Begin With the Quantity You Want to Know
A competing-risk analysis can produce absolute event probabilities, hazard ratios, and other summaries. These quantities answer different questions. Before selecting a model or writing a result, specify the estimand: the target quantity the analysis is intended to estimate. Naming the estimand first prevents a common error—reporting a model coefficient as though it were an event probability.
A useful specification states who the target population is, which event is of interest, what other events prevent that event from occurring, when follow-up begins, the time horizon, and the contrast or summary scale. It should also say whether the target is an absolute probability or a model parameter, and whether it is conditional on covariates or averaged over a population.
Use \(T\) for time to the first event and \(J\) for its type. For cause \(k\), the cumulative incidence function (CIF) is
It is the probability of having experienced cause \(k\) by time \(t\), before any competing event. In contrast, the cause-specific hazard for cause \(k\) is an instantaneous event rate among individuals who have not yet experienced any event. The subdistribution hazard for cause \(k\) is a hazard function defined in relation to the CIF, commonly modeled using the Fine–Gray model, whose risk-set construction retains people who have already had a competing event. The following distinction is central: a CIF is an absolute probability; either hazard is a rate, and a hazard ratio is a relative comparison of rates.
A Precise Specification
For a practical report, write the target in ordinary language before choosing an estimator. Include the following elements.
- Population: Who is the target group? For example, all eligible patients at treatment initiation, not merely those who completed follow-up.
- Event of interest: State the particular event whose probability or hazard is being studied.
- Competing events: Name the event types that can occur first and preclude the event of interest from occurring first.
- Time origin and horizon: Define when the clock starts and the time \(t\) at which the target is evaluated. Include units.
- Contrast: Specify the groups or conditions being compared and the scale, such as a risk difference or risk ratio at time \(t\), or a hazard ratio.
- Conditioning or averaging: Say whether results apply at specified covariate values or are averaged over a stated target population.
The same event and population can support more than one valid estimand. For instance, a two-year CIF in each treatment group is an absolute-risk target; the difference between those CIFs is a risk-difference target. A cause-specific hazard ratio comparing the groups is a different target, even if it comes from the same cohort.
Why the Absolute Risk Depends on Competing Events
Let \(\lambda_k(t)\) denote the cause-specific hazard for cause \(k\), and let \(S(t)\) be the probability of remaining free of every event through time \(t\). The probability of experiencing cause \(k\) in a short time interval around \(u\) is determined both by still being event-free just before \(u\) and by the cause-specific hazard at \(u\). If event times have continuous distributions and the cause-specific hazards are absolutely continuous, the CIF satisfies
Here \(S(u-)\) is the probability of having had no event before \(u\). This identity gives a short justification for why the event-specific hazard alone does not determine absolute risk: \(S(u-)\) is affected by the hazards of all event types. If another cause becomes more frequent, fewer individuals remain event-free and available to experience cause \(k\); the CIF for cause \(k\) can therefore change even if its cause-specific hazard does not.
A cause-specific proportional hazards model might specify
Under this model, \(\exp(\beta_k)\) is a cause-specific hazard ratio for a one-unit contrast in \(X\), conditional on the model covariates. The proportional-hazards assumption is that this ratio does not vary with time. A Fine–Gray proportional subdistribution hazards model instead specifies a subdistribution hazard ratio, often written \(\exp(\gamma_k)\). Its proportionality assumption concerns the subdistribution hazards, not the cause-specific hazards. Neither coefficient is an absolute risk, and the two model parameters generally need not agree.
To identify or estimate a target from observed follow-up, further conditions are needed. In particular, loss to follow-up must be handled under a defensible censoring assumption, such as independent censoring conditional on modeled covariates, or by an appropriate method that accounts for informative censoring. Model-based estimates also depend on the model being adequate for the intended target. If a contrast is described causally, rather than associationally, additional causal assumptions—such as control of confounding and consistency of the treatment definitions—are required. A model coefficient alone does not establish those assumptions.
Worked Example: Naming a Two-Year Absolute-Risk Contrast
Suppose a hypothetical study targets eligible adults beginning one of two treatments. The event of interest is first hospitalization for a specified complication; death before hospitalization is a competing event. The time origin is treatment initiation, and the horizon is two years. Suppose covariate-standardized model predictions for the target population give two-year CIFs of 0.06846 for Treatment A and 0.1098 for Treatment B.
Estimand. The population is the eligible adults beginning either treatment; the event is first hospitalization for the complication; death is the competing event; the horizon is two years after initiation; and the primary contrast is the Treatment A minus Treatment B difference in standardized two-year CIF.
Using the displayed CIFs, the risk difference is
Thus the estimated two-year absolute risk is 4.134 percentage points lower for Treatment A than Treatment B in this hypothetical target population. This is a contrast in predicted probabilities, not a statement that treatment caused the difference unless the design and analysis justify a causal interpretation.
If a risk ratio is also requested, the calculation is
The estimated two-year risk under Treatment A is about 0.624 times that under Treatment B. The denominator and numerator are absolute CIFs at the same horizon and for the same event. This risk ratio is not a hazard ratio.
Model Parameters Are Different Targets
A model-based estimand must name both the model parameter and the comparison it represents. For a cause-specific hazard ratio, the comparison is between instantaneous rates among people who remain free of every event. For a subdistribution hazard ratio, the comparison is between subdistribution hazards associated with the CIF. The Fine–Gray risk-set construction retains individuals with competing events, so its hazard has a different interpretation from the cause-specific hazard.
Neither ratio can be translated into an absolute probability by simply reading it as a percentage increase or decrease in risk. To obtain absolute risks from a fitted model, one needs enough information to predict the relevant CIFs in the target population, accounting for the event process and the covariate distribution. In particular, a cause-specific hazard ratio for one cause does not tell us how often competing causes occur.
Worked Example: Interpreting Two Different Hazard Ratios
In a hypothetical analysis of first relapse, death before relapse is a competing event. Suppose a cause-specific proportional hazards model reports a treatment hazard ratio of 0.70, and a Fine–Gray model reports a subdistribution hazard ratio of 0.78. Assume each model’s proportionality assumption is adequate for the stated comparison, and that both estimates are conditional on the covariates included in their respective models.
For the cause-specific model, the comparison is \(0.70\): among patients still free of relapse and death at a given time, the estimated instantaneous relapse hazard in the treatment group is 0.70 times that in the reference group, conditional on modeled covariates. Equivalently, it is 30% lower on the hazard scale because \(1-0.70=0.30\). This does not mean 30% fewer patients will relapse by a fixed time.
For the Fine–Gray model, the corresponding interpretation is a subdistribution hazard 0.78 times that in the reference group, or 22% lower on that scale because \(1-0.78=0.22\). This is not a claim that relapse risk is 22% lower. The two ratios differ because they describe different hazard quantities and risk sets; neither number alone gives the groups’ relapse probabilities.
A suitable report would name the model and parameter explicitly, for example, “the estimated cause-specific hazard ratio for relapse was 0.70,” rather than saying simply “the risk ratio was 0.70.” If the scientific question concerns the probability of relapse by a specified time, the relevant target is a CIF at that time, or a contrast between CIFs.
Worked Example: Turning a Vague Question Into an Estimand
A hypothetical clinical team asks, “Does Treatment A reduce relapse?” The target population is patients beginning treatment for a condition. The event of interest is first relapse; death before relapse is competing. The follow-up starts at treatment initiation and the clinical team considers three years relevant. Suppose standardized three-year relapse CIFs are 0.22 under Treatment A and 0.15 under Treatment B.
The question is not yet precise: “reduce” could refer to an absolute difference, a relative risk, or a model parameter. If the team wants the absolute difference, define it as Treatment A minus Treatment B:
The estimated three-year relapse risk is therefore 7 percentage points higher under Treatment A in this hypothetical comparison. If the team instead wants a risk ratio, the calculation is
The estimated three-year relapse risk under Treatment A is about 1.47 times the risk under Treatment B. Both calculations use the same population, event, competing event, and horizon, but answer different questions because the contrast scale differs. If treatment assignment was not randomized, or confounding has not otherwise been addressed, these results should be described as associations rather than causal effects.
Choosing Among the Main Targets
The choice depends on the scientific question, not on which model is most familiar or which output is easiest to obtain. An absolute-risk target is often the clearest choice when the question asks how many people experience an event by a clinically meaningful time. A hazard-model parameter may be useful when the question concerns relative event rates or when modeling is intended to support further prediction. The main alternatives have different interpretations.
| Target | What it describes | What it does not mean |
|---|---|---|
| CIF \(F_k(t)\) | Probability of cause \(k\) by time \(t\), with competing events present | A cause-specific hazard or hazard ratio |
| Cause-specific hazard ratio | Relative instantaneous cause-\(k\) hazard among those still event-free | Risk ratio or percentage change in cumulative risk |
| Subdistribution hazard ratio | Relative subdistribution hazard associated with the cause-\(k\) CIF | Risk ratio or percentage change in cumulative risk |
An estimand can also be conditional or marginal. A conditional CIF refers to people with specified covariate values. A standardized CIF averages predicted probabilities over a defined target covariate distribution. These need not be numerically identical, even when based on the same fitted model. State which one is intended; otherwise “adjusted risk” may be ambiguous.
Diagnostics and Common Mistakes
Several warning signs suggest that the reported result does not match the intended estimand:
- The report gives a “risk” but supplies only a hazard ratio, with no time horizon or absolute probability.
- The event of interest is named, but the competing event structure is not clear enough to tell what counts as occurring first.
- A contrast is reported without its direction or reference group, so the sign or meaning of the effect is unclear.
- A result is called a population risk even though it is conditional on a particular covariate profile, or vice versa.
- A causal phrase such as “reduced risk” is used without a design and assumptions supporting causal interpretation.
Common mistake: “The cause-specific hazard ratio was 0.70, so treatment lowered the three-year event risk by 30%.” This reasoning confuses a relative instantaneous hazard among those still event-free with an absolute probability by a fixed horizon. The correction is to report the hazard ratio as a hazard ratio and, if the question concerns three-year risk, estimate and report the three-year CIFs and their stated contrast.
The practical limitation is that no single summary answers every question. Absolute risks depend on the horizon and population; hazard ratios depend on a model and its assumptions. When a report includes more than one target, label each one and ensure the event, population, covariate handling, and contrast direction are consistent.
Check Your Understanding
For each item, identify the estimand or the interpretation error. Solutions follow the questions.
- A report gives a subdistribution hazard ratio of 0.80 and states that the event probability is 20% lower at five years. What is wrong with this interpretation?
- Write the essential components of an absolute-risk estimand for stroke by two years in a population where death before stroke is possible.
- Two groups have three-year CIFs of 0.12 and 0.09 for the same event. Calculate the first group minus second group risk difference and the risk ratio.
- Why can a cause-specific hazard ratio for relapse stay unchanged while the relapse CIF changes?
Solutions. (1) A subdistribution hazard ratio compares subdistribution hazards, not event probabilities. Report the five-year CIFs and the corresponding five-year risk contrast if absolute risk is the target. (2) Specify the target population, stroke as the event of interest, death before stroke as a competing event, the time origin, two-year horizon, and the desired summary or group contrast; state whether the risk is conditional or population-averaged. (3) The difference is \(0.12-0.09=0.03\), or 3 percentage points; the ratio is \(0.12/0.09=1.333\), approximately 1.33. (4) The CIF depends on both the cause-specific hazard for relapse and the probability of remaining free of all events. A change in the competing-event hazard can change that event-free probability and hence the relapse CIF, even if the relapse hazard ratio remains unchanged.