Comparing Cause-Specific and Fine-Gray Risk Predictions

Why Compare the Two Prediction Approaches?

Cause-specific Cox and Fine-Gray models describe competing-risk data using different hazards. Yet both can produce a prediction of the same quantity: the probability that a person experiences a specified event by a stated time, before experiencing a competing event. Comparing predictions is therefore more informative than comparing hazard-ratio coefficients alone. The models can produce similar absolute risks for one covariate profile and time horizon, but differ for another.

The goal is not to declare one model universally correct from a side-by-side table. Instead, calculate predictions for the same people, event of interest, and time points; examine where the predictions agree or diverge; and check whether the comparison is fair and the model assumptions are tenable. The earlier material on adjusted cumulative incidence and riskRegression provides the prediction tools; here the emphasis is interpreting predictions from the two model types together.

Two Models, One Absolute-Risk Target

Let \(T\) be time to the first event, \(J\) its type, and \(k\) the event of interest. The cumulative incidence function (CIF) for event \(k\), conditional on covariates \(x\), is \(F_k(t\mid x)=P(T\leq t,J=k\mid x)\). It is the probability of event \(k\) by time \(t\), with competing events precluding a later event of type \(k\) in this first-event framework.

Definition: A matched prediction comparison uses the same target event \(k\), prediction horizon \(t\), and covariate profile \(x\) for both models. It compares their predicted CIF values, not a cause-specific hazard from one model with a subdistribution hazard from the other.

In a cause-specific Cox approach, a separate cause-specific hazard model is fitted for each event type. For event type \(j\), the hazard among people who have not yet experienced any event is \(\lambda_j(t\mid x)=\lambda_{0j}(t)\exp(\beta_j^\mathsf{T}x)\). The cause-specific hazard describes the instantaneous event rate among those still event-free. To obtain the CIF for \(k\), predictions must account for all causes: the chance of remaining event-free depends on the hazards for every event type.

S(t\mid x)=\exp\left\{-\sum_j \Lambda_j(t\mid x)\right\}, \qquad F_k(t\mid x)=\int_0^t S(u-\mid x)\,d\Lambda_k(u\mid x) \quad \text{(continuous hazards)}. $$ For hazards with jumps: $$ S(t\mid x)=\exp\left\{-\sum_j \Lambda_j^c(t\mid x)\right\} \prod_{0<u\leq t}\left(1-\sum_j\Delta\Lambda_j(u\mid x)\right), \qquad F_k(t\mid x)=\int_0^t S(u-\mid x)\,d\Lambda_k^c(u\mid x) +\sum_{0<u\leq t}S(u-\mid x)\Delta\Lambda_k(u\mid x).

Here \(\Lambda_j\) is the cumulative cause-specific hazard, \(\Lambda_j^c\) its continuous component, and \(\Delta\Lambda_j(u\mid x)=\Lambda_j(u\mid x)-\Lambda_j(u-\mid x)\) its jump at time \(u\). The left limit \(S(u-\mid x)\) accounts for the event-free probability immediately before time \(u\); at a jump, the event-probability increment for cause \(k\) is \(S(u-\mid x)\Delta\Lambda_k(u\mid x)\).

A Fine-Gray model instead models the subdistribution hazard for the target event. Its risk-set construction retains people who have experienced a competing event, with the appropriate weighting, so the model directly represents the event-specific CIF. Under the proportional subdistribution hazards specification, \(\alpha_k(t\mid x)=\alpha_{0k}(t)\exp(\gamma_k^\mathsf{T}x)\). For a continuous cumulative subdistribution hazard \(A_k\), the corresponding CIF is \(F_k(t\mid x)=1-\exp\{-A_k(t\mid x)\}\). If \(A_k\) has jumps, the relationship is \(F_k(t\mid x)=1-\exp\{-A_k^c(t\mid x)\}\prod_{0<u\leq t}(1-\Delta A_k(u\mid x))\), where \(A_k^c\) is its continuous component and \(\Delta A_k(u\mid x)=A_k(u\mid x)-A_k(u-\mid x)\).

F_k(t\mid x) = 1-\exp\left\{-A_{0k}(t)\exp(\gamma_k^\mathsf{T}x)\right\}.

The cause-specific and subdistribution hazards have different risk sets and interpretations. Their regression coefficients are not interchangeable, and a larger coefficient in one model does not by itself establish that its absolute-risk prediction must be larger. The shared object to compare is the CIF.

Why the Predictions Can Differ

A cause-specific prediction for event \(k\) is affected by the fitted hazards for event \(k\) and the competing event types, because those hazards determine who remains event-free and available to experience \(k\). A Fine-Gray model represents the target CIF through its subdistribution hazard. These different constructions can yield different predicted CIFs, even when the models use the same covariates and cohort.

The comparison also depends on the covariate profile and prediction horizon. A profile can have a high cause-specific hazard for the target event but also a high competing-event hazard. The latter reduces the opportunity to experience the target event first. At short horizons, few competing events may have occurred, so the predictions can be close; at longer horizons, differences in competing-event patterns and model form can become more pronounced. Neither pattern is guaranteed: inspect the predictions rather than assuming how they must behave.

Worked Example: Calculating a Cause-Specific CIF

Consider a hypothetical individual profile and a one-unit prediction horizon. Suppose the fitted cause-specific model gives cumulative hazards \(\Lambda_1=0.20\) for the event of interest and \(\Lambda_2=0.30\) for a competing event by that horizon. For simplicity, assume both cause-specific hazards are constant over the interval. Calculate the cause-specific-model prediction of the event-1 CIF.

With constant hazards over an interval of length one, the event-1 hazard is \(0.20\) per unit and the total hazard is \(0.20+0.30=0.50\) per unit. Integrating the event-1 hazard against event-free survival gives

F_1(1) = \int_0^1 e^{-0.50u}(0.20)\,du = \frac{0.20}{0.50}\left(1-e^{-0.50}\right) = 0.4(0.3935) = 0.1574.

The predicted probability of event 1 by one unit is therefore about 15.74%. It is not \(1-e^{-0.20}=18.13\%\): that expression treats the event-1 hazard as if competing events could not remove people from the event-free state. If a Fine-Gray fit for this same profile and horizon predicted 0.1800, the absolute difference would be \(0.1800-0.1574=0.0226\), or 2.26 percentage points. That difference is a model-prediction contrast, not evidence by itself that either model is misspecified.

Comparisons Across Covariate Profiles and Horizons

A useful comparison uses a prediction grid: the same cohort or explicitly selected profiles, the same event of interest, and multiple clinically relevant times. For each profile, calculate both models’ CIF predictions at each time. Summarize absolute differences in probability units, and inspect whether the ordering of profiles changes. Averages alone can conceal substantial disagreement in subgroups.

Worked Example: Competing Events and Longer Horizons

Consider two hypothetical profiles under constant cause-specific hazards. For profile A, the cumulative hazards by time 1 are \(\Lambda_1=0.12\) and \(\Lambda_2=0.08\); for profile B they are \(\Lambda_1=0.30\) and \(\Lambda_2=0.45\). Assume these hazards remain constant per unit time, so at time \(t\) each cumulative hazard is its time-1 value multiplied by \(t\). Calculate the target-event CIFs at times 1 and 5.

For constant hazards, if the time-1 hazard rates are \(r_1\) and \(r_2\), then \(F_1(t)=\frac{r_1}{r_1+r_2}\{1-\exp[-(r_1+r_2)t]\}\). For profile A, the target-event share of the total hazard is \(0.12/0.20=0.60\). Thus

F_{1,A}(1)=0.60(1-e^{-0.20})=0.1088, \qquad F_{1,A}(5)=0.60(1-e^{-1.00})=0.3793.

For profile B, the target-event share is \(0.30/0.75=0.40\), so

F_{1,B}(1)=0.40(1-e^{-0.75})=0.2111, \qquad F_{1,B}(5)=0.40(1-e^{-3.75})=0.3906.

At time 1, profile B has the larger target-event probability: 21.11% versus 10.88%. By time 5, the predictions are much closer: 39.06% versus 37.93%. Under this constant-hazard illustration, the limiting CIFs as time grows are the target-event shares, 0.60 for A and 0.40 for B. The example shows why a profile contrast at one horizon need not describe the contrast at another. It does not imply that every fitted model has constant hazards or that risk rankings must converge in real applications.

Worked Example: Reading a Matched Prediction Table

Suppose both models are fitted in the same hypothetical cohort, with the same endpoint coding and covariates. Predictions are requested for event 1 for two profiles at two horizons. The table reports hypothetical predicted CIFs; the Fine-Gray values are model outputs, not values inferred from the cause-specific hazards above.

Profile	Horizon	Cause-specific Cox CIF	Fine-Gray CIF	Fine-Gray minus cause-specific
A	1 year	0.109	0.115	0.006
A	5 years	0.379	0.430	0.051
B	1 year	0.211	0.205	-0.006
B	5 years	0.391	0.360	-0.031

For profile A at five years, the absolute difference is \(0.430-0.379=0.051\), or 5.1 percentage points. For profile B at one year it is \(0.205-0.211=-0.006\), or -0.6 percentage points. The sign changes across these comparisons, so it would be inaccurate to say that Fine-Gray always predicts more risk. At one year, both models rank B above A. At five years, the Fine-Gray model ranks A above B, whereas the cause-specific model ranks B above A. This ranking change is a reason to examine profile-specific predictions and the fitted models rather than reporting only an overall mean difference.

A table like this is interpretable only if each row uses the same profile and horizon in both models. The examples’ numerical predictions do not establish which model is better; judging predictive performance requires additional evaluation, not just a difference between fitted predictions.

Practical Workflow and Diagnostics

When using software such as riskRegression to obtain predictions from fitted cause-specific and Fine-Gray models, make the comparison explicit before interpreting the output. Use a common prediction data set, ensure the target event is coded consistently, and request the same horizons. For person-level comparisons, retain a stable row identifier so that the predictions from the two fits are paired by person.

Set the target.
Specify the event of interest, time scale, and prediction horizons. Confirm which event codes represent competing events and censoring.

Match the prediction inputs.
Use the same people or covariate profiles for both fits. Check that factor levels, transformations, and units agree with the fitted models.

Compare on the probability scale.
For every profile and horizon, calculate the Fine-Gray predicted CIF minus the cause-specific predicted CIF. Report differences in percentage points as well as the predictions themselves.

Investigate patterns.
Look for differences that grow with time, vary by profile, or accompany a reversal in profile ranking. Check the model fits and data coding before assigning a substantive interpretation.

Basic checks catch many errors. Each predicted CIF should lie between 0 and 1 and should not decrease as the horizon increases. Predictions should refer to the intended event, and a person who has already had a competing event should not be treated as still available for the event of interest in a first-event prediction. If the outputs violate these checks, investigate event coding, prediction-time units, and software input conventions.

A large discrepancy can also be a sign of model limitations rather than a software error. Review the proportional-hazards assumptions for the cause-specific and Fine-Gray fits, including whether covariate effects vary over time. Check whether the prediction horizons extend far beyond observed follow-up, where the baseline hazard and risk-set information may be sparse. Examine whether the profiles lie within the covariate patterns represented in the cohort. These checks help identify unstable or poorly supported predictions; they do not by themselves prove that one model is superior.

Common Pitfall and Limits of the Comparison

Common mistake: “The Fine-Gray coefficient is larger, so Fine-Gray must predict a larger absolute risk.” This compares coefficients from different hazard definitions and risk sets. The correction is to calculate both models’ CIFs for the same event, profile, and horizon, then compare those probabilities.

Another frequent error is comparing predictions at different horizons or for different covariate profiles, or comparing a CIF with \(1-\)survival from a model that ignores competing events. Such comparisons do not answer whether the two approaches agree about the same event probability. Even a correctly matched difference is not a causal effect and does not establish which model is more accurate.

Approach	What is modeled directly?	How the target CIF is obtained	Key comparison caution
Cause-specific Cox models	Cause-specific hazards among individuals still free of all events	Combine target and competing cause-specific hazards to calculate the CIF	Model each relevant cause adequately; competing-event hazards affect the target CIF
Fine-Gray model	Subdistribution hazard for the target event	Transform the fitted cumulative subdistribution hazard to a CIF	Interpret coefficients as subdistribution-hazard effects, not cause-specific effects

The two approaches are useful for different modeling questions and can be complementary. Neither guarantees accurate predictions merely because its fitted values look plausible. A prediction comparison describes agreement between the fitted models for the profiles and horizons examined; it does not replace assessment of predictive accuracy or calibration.

Check Your Understanding

Answer each question, then check the brief solutions below.

What quantity should be compared when contrasting cause-specific Cox and Fine-Gray predictions?
For constant cause-specific hazards \(r_1=0.10\) and \(r_2=0.15\) per year, calculate the event-1 CIF at one year.
Why can a high cause-specific hazard for the event of interest coexist with a modest CIF?
A Fine-Gray prediction is 0.32 and a cause-specific prediction is 0.27 for the same profile and horizon. State their absolute difference in percentage points.
Name two checks to make if a predicted CIF decreases as the requested horizon increases.

Solutions. (1) Compare the CIF for the same target event, profile, and time horizon. (2) \(F_1(1)=\frac{0.10}{0.25}(1-e^{-0.25})=0.4(0.2212)=0.0885\), or 8.85%. (3) A high competing-event hazard can remove people from the event-free state before they experience the target event, reducing its CIF. (4) \(0.32-0.27=0.05\), or 5 percentage points. (5) Check event coding and prediction-time units, and confirm that the software is returning the intended event-specific CIF for the fitted models.