Interpreting Fine-Gray Subdistribution Hazard Ratios

What a Fine–Gray Coefficient Compares

A Fine–Gray regression coefficient describes how a covariate is associated with the occurrence of one specified event type in the presence of competing events. Its exponentiated value is a subdistribution hazard ratio (SHR). The word “hazard” can invite an overly familiar interpretation: an SHR is not the ratio of the probabilities of the event, and it is not the cause-specific hazard ratio for that event. Its denominator and risk-set meaning are different.

Let \(T\) be the time to the first event and \(J\) its type, with \(J=k\) denoting the event of interest. The cumulative incidence function (CIF) for type \(k\) is \(F_k(t)=P(T\leq t,J=k)\). It gives the probability of experiencing event \(k\) by time \(t\), before a competing event occurs. In continuous time, the subdistribution hazard is defined through the rate of change of this CIF relative to the probability of not yet having experienced event \(k\).

The denominator is the essential distinction. In a cause-specific hazard, individuals leave the risk set when they have any event, including a competing event. In the Fine–Gray construction, people who have had a competing event remain represented in the subdistribution risk set; the weighted partial likelihood accounts for censoring, rather than treating these people as if they could later experience the event of interest. This definition is about the model’s event-specific rate, not a literal rate among only people who remain event-free.

Model Assumptions and the Coefficient’s Meaning

The coefficient has a direct relative interpretation under the specified model, but that interpretation depends on the model and data assumptions. For a fixed covariate contrast, the proportional subdistribution-hazards model assumes that the SHR does not vary over time. It also requires an appropriate covariate form: for example, a linear term assumes a linear association on the log subdistribution-hazard scale. The model must be fitted for a clearly identified event type, with the competing event types coded distinctly from censoring.

Censoring must be independent of the event process, at least conditional on the covariates used in the censoring adjustment, and the censoring model must provide adequate support over the follow-up period. If people who are censored differ systematically in their unobserved future event experience, the weighted estimation may not remove the resulting bias. The usual regression interpretation also presumes independent observational units, unless dependence is explicitly accounted for.

Under these assumptions, \(\exp(\beta_j)\) compares the subdistribution hazards for a one-unit increase in \(x_j\), holding the other included covariates fixed, when \(x_j\) is entered as a linear main effect. For a binary covariate coded 1 versus 0, it compares the group coded 1 with the reference group coded 0. The comparison is an adjusted association conditional on the model’s covariates; it does not by itself establish a causal effect.

From a Subdistribution Hazard Ratio to Cumulative Incidence

The link between the model and absolute event probability makes clear why an SHR is not a risk ratio. Define the cumulative subdistribution hazard by \(\tilde{\Lambda}_k(t\mid x)=\int_0^t\tilde{\lambda}_k(u\mid x)\,du\). Since \(\tilde{\lambda}_k(t)=\frac{dF_k(t)/dt}{1-F_k(t)}\), integrating both sides gives the relationship between the cumulative subdistribution hazard and the CIF.

Thus \(F_k(t)=1-\exp\{-\tilde{\Lambda}_k(t)\}\). Under the proportional model, \(\tilde{\Lambda}_k(t\mid x)=\tilde{\Lambda}_{k0}(t)\exp(\beta^\top x)\), so the model-implied CIF is \(1-\exp\{-\tilde{\Lambda}_{k0}(t)\exp(\beta^\top x)\}\). This is a nonlinear transformation. The SHR multiplies the cumulative subdistribution hazard, whereas the risk at a horizon depends on both that multiplier and the baseline cumulative subdistribution hazard at that horizon.

Interpreting Direction, Scale, and Uncertainty

For a positive coefficient, the SHR exceeds 1 and indicates a higher modeled subdistribution hazard for the stated covariate contrast. For a negative coefficient, it is below 1 and indicates a lower modeled subdistribution hazard. The percentage description is obtained from \((\mathrm{SHR}-1)\times100\%\): for example, an SHR of 0.80 corresponds to a 20% lower subdistribution hazard, not a 20% lower event probability. Always state the contrast, the event type, and the adjustment context.

A confidence interval for \(\beta\) is commonly transformed by exponentiating both endpoints to obtain an interval for the SHR. A 95% confidence interval describes a procedure that, under repeated sampling and the model assumptions, would cover the true parameter in 95% of samples. It is not a 95% probability statement about the fixed parameter after observing the data. If the interval for the SHR includes 1, the data are compatible with no subdistribution-hazard contrast at the corresponding confidence level; this does not prove the contrast is zero.

How This Differs From Other Effect Measures

The cause-specific hazard ratio and the SHR can both be reported as exponentiated regression coefficients, but they answer different questions. The cause-specific hazard ratio compares instantaneous event rates among people still free of every event type. The SHR compares subdistribution hazards, whose risk-set construction retains representation of people who have experienced a competing event. Neither coefficient is automatically a risk ratio.

An SHR is often useful when the scientific question concerns how covariates relate to the cumulative incidence of a particular event in the presence of competing events. But it remains a relative model parameter. To report an absolute probability, calculate or estimate the CIF at a specified horizon for clearly stated covariate values; do not substitute the SHR for that probability or for a risk ratio.

Common Interpretive Errors and Limitations

• Calling an SHR a risk ratio. A ratio of 1.5 does not mean a 50% higher probability by a chosen time. Use the model-implied or estimated CIFs to calculate a horizon-specific risk contrast.
• Calling an SHR a cause-specific hazard ratio. The cause-specific risk set removes people after any event; the subdistribution risk set retains competing-event experience. The measures have different definitions.
• Leaving the contrast unspecified. “The SHR is 0.80” is incomplete unless the covariate change, reference group, event type, and adjustment context are stated.
• Interpreting an association causally. A fitted coefficient alone does not establish that changing the covariate would change the event outcome.
• Assuming a single constant SHR fits every time pattern. The basic Fine–Gray model assumes proportional subdistribution hazards. If that structure is unsuitable, the reported coefficient may summarize an inadequate model rather than a stable relative effect.

The SHR is also not a direct description of each individual’s probability or of the physical event rate among currently event-free individuals. Its interpretation relies on the model, covariate coding, censoring assumptions, and specified event type. A numerically precise estimate cannot compensate for a poorly chosen model or unsupported censoring assumptions.