Reporting Fine-Gray Estimates Without Overclaiming

What a Fine-Gray Estimate Says

A Fine-Gray regression coefficient is often reported as a subdistribution hazard ratio (SHR). The number is compact, but it is easy to overstate what it means. An SHR is neither a ratio of cumulative incidence probabilities nor, by itself, evidence that a covariate causes an outcome. A sound report names the event of interest, identifies the covariate contrast and adjustment set, gives uncertainty, and describes the estimate on the subdistribution-hazard scale.

For event type \(k\), let \(F_k(t)=P(T\leq t,J=k)\) denote the cumulative incidence function (CIF): the probability of experiencing event \(k\) by time \(t\) before a competing event. The subdistribution hazard for event \(k\) is defined by

\widetilde{\lambda}_k(t) = \lim_{\Delta t\downarrow 0} \frac{P(t\leq T<t+\Delta t,\ J=k\mid T\geq t\ \text{or}\ (T<t,\ J\ne k))}{\Delta t}.

This definition uses a special risk set: people who have already experienced a competing event remain represented in it. Consequently, the subdistribution hazard is not the ordinary instantaneous event rate among people who are still free of every event. The Fine-Gray proportional subdistribution hazards model is commonly written

\widetilde{\lambda}_k(t\mid \mathbf{x}) = \widetilde{\lambda}_{k0}(t)\exp(\boldsymbol{\beta}^{\mathsf T}\mathbf{x}).

Here \(\mathbf{x}\) is the covariate vector, \(\widetilde{\lambda}_{k0}(t)\) is the baseline subdistribution hazard, and \(\boldsymbol{\beta}\) contains regression coefficients. For a one-unit increase in a covariate, holding the other modeled covariates fixed, the corresponding SHR is \(\exp(\beta)\). For a categorical covariate, it compares a specified category with its reference category, conditional on the other modeled covariates.

Definition: An SHR is a ratio of subdistribution hazards for a stated covariate contrast under a Fine-Gray model. Under proportional subdistribution hazards, this ratio is constant over time. It is not itself a risk ratio, a difference in cumulative incidence, or a causal effect.

What to Include in the Results

A useful report lets the reader identify the outcome, comparison, time context, model, and uncertainty without having to infer them from a table. State which event is the event of interest and how competing events were treated. For every covariate estimate, name the contrast or unit, identify the reference group when relevant, and state which other covariates were included in the model. Report the SHR with a confidence interval, normally at a stated confidence level, and include a p-value only when it adds useful information.

For a continuous covariate, a one-unit contrast may be too small or otherwise awkward to interpret. You may report an SHR for a clinically meaningful increment, such as 10 years of age, but state that increment explicitly. For a categorical variable, provide the reference category and make clear whether the estimate is adjusted. Avoid a table heading such as “risk ratio” for an SHR: that label describes a different quantity.

Worked Example: Reporting an Adjusted SHR and Its Interval

In a hypothetical cohort, a Fine-Gray model for cardiovascular death treats non-cardiovascular death as a competing event. The adjusted coefficient for a treatment indicator is \(\hat{\beta}=0.4055\), with \(SE=0.1800\). The model includes age and baseline disease severity. The treatment group is compared with usual care.

Estimate. Exponentiating the coefficient gives

\widehat{SHR}=\exp(0.4055)=1.500.

Confidence interval. Using the Wald interval on the log scale, the 95% confidence limits are \(0.4055\pm1.96(0.1800)\). The margin is \(1.96(0.1800)=0.3528\), so the limits on that scale are \(0.0527\) and \(0.7583\). Exponentiating them gives

\left(\exp(0.0527),\exp(0.7583)\right) \approx(1.054,\ 2.135).

A suitable report is: “After adjustment for age and baseline disease severity, the treatment group had an estimated subdistribution hazard for cardiovascular death 1.50 times that of the usual-care group (95% CI 1.05 to 2.13).” The interval excludes 1, which is consistent with a nonzero coefficient at the two-sided 5% level under the Wald approximation. The wording does not claim that treatment raises the probability of cardiovascular death by 50%.

Why an SHR Is Not a Risk Ratio

A hazard ratio compares hazards; a risk ratio compares probabilities over a specified time horizon. Even under the proportional subdistribution hazards model, converting an SHR into a CIF comparison requires a baseline CIF at the horizon and the model’s proportionality relationship. In the single-covariate contrast form, that relationship is

F_1(t)=1-\{1-F_0(t)\}^{\exp(\beta)},

where \(F_0(t)\) is the CIF for the reference covariate value and \(F_1(t)\) is the model-implied CIF for the comparison value, with other covariates held fixed. This equation shows why the SHR is not generally equal to \(F_1(t)/F_0(t)\): the conversion depends on \(F_0(t)\), and therefore on the time horizon and baseline risk.

Worked Example: A Subdistribution Hazard Ratio and a CIF Contrast

Suppose, hypothetically, the reference group has a CIF of \(0.20\) for the event of interest by a prespecified time, and the fitted proportional subdistribution hazard ratio for the comparison group is \(1.50\). Under the model relationship above, the comparison-group CIF is

F_1(t)=1-(1-0.20)^{1.50} =1-0.8^{1.50} \approx1-0.715542 =0.284458.

Thus the model-implied comparison CIF is about \(0.2845\), or \(28.45\%\), versus \(20.0\%\) in the reference group. Their risk ratio at this horizon is

\frac{0.284458}{0.20}\approx1.422,

not 1.50. The CIF difference is about \(0.0845\), or 8.45 percentage points. These are model-implied absolute-risk summaries for the stated horizon and contrast, not the SHR itself. They do not establish a causal effect. In an applied report, absolute-risk predictions and their uncertainty should be presented using an appropriate prediction procedure; do not manufacture them by relabeling the SHR.

Assumptions, Diagnostics, and Limits

A report should not present the fitted SHR as meaningful without considering whether the model and data support it. The usual interpretation relies on a correctly specified proportional subdistribution hazards model for the covariates being described. It also relies on suitable handling of censoring: censoring must be independent of the event process conditional on modeled information, or the method used must otherwise account for the relevant censoring mechanism. Event causes and censoring must be classified consistently, and the observed data must provide enough information to estimate the contrasts.

The proportionality assumption says that the subdistribution hazard ratio for a covariate contrast does not vary over time. It is worth checking time-varying patterns or other relevant model diagnostics rather than treating the assumption as guaranteed. Also inspect the number of events of interest and competing events, the covariate distributions, convergence warnings, and whether the functional form chosen for continuous covariates is plausible. Sparse data can produce unstable estimates and wide intervals; model fitting alone does not make a contrast precise.

If proportionality or the covariate form is doubtful, a single reported SHR may conceal important time variation or misspecification. The analysis may need an alternative specification or a more direct presentation of time-specific CIFs. Such checks do not convert the SHR into a causal effect. Causal interpretation requires a suitable causal design and assumptions beyond those required to fit and report this regression.

Name the estimand.
Specify the event of interest and identify the Fine-Gray subdistribution hazard as the modeled quantity.

State the contrast and adjustment.
Give the unit or reference category, the covariates included, and whether the estimate is adjusted.

Report estimate and uncertainty.
Give the SHR and confidence interval, using the same contrast and scale in both. Include a p-value only with its appropriate interpretation.

Check the claim against the estimand.
Use “subdistribution hazard,” not “risk,” “probability,” or “causal effect,” unless separately reporting a justified absolute-risk or causal estimand.

Choosing Accurate Language

Cause-specific Cox regression addresses a different hazard: the instantaneous rate of the event of interest among individuals who remain free of all event types. A Fine-Gray model instead targets the subdistribution hazard linked to the CIF. The choice depends on the scientific question, and the two estimates should not be substituted for one another. Unadjusted CIF estimates answer an absolute-probability question for the groups observed; they are not regression coefficients. The earlier material on cause-specific Cox estimates uses the cause-specific hazard scale, so its wording should not be carried over unchanged to a Fine-Gray result.

A concise, defensible sentence might read: “In the adjusted Fine-Gray model for [event of interest], [group] had an estimated subdistribution hazard [higher/lower] than [reference group] (SHR [value], 95% CI [limits]), conditional on the covariates listed.” If an interval includes 1, avoid describing the result as evidence of a clear increase or decrease. State the estimate and interval, and let their uncertainty remain visible. A p-value is the probability, under the null model and the test assumptions, of a result at least as extreme as the observed statistic; it is not the probability that the null hypothesis is true.

Worked Example: Making the Covariate Scale Explicit

Consider a hypothetical adjusted Fine-Gray model for hospitalization, with death before hospitalization as a competing event. The estimated SHR for age, entered per 10-year increase, is \(1.08\) with a 95% confidence interval of \(0.98\) to \(1.19\). The model also includes baseline comorbidity.

The contrast is a 10-year age difference, not a one-year difference. A suitable interpretation is: “At the same modeled comorbidity level, each 10-year increase in age was associated with an estimated 1.08-fold subdistribution hazard of hospitalization (95% CI 0.98 to 1.19).” The interval includes 1, so the data do not provide strong evidence of a non-null coefficient at the conventional two-sided 5% level. This is not evidence that the true SHR is exactly 1, and it does not imply that the hospitalization probability is 8% higher.

The statement identifies the outcome, competing event, age increment, adjustment, estimate, and uncertainty. It describes an association on the model’s scale and does not claim that age causes the outcome.

Common Mistakes

Calling the SHR a risk ratio. An SHR compares subdistribution hazards. A risk ratio compares CIFs at a specified horizon; the two need not be equal.
Giving a unit-free continuous effect. “SHR 1.08 for age” is incomplete if the reader cannot tell whether the contrast is per year, 10 years, or another increment.
Using causal language from an adjusted model alone. Adjustment does not by itself establish exchangeability, eliminate unmeasured confounding, or justify a causal conclusion.
Omitting uncertainty or the competing event. A point estimate alone hides precision, and an estimate without its event definition is difficult to interpret.
Interpreting a confidence interval as a probability statement about the parameter. A 95% confidence procedure has its stated long-run coverage under its assumptions; it does not mean there is a 95% probability that this fixed parameter lies in the observed interval.

Key takeaway: Report an SHR as a covariate contrast on the subdistribution-hazard scale, with its interval, covariate units, reference group, and adjustment set. It can inform how the fitted model relates covariates to the CIF, but it is not itself a risk ratio or a causal effect.

Check Your Understanding

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

A Fine-Gray model reports an SHR of 0.72 for treatment versus usual care. What does the number compare, and what must you not call it?
A continuous biomarker has an SHR of 1.10, but the report does not state its unit. What information is missing?
At a fixed horizon, the reference CIF is 0.10 and the SHR is 2.00. Using the proportional subdistribution hazards relationship, calculate the comparison CIF.
Does adjustment for several measured covariates make an SHR a causal effect? Explain briefly.

Solutions. (1) It compares the modeled subdistribution hazards for the stated treatment contrast, conditional on the other modeled covariates. It must not be called a risk ratio or an absolute increase in probability. (2) The report must state the increment represented by the estimate, such as per 1-unit or per 5-unit increase, and identify the biomarker’s measurement scale. (3) \(F_1=1-(1-0.10)^2=1-0.81=0.19\), so the model-implied comparison CIF is 19% at that horizon; the risk ratio is \(0.19/0.10=1.90\), not 2.00. (4) No. Causal interpretation requires additional design and causal assumptions; covariate adjustment alone is not sufficient.