Why Competing Events Change the Question
Suppose an omics feature is associated with time to disease progression, but some participants die before progression. Death is not merely missing information about a later progression time: it prevents progression from occurring first. An analysis that treats death as ordinary censoring can therefore answer a different question from one that describes the probability of progression in the presence of death.
The choice is between distinct estimands, not simply between two ways to fit a regression. A cause-specific hazard describes the instantaneous rate of one event among people who are still free of every event. A subdistribution hazard is linked directly to the cumulative incidence of one event in a setting where competing events occur. Neither is automatically a causal effect, and neither is universally preferable. The right choice depends on the scientific question.
The risk set in the cause-specific hazard is restricted to people who are event-free immediately before time \(t\). A person who has already experienced a competing event is no longer at risk of experiencing the event of interest first. The cause-specific hazard is therefore a rate in the current event-free group, not the probability of the event by a given time.
How the Hazards Determine Cumulative Incidence
For competing causes \(1,\ldots,K\), define the event-free survival probability as \(S(t\mid x)=P(T>t\mid x)\). With continuous event times, the total hazard is the sum of the cause-specific hazards. If the hazards are integrable, this gives
The integrand in the CIF formula is the probability of still being event-free at time \(u\), multiplied by the instantaneous rate of cause \(k\) among those still event-free. The formula shows why a cause-specific hazard alone does not determine the probability of the event: hazards for competing causes also affect \(S(u\mid x)\). This is an important point when an omics feature is related to more than one event type.
Because \(dF_k(t\mid x)/dt=S(t\mid x)\lambda_k(t\mid x)\), the two hazards are related by
Thus a subdistribution hazard depends on both the cause-specific hazard for the event of interest and the event-free survival shaped by all competing causes. It is not a rescaled cause-specific hazard in general.
Worked Example: A Higher CIF Despite a Lower Progression Hazard
Worked Example: A Higher CIF Despite a Lower Progression Hazard
Consider two hypothetical groups with constant cause-specific hazards per year. For group A, the progression hazard is \(0.07\) and the competing-death hazard is \(0.22\). For group B, the progression hazard is \(0.06\) and the competing-death hazard is \(0.04\). Assume these are the only event types, there is no loss to follow-up, and the hazards are constant over the three years.
For constant hazards, the event-free probability at time \(t\) is \(S(t)=\exp[-(\lambda_{\text{progression}}+\lambda_{\text{death}})t]\). Substituting this into the CIF integral gives
At three years, group A has total hazard \(0.07+0.22=0.29\) per year. Its progression CIF is
That is about \(14.02\%\) by three years. Group B has total hazard \(0.06+0.04=0.10\) per year, so
That is about \(15.55\%\) by three years. Group B has the lower cause-specific progression hazard, \(0.06\) rather than \(0.07\) per year, yet the higher progression CIF. Its much lower competing-death hazard leaves more people event-free long enough to experience progression. Both calculations use the stated constant hazards; neither percentage is the cause-specific hazard expressed as a probability.
Regression Models and Their Interpretations
A cause-specific Cox model for cause \(k\) commonly specifies
Under this model, a one-unit increase in \(x\) multiplies the cause-specific hazard by \(\exp(\beta_k)\), conditional on other model covariates, at times when the compared individuals remain event-free. For an omics measurement standardized to have a one-standard-deviation unit, the hazard ratio compares event-free individuals differing by one standard deviation in that measurement. Competing events remove people from this cause-specific risk set. Censoring before an event is handled under an appropriate non-informative censoring assumption conditional on modeled variables.
A proportional subdistribution hazards model specifies
Here \(\exp(\gamma_k)\) is the ratio of subdistribution hazards for a one-unit increase in \(x\). It is a relative rate on the subdistribution-hazard scale, not a risk ratio and not a cause-specific hazard ratio. The model is commonly fitted using the Fine–Gray approach, which constructs a modified risk set that retains people with competing events and uses censoring weights. The retained participants are a modeling device, not people who can still experience the event of interest first.
Worked Example: Translating Two Hypothetical Model Estimates
Suppose a hypothetical analysis measures an RNA score in standard-deviation units and reports a cause-specific progression hazard ratio of \(1.50\) per standard deviation, with a 95% confidence interval from \(1.12\) to \(2.01\). A separate proportional subdistribution hazards analysis reports a subdistribution hazard ratio of \(1.18\), with a 95% confidence interval from \(1.01\) to \(1.38\).
On the cause-specific scale, the fitted model says that among participants still free of all events, each one-standard-deviation higher score is associated with a \(1.50\)-fold progression hazard, conditional on the model covariates. The confidence interval expresses uncertainty about this hazard ratio under the model and its sampling assumptions.
On the subdistribution scale, each one-standard-deviation higher score is associated with a \(1.18\)-fold subdistribution hazard for progression. This is evidence of a positive association on that scale, but it does not mean the progression probability is 18% higher at every follow-up time. The CIF comparison depends on the baseline CIF and the model, and the competing-event experience also matters.
The estimates \(1.50\) and \(1.18\) need not agree: they describe different quantities and use different risk-set constructions. Neither estimate alone establishes that changing the RNA score would change progression or death. A causal interpretation would require a suitable causal design and assumptions beyond these hazard models.
Assumptions, Estimation, and Diagnostics
The estimand must be chosen before the model. If the question is whether an omics feature is associated with the instantaneous progression rate among those still event-free, the cause-specific hazard is aligned with it. If the question concerns the probability of progression by a specified time in the presence of competing death, the CIF is the direct target; a subdistribution hazards model is one way to model that target.
Both regression approaches require correct event classification, an appropriate functional form for the omics predictor and other covariates, and censoring that is independent of the event process conditional on the variables used in the analysis. A proportional hazards model additionally assumes its relevant hazard ratio is constant over time. The earlier material on time-varying effects is relevant when that condition is doubtful. Fine–Gray estimation also relies on appropriate handling of censoring in its weighted risk-set construction; informative censoring not adequately accounted for can bias results.
For descriptive CIF estimation, the Aalen–Johansen estimator combines the estimated probability of remaining event-free with the observed cause-specific event increments. At event times \(u\), its form is
where \(Y(u)\) is the number at risk just before \(u\), \(d_k(u)\) is the number of cause-\(k\) events at \(u\), and \(\widehat S(u-)\) is the estimated event-free probability immediately before \(u\). This estimator respects competing events rather than treating them as ordinary independent censoring when estimating the CIF.
Useful diagnostics include checking event counts by cause and predictor group, inspecting whether censoring differs substantially across groups, and examining time patterns in fitted effects. Very few events of a given cause lead to unstable estimates even when the overall sample is large. For Fine–Gray models, scrutinize the censoring distribution underlying the weights; extreme weights can make estimates imprecise or sensitive to a few observations. For either model, compare predicted CIFs with nonparametric CIF estimates at clinically relevant times when feasible, and inspect whether conclusions change under reasonable covariate or functional-form choices.
Choosing Between the Approaches
| Feature | Cause-specific hazard | Subdistribution hazard |
|---|---|---|
| Risk-set idea | Participants still free of every event | Modified risk set also retains those with competing events |
| Direct interpretation | Instantaneous rate of cause \(k\) among event-free participants | Rate tied mathematically to the CIF for cause \(k\) |
| Common model | Cause-specific Cox model | Proportional subdistribution hazards (Fine–Gray) model |
| Useful question | Is the feature associated with the event’s rate among those still event-free? | Is the feature associated with the event’s cumulative incidence in the presence of competing events? |
| Key caution | The hazard ratio alone does not give the event probability | The subdistribution hazard ratio is not a risk ratio or a biological rate among only event-free people |
A cause-specific model can be useful for etiologic questions about event rates among those currently event-free. To obtain a CIF from cause-specific models, the relevant hazards for all competing causes must be combined; modeling only the hazard of interest is not sufficient. A subdistribution model is convenient when the target is the CIF association, but its coefficient can be less intuitive as a biological rate. Directly estimating CIFs is another option when the goal is descriptive risk at specified times rather than a regression coefficient.
Neither approach eliminates the need to state the target population, follow-up horizon, event definitions, and covariates. For omics analyses, the feature’s measurement scale and preprocessing also affect what a one-unit contrast means. Associations may reflect confounding or selection as well as biological relationships. Prediction questions require assessment of predicted risks and their performance, not just a statistically significant hazard coefficient.
Common Pitfalls
- Calling a competing event ordinary censoring for a CIF analysis. This can make the analysis imply that participants who die could later progress. For the real-world probability of progression before death, use a competing-risk CIF estimator or an appropriate model.
- Reading a cause-specific hazard ratio as a risk ratio. A hazard ratio compares instantaneous rates among those still event-free. Report a CIF at a stated time if the question is about probability by that time.
- Treating the Fine–Gray risk set as a biological risk set. Competing-event participants are retained by construction to model the subdistribution. They are not literally still able to have the first event of interest.
- Assuming the two coefficients should match. Different risk sets and estimands can produce different associations, particularly when the omics feature is related to competing events.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In a study of progression and death, whose experience defines the cause-specific progression hazard risk set at time \(t\)?
- Why can a group with a lower cause-specific progression hazard nevertheless have a higher progression CIF?
- A subdistribution hazard ratio of \(0.80\) per standard-deviation increase in an omics score is reported. What does it compare, and what does it not mean?
- What additional cause-specific hazard information is needed to calculate a progression CIF from a cause-specific model?
- Name one diagnostic concern specific to the Fine–Gray weighted risk-set construction.
Solutions. (1) It includes participants who have not yet experienced progression or any competing event and remain under observation at \(t\). (2) The group may have fewer competing events, leaving more participants event-free long enough to progress; the CIF depends on hazards for all causes. (3) It compares subdistribution hazards for the modeled contrast; it is not an 20% lower progression probability or a cause-specific hazard ratio. (4) The hazards for the competing causes are needed to obtain event-free survival, which enters the CIF integral. (5) Inspect the censoring weights for extreme values or sensitivity to a few observations; inadequate handling of informative censoring is also a concern.