Causal Interpretation of Fine-Gray Coefficients

What a Fine–Gray Treatment Coefficient Summarizes

A Fine–Gray regression coefficient is often reported as a treatment effect, but that phrase needs qualification. The coefficient is on the subdistribution hazard scale. It can summarize a treatment contrast in the cumulative incidence process under a specified model, but it is not itself a difference in event probabilities. Giving it a causal interpretation also requires assumptions about treatment assignment, censoring, and the model.

Let \(T\) be time to the first event and \(J\) its type, with \(J=k\) denoting the event of interest. The cumulative incidence function (CIF) for cause \(k\) is \(F_k(t)=P(T\le t,J=k)\). For a treatment \(A\) and baseline covariates \(L\), write the corresponding conditional CIF as \(F_k(t\mid A,L)\).

Definition: When the CIF is differentiable, the cause-\(k\) subdistribution hazard is \(\tilde\lambda_k(t\mid A,L)=\{dF_k(t\mid A,L)/dt\}/\{1-F_k(t\mid A,L)\}\). Its denominator is the probability of not yet having had the cause-\(k\) event, so people who have already experienced a competing event remain part of the subdistribution risk-set construction. It is not the instantaneous cause-\(k\) event rate among people who are alive and free of every event.

A proportional subdistribution hazards model with a binary treatment and baseline covariates can be written \(\tilde\lambda_k(t\mid A,L)=\tilde\lambda_{k0}(t)\exp(\beta A+\gamma^\mathsf{T}L)\). Here \(\tilde\lambda_{k0}(t)\) is an unspecified baseline subdistribution hazard, \(\beta\) is the treatment coefficient, and \(\gamma\) is the covariate coefficient vector. The model assumes that covariate effects multiply the subdistribution hazard by constant factors over time; in particular, the treatment subdistribution hazard ratio is constant over time conditional on \(L\).

Under this model, \(\exp(\beta)\) is the conditional subdistribution hazard ratio comparing \(A=1\) with \(A=0\), at the same covariate values. The interpretation is conditional on the included covariates and on the proportional subdistribution hazards model. It is not a ratio of probabilities, and it does not state how many additional people experience the event by a particular time.

Why the Coefficient Is Not an Absolute-Risk Effect

The relation between the subdistribution hazard and the CIF makes the distinction precise. If \(\tilde\Lambda_k(t\mid A,L)\) is the cumulative subdistribution hazard, then \(F_k(t\mid A,L)=1-\exp\{-\tilde\Lambda_k(t\mid A,L)\}\). Under the proportional model, \(\tilde\Lambda_k(t\mid A,L)=\tilde\Lambda_{k0}(t)\exp(\beta A+\gamma^\mathsf{T}L)\). Thus the coefficient changes the cumulative subdistribution hazard multiplicatively, while the CIF is obtained through a nonlinear transformation.

\frac{F_k(t\mid A=1,L)}{F_k(t\mid A=0,L)} = \frac{1-\exp\{-\tilde\Lambda_{k0}(t)\exp(\beta+\gamma^\mathsf{T}L)\}} {1-\exp\{-\tilde\Lambda_{k0}(t)\exp(\gamma^\mathsf{T}L)\}}

The right-hand side depends on the baseline cumulative subdistribution hazard, the covariate values, and time. It generally is not equal to \(\exp(\beta)\). The same coefficient can therefore correspond to different risk differences and risk ratios at different follow-up times or in populations with different covariate distributions.

Worked Example: Translating a Coefficient Into Risks

Suppose a correctly specified Fine–Gray model has \(\beta=\log(1.5)\). At a chosen follow-up time and fixed covariate values, suppose the baseline cumulative subdistribution hazard is \(0.20\), with the covariate contribution set to zero. Assume the proportional subdistribution hazards model holds at these covariate values and the time point. Calculate both model-implied CIFs and their contrasts.

For control, the cumulative subdistribution hazard is \(0.20\), so \(F_k(t\mid A=0,L)=1-\exp(-0.20)=0.1813\).

For treatment, it is \(0.20\exp(\beta)=0.20(1.5)=0.30\), giving \(F_k(t\mid A=1,L)=1-\exp(-0.30)=0.2592\).

The model-implied risk difference is \(0.2592-0.1813=0.0779\), or about 7.79 percentage points. The risk ratio is \(0.2592/0.1813=1.430\). The subdistribution hazard ratio is 1.5, but neither risk contrast is 1.5. These are conditional, model-implied risks at the stated time and covariate values, not population-average effects.

The CIF is an absolute probability of experiencing the event of interest by time \(t\) before a competing event. Since the CIF reflects the observed competing-event process, a treatment association with the competing event can also matter for the cause-\(k\) CIF. A Fine–Gray coefficient does not isolate a mechanism acting only on cause \(k\), nor does it describe the outcome under removal of competing events.

When a Causal Interpretation Is Defensible

Write \(T^a,J^a\) for the potential event time and event type under treatment assignment \(a\). The causal cause-\(k\) risk by time \(t\) under that assignment is \(F_k^a(t)=P(T^a\le t,J^a=k)\). A regression coefficient is not automatically a contrast between these potential-outcome quantities. For a causal interpretation, the analysis needs a treatment-assignment basis for identifying the relevant counterfactual distributions.

For baseline treatment and covariates, common identification conditions are consistency, conditional exchangeability, and positivity. Consistency means that a person's observed event time and cause under their received treatment agree with the potential event time and cause under that treatment. Conditional exchangeability means \((T^0,J^0,T^1,J^1)\) is independent of \(A\) given \(L\); random assignment can support this condition, while an observational study requires adequate measurement and adjustment of common causes of treatment and outcome. Positivity requires a positive probability of each treatment level at covariate values represented in the target population.

Censoring must also be handled under an appropriate assumption, such as independent censoring conditional on treatment and the modeled baseline covariates, with adequate support for censoring adjustment. In addition, the Fine–Gray regression form must adequately describe the conditional subdistribution hazards, including proportionality and any needed covariate or treatment interactions. These conditions are distinct: satisfying causal identification assumptions does not guarantee that the regression model is correctly specified.

If these conditions hold, the fitted model can describe a causal conditional subdistribution hazard contrast under treatment assignment. But the coefficient remains a conditional effect on that particular scale. To obtain marginal causal risks, predict the CIF under each treatment for members of a specified target population, then average those predictions. This standardization yields risks and risk contrasts in interpretable probability units.

Worked Model Predictions and Standardization

Worked Example: From Conditional Predictions to Population Risks

Consider a hypothetical randomized study with two equally sized baseline-risk strata. At a chosen follow-up time, a Fine–Gray model that allows the treatment contrast to vary by stratum gives the following unrounded predicted CIFs: control risks \(0.09518\) and \(0.329662\); treatment risks \(0.13058\) and \(0.428852\). Assume the model is adequate, censoring is handled appropriately, treatment assignment is randomized, and the target population gives equal weight to the two strata. The randomized design supports exchangeability, and both treatments occur in both strata, so positivity holds.

Average the predictions separately under each treatment. The standardized control risk is \((0.09518+0.329662)/2=0.212421\), which rounds to \(0.2124\). The standardized treatment risk is \((0.13058+0.428852)/2=0.279716\), which rounds to \(0.2797\).

Calculate contrasts using the unrounded averages, then round the results. The standardized risk difference is \(0.279716-0.212421=0.067295\), or \(6.73\) percentage points. The standardized risk ratio is \(0.279716/0.212421=1.31680\), which rounds to \(1.317\).

These are model-based, population-standardized causal risk contrasts at the specified follow-up time, provided the assumptions and model are adequate. They are not the treatment coefficient. Averaging rounded stratum-specific predictions or treating the conditional coefficient as the marginal risk ratio can produce incorrect contrasts.

The target population matters: different stratum weights can produce different standardized risks and contrasts even when the fitted conditional model is unchanged. A model without treatment-by-covariate interactions also imposes a common conditional subdistribution hazard ratio, which may be inappropriate if treatment associations vary across covariate strata. Model specification should reflect the scientific question and available data rather than be selected solely for a single summary coefficient.

Diagnostics, Alternatives, and Common Pitfalls

Diagnostics should address both the regression model and the causal design. Examine whether fitted subdistribution hazards appear proportional over time, whether treatment effects vary across important baseline covariates, and whether predictions are plausible across the covariate range. Check treatment overlap and the extent of censoring; sparse treatment support or heavy censoring can make estimates unstable. Sensitivity to reasonable model specifications is also informative. A good fit alone does not establish absence of unmeasured confounding.

The cause-specific hazard approach answers a different question: its hazard is conditional on remaining free of all event types, whereas the Fine–Gray subdistribution hazard is linked directly to the cause-specific CIF. Neither hazard ratio is an absolute risk contrast. For a question about event probabilities by a specified time, treatment-specific CIFs and their standardized differences or ratios are more direct summaries. The choice should follow the estimand, not a preference for one regression coefficient.

Common pitfall: “The Fine–Gray treatment hazard ratio is 1.5, so treatment raises the probability of the event by 50%.” This is incorrect: 1.5 is a ratio of conditional subdistribution hazards under the model, not a risk ratio. Calculate treatment-specific CIFs at a stated time and, for a population-level causal contrast, standardize them to the target covariate distribution.

A causal label is also not justified just because a regression includes many covariates or because the coefficient is statistically distinguishable from zero. Confounding control depends on whether the adjustment set and design support exchangeability; the model's functional form and censoring assumptions still matter. Finally, the coefficient does not describe an intervention that prevents competing events. It concerns the treatment contrast for the observed competing-risk outcome process.

Key takeaway: A Fine–Gray treatment coefficient summarizes a conditional subdistribution hazard contrast. Causal interpretation requires design and modeling assumptions; absolute causal risks and risk contrasts require treatment-specific CIF predictions standardized to a defined target population.

Worked Example: Confounding Can Create an Apparent Treatment Contrast

Suppose a hypothetical observational cohort has a binary baseline risk group. At a fixed time, the cause-\(k\) risks are \(0.30\) in the high-risk group and \(0.05\) in the low-risk group under either treatment. Assume treatment has no effect within either group. However, 80% of treated people are high risk, compared with 20% of untreated people. These risks are assumed to be correctly measured, but treatment is not exchangeable without adjustment for risk group.

The crude risk among treated people is \(0.80(0.30)+0.20(0.05)=0.25\). Among untreated people it is \(0.20(0.30)+0.80(0.05)=0.10\). The crude risk ratio is \(0.25/0.10=2.5\), despite equal treatment-specific risks within each stratum. The difference arises because the treatment groups have different risk-group distributions. A coefficient from an inadequately adjusted Fine–Gray model would not repair this causal problem merely by being a Fine–Gray coefficient.

Adjustment can address this measured imbalance if the model is adequate and both treatments are represented within each risk group. It cannot, by itself, eliminate bias from unmeasured confounding. This example illustrates why the causal status of a coefficient depends on the treatment-assignment assumptions, not on the regression method alone.

Check Your Understanding

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

In a Fine–Gray model with treatment coefficient \(\beta\), what does \(\exp(\beta)\) compare?
Why does a subdistribution hazard ratio of 1.5 not imply a 50% increase in the CIF?
Name three assumptions that support a causal interpretation of treatment-specific CIFs in a baseline-treatment study.
Why average treatment-specific predictions before calculating a marginal risk difference?
Can a good proportional subdistribution hazards fit rule out unmeasured confounding? Explain.

Solutions. (1) It is the conditional treatment subdistribution hazard ratio, comparing \(A=1\) with \(A=0\) at the same modeled covariate values, under the fitted model. (2) The CIF is \(1-\exp(-\tilde\Lambda_k)\), a nonlinear function of cumulative subdistribution hazard, so the resulting risk ratio depends on baseline risk and follow-up time. (3) Consistency, conditional exchangeability, and positivity are standard conditions; appropriate censoring assumptions and a correctly specified model are also needed for the proposed estimation. (4) Averaging the predictions under each treatment gives risks for the chosen target population; their difference is the marginal risk difference for that population. (5) No. Model fit concerns the outcome model and does not establish that treatment groups are exchangeable or that all confounders were measured.