The Fine-Gray Model for a Subdistribution Hazard

What the Fine-Gray Model Describes

In competing-risks data, a subject can experience one of several mutually exclusive first events. For a selected event type, called the target cause, the scientific question is often about its cumulative incidence: the probability that this event has occurred by a given time, before any competing event. The Fine-Gray model describes covariate associations with this target cumulative incidence through a corresponding subdistribution hazard.

This is a different target from the cause-specific hazard. A cause-specific hazard concerns the instantaneous rate of the target event among subjects who have experienced no event yet. The subdistribution hazard is defined from the target cause’s cumulative incidence function. Its denominator therefore does not represent only subjects who remain free of every event: subjects who have already had a competing event remain part of the mathematical construction. This distinction is central to interpreting the model.

Definition: Let $T$ denote time to the first event, $J$ its cause, and $k$ the target cause. For covariate profile $z$, the cause-specific cumulative incidence function is $F_k(t\mid z)=P(T\leq t,J=k\mid z)$. If this function has density $f_k(t\mid z)$, the subdistribution hazard for cause $k$ is $$ \tilde{\lambda}_k(t\mid z)=\frac{f_k(t\mid z)}{1-F_k(t\mid z)}. $$ The cumulative subdistribution hazard is $\tilde{\Lambda}_k(t\mid z)=\int_0^t\tilde{\lambda}_k(u\mid z)\,du$, when the integral is defined.

Because $F_k$ concerns only the target event, $1-F_k(t\mid z)$ includes both subjects who have not yet had any event and subjects who have already had a competing event by time $t$. This is a definition based on the cumulative incidence, not a claim that a subject with a competing event remains physically at risk of the target event. The model uses this constructed subdistribution hazard to express how covariates relate to the target event’s cumulative incidence.

Model Specification and Coefficient Meaning

Let $Z$ be a vector of covariates measured at baseline, and let $\beta$ be the corresponding coefficient vector. The Fine-Gray proportional subdistribution hazards model for cause $k$ is

\tilde{\lambda}_k(t\mid Z)=\tilde{\lambda}_{k0}(t)\exp(\beta^{\mathsf T}Z).

Here, $\tilde{\lambda}_{k0}(t)$ is an unspecified baseline subdistribution hazard: it may vary freely with time, but is common to covariate profiles when $Z=0$. The model specifies a log-linear covariate effect and assumes that the covariate multiplier does not change over time. That constant-multiplier condition is the proportional subdistribution hazards assumption.

Interpretation: Holding the other modeled covariates fixed, a one-unit increase in covariate $Z_j$ multiplies the target cause’s subdistribution hazard by $\exp(\beta_j)$. More generally, comparing profiles $z_a$ and $z_b$, their subdistribution-hazard ratio is $\exp\{\beta^{\mathsf T}(z_a-z_b)\}$, at every time under the model.

The multiplier is dimensionless; each subdistribution hazard has units of inverse time. The coefficient itself is on the log subdistribution-hazard scale. For a binary covariate coded 0 and 1, $\exp(\beta_j)$ compares the covariate-1 profile with the covariate-0 profile, conditional on other included covariates. For a continuous covariate, the one-unit interpretation depends on its measurement scale, so rescaling or centering can make the comparison more useful.

Why the Model Implies a Cumulative-Incidence Relation

The link between the subdistribution hazard and cumulative incidence follows directly from the definition. Write $S_k^*(t\mid z)=1-F_k(t\mid z)$. Since $f_k(t\mid z)=dF_k(t\mid z)/dt$, the definition gives

\frac{dF_k(t\mid z)}{dt} =\tilde{\lambda}_k(t\mid z)\{1-F_k(t\mid z)\}, \qquad \frac{dS_k^*(t\mid z)}{dt} =-\tilde{\lambda}_k(t\mid z)S_k^*(t\mid z).

For $S_k^*(0\mid z)=1$, divide the second equation by $S_k^*(t\mid z)$ and integrate from 0 to $t$. This yields

\log S_k^*(t\mid z)=-\int_0^t\tilde{\lambda}_k(u\mid z)\,du =-\tilde{\Lambda}_k(t\mid z), \qquad F_k(t\mid z)=1-\exp\{-\tilde{\Lambda}_k(t\mid z)\}.

Under the Fine-Gray model, the cumulative subdistribution hazard is $\tilde{\Lambda}_{k0}(t)\exp(\beta^{\mathsf T}z)$, so the model-implied cumulative incidence is

F_k(t\mid z)=1-\exp\left\{-\tilde{\Lambda}_{k0}(t)\exp(\beta^{\mathsf T}z)\right\}.

This shows why a coefficient is a multiplicative effect on the subdistribution hazard, not generally a multiplicative effect on cumulative incidence. The exponential transformation from cumulative subdistribution hazard to cumulative incidence is nonlinear.

Worked Example: One-Unit Increase in a Continuous Covariate

Suppose a hypothetical fitted model for a target event has coefficient $\hat{\beta}=0.240$ for a baseline biomarker measured in units of 10 points. At a specified time, the estimated baseline cumulative subdistribution hazard is $0.120$. Compare profiles whose biomarker values differ by one model unit, with all other covariates held fixed.

Subdistribution-hazard multiplier. The estimated multiplier is $\exp(0.240)=1.271$. Thus the profile one unit higher has an estimated target subdistribution hazard 1.271 times that of the lower profile at every time under the proportional model. This is a 27.1% higher subdistribution hazard, not a 27.1% higher cumulative incidence.

Cumulative-incidence calculation at the specified time. The higher profile’s estimated cumulative subdistribution hazard is $0.120\exp(0.240)=0.120(1.2712492\ldots)=0.1525499\ldots$, which rounds to 0.1525 to four decimal places. Its model-implied cumulative incidence is $1-\exp(-0.1525499)\approx0.1415$, or about 14.15%.

For comparison, the lower profile has cumulative incidence $1-\exp(-0.120)\approx0.1131$, or about 11.31%. The estimated difference is about 2.84 percentage points at this time. These two incidences are not in the ratio 1.271; the ratio applies to the subdistribution hazards.

Assumptions and Conditions for Interpretation

A coefficient has the interpretation above only in a model that is appropriate for the data and the question. The main conditions to state are:

Competing-event definition: Event types are mutually exclusive first-event categories, and the target cause is correctly identified. The time origin and event-time definitions are meaningful and consistently applied.
Proportional subdistribution hazards: The covariate effects are represented by the specified exponential multiplier and do not vary with time. A constant coefficient is not an adequate summary if the effect changes materially over follow-up.
Covariate specification: The chosen functional forms and coding describe the intended contrasts. A linear term on the log-hazard scale, for example, represents a constant multiplicative change per unit, not an arbitrary nonlinear relationship.
Censoring and observation: Estimation requires censoring to be handled under suitable conditions, commonly independent censoring conditional on modeled covariates, together with adequate follow-up and support for the prediction times of interest. The data must contain enough information to estimate the target association.

These conditions do not make the model causal. In observational data, a coefficient describes a covariate association conditional on the included model terms; unmeasured confounding, selection, or measurement error can remain. The proportionality assumption and time-varying effects require dedicated assessment; they should not be inferred merely because a software routine returns a coefficient.

Worked Example: A Binary Covariate and a Baseline Rate

Consider hypothetical data on time to a target complication, with death before the complication as a competing event. A fitted model uses $Z=1$ for an exposure and $Z=0$ for no exposure. Suppose the estimated coefficient is $0.700$, and the estimated baseline subdistribution hazard at a particular time is $0.080$ per year. Assume the event coding is correct, censoring is adequately handled, and the proportional subdistribution hazards model is being used for illustration.

The estimated exposure multiplier is $\exp(0.700)=2.014$. At the stated time, the estimated subdistribution hazard for $Z=1$ is $0.080\exp(0.700)=0.080(2.01375)=0.1611$ per year. The corresponding value for $Z=0$ is $0.080$ per year. The comparison is conditional on any other modeled covariates, if present.

This result says that the exposure profile has about 2.014 times the target subdistribution hazard at that time under the fitted model. It does not say that exposed subjects have twice the instantaneous rate of the target event among those still event-free; that is a cause-specific hazard comparison. Nor does it say that exposure causes the target event.

Worked Example: Combining Two Covariate Contrasts

Suppose a hypothetical Fine-Gray model contains two baseline covariates, $Z_1$ and $Z_2$, with coefficients $\hat{\beta}_1=0.400$ and $\hat{\beta}_2=-0.300$. Compare profile A, with $(Z_1,Z_2)=(0,0)$, to profile B, with $(Z_1,Z_2)=(1,2)$. Assume the fitted model has no interaction and its proportionality and covariate-form assumptions are suitable for these profiles.

The linear predictor difference is $0.400(1-0)-0.300(2-0)=0.400-0.600=-0.200$. Therefore the subdistribution-hazard ratio for B versus A is $\exp(-0.200)=0.8187$. Under the model, profile B has an estimated target subdistribution hazard about 18.13% lower than profile A at every time.

If the baseline subdistribution hazard at a particular time is $0.050$ per month, then profile B’s model-implied value is $0.050\exp(-0.200)=0.050(0.8187)=0.04094$ per month, compared with $0.050$ per month for profile A. This calculation depends on the specified covariate values and the no-interaction model; an interaction term would change the contrast formula.

How This Differs from Cause-Specific Cox Regression

Cause-specific Cox regression and Fine-Gray regression address different quantities. In the cause-specific approach, a subject who experiences a competing event leaves the risk set for the target cause. In Fine-Gray regression, the model targets the subdistribution hazard defined from the target cumulative incidence. Both approaches can be useful, but their coefficients should not be exchanged or interpreted as if they described the same rate.

Feature	Cause-specific Cox model	Fine-Gray model
Modeled quantity	Cause-specific hazard among subjects still event-free	Subdistribution hazard defined from the target cause’s cumulative incidence
Competing event in the target construction	Removes the subject from the target cause’s event-free risk set	Is accounted for through the subdistribution definition
Coefficient’s direct multiplier	Cause-specific hazard	Target subdistribution hazard
Primary descriptive connection	Instantaneous target-event rate among event-free subjects	Target cause’s cumulative incidence over time

A covariate can have different associations with the cause-specific hazard and the subdistribution hazard. For example, a covariate might increase the target event’s cause-specific hazard but also strongly increase a competing event’s hazard. The cumulative incidence of the target event reflects both processes; its Fine-Gray coefficient is not determined by the target cause-specific coefficient alone.

Diagnostics, Limitations, and a Common Mistake

Before interpreting a fitted coefficient, verify the target cause, event coding, time scale, covariate reference levels, and units. Check that the intended covariate contrast is represented by the model, that the data support the time horizon being discussed, and that censoring is not so extensive in relevant covariate groups that estimation becomes unstable. Sparse target events or sparse covariate patterns can yield imprecise estimates even when the model can be fit.

The proportionality assumption is substantive: it requires a constant covariate multiplier over time. If it fails, one coefficient can conceal distinct early and late associations. The model also relies on the chosen functional form for continuous covariates and may be sensitive to coding or influential observations. A fitted model does not guarantee reliable extrapolation beyond the times and covariate profiles supported by the observed data.

Common mistake: “The Fine-Gray coefficient is positive, so the covariate raises the target event’s cause-specific hazard by $\exp(\beta)$, and the cumulative incidence is multiplied by the same amount.” This reasoning assigns the coefficient to the wrong hazard and then treats a nonlinear probability as a rate. The correction is to interpret $\exp(\beta)$ as a multiplicative effect on the target subdistribution hazard under the model. Obtain the cumulative incidence through $1-\exp\{-\tilde{\Lambda}_k(t\mid z)\}$; do not infer a constant cumulative-incidence ratio from the coefficient alone.

Key takeaway: Fine-Gray regression models the target cause’s subdistribution hazard as a baseline function of time multiplied by $\exp(\beta^{\mathsf T}Z)$. A coefficient describes a covariate’s multiplicative association with that hazard; the model-implied cumulative incidence follows through a nonlinear transformation.

Check Your Understanding

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

If a coefficient is $-0.400$, what is the subdistribution-hazard multiplier for a one-unit increase in its covariate?
Why can a subject with a competing event still contribute to the subdistribution-hazard definition?
A model gives a multiplier of 1.5. Does this imply that the cumulative incidence is 1.5 times as large at every time? Explain.
For coefficients $\beta_1=0.200$ and $\beta_2=-0.100$, calculate the subdistribution-hazard ratio comparing profiles differing by $+2$ units in $Z_1$ and $+1$ unit in $Z_2$.

Solutions. (1) $\exp(-0.400)=0.6703$, so the modeled subdistribution hazard is multiplied by about 0.6703 per unit increase, holding other covariates fixed. (2) The denominator is $1-F_k(t\mid z)$, defined from the target cause’s cumulative incidence; it includes subjects who have not had the target event, including those with a prior competing event. This is a mathematical construction, not a claim that they remain physically at risk for the target event. (3) No. The multiplier applies to the subdistribution hazard; cumulative incidence is $1-\exp\{-\tilde{\Lambda}_k(t\mid z)\}$, a nonlinear transformation. (4) The linear predictor difference is $0.200(2)-0.100(1)=0.300$, so the ratio is $\exp(0.300)=1.350$.