Competing-Event Rates and Cause-Specific Cox Coefficients

Conditional Rates and Absolute Event Probabilities

A cause-specific Cox coefficient answers a conditional question: among individuals who have not yet experienced any event, how does a covariate relate to the instantaneous rate of the target event? An absolute probability asks a different question: what proportion will experience that event by a specified time? These quantities are connected, but they are not interchangeable. In particular, the probability of a target event depends on the rates of all competing events, because those events can remove individuals from the event-free group.

The distinction matters when comparing groups. A group can have a higher target-event hazard and yet a similar, or even lower, target-event probability if its competing-event hazard is also higher. A cause-specific Cox coefficient alone therefore does not determine an absolute event probability.

Definition: Let $T$ be the time to the first event and $J\in\{1,\ldots,K\}$ its cause. The cause-specific hazard for cause $k$, conditional on covariates $X=x$, is $$ \lambda_k(t\mid x)=\lim_{\Delta t\to 0} \frac{P(t\leq T<t+\Delta t,\ J=k\mid T\geq t,\ X=x)}{\Delta t}. $$ It is an instantaneous rate, with units of inverse time, among individuals who remain free of every event just before time $t$. The event-free survival function is $S(t\mid x)=P(T>t\mid X=x)$, and the cumulative incidence function for cause $k$ is $F_k(t\mid x)=P(T\leq t,\ J=k\mid X=x)$.

What a Cause-Specific Cox Coefficient Describes

For a target cause $k$, a cause-specific Cox model has the form

\lambda_k(t\mid X)=\lambda_{k0}(t)\exp(\beta_k^\mathsf{T}X).

Here $\lambda_{k0}(t)$ is the unspecified baseline hazard for cause $k$, and $\beta_k$ is its coefficient vector. For two covariate values $x_a$ and $x_b$, the model implies

\frac{\lambda_k(t\mid x_a)}{\lambda_k(t\mid x_b)} =\exp\{\beta_k^\mathsf{T}(x_a-x_b)\}.

For a one-unit increase in a continuous covariate, or for a specified category relative to a reference category, $\exp(\beta_k)$ is the cause-specific hazard ratio, provided the proportional-hazards model is appropriate. It compares instantaneous target-event rates at time $t$ for individuals with those covariate values who are still event-free at $t$. It is not, by itself, a ratio of cumulative incidence probabilities.

The cause-specific model is fitted using the target event as the event of interest; a different event type ends that individual’s time in the event-free risk set. This risk-set construction estimates the target cause-specific hazard under appropriate assumptions about censoring and the model. It does not mean that a competing event is ordinary loss to follow-up when the goal is to calculate absolute probabilities. For that calculation, the competing cause must remain part of the event process.

Worked Example: Interpreting a Cause-Specific Cox Estimate

In a hypothetical study, a binary treatment indicator has estimated coefficient $\hat\beta_1=0.405$ in a Cox model for the target event, with estimated standard error $SE=0.200$. Assume the proportional-hazards form is adequate for this target cause and that treatment is coded 1 versus 0.

The estimated cause-specific hazard ratio is $$ \exp(\hat\beta_1)=\exp(0.405)=1.499. $$

Thus, at a given time, among individuals still free of every event, the estimated instantaneous target-event hazard in the treated group is about 1.50 times that in the untreated group. This is a conditional rate comparison, not a statement that 50% more treated individuals will experience the target event.

For illustration, a Wald 95% confidence interval for the coefficient is $0.405\pm1.96(0.200)=[0.013,0.797]$. Exponentiating its endpoints gives an approximate hazard-ratio interval of $[1.013,2.219]$. Under the model and large-sample approximation, the interval procedure has 95% coverage across repeated samples. Here the interval is compatible with a positive conditional association, but it does not give a confidence interval for the difference in target-event probabilities.

From Cause-Specific Hazards to Cumulative Incidence

The event-free group is depleted by every cause. With cause-specific hazards $\lambda_1,\ldots,\lambda_K$, the total hazard is their sum, and the event-free survival function is

S(t\mid x)=\exp\left\{-\sum_{j=1}^{K}\Lambda_j(t\mid x)\right\}, \qquad \Lambda_j(t\mid x)=\int_0^t\lambda_j(u\mid x)\,du.

At time $u$, the probability of still being event-free is $S(u\mid x)$. Multiplying this by the target cause’s instantaneous hazard gives the rate of accumulating target events in the original population. Integrating over time yields

F_k(t\mid x)=\int_0^t S(u\mid x)\lambda_k(u\mid x)\,du.

This identity is also a direct justification of the distinction: a competing hazard changes $S(u\mid x)$, and therefore can change $F_k(t\mid x)$, even if it leaves $\lambda_k(u\mid x)$ unchanged. The derivation uses mutually exclusive first-event causes, hazards defined among individuals still event-free, and a well-defined event-time distribution. It does not require the causes to be statistically independent latent event times.

Worked Example: Same Target Hazard, Different Competing Hazard

Consider two hypothetical groups followed for five years. Suppose their hazards are constant over time. In both groups, the target-event hazard is $0.080$ per year. The competing-event hazard is $0.120$ per year in Group A and $0.320$ per year in Group B. These assumptions give a cause-specific target hazard ratio of 1.00 between groups.

For constant hazards $\lambda_1$ and $\lambda_2$, the target cumulative incidence by time $t$ is $$ F_1(t)=\frac{\lambda_1}{\lambda_1+\lambda_2} \left[1-\exp\{-(\lambda_1+\lambda_2)t\}\right]. $$

For Group A, the total hazard is $0.080+0.120=0.200$ per year, so $$ F_{1,A}(5)=\frac{0.080}{0.200}\left[1-\exp\{-(0.200)(5)\}\right] =0.400(1-\exp(-1))=0.2528. $$

For Group B, the total hazard is $0.080+0.320=0.400$ per year, giving $$ F_{1,B}(5)=\frac{0.080}{0.400}\left[1-\exp\{-(0.400)(5)\}\right] =0.200(1-\exp(-2))=0.1729. $$

The target-event hazard is identical, yet the five-year target-event probability is about 25.3% in Group A and 17.3% in Group B. The higher competing hazard in Group B removes more people from the event-free group before they can experience the target event. The probability of any first event by five years is $1-\exp(-1)=0.6321$ in Group A and $1-\exp(-2)=0.8647$ in Group B; those totals are not target-event probabilities.

Why a Higher Target Hazard May Not Mean Higher Risk

The same logic applies when the target-event hazard differs across groups. A positive cause-specific coefficient can coexist with a lower target cumulative incidence if the competing-event hazard is sufficiently elevated. The next example calculates this explicitly.

Worked Example: Higher Target Hazard but Lower Target Incidence

Suppose two hypothetical groups have constant hazards over a five-year period. The reference group has target hazard $0.080$ per year and competing hazard $0.120$ per year. The comparison group has target hazard $0.120$ per year and competing hazard $0.500$ per year. The comparison-group cause-specific target hazard ratio is $0.120/0.080=1.50$.

For the reference group, $$ F_{1,0}(5)=\frac{0.080}{0.080+0.120} \left[1-\exp\{-(0.080+0.120)5\}\right] =0.400(1-\exp(-1))=0.2528. $$

For the comparison group, $$ F_{1,1}(5)=\frac{0.120}{0.120+0.500} \left[1-\exp\{-(0.120+0.500)5\}\right] =\frac{0.120}{0.620}(1-\exp(-3.10))=0.1848. $$

The comparison group has a 50% higher target cause-specific hazard but a lower five-year target-event probability, about 18.5% versus 25.3%. These results are not contradictory: one compares instantaneous target rates among those currently event-free, and the other accumulates target events in the presence of both causes. The hazard values are hypothetical and constant by assumption; real analyses would need to estimate the hazard functions and account for uncertainty.

A Short Justification for the Competing-Hazard Effect

For two causes with constant hazards, write the target hazard as $\lambda_1$ and the competing hazard as $\lambda_2$. The target cumulative incidence can be written as

F_1(t)=\int_0^t \exp\{-(\lambda_1+\lambda_2)u\}\lambda_1\,du.

Holding $\lambda_1$ fixed, increasing $\lambda_2$ by an amount $\delta>0$ changes the integrand at each $u>0$ by the factor $\exp(-\delta u)<1$. More explicitly, the derivative is

\frac{\partial F_1(t)}{\partial\lambda_2} =-\int_0^t u\,\lambda_1\exp\{-(\lambda_1+\lambda_2)u\}\,du<0 \qquad (t>0,\ \lambda_1>0).

Thus, in this setting, raising the competing hazard lowers the target cumulative incidence while leaving the target cause-specific hazard unchanged. With time-varying hazards, the same survival-weighting mechanism applies through the integral formula: a larger competing hazard over some interval reduces the event-free probability available for later target events.

Assumptions, Diagnostics, and Alternatives

A cause-specific Cox coefficient is interpretable as a constant hazard ratio over time only if the proportional-hazards specification for that cause is reasonable. The model also requires an adequate covariate specification and appropriate handling of the observation process. In particular, censoring must be independent of the target event process conditional on the modeled information for the usual analysis to identify the cause-specific hazard. A competing event is observed as an event type, not treated as unobserved censoring when estimating absolute cumulative incidence.

To estimate a group-specific cumulative incidence from cause-specific models, one needs all relevant cause-specific hazards, not just the target model. Each hazard must be estimated adequately for the covariate pattern of interest. If covariates vary over time, the hazard prediction must use a defined covariate history; simply inserting a baseline value may not represent the intended prediction. Predicted risks also depend on the chosen time horizon and on how covariates are held fixed or averaged over a population.

Useful checks include examining whether the target cause-specific hazard ratio changes over time, checking functional forms and interactions, and assessing whether predicted event-free survival and cause-specific event probabilities are plausible. A notable mismatch between observed and predicted cumulative incidence can arise from misspecified hazards for the target or competing causes, not only from the target model. Sparse numbers at risk late in follow-up also make hazard estimates and resulting absolute risks unstable.

The Fine-Gray model is a different regression approach that relates covariates to a subdistribution hazard associated with the cumulative incidence function. Its coefficient does not have the same risk-set interpretation as a cause-specific Cox coefficient, and it is not a replacement label for the same estimand. Cause-specific models are useful when the question concerns instantaneous rates among those still event-free or the separate event processes; Fine-Gray regression can be useful when modeling covariate associations with cumulative incidence directly. For absolute probabilities from cause-specific models, all causes must be combined through event-free survival and the cumulative-incidence integral.

Common pitfall: “The cause-specific hazard ratio is 1.5, so the target-event risk is 50% higher.” This treats a conditional instantaneous-rate ratio as a probability ratio. The correction is to state the hazard-ratio interpretation among individuals still event-free, then calculate or estimate the target cumulative incidence using the hazards for every competing cause and a specified time horizon.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A Cox model gives a cause-specific hazard ratio of 0.80 for a treatment. State its interpretation and one conclusion it does not establish.
With constant target and competing hazards $0.10$ and $0.15$ per year, respectively, calculate the target cumulative incidence by two years.
Two groups have the same target hazard, but one has a larger competing hazard. What happens to target cumulative incidence under constant hazards, and why?
Why is a target cause-specific Cox model alone insufficient to calculate a target-event probability by five years?

Solutions. (1) At a given time, among individuals still free of any event, the treatment group’s instantaneous target-event hazard is estimated to be 0.80 times the reference group’s, under the model. This does not establish that treatment lowers the five-year target-event probability by 20%; that probability also depends on competing hazards. (2) The total hazard is $0.10+0.15=0.25$ per year, so $$ F_1(2)=\frac{0.10}{0.25}\left[1-\exp\{-(0.25)(2)\}\right] =0.400(1-\exp(-0.50))=0.1574. $$ The target-event probability is about 15.7% by two years. (3) It decreases, because the larger competing hazard lowers event-free survival and leaves fewer individuals available to experience the target event. (4) The target model supplies only the target cause-specific hazard. The probability calculation also needs the competing cause-specific hazards to determine event-free survival, as well as a specified covariate pattern and time horizon.