Written and maintained by CASRAI Editorial Board
Last updated
When a study follows patients for a time-to-event outcome, it is common for more than one kind of event to be possible — relapse or death without relapse, graft failure or death, device failure or death from another cause. These alternative endpoints are competing risks: the occurrence of one event removes a subject from being able to experience the other, and treating them as ordinary censoring in a standard Kaplan-Meier analysis produces a biased answer. This guide covers what a competing risk actually is, why the naive 1 − Kaplan-Meier estimate overstates the real-world probability of the event you care about, the cumulative incidence function that fixes the estimation problem, and the Fine-Gray model — the regression approach built specifically to model that cumulative incidence directly.
What Makes an Event a Competing Risk
An event is a competing risk relative to your event of interest when its occurrence precludes the event of interest from ever happening to that subject, not merely delays observing it. In a bone-marrow transplant study following time to relapse, death in remission is a competing risk for relapse — a patient who dies without relapsing can never subsequently relapse, so that death is not equivalent to being lost to follow-up. Contrast this with administrative censoring (the study ends before the patient’s outcome is known) or loss to follow-up (the patient stops attending visits): both genuinely leave the event of interest still possible, just unobserved. Competing risks are common wherever the study population faces meaningful all-cause mortality alongside a disease-specific or treatment-specific endpoint: cardiovascular trials (cardiac death vs. non-cardiac death), oncology (relapse vs. treatment-related mortality), and device studies (device failure vs. patient death from unrelated causes) are the settings where this shows up most often.
Why 1 − Kaplan-Meier Overstates the Risk
The standard Kaplan-Meier estimator treats every exit from the risk set that is not the event of interest — including a competing event — as if the subject had simply been censored. That assumption (that a censored subject remains equally likely to have the event of interest as everyone still at risk) is exactly what a competing event violates: a patient who died in remission has a zero, not an unknown, probability of subsequently relapsing. Because the Kaplan-Meier estimator implicitly imagines those competing-risk exits as still “at risk” for the event of interest, 1 − KM systematically overestimates the true cumulative probability of the event of interest, and the overestimation grows with the competing event’s own rate. In a population with substantial competing mortality, this is not a minor rounding difference — published comparisons in oncology and transplant cohorts have shown 1 − KM inflating the estimated cumulative incidence by several percentage points to over ten, depending on how common the competing event is. The correct nonparametric estimator for “the probability of experiencing this specific event by time t, accounting for the fact that a competing event can happen first” is the cumulative incidence function (CIF), not 1 − KM.
The Cumulative Incidence Function
The CIF for a given event type k, CIFk(t), is the probability of experiencing event type k by time t in the presence of the competing events. It is calculated as a weighted sum across event times: at each observed occurrence of event k, the increment added to the CIF is the cause-specific hazard for k at that instant, multiplied by the overall (all-cause) survival probability up to just before that instant — not by the cause-specific survival for k alone, which is what the (biased) 1 − KM-for-cause-k calculation implicitly uses. Summing CIFk(t) across every competing event type at a given t recovers 1 minus the overall (all-cause) Kaplan-Meier survival at that t — the CIFs partition the total probability of “something happened” correctly, which is the internal-consistency check that a competing-risks analysis should satisfy and a naive per-cause 1 − KM analysis does not.
Two Different Hazards Answer Two Different Questions
Competing-risks regression is not one model with one right answer — it is two different hazard definitions, each valid for a different question, and the field’s most common analysis mistake is picking one when the study question calls for the other.
| Cause-specific hazard (standard Cox) | Subdistribution hazard (Fine-Gray) | |
|---|---|---|
| What it conditions on | Subjects still event-free for any reason at time t (competing-event subjects are removed from the risk set the instant they experience the competing event) | Subjects who have not yet had the event of interest — competing-event subjects stay in the risk set indefinitely, with time frozen at their competing-event time |
| Question it answers | “Among people still free of all events, how does this covariate change the instantaneous rate of event k?” — an etiologic / mechanistic question | “How does this covariate change the predicted cumulative incidence (absolute risk) of event k by time t?” — a prognostic / absolute-risk question |
| Correct use case | Understanding biological or causal mechanism; each cause-specific Cox model is fit independently per event type | Building a risk-prediction model, a clinical risk calculator, or any statement of the form “X% of patients like this will experience event k by year 5” |
| A hazard ratio > 1 means | Higher instantaneous rate of event k among those still at risk for everything | Higher cumulative incidence of event k over time — consistent with, but not numerically identical to, the cause-specific interpretation |
Because a covariate can genuinely raise the cause-specific hazard for event k while lowering event k’s cumulative incidence (if that same covariate raises the competing event’s hazard enough to remove people from the risk pool faster than event k can act on them), the two models can point in different directions on the same data without either being wrong — they are answering different questions. Reporting only one, without naming which, is a common and avoidable source of confusion when readers try to reconcile a paper’s Cox table against its cumulative-incidence figure.
The Fine-Gray Model
The Fine-Gray model, introduced by Fine and Gray (1999) in the Journal of the American Statistical Association (94(446): 496–509, DOI 10.1080/01621459.1999.10474144), is a proportional-hazards regression directly on the subdistribution hazard — the quantity whose relationship to the CIF is analogous to the relationship between the ordinary hazard and ordinary survival in ordinary time-to-event regression. Its defining trick is how it constructs the risk set: a subject who experiences the competing event is not removed from the risk set at that point (as a cause-specific Cox model would remove them) but is instead retained in the risk set indefinitely, with their observed time held fixed at their (already-occurred) competing-event time. This “artificial” retention is precisely what makes the resulting hazard ratio interpretable directly against the cumulative incidence function rather than against an “everyone else still fully at risk” population that, in reality, no longer exists once competing events have started occurring. The model is fit as a weighted Cox regression, where the weights (typically based on the Kaplan-Meier estimate of the censoring distribution) correct for the fact that retained subjects are not genuinely still under observation.
Worked Illustration
The following is a hypothetical, illustrative scenario for teaching the calculation — it is not drawn from a real published dataset or attributed to any specific trial or institution.
Consider a cohort of 500 patients followed for five years after a cardiac procedure, with two possible outcomes: cardiac-cause death (the event of interest) and non-cardiac death (the competing risk). Suppose the naive 1 − KM calculation, treating non-cardiac deaths as censored, estimates a 5-year cardiac-death probability of 22%. A correct CIF calculation on the same data, properly accounting for the 15% of patients who died of non-cardiac causes before they could have a cardiac death, estimates the 5-year cardiac-death probability at 17% — the naive estimate overstated the real risk by roughly five percentage points because it implicitly imagined the non-cardiac deaths as still “at risk” for a cardiac death that, having already died of another cause, they could no longer have. A Fine-Gray regression on this same cohort might report a subdistribution hazard ratio of 1.6 (95% CI 1.1–2.3) for a comorbidity index predictor, read as: patients with the higher comorbidity burden have 60% higher cumulative incidence of cardiac death over the follow-up period, holding the model’s other covariates fixed — a direct absolute-risk statement, not a cause-specific rate statement.
Fitting a Fine-Gray Model in R, Stata, and SAS
R offers two standard routes. The dedicated cmprsk package’s crr() function fits the Fine-Gray model directly against a status/time/covariate structure. The general-purpose survival package (from version 3.0 onward) takes a different implementation path: its finegray() function reshapes the data into the appropriate risk-set-with-weights structure, after which an ordinary weighted coxph() call on that reshaped data reproduces the Fine-Gray fit — useful because it lets a Fine-Gray analysis reuse the same diagnostics and prediction machinery already built for coxph objects. Stata‘s stcrreg command fits competing-risks regression on Fine and Gray’s subdistribution-hazard model natively, with syntax closely mirroring stcox. SAS‘s PROC PHREG fits the Fine and Gray (1999) model when the EVENTCODE= option is specified on the MODEL statement to flag which outcome code is the event of interest (with the other event codes automatically treated as competing risks), and the BASELINE statement with PLOTS=CIF produces the corresponding cumulative-incidence curves.
Reporting a Subdistribution Hazard Ratio
Because a subdistribution HR is easy to misread as a cause-specific HR, state explicitly in methods and results which model produced the reported ratio — “subdistribution hazard ratio (Fine-Gray)” or “cause-specific hazard ratio (Cox),” not a bare “hazard ratio.” Report the cumulative incidence curves alongside the regression table when the paper’s framing is prognostic (absolute risk), since a subdistribution HR alone does not convey the absolute magnitude of risk the way a labeled CIF plot does. If the analysis question genuinely spans both etiologic and prognostic interest, reporting both the cause-specific and subdistribution models side by side — rather than picking one — is standard practice in the methodological literature and avoids readers wrongly assuming the two would necessarily agree.
Choosing Between the Two Models
The choice is determined by the study question, not by which software is more familiar. Use the cause-specific Cox model when the question is mechanistic or etiologic — what changes the instantaneous rate of event k among those who remain free of every event. Use the Fine-Gray model when the question is about absolute, real-world risk — what a patient’s or population’s actual probability of experiencing event k by a given time is, which is what most clinical risk-prediction, risk-calculator, and patient-counseling applications require. When in doubt, report both, since they are not competing estimates of the same quantity but answers to two genuinely different questions asked of the same data.
Related CASRAI Guides
For the estimator this guide corrects, see How to Read a Kaplan-Meier Curve and Survival Analysis: Choosing Between Kaplan-Meier, Cox, and Parametric Models. For the standard (non-competing-risks) regression this model is built on top of, see The Cox Proportional Hazards Model and How to Interpret a Hazard Ratio. For sample-size planning on a time-to-event outcome, see Sample Size for Survival Analysis. Related time-to-event extensions covered elsewhere on CASRAI: Recurrent Event Analysis (when the event of interest can happen more than once), Frailty Models for Clustered Survival Data (unobserved heterogeneity), and Landmark Analysis (a practical fix for immortal time bias, a distinct but often co-occurring design problem). See also the Research Methods pillar for the full quantitative-analysis subcluster.
Frequently Asked Questions
Is a competing risk the same thing as censoring?
No. Ordinary censoring (administrative end-of-study, loss to follow-up) leaves the event of interest genuinely still possible for that subject, just unobserved. A competing risk removes the possibility entirely — the subject cannot subsequently experience the event of interest because a different, mutually exclusive event already happened to them.
Can I just run separate Kaplan-Meier curves for each event type?
You can generate separate 1 − KM curves per event type, but each one will be biased upward for the reason explained above, and the curves will not sum correctly against the all-cause Kaplan-Meier survival. Use the cumulative incidence function instead — it is the estimator built specifically to remain internally consistent across all the competing event types.
Does the Fine-Gray model replace the cause-specific Cox model?
No — they answer different questions (see the comparison table above) and are commonly reported together, not as alternatives to choose between once and discard the other.
What software is easiest to start with?
If you already use R’s survival package for standard Cox models, finegray() plus coxph() is usually the smoothest path, since it reuses familiar model-object machinery. cmprsk::crr(), Stata’s stcrreg, and SAS’s PROC PHREG with EVENTCODE= are equally valid, purpose-built alternatives.
