Proportional Subdistribution Hazards in Fine-Gray Models

Why Proportionality Has a Different Meaning Here

A Fine–Gray regression coefficient describes a covariate association with the subdistribution hazard for a specified event cause. The proportionality assumption says that, for a fixed covariate contrast, the ratio of these hazards does not change over time. This is a statement about subdistribution hazards—not about the ratio of cumulative incidences, which generally does change over time.

The distinction matters because a subdistribution hazard is defined using the cumulative incidence of the event of interest. Its risk-set construction retains people who have already experienced a competing event, with appropriate weighting in estimation. Thus, a Fine–Gray coefficient has a different target from a cause-specific Cox coefficient, even though both models use a proportional hazards form.

Definition: For event cause $k$, let $F_k(t\mid z)=P(T\leq t,J=k\mid Z=z)$ be the cumulative incidence function (CIF), where $T$ is event time, $J$ is event type, and $Z=z$ is the covariate vector. The cause-$k$ subdistribution hazard is $$ \widetilde{\lambda}_k(t\mid z) = \frac{f_k(t\mid z)}{1-F_k(t\mid z)}, $$ where $f_k(t\mid z)$ is the cause-$k$ event density. In the Fine–Gray proportional subdistribution hazards model, $$ \widetilde{\lambda}_k(t\mid z) = \widetilde{\lambda}_{k0}(t)\exp(\beta^\mathsf{T}z). $$ Here $\widetilde{\lambda}_{k0}(t)$ is an unspecified baseline subdistribution hazard and $\beta$ is a fixed coefficient vector.

The denominator is $1-F_k(t\mid z)$, not the probability of remaining free of every event. A person who has had a competing event has not had event $k$, so that person remains part of the subdistribution-hazard construction. This definition is why the subdistribution hazard is not simply the instantaneous rate of event $k$ among people currently event-free.

Assumption and Derivation of the Constant Ratio

The proportionality assumption is that covariates multiply the cause-$k$ baseline subdistribution hazard by a factor that does not vary with time. For two covariate vectors $z_A$ and $z_B$, the model uses the same baseline function and coefficient vector at every time. The resulting ratio follows directly by dividing the two model expressions.

\frac{\widetilde{\lambda}_k(t\mid z_A)} {\widetilde{\lambda}_k(t\mid z_B)} = \frac{\widetilde{\lambda}_{k0}(t)\exp(\beta^\mathsf{T}z_A)} {\widetilde{\lambda}_{k0}(t)\exp(\beta^\mathsf{T}z_B)} = \exp\{\beta^\mathsf{T}(z_A-z_B)\}.

The baseline cancels, so the ratio has no $t$ in it. If $z_A-z_B$ is fixed and $\beta$ is fixed, the model therefore implies a time-constant subdistribution hazard ratio. For a binary covariate $Z$, comparing $Z=1$ with $Z=0$ gives $\exp(\beta)$. For a continuous covariate, the ratio depends on the size of the contrast: a one-unit increase gives $\exp(\beta)$, while a $d$-unit increase gives $\exp(d\beta)$, assuming a linear covariate term.

This conclusion is conditional on the model’s covariates and functional form. It does not assert that the subdistribution hazard itself is constant over time: the baseline $\widetilde{\lambda}_{k0}(t)$ may vary freely. Nor does it say that every possible covariate contrast has the same ratio; the contrast determines the exponent.

Assumptions for the model interpretation: The event cause and time origin are defined consistently; the specified covariates and their functional forms are appropriate; the log subdistribution-hazard effect is constant over time; and censoring is independent of the event process conditional on the modeled covariates, or is handled with a valid method that accounts for the censoring mechanism. The baseline subdistribution hazard need not follow a parametric shape. As with other regression associations, a fitted coefficient alone does not establish a causal effect.

From Subdistribution Hazards to Cumulative Incidence

The constant ratio is not generally a constant ratio of CIFs. The connection can be derived from the definition. Since $f_k(t\mid z)=\widetilde{\lambda}_k(t\mid z)\{1-F_k(t\mid z)\}$ and $f_k$ is the derivative of $F_k$, we have

\frac{dF_k(t\mid z)}{dt} = \widetilde{\lambda}_k(t\mid z)\{1-F_k(t\mid z)\}.

Rearrange and integrate from time 0, where $F_k(0\mid z)=0$, to time $t$:

\frac{dF_k(t\mid z)}{1-F_k(t\mid z)} = \widetilde{\lambda}_k(t\mid z)\,dt, \qquad -\log\{1-F_k(t\mid z)\} = \widetilde{\Lambda}_k(t\mid z),

where $\widetilde{\Lambda}_k(t\mid z)=\int_0^t\widetilde{\lambda}_k(u\mid z)\,du$ is the cumulative subdistribution hazard. Thus,

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

Even when two subdistribution hazards have a constant ratio, applying $1-\exp(-x)$ to their cumulative hazards does not preserve that ratio. CIF differences and ratios can consequently vary with time. The Fine–Gray proportionality assumption is a model for the subdistribution hazard; it is not a claim that covariates multiply the CIF by a constant.

Worked Example: A Binary Covariate and a Constant Ratio

Suppose a hypothetical Fine–Gray model for relapse (cause $k$) uses a binary treatment indicator $Z$, with $Z=1$ for the treatment group and $Z=0$ for the comparison group. The fitted coefficient is $\hat{\beta}=0.4055$. Interpret the modeled contrast.

Model. The assumed subdistribution hazard is $\widetilde{\lambda}_k(t\mid Z)=\widetilde{\lambda}_{k0}(t)\exp(\beta Z)$. The model specifies one fixed coefficient, and therefore assumes proportional subdistribution hazards for the two groups.

Ratio. Substituting the treatment and comparison values gives $$ \frac{\widetilde{\lambda}_k(t\mid Z=1)} {\widetilde{\lambda}_k(t\mid Z=0)} = \exp\{0.4055(1-0)\} = \exp(0.4055) \approx 1.500. $$

Under the fitted model, the treatment group’s subdistribution hazard for relapse is 1.500 times the comparison group’s at each modeled time. This is a 50.0% higher subdistribution hazard, not a 50.0% higher relapse probability. The interpretation is an association conditional on the model covariates; it is not, by itself, a causal treatment effect.

Worked Example: Constant Hazard Ratio, Different CIF Ratio

Continue with the binary-covariate model above. At a particular follow-up time, suppose the estimated baseline cumulative subdistribution hazard is $0.2000$. Calculate the model-implied CIFs for $Z=0$ and $Z=1$, then compare their ratio with the subdistribution hazard ratio.

For the comparison group, the cumulative subdistribution hazard is $0.2000$, so $$ F_k(t\mid Z=0) = 1-\exp(-0.2000) \approx 0.1813. $$

For the treatment group, the cumulative subdistribution hazard is multiplied by $\exp(0.4055)\approx1.500$, giving $0.2000(1.500)=0.3000$. Therefore $$ F_k(t\mid Z=1) = 1-\exp(-0.3000) \approx 0.2592. $$

At this time, the CIF ratio is $0.2592/0.1813\approx1.430$, whereas the subdistribution hazard ratio is $1.500$. The difference is expected: the constant ratio applies to the hazards, while the CIFs are obtained through a nonlinear transformation of the cumulative hazards. At other times, the CIF ratio may differ again as the baseline cumulative subdistribution hazard changes.

Worked Example: A Continuous Covariate Contrast

In a hypothetical model, age is entered in years with coefficient $\hat{\beta}=0.0400$ per year. Find the subdistribution hazard ratio for a 5-year age difference, and state the proportionality implication.

For an older-versus-younger contrast of $d=5$ years, the modeled ratio is $$ \exp(d\hat{\beta}) = \exp(5\times0.0400) = \exp(0.2000) \approx1.221. $$

The model therefore associates a 5-year increase in age with a subdistribution hazard 1.221 times as large, or about 22.1% higher, at each time, conditional on other modeled covariates. A one-year contrast would instead have ratio $\exp(0.0400)\approx1.041$. Both interpretations rely on the assumed linear effect of age on the log subdistribution hazard and on proportionality over time.

What Proportionality Does—and Does Not—Require

The assumption concerns the relative effect over time, not a flat hazard curve. The baseline can rise, fall, or vary irregularly while the covariate multiplier remains fixed. Conversely, a model may have a plausible baseline shape but fail proportionality if the covariate contrast changes over follow-up.

In practice, concern is warranted when the scientific mechanism suggests an effect that strengthens, weakens, or reverses with time, or when estimated covariate contrasts across follow-up appear inconsistent with a stable multiplier. Sparse events late in follow-up can make apparent changes highly unstable, so such patterns need uncertainty considered rather than being treated as decisive evidence. Model adequacy also depends on the covariate scale: a fixed coefficient for a linear age term does not compensate for a substantially nonlinear age association.

Formal assessment requires methods designed for Fine–Gray regression. A pattern in predicted CIFs alone is not evidence against proportional subdistribution hazards, because CIF ratios need not be constant even when the assumption holds. Likewise, a time-varying CIF difference does not by itself imply a time-varying coefficient. The central diagnostic question is whether the covariate effect on the log subdistribution hazard is plausibly stable over time.

Model feature	Fine–Gray proportionality means	It does not mean
Covariate effect	A fixed covariate contrast has a constant subdistribution hazard ratio.	The cause-specific hazard ratio is the same as the subdistribution hazard ratio.
Baseline	The baseline subdistribution hazard is common to covariate profiles.	The baseline hazard must be constant or follow a specified parametric curve.
Predicted risk	The CIF is linked to the cumulative subdistribution hazard by $F_k=1-\exp(-\widetilde{\Lambda}_k)$.	The CIF ratio is constant over time or equals the subdistribution hazard ratio.

Comparison, Limitations, and a Common Pitfall

A cause-specific Cox model describes the instantaneous rate of a particular cause among individuals who remain free of all event types. A Fine–Gray model instead targets the subdistribution hazard linked to the CIF for a particular cause. Their risk-set definitions and interpretations differ, so their coefficients need not agree. Choosing between them depends on the scientific question: an etiologic question about the rate among those still event-free and a prediction question about cumulative incidence are not interchangeable.

The proportional subdistribution hazards model is useful when a constant relative subdistribution-hazard effect is a defensible summary and the CIF is of interest. Its limitations include sensitivity to a poor covariate form, inadequate handling of censoring, and an effect that genuinely changes over time. A single coefficient can also hide important differences in absolute risk: the same relative hazard multiplier can correspond to different CIF differences depending on baseline risk.

Common mistake: “The fitted subdistribution hazard ratio is 1.5, so the event probability is 50% higher at every time.” This confuses a hazard ratio with a probability ratio. The correct interpretation is that the model multiplies the cause-specific subdistribution hazard by 1.5 at each time for the stated covariate contrast. The CIF must be calculated from the cumulative subdistribution hazard and will not generally have a constant ratio.

Key takeaway: With a fixed Fine–Gray coefficient and a fixed covariate contrast, the subdistribution hazard ratio is $\exp\{\beta^\mathsf{T}(z_A-z_B)\}$, constant over time under the model. This proportionality statement is about the subdistribution hazard, not the CIF or the cause-specific hazard.

Check Your Understanding

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

A binary covariate has Fine–Gray coefficient $-0.2231$. What is the modeled subdistribution hazard ratio for $Z=1$ versus $Z=0$, and how should its direction be described?
A model has coefficient $0.0600$ per year for a continuous covariate. What is the subdistribution hazard ratio for a 10-year contrast?
If proportional subdistribution hazards hold, must the CIF ratio for two covariate groups be constant over time? Explain.
State one conceptual reason a cause-specific Cox coefficient and a Fine–Gray coefficient can differ.

Solutions. (1) $\exp(-0.2231)\approx0.8000$. The $Z=1$ group has a modeled subdistribution hazard 20.0% lower than the $Z=0$ group at each time, conditional on the model. (2) $\exp(10\times0.0600)=\exp(0.6000)\approx1.822$; the modeled subdistribution hazard is 1.822 times as large for a 10-year increase. (3) No. The CIF is $1-\exp(-\widetilde{\Lambda}_k)$, a nonlinear function of cumulative subdistribution hazard, so a constant hazard ratio does not generally yield a constant CIF ratio. (4) The cause-specific Cox model concerns the event rate among those still free of all event types, whereas Fine–Gray regression targets a subdistribution hazard tied to the cause-specific CIF and uses a different risk-set construction.