Choosing Between Cause-Specific and Fine-Gray Regression

Begin With the Scientific Question

Cause-specific Cox regression and Fine-Gray regression are not interchangeable ways to estimate one universal “effect” in competing risks. They describe different quantities. The first concerns the instantaneous rate of a particular event among people who have not yet experienced any event. The second concerns a subdistribution hazard constructed to model that event’s cumulative incidence. The right choice follows from the scientific question, not from which coefficient looks more compelling or has a smaller p-value.

Suppose $T$ is time to the first event and $J$ identifies its type, with $J=k$ denoting the event of interest. Let $F_k(t\mid x)=P(T\leq t,J=k\mid x)$ be its cumulative incidence by time $t$, given covariates $x$. A competing event is a different event type that prevents the event of interest from occurring first. For example, death from another cause competes with death from the cause under study.

Definition: The cause-specific hazard for event $k$, $\lambda_k(t\mid x)$, is the instantaneous rate of event $k$ at time $t$, conditional on covariates $x$ and on being free of all event types just before $t$. The subdistribution hazard for event $k$, $\alpha_k(t\mid x)$, is the rate defined by the event-$k$ cumulative incidence and its complement: $$ \alpha_k(t\mid x)=\frac{f_k(t\mid x)}{1-F_k(t\mid x)}, $$ where $f_k(t\mid x)$ is the event-$k$ cumulative-incidence density when it exists. It is the hazard associated with the cumulative incidence, not the ordinary rate among people still free of all events.

A cause-specific Cox model is commonly used to relate covariates to $\lambda_k(t\mid x)$. A Fine-Gray model relates covariates to $\alpha_k(t\mid x)$. In the latter model, people who have already had a competing event remain represented in a modified, weighted risk set; they are not being treated as if they remain physically event-free. This construction is what makes the model’s target the cumulative incidence.

Why the Targets Differ

The event-$k$ cumulative incidence depends both on the event-$k$ hazard and on survival free of every event. If there are $K$ event types, write $S(t\mid x)=P(T>t\mid x)$. The standard competing-risks relationship is

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

The notation $S(u-\mid x)$ means the probability of remaining free of all event types just before $u$. Thus a covariate can change $F_k$ by changing $\lambda_k$, by changing one or more competing-event hazards, or by changing both. This is the central reason a cause-specific hazard ratio and a subdistribution hazard ratio need not agree.

The formula also gives a practical distinction. If the question is “among people still free of any event, how is the instantaneous rate of event $k$ associated with this covariate?”, the cause-specific hazard is directly aligned. If the question is “how does the covariate relate to the probability of having event $k$ by a specified time, given the competing events that occur in this population?”, cumulative incidence is the target; Fine-Gray regression is one way to model it.

Worked Example: An Etiologic Question About an Event-Specific Rate

In a hypothetical cohort, investigators study whether a medication is associated with the instantaneous rate of cardiovascular death among patients who remain free of any death. Other-cause death is a competing event. The scientific question is about the event-specific rate in the event-free group, rather than the probability of cardiovascular death by a particular horizon.

A cause-specific Cox model gives a hypothetical medication hazard ratio of $0.62$, with a 95% confidence interval from $0.43$ to $0.90$. The target is the cardiovascular-death cause-specific hazard. The point estimate says that, at a given time and among patients still free of death, the fitted cardiovascular-death rate in the medication group is 0.62 times the rate in the reference group, under the model.

Because the confidence interval is entirely below 1, these hypothetical data are inconsistent with a null cause-specific hazard ratio at the conventional two-sided 5% level under the model and its assumptions. That result is an association, not proof that the medication causes a reduction. In particular, the model’s risk-set comparison is conditional on remaining event-free, and confounding or selection processes may affect that comparison.

A Fine-Gray model could instead address how medication use relates to the cardiovascular-death cumulative incidence. That is a different target: it incorporates the observed competing-event process rather than isolating the event-specific rate among those still event-free.

A Numerical Illustration of Competing Events

A simple constant-hazard model makes the distinction explicit. Assume two event types, constant cause-specific hazards $\lambda_1$ and $\lambda_2$, no delayed entry, and follow-up from time zero. The event-free survival is $S(t)=\exp\{-(\lambda_1+\lambda_2)t\}$. Integrating the event-1 hazard against that survival gives

F_1(t)=\int_0^t e^{-(\lambda_1+\lambda_2)u}\lambda_1\,du =\frac{\lambda_1}{\lambda_1+\lambda_2} \left\{1-e^{-(\lambda_1+\lambda_2)t}\right\}.

The calculation assumes that each cause-specific hazard is constant over time within each group. It does not assume that the two causes are independent latent event times; it specifies the observable first-event hazards. The resulting cumulative incidence changes with either cause-specific hazard.

Worked Example: Same Cause-Specific Rate, Different Event Risk

Consider two hypothetical groups followed for five years. In both, the cause-specific hazard for event 1 is $0.10$ per year. The competing-event hazard is $0.05$ per year in group A and $0.15$ per year in group B. Assume the hazards are constant within each group.

For group A, the total event hazard is $0.10+0.05=0.15$ per year. Therefore,

F_{1,A}(5)=\frac{0.10}{0.15}\left(1-e^{-0.15(5)}\right) =0.6667(1-e^{-0.75}) =0.6667(0.5276) \approx 0.3518.

The model gives about 35.2% cumulative incidence of event 1 by five years in group A. For group B, the total hazard is $0.10+0.15=0.25$ per year, so

F_{1,B}(5)=\frac{0.10}{0.25}\left(1-e^{-0.25(5)}\right) =0.4000(1-e^{-1.25}) =0.4000(0.7135) \approx 0.2854.

The event-1 cause-specific hazard ratio for B versus A is $0.10/0.10=1$, but the five-year event-1 cumulative incidences are about 28.5% and 35.2%. The higher competing-event hazard in B leaves fewer people available to experience event 1 first. The difference in cumulative incidence is therefore compatible with no difference in the event-1 cause-specific hazard.

This is not a paradox and does not mean one model is wrong. The models answer different questions. The cause-specific hazard describes an event-specific rate in the event-free risk set. The cumulative incidence describes the probability of that event being the first event by a given time in the presence of competing events.

Choosing a Model in Practice

Make the choice by writing the estimand in words before fitting a regression. Specify the event, the population, the time scale or horizon, and whether the question concerns a rate among event-free people or a cumulative probability in the presence of competing events. Then choose the model whose target corresponds to that statement.

State the outcome and competing events.
Define event $k$ and identify which other first events preclude it. Check that event coding matches the scientific definitions.

Write the question as a target.
Use a cause-specific hazard for an event-specific rate among those still event-free. Use cumulative incidence, potentially modeled with Fine-Gray regression, for prognosis about occurrence by a specified horizon.

Check whether the model can support the intended summary.
Consider covariate adjustment, censoring, event counts, and whether a proportional-hazards summary is plausible over the time range being reported.

Report the target and its limits.
Name the hazard or cumulative-incidence quantity, the comparison, and the time context. Do not describe a hazard ratio as a risk ratio or a causal effect without additional justification.

For either regression, the event types and times must be measured and coded correctly. Censoring must be noninformative for the relevant analysis, at least conditional on the covariates used and other modeled information: in broad terms, censoring should not selectively hide event experience in a way left unaccounted for. The covariate specification must be adequate for the intended comparison. A constant hazard-ratio coefficient also relies on a proportional-hazards assumption for the relevant hazard; if that is implausible, one constant ratio may not summarize the association over follow-up.

For a prognosis question, a hazard ratio alone may not answer what a clinician or patient needs. A predicted cumulative incidence at a named horizon is often more directly interpretable, and its calibration should be checked against observed outcomes in an appropriate validation setting. For an etiologic description, inspect whether the chosen event-free risk set and adjustment set represent the intended comparison. Sparse events, coding errors, influential observations, or poorly supported covariate patterns can undermine either model. These checks do not change the estimand; they assess whether the fitted model is a credible way to estimate it.

Worked Example: Matching a Prognosis Question to Fine-Gray Regression

A hypothetical clinical team wants to compare the probability of non-cardiovascular death within three years for two patient profiles, accounting for cardiovascular death as a competing event. The requested quantity is explicitly a three-year event probability, not the instantaneous rate among patients still alive.

Fine-Gray regression is aligned with this question because it models the subdistribution hazard for non-cardiovascular death, which is linked to that cause’s cumulative incidence. Suppose the fitted model reports a subdistribution hazard ratio of $1.30$ for the profile with a specified risk factor versus the reference profile. The appropriate interpretation is that the fitted non-cardiovascular-death subdistribution hazard is 1.30 times as high, under the model, for that comparison. It is not correct to say that the three-year probability is 30% higher: a subdistribution hazard ratio is not a risk ratio.

To answer the team’s probability question, use the fitted model to obtain predicted three-year cumulative incidences. Suppose these are 0.12 for the reference profile and 0.17 for the risk-factor profile. The estimated absolute difference is $0.17-0.12=0.05$, or 5 percentage points by three years. This is the requested probability-scale comparison, subject to model adequacy and appropriate uncertainty quantification. The two estimates are hypothetical predictions, not observed facts about a real population.

If the team instead asked whether the risk factor is associated with the instantaneous non-cardiovascular-death rate among patients still alive, the cause-specific hazard would match that different question. Choosing Fine-Gray here is justified by the stated prognosis target, not by a general rule that Fine-Gray is preferable for clinical data.

When Both Models Are Useful—and What They Cannot Establish

The two approaches can be complementary when a study has both an etiologic question about event-specific rates and a prediction question about event probabilities. Fitting both does not make their coefficients competing estimates of the same parameter. Each should be labeled and interpreted on its own scale. Conversely, fitting both by default and highlighting whichever result is statistically significant is not a principled way to choose a target.

Neither cause-specific nor Fine-Gray regression automatically identifies a causal effect. A regression coefficient is an association under the fitted model. Causal interpretation requires a well-defined intervention or exposure contrast and suitable assumptions, including control of confounding and appropriate handling of selection and censoring. In particular, a cause-specific hazard comparison conditions on remaining event-free, a status that can itself be affected by covariates. A Fine-Gray coefficient is also not a direct causal effect on the cumulative incidence merely because that outcome is clinically meaningful.

A common mistake is to argue, “The cause-specific model removes people with competing events, so it ignores them; therefore Fine-Gray must be the correct model.” The correction is that leaving the cause-specific risk set is part of the definition of its target, not automatically a defect. If the question concerns the event-specific rate among people still event-free, that is exactly the quantity of interest. If the question concerns the probability of the event by a horizon in a world where competing events occur, a cumulative-incidence target is needed. The scientific question decides which quantity matters.

Key takeaway: Choose cause-specific regression for associations with an event-specific rate among those still event-free; choose a cumulative-incidence target, often modeled with Fine-Gray regression, for prognosis about the probability of an event by a specified time. State the target explicitly, and do not treat either coefficient as a causal effect without further assumptions.

Check Your Understanding

Answer each question, then compare with the brief solutions.

A study asks whether a biomarker is associated with the instantaneous rate of relapse among patients who have had neither relapse nor death. Which regression target aligns with the question?
A treatment leaves the cause-specific hazard for an event unchanged but lowers a competing-event hazard. Could the event’s cumulative incidence change? Explain.
A Fine-Gray model gives a subdistribution hazard ratio of $0.80$. Is it correct to conclude that the event probability is 20% lower by every follow-up time? Why or why not?
Give one reason why a statistically significant cause-specific hazard ratio does not establish that a treatment causally changes the event rate.

Solutions. (1) The cause-specific hazard, because the target is the instantaneous event rate among people still free of all events. (2) Yes. The cumulative incidence depends on the competing-event hazards through event-free survival, so changing a competing hazard can change the probability of the event of interest. (3) No. $0.80$ is a ratio of subdistribution hazards under the model, not a 20% relative reduction in cumulative-incidence probability at every time. (4) For example, confounding may remain, or conditioning on remaining event-free may induce selection; statistical significance does not address these causal assumptions.