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Competing Risks

Event probabilities when one event type prevents another

Statistics
Advanced Modeling
Survival Analysis
Cause-specific hazards, cumulative incidence, competing-event estimands, regression, diagnostics, and design-aware bioinformatics interpretation.
Author

Jiahong Sun

Published

November 7, 2026

Modified

November 7, 2026

1 More Than One Mutually Exclusive Event

A competing-risks process has a first-event time \(T\) and event type \(J\in\{1,\ldots,K\}\). Occurrence of one type prevents observation of the others as first events. For example, death without molecular relapse prevents a later relapse from being observed as the first endpoint.

The central scientific targets differ:

  • Etiologic or rate question: how does a covariate relate to the instantaneous rate of cause \(k\) among subjects currently event-free?
  • Prognostic probability question: what proportion of a population will experience cause \(k\) by time \(t\) in the presence of all other causes?
  • Decision question: how would a treatment change the whole multistate event process and its absolute consequences?

These questions use different estimands. Treating competing events as if they were ordinary loss to follow-up generally does not estimate the observed-world event probability.

2 Cause-specific Hazards and Cumulative Incidence

The cause-specific hazard for event type \(k\) is

\[ \lambda_k(t\mid x) =\lim_{\Delta t\downarrow0} \frac{P(t\le T<t+\Delta t,J=k\mid T\ge t,x)}{\Delta t}. \]

The overall hazard is the sum

\[ \lambda(t\mid x)=\sum_{k=1}^K\lambda_k(t\mid x), \]

so event-free survival is

\[ S(t\mid x)= \exp\left\{-\sum_{k=1}^K\Lambda_k(t\mid x)\right\}, \qquad \Lambda_k(t)=\int_0^t\lambda_k(u)\,du. \]

The cumulative incidence function (CIF) for cause \(k\) is

\[ \boxed{F_k(t\mid x)=P(T\le t,J=k\mid x) =\int_0^t S(u-\mid x)\lambda_k(u\mid x)\,du.} \]

This equation is the key to competing risks: cause-\(k\) incidence depends on both its own hazard and every competing hazard through \(S(u-)\). A treatment can lower the hazard of relapse yet increase the observed relapse probability if it lowers the competing mortality hazard even more. Hazard effects and probability effects therefore need not have the same direction.

3 Why One Minus Kaplan–Meier Is Wrong Here

If competing events are censored and a Kaplan–Meier curve is calculated for cause \(k\), \(1-\widehat S_{KM}(t)\) estimates a hypothetical probability under an operation that removes competing causes while preserving the cause-\(k\) hazard. It does not estimate the crude probability of cause \(k\) in the observed process. It typically overestimates that probability because subjects who already had a competing event remain represented through a censoring assumption they cannot satisfy.

The nonparametric estimator of \(F_k\) instead sums cause-specific event increments weighted by the probability of remaining event-free:

\[ \widehat F_k(t) =\sum_{t_j\le t}\widehat S(t_j-) \frac{d_{kj}}{Y_j}, \]

where \(Y_j\) is the risk-set size and \(d_{kj}\) is the number of type-\(k\) events at time \(t_j\). With all transition types assembled, this is a special case of the Aalen–Johansen estimator.

4 Regression Targets

4.1 Cause-specific hazard regression

A Cox model for cause \(k\) specifies

\[ \lambda_k(t\mid x)=\lambda_{k0}(t)\exp(x^\mathsf{T}\beta_k). \]

Other event types are removed from that cause-specific risk set when they occur. This is valid risk-set construction for estimating the cause-specific hazard; it does not make competing events ordinary independent censoring for estimating a CIF. To predict absolute incidence, models for all relevant cause-specific hazards must be combined through the CIF equation.

\(e^{\beta_{kj}}\) is a cause-specific hazard ratio among subjects who remain free of every event. Because membership in that risk set evolves after baseline, it is not generally a marginal risk ratio or a causal treatment effect.

4.2 Fine–Gray subdistribution hazard regression

Fine and Gray model the subdistribution hazard associated with the CIF:

\[ \widetilde\lambda_k(t\mid x) =\widetilde\lambda_{k0}(t)\exp(x^\mathsf{T}\gamma_k). \]

Its constructed risk set retains, with appropriate weighting, subjects who had a competing event. This permits direct regression on the CIF. The subdistribution hazard ratio \(e^{\gamma_{kj}}\) is a model parameter describing relative subdistribution rates; it is not an instantaneous rate among currently event-free subjects and is not itself an event-probability ratio.

Use cause-specific models when transition rates, mechanisms, or multistate prediction are central. A Fine–Gray model can be useful when a direct covariate model for one cause’s cumulative incidence is the prespecified target. For communication, show standardized CIFs or risk differences at meaningful times alongside either regression coefficient.

5 Estimands, Estimators, and Observed Estimates

For a target population and time \(t^*\), an estimand might be

\[ \Delta_k(t^*) =F_k(t^*\mid A=1)-F_k(t^*\mid A=0), \]

possibly standardized over baseline covariates. The Aalen–Johansen estimator, a collection of fitted cause-specific models, or a Fine–Gray model supplies an estimator under its assumptions. The number printed after fitting the observed data is an estimate.

Uncertainty should match the complete estimator. Analytic influence-function standard errors or a bootstrap can be used, but resampling must preserve the independent experimental unit, clustering, and all model-selection steps. Confidence bands for entire CIF curves are different from pointwise intervals.

Gray’s test compares CIFs between groups under the null of equal cause-specific cumulative incidence functions, using weighted differences over time. It does not estimate an effect size, adjust automatically for confounding, prove equality at every time when nonsignificant, or reveal which competing hazard produced a difference.

6 Censoring, Truncation, and Identification

Competing events are observed outcomes, whereas censoring means the relevant future event process becomes unobserved. Standard CIF estimation assumes censoring is independent of the future competing-risks process, unconditionally or after adequate conditioning. Delayed entry additionally requires correct risk sets and an assumption connecting entry to future outcomes.

The observed data identify cause-specific hazards and CIFs under these observation assumptions. They do not nonparametrically identify the latent time to each cause that would have occurred if all other causes were disabled. Dependence among those latent failure times cannot be learned from ordinary competing-risks data alone. Statements about “net risk if death were impossible” therefore require a different estimand and untestable assumptions.

For causal treatment effects, additional requirements include consistency, exchangeability, positivity, correct time ordering, and an explicit strategy for post-treatment competing events. Eliminating, holding fixed, or allowing a competing event corresponds to different hypothetical questions; no regression coefficient chooses among them automatically.

7 Multistate View

Competing risks are the simplest multistate model: one initial state and several absorbing destination states.

Event-free ──→ event of interest
     │
     ├──────→ competing event 1
     │
     └──────→ competing event 2

If later transitions matter—for example, remission to relapse to death—then a larger multistate model preserves that sequence instead of collapsing it into a single first event. Transition hazards describe local movement; transition probabilities combine all routes through the system. The Markov assumption says future transition rates depend on the current state and covariates but not the earlier path; semi-Markov or history-dependent models relax it.

8 Diagnostics and Sensitivity

  1. Define the event system. Predefine event types, time zero, simultaneous event rules, delayed entry, recurrent events, and whether states are absorbing.
  2. Audit ascertainment. Compare follow-up schedules, loss to follow-up, adjudication, and cause classification across centers and covariate groups.
  3. Plot every important CIF. A cause cannot be interpreted without its competing causes, event counts, and risk-set support.
  4. Check regression structure. Assess nonlinear covariate functions, proportionality for each cause-specific or subdistribution model, influence, and calibration of predicted CIFs.
  5. Inspect sparsity. Few cause-specific events cannot be rescued by many competing events or molecular features. Use shrinkage or simpler prespecified models when appropriate.
  6. Vary defensible definitions. Examine ambiguous causes, composite endpoints, time horizons, censoring weights, and model families.
  7. Validate the full prediction pipeline. Repeat feature processing and model fitting within resampling; assess cause-specific calibration at relevant times.

Cause adjudication error can move observations directly between CIFs. Sensitivity analyses that reclassify uncertain events or use probabilistic classification may be more informative than extra decimal places in a model coefficient.

9 Bioinformatics Example: Relapse and Non-relapse Mortality

Consider 500 transplant recipients followed from transplantation to relapse, non-relapse death, or censoring. The target is two-year relapse incidence by a prespecified genomic-risk group. Death without relapse is competing because it prevents later relapse as a first event.

The recipient is the experimental unit. Longitudinal blood draws, cell types, technical libraries, and reads are nested observations, not independent event histories. Donor may be an additional clustering level if donors contribute to multiple recipients. Batch, center, treatment, graft source, and surveillance frequency can affect measured genomic covariates, event rates, or observation.

A defensible analysis reports relapse and non-relapse-mortality CIFs, standardized two-year risk differences with uncertainty, and cause-specific models if mechanism is of interest. If the genomic score was selected from the same outcomes, the complete selection procedure belongs inside validation.

Statistically, a higher relapse CIF means more observed first relapses by two years in the target population. Biologically, that difference can arise through the relapse process, survival long enough to relapse, treatment allocation, measurement error, or population composition. It does not by itself show that the score causes relapse.

10 Common Misinterpretations

Claim Why it fails Better statement
“Death was censored, so Kaplan–Meier estimates relapse risk” Death makes later first relapse impossible Estimate the relapse CIF with death as a competing event
“The cause-specific hazard ratio is the risk ratio” A hazard is conditional on remaining event-free Report the coefficient’s scale and absolute CIF contrasts
“Fine–Gray adjusts for competing risks” It models a different hazard on a constructed risk set Choose it because the CIF is the target, not as an automatic correction
“Lower relapse incidence proves relapse biology improved” More early competing deaths can lower relapse incidence Examine every cause and clinically relevant composite outcomes
“Each cell is a replicate” Cells share a recipient and event history Base inference and resampling on independent recipients
“The data reveal what would happen without death” Latent event times under elimination are not identified State the hypothetical intervention and added assumptions

11 Connections and Take-home Messages

The CIF is the actual probability of a particular first event in a world where all competing events operate. It combines every cause-specific hazard, so a covariate can affect it through several pathways. Cause-specific and Fine–Gray models answer different questions; neither coefficient is an absolute risk. Valid interpretation begins with event definitions, experimental units, and the scientific estimand, then reports all consequential causes together.

12 References

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