Cause-Specific And Subdistribution Hazards

Why Two Hazards?

When more than one type of event can end follow-up, the event types compete: experiencing one prevents a person from experiencing another as the first event. For example, a patient may experience disease recurrence or die before recurrence. A model for recurrence must account for deaths, but there is more than one sensible question to ask. One question concerns the instantaneous recurrence rate among patients who remain event-free. Another concerns how recurrence accumulates in the original population, including the fact that some patients have already experienced a competing event.

These questions lead to the cause-specific hazard and the subdistribution hazard. They use different risk sets and describe different quantities. Their regression coefficients therefore do not answer the same question, even when both models use the same covariates.

Definition: Let $T$ be time to the first event, $J$ its cause, and $J=k$ the event type of interest. Given covariates $X=x$, the cause-specific hazard for cause $k$ is $$ \lambda_k(t\mid x)=\lim_{\Delta t\downarrow 0} \frac{P(t\leq T<t+\Delta t,\ J=k\mid T\geq t,\ X=x)}{\Delta t}. $$ It is the instantaneous rate of cause $k$ at time $t$, conditional on having had no event of any cause by $t$. The cumulative incidence function for cause $k$ is $F_k(t\mid x)=P(T\leq t,\ J=k\mid X=x)$. When this function is differentiable, the subdistribution hazard is $$ \lambda_k^*(t\mid x)=\frac{f_k(t\mid x)}{1-F_k(t\mid x)}, $$ where $f_k(t\mid x)=dF_k(t\mid x)/dt$. Its denominator is the probability of not yet having experienced cause $k$, including people who have already experienced a competing cause.

The Risk Sets and Their Estimands

At time $t$, the cause-specific hazard uses people who are still free of all event types. A person whose first event was a competing cause has left this risk set. The subdistribution hazard instead describes the rate at which the cumulative incidence of cause $k$ increases relative to the remaining probability $1-F_k(t)$. People with a prior competing event remain represented in that denominator: they cannot later experience cause $k$ as their first event, but their competing event contributes to the population’s probability of not having had cause $k$.

This difference also appears in model-based risk sets. In a cause-specific Cox analysis of cause $k$, a competing event ends a person’s time at risk for that cause, just as an event of interest does. Fine–Gray estimation uses a subdistribution risk set that retains people with competing events. With right-censored observations, estimation commonly uses weights to account for censoring. Such estimation requires an appropriate independent-censoring assumption conditional on the covariates used for the censoring model, as well as adequate probability of remaining uncensored over the times being analyzed.

A proportional cause-specific hazards model has the form $\lambda_k(t\mid x)=\lambda_{0k}(t)\exp(\beta^\mathsf{T}x)$. A proportional subdistribution hazards model has the form $\lambda_k^*(t\mid x)=\lambda_{0k}^*(t)\exp(\gamma^\mathsf{T}x)$. In each, the proportionality assumption concerns that model’s own hazard. The two baseline hazards and coefficients need not agree.

Worked Example: Comparing Risk Sets in Two Intervals

Consider a hypothetical cohort of 100 people followed without censoring. By the end of year 1, 20 have experienced event A, the event of interest, 10 have experienced competing event B, and 70 remain event-free. During year 2, 7 of those 70 experience event A.

Cause-specific view. At the start of year 2, only the 70 people still free of both events are at risk. The observed year-2 conditional probability of event A among them is $7/70=0.10$, or 10%. This is a discrete-interval probability, not a continuous-time hazard rate.

Subdistribution view. By the end of year 1, the cumulative incidence of A is $F_A(1)=20/100=0.20$. The probability of not having experienced A is $1-F_A(1)=0.80$, corresponding to 80 people in the subdistribution risk set: the 70 event-free people and the 10 who experienced B. The increase in A’s cumulative incidence during year 2 is $7/100=0.07$. The discrete-interval subdistribution hazard is therefore $0.07/0.80=0.0875$, or 8.75%.

The calculations differ because the denominators answer different questions. The cause-specific quantity is conditional on being free of every event at the start of the interval. The subdistribution quantity measures the increase in A’s cumulative incidence relative to the probability of not yet having experienced A.

How Cause-Specific Hazards Build Cumulative Incidence

The cause-specific hazard is not itself the probability of the event by a given time. To obtain the cumulative incidence of cause $k$, one must account both for the rate of cause $k$ and for the probability of still being free of every cause. For continuous event times, write $S(t\mid x)=P(T>t\mid X=x)$ for the probability of remaining event-free. If the possible causes are indexed by $j$, then

S(t\mid x)=\exp\left\{-\int_0^t\sum_j\lambda_j(u\mid x)\,du\right\}, \qquad F_k(t\mid x)=\int_0^t S(u\mid x)\lambda_k(u\mid x)\,du.

The first expression follows because the total instantaneous event rate among people still event-free is the sum of the cause-specific hazards. The second expression adds the probability of an event of type $k$ at each time $u$: a person must first still be event-free at $u$, with probability $S(u\mid x)$, and then experience cause $k$, at rate $\lambda_k(u\mid x)$. These relationships show why a cause-specific hazard alone does not determine the observed cumulative incidence: the hazards of the competing causes also affect how many people remain event-free and available to experience cause $k$.

Worked Example: From Constant Cause-Specific Hazards to Risk

Suppose, hypothetically, that two event types have constant cause-specific hazards: $\lambda_A=0.08$ per year and $\lambda_B=0.12$ per year. There is no additional censoring in this illustration. Calculate the five-year cumulative incidence of A.

The total hazard is $0.08+0.12=0.20$ per year, so the event-free survival probability is $S(t)=\exp(-0.20t)$. Substituting into the cumulative-incidence formula gives

F_A(5)=\int_0^5 0.08\exp(-0.20u)\,du =\frac{0.08}{0.20}\left(1-\exp(-0.20\times5)\right) =0.4(1-e^{-1}) \approx 0.2528.

The five-year probability of experiencing A first is approximately 0.2528, or 25.28%. The calculation uses both hazards: B reduces the event-free population that could later experience A. For contrast, $1-\exp(-0.08\times5)\approx0.3297$ is the risk under a model with only A’s hazard and no competing event. It is not the observed-world cumulative incidence of A in the two-cause setting.

At year 5, the cause-specific hazard of A remains 0.08 per year. The subdistribution hazard at that time is different. The density of A events is $f_A(5)=S(5)\lambda_A=e^{-1}(0.08)\approx0.02943$ per year, while $1-F_A(5)\approx0.7472$. Thus

\lambda_A^*(5)=\frac{0.02943}{0.7472}\approx 0.03939\text{ per year}.

This is a time-specific subdistribution hazard, not a second estimate of the cause-specific rate. Its denominator includes people who experienced B before year 5 but have not experienced A.

Interpreting Regression Coefficients

For a cause-specific Cox model, $\exp(\beta_r)$ is the multiplicative change in the cause-specific hazard associated with a one-unit increase in covariate $X_r$, holding the other modeled covariates fixed. The comparison is among people who are still free of all event types at time $t$. It is an instantaneous rate comparison in that selected risk set, not a comparison of cumulative-incidence probabilities.

In a Fine–Gray proportional subdistribution hazards model, $\exp(\gamma_r)$ is the multiplicative change in the subdistribution hazard associated with a one-unit increase in $X_r$, holding other modeled covariates fixed. It describes the relative rate of increase in the event’s cumulative incidence on the subdistribution scale, with competing-event cases retained in the risk set. It is not a risk ratio at a specified time, and it does not directly give an absolute change in cumulative incidence.

Worked Example: Reading Two Different Coefficients

Suppose hypothetical fitted models for event A report a treatment coefficient of $-0.40$ in the cause-specific Cox model and $0.25$ in the Fine–Gray model. Assume each model’s proportionality assumption is suitable for the intended analysis, covariates are specified appropriately, and censoring assumptions are satisfied.

In the cause-specific model, the treatment hazard ratio is $\exp(-0.40)\approx0.6703$. Conditional on still being free of all events, the treated group’s instantaneous cause-A hazard is estimated to be about 0.67 times that of the comparison group at a given time, under the model. This is about a 33.0% lower cause-specific hazard, not necessarily a 33.0% lower cumulative incidence of A.

In the Fine–Gray model, the subdistribution hazard ratio is $\exp(0.25)\approx1.284$. The treated group’s subdistribution hazard is estimated to be about 1.28 times that of the comparison group, under the model. It would be incorrect to interpret this as a 28.4% higher probability of A by any particular time.

Opposite coefficient signs are possible. A covariate can be associated with a lower cause-specific rate for A among people still event-free while also being associated with changes in competing events that alter the number and composition of people who remain free of A. These model outputs describe distinct estimands; they do not, by themselves, establish a causal effect.

A Short Derivation of the Subdistribution Hazard Relationship

The definition of the subdistribution hazard can be connected directly to the cumulative incidence. Since $1-F_k(t\mid x)$ is the probability of not yet having experienced cause $k$, differentiating its logarithm gives

\frac{d}{dt}\{-\log[1-F_k(t\mid x)]\} =\frac{f_k(t\mid x)}{1-F_k(t\mid x)} =\lambda_k^*(t\mid x).

Integrating from 0 to $t$, with $F_k(0\mid x)=0$, yields

\int_0^t\lambda_k^*(u\mid x)\,du =-\log[1-F_k(t\mid x)], \qquad F_k(t\mid x)=1-\exp\left\{-\int_0^t\lambda_k^*(u\mid x)\,du\right\}.

Thus, the subdistribution hazard determines the cumulative incidence for that cause through its cumulative value. This relationship does not make a subdistribution hazard a probability: the hazard has units of inverse time, while $F_k(t\mid x)$ is a probability.

Choosing and Reporting the Model

The choice follows the scientific estimand. A cause-specific model is appropriate when the target is the event-specific instantaneous rate among individuals who remain event-free, such as understanding how covariates relate to the rate of recurrence before any competing event. A Fine–Gray model is useful when the target is how covariates relate to the cumulative incidence of a particular event in the presence of competing events. Neither is universally preferable; the model should match the question.

A cause-specific analysis treats competing events as exits from the risk set for the cause of interest. This does not require assuming that the competing event is independent of the event of interest in a latent-time sense. It defines and estimates a cause-specific hazard among those still event-free. However, interpreting the resulting cause-specific coefficient as a causal effect requires appropriate causal assumptions beyond the hazard model itself.

Reports should name the event type, the hazard being modeled, the comparison, and the relevant risk set. For a Fine–Gray analysis, also report cumulative-incidence estimates at substantively meaningful times when those are central to the question. A hazard ratio alone does not provide an absolute risk difference.

Common Mistake

A frequent error is to report a cause-specific hazard ratio and a subdistribution hazard ratio as though both were alternative estimates of the same “risk ratio,” then select whichever supports the preferred conclusion. They are not interchangeable. The cause-specific ratio compares instantaneous rates among people free of every event. The subdistribution ratio compares the rate governing cumulative incidence while retaining competing-event cases in its risk set. State the estimand first, use the corresponding model, and interpret its coefficient on that model’s scale.

Key takeaway: Cause-specific hazards describe event-type-specific rates in the all-event-free risk set; subdistribution hazards describe the accumulation of a cause-specific cumulative incidence with competing-event cases retained in the risk set. Their coefficients answer different questions.

Check Your Understanding

Answer each question by identifying the risk set or estimand involved.

At time $t$, who contributes to the cause-specific risk set for event A, and who is additionally retained in the Fine–Gray subdistribution risk set?
Why can a cause-specific hazard ratio differ in direction from a subdistribution hazard ratio?
In the constant-hazard example, why is $1-\exp(-0.08\times5)$ not the five-year cumulative incidence of A when event B is also possible?
What does a Fine–Gray subdistribution hazard ratio of 1.28 mean, and what probability statement does it not mean?
Does estimating a cause-specific hazard require assuming that the latent times to competing event types are independent? Explain briefly.