization first |
| B | Dies at month 3 without prior hospitalization | Death first |
| C | No hospitalization or death before last contact at month 5 | Censored event-free at month 5 |
| D | Hospitalized at month 4; alive at last contact at month 8 | Hospitalization first |
| E | Dies at month 1; a hypothetical later record would show hospitalization at month 6 | Death first |
Patients A and D have hospitalization as their first event. B and E have death as their competing first event. C has no observed first event before censoring. Patient E illustrates why the endpoint matters: a later hospitalization does not become a first hospitalization after death has already occurred. In a broader study of all hospitalizations, later events might be recorded, but that is a different endpoint and requires a different analysis.
What Each Risk Set Means
For the cause-specific hazard, a patient contributes to the hospitalization risk set at time \(t\) only while still free of both hospitalization and death. Once death occurs first, that patient is no longer eligible to have a future first hospitalization. This is why the cause-specific hazard has a direct rate interpretation for the currently event-free population.
For the subdistribution hazard, a patient who had death first remains in the risk set for the hospitalization event. This is a mathematical device in the subdistribution definition, not a claim that the patient can later experience hospitalization as a first event. The patient remains in the denominator because the target is the distribution of first hospitalizations in the original population, with competing deaths reflected in that distribution.
Worked Example: Approximate Rates in a Short Interval
In a hypothetical cohort of 1,000 patients, suppose that just before month 6, 600 patients are event-free, 120 have already died before hospitalization, and 280 have already been hospitalized first. During the following 0.1 month, 12 of the 600 event-free patients have a first hospitalization. Assume for this short interval that the risk-set counts are approximately constant and that there is no censoring. Calculate the two approximate hospitalization rates.
Cause-specific rate. Only the 600 event-free patients contribute. Their approximate person-time is \(600(0.1)=60\) patient-months. The rate is $$ \frac{12\ \text{first hospitalizations}}{60\ \text{patient-months}} =0.2000\ \text{first hospitalizations per patient-month}. $$ This estimates the short-interval cause-specific rate among patients who were still free of either first event.
Subdistribution rate. At the interval’s start, the subdistribution risk set includes the 600 event-free patients plus the 120 patients who previously died first. It excludes the 280 patients who already had hospitalization first. Its approximate risk-set time is \((600+120)(0.1)=72\) patient-months. Therefore, $$ \frac{12\ \text{first hospitalizations}}{72\ \text{patient-months}} =0.1667\ \text{first hospitalizations per patient-month}. $$ This is an approximate subdistribution rate based on the specified short-interval risk set. The difference from 0.2000 arises from the 120 prior competing deaths retained in that denominator. Neither number is a 6-month hospitalization probability.
The calculation uses a short-interval approximation: with changing risk-set membership or substantial censoring during the interval, person-time and event counts would need to be handled more carefully. The population definitions of the hazards themselves do not require constant risk-set counts.
A Relationship Between the Hazards
A useful justification links the cause-specific hazard to the density of first hospitalizations. Let \(S(t)=P(T\geq t)\) be the probability of remaining free of both first events up to time \(t\). In a continuous-time model, the probability of still being event-free at \(t\), multiplied by the instantaneous hospitalization rate among event-free people, gives the density of first hospitalization:
This identity follows from the small-interval definition. Among people event-free at \(t\), the probability of first hospitalization in \([t,t+\Delta t)\) is approximately \(\lambda_H(t)\Delta t\). Multiplying by the probability \(S(t)\) of being event-free gives the population probability of a first hospitalization in that interval. Dividing by \(\Delta t\) and taking the limit yields the density.
Since \(F_H(t)\) is the probability of a first hospitalization by \(t\), the probability of not yet having had hospitalization first is \(1-F_H(t)\). That group includes people who had death first. Substituting the density relationship into the definition of the subdistribution hazard gives
Thus, the subdistribution hazard is not generally the cause-specific hazard with a different label. It depends both on the current cause-specific rate and on how many people remain without a first hospitalization, including those removed from the event-free population by competing death.
Worked Example: Interpreting Both Hazards at Six Months
Consider a hypothetical continuous-time model with constant cause-specific hospitalization rate \(\lambda_H(t)=0.08\) per month and death rate \(\lambda_D(t)=0.04\) per month. Assume these are the only first-event types and that the rates are constant for everyone in this illustrative model. Find the two hospitalization hazards at month 6.
The probability of remaining free of both events through time \(t\) is $$ S(t)=\exp[-(0.08+0.04)t]. $$ At month 6, \(S(6)=\exp(-0.72)=0.4868\). The first-hospitalization density is therefore $$ f_H(6)=S(6)\lambda_H(6) =0.4868(0.08)=0.03894\ \text{per month}. $$ The hospitalization cumulative incidence by time \(t\) is the integral of that density: $$ F_H(t)=\int_0^t 0.08\exp(-0.12u)\,du =\frac{0.08}{0.12}\left(1-\exp(-0.12t)\right). $$ Thus \(F_H(6)=\frac{2}{3}(1-\exp(-0.72))=0.3422\), and \(1-F_H(6)=0.6578\). The month-6 subdistribution hazard is $$ \lambda_H^*(6)=\frac{0.03894}{0.6578} =0.05919\ \text{per month}. $$ The cause-specific hazard is \(0.08000\) first hospitalizations per event-free patient-month. The subdistribution hazard is \(0.05919\) per month under its larger, differently defined risk set. These values are rates in the model, not probabilities of hospitalization at month 6. The calculation also shows that the hazards need not be equal, even though the underlying cause-specific rates are constant.
Assumptions, Diagnostics, and Alternatives
The definitions require a well-specified time origin, an event of interest, and competing first-event types that are classified consistently. The continuous-time formulas above assume event times have a continuous distribution, so exact ties have probability zero. If the recorded time scale makes simultaneous event types possible, the study needs a prespecified rule for classifying tied events; otherwise the event labels and resulting hazards are ambiguous.
The hazard definitions describe the underlying event process. Estimating them from censored follow-up additionally requires appropriate handling of censoring. In particular, censoring must not selectively remove people with different future event experiences after accounting for the variables used in the analysis. Check that follow-up and censoring are recorded consistently across event types, that event times and causes are plausible, and that the chosen first-event definition matches the clinical question.
A useful first diagnostic is to write down who is in each risk set at the time being interpreted. If deaths are excluded from the cause-specific risk set but inadvertently removed from the subdistribution risk set, the two calculations have been made the same when they should not be. If people with prior hospitalization remain in the subdistribution risk set for first hospitalization, the denominator is also wrong. The next diagnostic is to check the time unit: a rate per month cannot be interpreted as a rate per year without conversion.
| Quantity | Who is in its risk set? | Interpretation for hospitalization |
|---|---|---|
| Cause-specific hazard \(\lambda_H(t)\) | Event-free of both first events just before \(t\) | Instantaneous first-hospitalization rate among those currently event-free |
| Subdistribution hazard \(\lambda_H^*(t)\) | No first hospitalization yet, including people with a prior competing death | Rate linked to the distribution of first hospitalizations in the original population |
The main alternative is therefore not another formula for the same quantity, but a different target. Use the cause-specific hazard when the question concerns the current event-free patients’ instantaneous rate. Use the subdistribution hazard when the question concerns a rate parameter tied to the population distribution of first hospitalization in the presence of competing death. For a direct probability statement—such as the chance of hospitalization first by month 6—the relevant quantity is \(F_H(6)\), not either hazard alone.
These rates do not, by themselves, establish a causal effect of a treatment or explain why groups differ. A comparison can reflect differences in the populations remaining event-free, in competing death, or in other measured or unmeasured characteristics. Avoid translating a hazard contrast into a probability contrast or a causal claim without the additional design and modeling assumptions needed for that interpretation.
Common Mistake
Wrong reasoning: “Death is a competing event, so it should be treated as ordinary censoring for every hospitalization analysis. Therefore, both hazards use only patients who are still alive and hospitalization-free.”
Correction: Death removes a patient from the cause-specific risk set because that patient can no longer have hospitalization as the first event. But the subdistribution risk set retains patients who died first, because its denominator is \(1-F_H(t)\), not the probability of remaining free of all first events. Treating those two denominators as interchangeable erases the distinction the hazards are designed to express.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A patient dies at month 2 and would have been hospitalized at month 5 if followed. What is the patient’s first-event label?
- At a given time, 400 patients remain event-free, 100 have died first, and 150 have been hospitalized first. Which of these groups belongs in the subdistribution risk set for first hospitalization?
- In the constant-rate example, why is the subdistribution hazard at month 6 not the same as the cause-specific hazard?
- What quantity directly represents the probability of first hospitalization by month 6?
- Give one reason that a short-interval rate calculated as events divided by risk-set person-time may only approximate an instantaneous hazard.
Solutions. (1) Death first; a later hypothetical hospitalization cannot be the first event. (2) The 400 event-free patients and the 100 patients who died first belong in the risk set; the 150 who already had hospitalization first do not. (3) Its denominator is \(1-F_H(6)\), which retains prior competing deaths, whereas the cause-specific hazard conditions only on being event-free; in the example the rates are \(0.05919\) and \(0.08000\) per month. (4) \(F_H(6)=P(T\leq 6,J=H)\), the hospitalization cumulative incidence at six months. (5) Risk-set membership or censoring can change during the interval, and the rate calculation treats the interval as sufficiently short that the stated risk-set count approximates its person-time.