Predicting Competing-Risk Probabilities with riskRegression

From a fitted model to an individual risk

A regression coefficient is not itself a patient’s probability of experiencing an event by a clinically meaningful time. For that, an analyst needs an estimate of the event-specific cumulative incidence function (CIF), evaluated at the patient’s covariate values and at a chosen horizon. In R, riskRegression::FGR fits a Fine–Gray model, and predictRisk turns that fit into absolute-risk predictions.

The model targets one event type at a time. For example, a model for relapse estimates the probability of relapse by one year while accounting for the fact that death before relapse prevents relapse from occurring. The prediction is not the probability of “relapse or death”; nor is it a cause-specific hazard.

A Fine–Gray model specifies the subdistribution hazard for the chosen event type. The subdistribution risk set is constructed so that people who have already experienced a competing event remain represented in a weighted way; the model therefore connects covariates directly to the CIF. This differs from fitting a cause-specific Cox model, which models a cause-specific hazard and requires combining cause-specific hazard estimates to obtain a CIF.

The CIF formula follows from the definition of the subdistribution hazard: \(\tilde{\lambda}_k(t\mid z)=f_k(t\mid z)/\{1-F_k(t\mid z)\}\), where \(f_k\) is the corresponding event-specific density. Thus \(dF_k/(1-F_k)=\tilde{\lambda}_k\,dt\). Integrating from zero to \(t\) gives \(-\log\{1-F_k(t\mid z)\}=\tilde{\Lambda}_k(t\mid z)\); solving for \(F_k\) yields the displayed expression. Since \(\tilde{\Lambda}_k(t\mid z)=\tilde{\Lambda}_{k0}(t)\exp(\beta^\mathsf{T}z)\) under proportional subdistribution hazards, the model produces an absolute risk for each covariate profile.

Fit the model and request predictions

Suppose a data frame dat contains one row per patient, follow-up time ftime, event indicator status, age in decades above a reference age as age10, and treatment coded as 0 or 1. Here status is 0 for right censoring, 1 for the event of interest, and 2 for a competing event. The time and status must describe the same first-event follow-up for each person. A Fine–Gray fit for event 1 is:

The line breaks above are for readability; in R, enter the call as a single expression or use ordinary R line breaks. Specify the event of interest with cause; do not recode competing events as censored observations before fitting. For new patients, supply the same covariates, with the same coding and units used in the fitted model:

The requested horizons here are 365 and 1,095 days. The result is a matrix of predicted risks, with one row per new patient and one column per requested time. Index it by row and column position when extracting values, rather than relying on how a software version labels its columns. Each entry estimates the probability of event 1 by that time for the covariate profile in that row. It is a cumulative probability, not the probability of an event occurring exactly on that day.

Assumptions and a practical workflow

The prediction is meaningful only if the data and model support it. Before fitting, confirm that follow-up times are positive and correctly measured, status codes identify censoring and every event type consistently, and the cause supplied to FGR is the intended event. Check that each covariate in newdata has the same name, coding, scale, and factor levels as in the fitting data. A binary treatment coded 0/1 cannot be supplied as “control” and “active” text unless the fitted model used that factor coding.

The central model assumption is proportional subdistribution hazards: covariate effects on the subdistribution hazard are constant over follow-up time. The model also needs an adequate covariate specification, including appropriate functional forms and relevant interactions, and independent censoring under the assumptions of the fitting procedure. These assumptions do not follow from successful software execution. Use the earlier material on checking Fine–Gray fits with residuals and time interactions to investigate proportionality and model form. If important time variation remains, a single constant coefficient may give poor predictions at some horizons.

A concise workflow is:

1. Verify the time, event, and censoring coding, and identify the event type to predict.
2. Fit FGR using the analysis data and a formula that matches the intended covariate model.
3. Construct new-patient rows using the exact covariate names, units, and coding from the fit.
4. Call predictRisk at prespecified, clinically relevant time horizons, giving the target cause.
5. Check the returned dimensions and values, then report the event, population profile, horizon, and risk together.

Checks, alternatives, and limitations

After prediction, confirm that each value lies between 0 and 1. For the same patient and event, estimated CIFs should ordinarily be nondecreasing across increasing horizons; a decrease suggests a time-ordering, extraction, or software-use problem. Check that the requested horizons are expressed in the same units as the fitting times and lie within a follow-up range with adequate information. Predictions far beyond observed follow-up rely heavily on model extrapolation and should not be presented as equally well supported.

Also inspect event counts by cause, missingness, sparse covariate patterns, and whether the new-patient profiles are represented in the fitting data. A valid-looking number can still be unreliable if few people remain informative near that horizon, if a covariate pattern is outside the observed range, or if the proportionality assumption fails. Predictions are conditional on the fitted model and data; software does not establish causality, guarantee calibration, or account automatically for every source of uncertainty in the analysis.

Fine–Gray prediction is useful when the target is an event-specific absolute probability and a direct regression model for the CIF is appropriate. The main alternative in this context is to fit cause-specific Cox models for each event type and combine their estimated hazards to obtain a CIF. The two approaches parameterize covariate effects differently and need not produce identical predictions. Which is preferable depends on the scientific question, model adequacy, and prediction performance; comparing their predictions is a separate analysis.

A frequent mistake is to interpret \(1-\hat{F}_k(t)\) as the probability of remaining alive and event-free. The correction is that \(1-\hat{F}_k(t)\) is the probability of not yet having experienced event \(k\); it can include people who have already experienced a competing event. The probability of remaining free of every event is a different quantity. Likewise, a predicted event-specific CIF is not an individual guarantee or a causal treatment effect.