Absolute Risk From Fine-Gray Fits

From a Fine–Gray Fit to an Absolute Risk

A Fine–Gray model describes the association between covariates and the cumulative incidence of a specified event cause. To use it for prediction, the regression coefficient alone is not enough: a horizon-specific probability also requires the fitted baseline subdistribution hazard. Combining these quantities gives a predicted cumulative-incidence function for a chosen covariate profile.

The target is the probability that the event of interest has occurred by time $t$, in the presence of competing events. It is not the probability of that event in a hypothetical world where competing events cannot happen. This distinction is essential when communicating a prediction as an absolute risk.

Definition: For cause $k$, let $F_k(t\mid z)$ be the cumulative-incidence function at time $t$ for covariate profile $z$. The Fine–Gray subdistribution hazard is $$ \tilde{\lambda}_k(t\mid z)=\frac{dF_k(t\mid z)/dt}{1-F_k(t\mid z)}. $$ A proportional subdistribution-hazards model specifies $$ \tilde{\lambda}_k(t\mid z)=\tilde{\lambda}_{k0}(t)\exp(\beta^\mathsf{T}z), $$ where $\tilde{\lambda}_{k0}(t)$ is the baseline subdistribution hazard and $\beta$ is the coefficient vector for the cause of interest.

The subdistribution hazard is constructed for modelling the cumulative incidence, not as the ordinary instantaneous event rate among people currently free of every event. In the Fine–Gray risk-set construction, individuals who experience a competing event remain represented in the subdistribution risk set. Consequently, $\exp(\beta_j)$ is a subdistribution hazard ratio for a one-unit increase in covariate $j$, conditional on the other model covariates. It is not itself an absolute risk or, in general, a risk ratio.

The Prediction Formula and Its Justification

Let $\tilde{H}_{k0}(t)$ denote the cumulative baseline subdistribution hazard through time $t$. Under the proportional subdistribution-hazards model, the fitted cumulative-incidence prediction for profile $z$ is

\hat{F}_k(t\mid z) = 1-\exp\left\{-\hat{\tilde{H}}_{k0}(t)\exp(\hat{\beta}^{\mathsf{T}}z)\right\}.

The fitted baseline term and regression coefficients must come from the same model fit and use the same covariate coding. The prediction is evaluated at the requested horizon $t$; changing either the horizon or the covariate profile changes the resulting probability.

Justification: By definition, $dF_k(t\mid z)/dt=\tilde{\lambda}_k(t\mid z)\{1-F_k(t\mid z)\}$. Therefore $$ \frac{d}{dt}\{-\log[1-F_k(t\mid z)]\} = \frac{dF_k(t\mid z)/dt}{1-F_k(t\mid z)} = \tilde{\lambda}_k(t\mid z). $$ Integrating from time $0$, where $F_k(0\mid z)=0$, to $t$ gives $-\log[1-F_k(t\mid z)]=\tilde{H}_{k0}(t)\exp(\beta^\mathsf{T}z)$. Solving for $F_k(t\mid z)$ gives the stated formula. In a fitted model, the baseline cumulative subdistribution hazard and coefficients are replaced by their estimates.

This representation is the continuous-time proportional-hazards form. With event-time data, software estimates the baseline using a specified tie and baseline-hazard convention; use the fitted baseline quantity and prediction convention supplied by the software rather than mixing estimates from different methods. For horizons beyond the time range with adequate follow-up, the formula may produce a number but does not make that prediction reliable.

Worked Example: A Three-Year Prediction for One Profile

Suppose a hypothetical Fine–Gray fit for a specified cause has estimated baseline cumulative subdistribution hazard $\hat{\tilde{H}}_{k0}(3)=0.080$ at three years. A binary covariate $x$ has fitted coefficient $\hat{\beta}=0.700$, and the profile of interest has $x=1$. Assume the proportional subdistribution-hazards model is the model being used for this prediction.

The linear predictor is $\hat{\beta}x=0.700(1)=0.700$, so the multiplicative factor is $\exp(0.700)=2.014$. The fitted cumulative hazard for this profile is $0.080(2.014)=0.1611$. Thus

\hat{F}_k(3\mid x=1) =1-\exp[-0.080\exp(0.700)] =1-\exp(-0.1611) \approx 0.1488.

The model predicts an approximately 14.9% probability of experiencing the specified cause by three years for this profile, with competing events present. This is a predicted cumulative incidence, not a 2.014-fold probability. The value 2.014 is the fitted subdistribution hazard ratio associated with $x=1$ versus $x=0$.

Working With Profiles and Horizons

A practical prediction requires four items: the cause being predicted, the horizon, the fitted coefficient vector, and the fitted baseline cumulative subdistribution hazard at that horizon. For a profile with several covariates, first calculate the linear predictor $\hat{\eta}=\hat{\beta}^{\mathsf{T}}z$, then calculate $\exp(\hat{\eta})$, multiply by the baseline cumulative hazard, and finally apply $1-\exp(-\,\cdot\,)$. Keep the reference categories and units used in the model when constructing $z$.

Worked Example: Comparing Two Covariate Profiles

Consider a hypothetical fitted model with one binary covariate $x$, coefficient $\hat{\beta}=-0.450$, and baseline cumulative subdistribution hazard $\hat{\tilde{H}}_{k0}(2)=0.120$ at two years. Calculate the two-year predictions for $x=0$ and $x=2$. The latter value is used only as a numerical illustration of a two-unit covariate contrast.

For $x=0$, the multiplier is $\exp(-0.450(0))=1$. Therefore

\hat{F}_k(2\mid x=0) =1-\exp[-0.120(1)] =1-\exp(-0.120) \approx 0.1131.

For $x=2$, the multiplier is $\exp[-0.450(2)]=\exp(-0.900)=0.4066$. Thus

\hat{F}_k(2\mid x=2) =1-\exp[-0.120(0.4066)] =1-\exp(-0.04879) \approx 0.04762.

The fitted two-year cumulative incidences are about 11.3% and 4.76%, respectively. Under this fitted model, the profile with $x=2$ has a lower predicted incidence of the specified cause by two years. The coefficient describes a subdistribution-hazard contrast; the two probabilities are obtained only after combining that contrast with the baseline and applying the prediction formula.

Predictions across horizons should be nondecreasing for a fixed profile because a cumulative-incidence function cannot decrease as time passes. This provides a simple numerical check when obtaining predictions at multiple times. The predicted probability must also lie between zero and one. A model for one cause does not by itself provide the event-free probability or the cumulative incidence for every competing cause.

Worked Example: Checking Predictions at Two Horizons

Suppose a hypothetical fit has $\hat{\beta}=\log(1.5)$ for a binary covariate $x$, with baseline cumulative subdistribution hazards of 0.050 at year 1 and 0.200 at year 5. For a profile with $x=1$, $\exp(\hat{\beta}x)=1.5$. The year-1 prediction is

\hat{F}_k(1\mid x=1) =1-\exp[-0.050(1.5)] =1-\exp(-0.075) \approx 0.07226.

At year 5, the calculation is

\hat{F}_k(5\mid x=1) =1-\exp[-0.200(1.5)] =1-\exp(-0.300) \approx 0.2592.

The predicted cumulative incidence rises from 7.23% by year 1 to 25.9% by year 5, as required for a cumulative probability. The difference, about 18.7 percentage points, is the increase in the unconditional cumulative-incidence probability between these horizons. It is not the conditional probability of the event during years 1 to 5 among people who were event-free at year 1.

Assumptions, Diagnostics, and Alternatives

The prediction formula is meaningful only in relation to the model and data conditions supporting the fit. In particular, the model assumes proportional subdistribution hazards over the time range of interest: covariate effects on the subdistribution-hazard scale are constant over time. It also relies on an adequately specified covariate form and appropriate handling of censoring. Informally, after accounting for model covariates, censoring must not systematically conceal event experience in a way that biases estimation. The desired horizon should have sufficient follow-up and support in the data.

Useful checks include reviewing the fitted model’s proportionality diagnostics, checking whether the covariate coding and functional forms are plausible, and comparing predicted probabilities with observed cumulative incidence in suitable validation data. A sharp discrepancy between predictions and observed incidence indicates poor calibration, even if a covariate association appears strong. Also check that baseline cumulative hazards are nonnegative and nondecreasing, that predictions are within $[0,1]$, and that predictions for a fixed profile do not decrease over time. Such checks can expose coding errors, inconsistent baseline estimates, or inappropriate use of a horizon with little follow-up; passing them does not establish that the model is correct.

Fine–Gray regression is one approach to absolute-risk prediction. Aalen–Johansen estimates provide a nonparametric route to cumulative incidence, while cause-specific Cox fits can be combined to obtain risks through the cause-specific hazards. Fine–Gray modelling directly parameterizes the target cause’s cumulative incidence through its subdistribution hazard, which can make covariate associations with that target convenient to express. The alternatives answer related but not identical modelling questions, and their predictions can differ when model assumptions or specifications differ.

Fine–Gray predictions also have limits. A fitted model for one cause does not supply a complete probability distribution across all outcomes. A subdistribution hazard ratio should not be described as a causal effect without a design and assumptions that support causal interpretation. And extrapolating the fitted baseline or proportional effect far beyond the observed follow-up is not justified merely because the formula can be evaluated there.

A Common Interpretation Error

A frequent mistake is to multiply a baseline cumulative incidence by the fitted subdistribution hazard ratio and call the result the predicted risk. For example, taking a baseline risk of 0.10 and multiplying by 2 gives 0.20. This is generally wrong: the Fine–Gray proportional-hazards formula acts on the cumulative subdistribution hazard scale, not by direct multiplication on the probability scale. The correction is to obtain the fitted baseline cumulative subdistribution hazard, apply the exponential covariate multiplier to that quantity, and transform back using $1-\exp(-\,\cdot\,)$.

Key takeaway: For a fitted Fine–Gray model, an absolute-risk prediction at horizon $t$ is $1-\exp\{-\hat{\tilde{H}}_{k0}(t)\exp(\hat{\beta}^{\mathsf{T}}z)\}$. State the target cause, horizon, and covariate profile, and interpret the result as a cumulative incidence in the presence of competing events.

Check Your Understanding

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

A fitted model gives $\hat{\tilde{H}}_{k0}(4)=0.10$ and $\hat{\beta}^{\mathsf{T}}z=0$. What is the predicted four-year cumulative incidence?
For a profile, $\hat{\tilde{H}}_{k0}(t)=0.15$ and $\exp(\hat{\beta}^{\mathsf{T}}z)=2$. Calculate the predicted risk and explain why it is not simply twice 0.15.
Why is a subdistribution hazard ratio not itself an absolute-risk prediction?
A fitted prediction for the same profile is 0.18 at year 3 and 0.14 at year 6. What should be checked?

Solutions. (1) The multiplier is $\exp(0)=1$, so $\hat{F}_k(4\mid z)=1-\exp(-0.10)\approx 0.09516$, or 9.52%. (2) The prediction is $1-\exp[-0.15(2)]=1-\exp(-0.30)\approx 0.2592$, or 25.9%; the model multiplies cumulative subdistribution hazard, then transforms to probability. (3) The ratio is a relative subdistribution-hazard measure; an absolute risk also requires the baseline cumulative subdistribution hazard and a specified horizon and profile. (4) A cumulative incidence should not decrease. Check the time ordering and baseline estimates, covariate inputs, and prediction implementation; the decrease signals an error or an inappropriate quantity being compared.