Competing Risks in Survival Analysis: When Standard Kaplan–Meier Risk Estimates Can Mislead

The Core Problem: A Competing Event Is Not Ordinary Loss to Follow-Up

Rosner defines a censored observation as an individual followed to a particular time without having failed, whose actual subsequent failure time is unknown. Kaplan–Meier estimation uses such censored observations up to their censoring times and removes them from subsequent risk sets (Rosner, 2016).

A competing event has a different scientific meaning.

If a participant is merely lost to follow-up, the event of interest may still occur later; it is simply no longer observed. If a participant experiences a genuinely competing event, the target event is precluded by that event.

That distinction matters when interpreting survival probabilities.

Steyerberg distinguishes a cause-specific or “actuarial” perspective, in which other event types are censored, from an “actual risk” perspective, which acknowledges that competing events can prevent the event of interest. The former considers an event-specific process in a setting where competing risks are effectively set aside; the latter concerns the probability of the event under the observed competing-event structure (Steyerberg, 2019).

Why Ordinary Kaplan–Meier Risk Can Mislead

The Kaplan–Meier estimator is designed to estimate a survival function with censored observations. Rosner describes the product-limit estimator as incorporating individuals into survival estimation until their censoring time and then removing them from later risk sets (Rosner, 2016).

That machinery does not make every event appropriately treated as ordinary censoring.

Suppose event A is the event of interest and event B prevents A from occurring. If event B is coded as censored, an ordinary cause-specific Kaplan–Meier analysis removes those individuals from subsequent observation of A rather than counting B as part of the real-world event process that determines whether A can ever occur.

Harrell shows the consequence explicitly: when the goal is the probability of failure from a particular cause, ordinary Kaplan–Meier calculations that treat other causes as censored do not estimate the corresponding cumulative incidence probability. He instead defines the cause-specific cumulative incidence through the cause-specific hazard together with overall event-free survival (Harrell, 2015).

Steyerberg makes the prediction implication particularly clear: traditional Kaplan–Meier curves that censor competing events are inappropriate for presenting absolute risk when meaningful competing risks are present (Steyerberg, 2019).

Event-Specific Processes and Absolute Event Probability Are Different Targets

Competing-risk analyses become easier to organize when the scientific target is specified before the statistical model.

The Decision Guide: Start With the Scientific Target

The table deliberately does not identify one universally preferred competing-risk model. The appropriate approach depends on the target quantity.

Cumulative Incidence: The Natural Quantity for Actual Event Probability

For cause m, Harrell expresses the cumulative incidence function as

F_m(t) = ∫₀^t λ_m(u)S(u) du

where λ_m(u) is the cause-specific hazard and S(u) represents survival free from all causes under consideration (Harrell, 2015).

This formula captures the central logic of competing risks.

The probability of experiencing the target event at a particular time depends both on:

1. the event-specific hazard at that time; and
2. whether the individual has remained free of all competing events up to that time.

This is why a target event's cause-specific hazard alone does not determine its absolute cumulative incidence. A predictor can also influence absolute event probability through its relationship with competing events.

For practical reporting, when the scientific statement is:

Cause-Specific Hazards: Useful, but Answering a Different Question

Cause-specific hazard models remain important.

Harrell defines the cause-specific hazard explicitly as an event-rate quantity conditional on being alive/event-free at the relevant time (Harrell, 2015).

A cause-specific hazard ratio therefore describes an association on the hazard scale. It should not automatically be translated into the same numerical statement about cumulative incidence.

This is a broader survival-analysis principle: hazard and probability are not interchangeable quantities.

In competing-risk settings, the distinction becomes especially important because the cumulative incidence for one event depends on the entire event system. The hazard of the target event may change in one direction while the frequency of competing events also changes, altering the resulting absolute probability.

Absolute-Risk Prediction Requires the Competing Process

Steyerberg emphasizes this distinction from the prediction perspective.

For absolute-risk prediction—particularly over longer follow-up or in settings where competing mortality is substantial—ignoring competing events is inappropriate. He notes that standard Cox regression and Kaplan–Meier approaches that simply censor competing events have often been used for absolute-risk prediction, but that this approach is adequate only when competing risks are rare (Steyerberg, 2019).

One technically valid strategy is to model the cause-specific hazard for the target event and the competing event process and then combine those models to obtain absolute risk. Steyerberg explicitly describes this approach (Steyerberg, 2019).

The second question cannot generally ignore other events that remove patients from the possibility of experiencing the target event.

Where the Fine–Gray Model Fits

Steyerberg discusses the Fine–Gray approach as a commonly used regression model for competing-risk data and distinguishes it from cause-specific modeling. In his comparison, cause-specific hazard ratios are obtained while censoring the competing event, whereas the Fine–Gray approach models a subdistribution hazard and retains individuals with the competing event in its modified risk-set construction (Steyerberg, 2019).

This means a subdistribution hazard ratio (sHR) is not the same estimand as a cause-specific hazard ratio.

The two coefficients should therefore not be interpreted as interchangeable measures of the same event process.

The Fine–Gray framework is closely connected to cumulative incidence and provides a regression approach for the competing-risk setting. But this does not make it the automatic answer to every competing-risk question.

If the research question concerns the cause-specific event mechanism or association, a cause-specific hazard model may be the more directly aligned quantity. If the question concerns absolute occurrence in the presence of competing events, cumulative incidence must be central to interpretation, with the regression strategy chosen accordingly.

Why Cause-Specific and Subdistribution Associations Can Differ

A predictor can affect more than one event process.

Steyerberg's presentation of competing-risk models demonstrates that predictors can have one relationship with the cause-specific hazard of the event of interest and another relationship with the hazard of the competing event. The resulting subdistribution association can consequently differ from the target event's cause-specific hazard association (Steyerberg, 2019).

That distinction is not a technical nuisance. It reflects the scientific structure of the problem.

Absolute probability is influenced not only by how rapidly the target event occurs among people still at risk, but also by how rapidly people leave that risk set through competing events.

This is another reason not to ask simply:

“Which competing-risk model should I run?”

Ask instead:

“Which quantity represents the scientific question I need to answer?”

When Are Competing Events Too Important to Ignore?

There is no universal numerical threshold in the mandatory sources that makes a competing event automatically “important.”

The decision should instead be driven by the target and the event process.

Steyerberg gives particularly clear guidance for prediction: ignoring competing risks is only adequate for absolute-risk prediction when competing risks are rare, and he specifically highlights long-term prediction and prediction in older populations as situations where competing risks deserve attention (Steyerberg, 2019).

Competing events should therefore receive explicit consideration when:

• the competing event genuinely prevents subsequent occurrence of the target event;
• absolute event probability is the scientific or clinical target;
• competing events are not rare over the prediction horizon;
• long-term risk is being estimated;
• the population has substantial competing mortality or another substantial competing endpoint; or
• predictors may relate differently to the target and competing event processes.

The important question is not whether a competing event occurs at all, but whether ignoring it changes the meaning of the quantity being reported.

A Practical Analysis Workflow

1. Define the target event precisely.

   State exactly what counts as the event of interest.

   Do not begin with the statistical model.

2. Identify events that preclude the target event.

   Ask whether another observed endpoint makes subsequent occurrence of the target event impossible.

   Distinguish this from ordinary administrative censoring or loss to follow-up.

3. Specify the scientific target.

   Decide whether the primary target is:

   • a cause-specific association or event process, or
   • the absolute probability of the event in the presence of competing events.

   This decision determines how the analysis should be interpreted.

4. Match the estimand to the method.

   For an event-specific hazard question, consider a cause-specific hazard model.

   For absolute event probability, use a framework that represents cumulative incidence and the competing-event structure. This may involve combining cause-specific hazard models or using an appropriate competing-risk regression framework such as the Fine–Gray approach when its estimand matches the objective (Harrell, 2015; Steyerberg, 2019).

5. Match the reported language to the estimand.

   Do not report:

   “The 5-year probability of event A was 1 − KM”

   when competing events were censored and the intended interpretation is actual real-world incidence.

   Likewise, do not translate a cause-specific hazard ratio or subdistribution hazard ratio directly into an absolute percentage-point change in event probability.

6. For prediction, evaluate actual predicted probabilities.

   If the intended output is an individual's absolute risk by a specified horizon, the model must produce a probability aligned with the competing-event structure. The model should subsequently be evaluated as a prediction model rather than judged solely from regression coefficients (Steyerberg, 2019).

Common Mistakes

Reporting Checklist for Competing Risks Survival Analysis

Before reporting results, verify that the manuscript makes clear:

• what the target event is;
• which events are considered competing events;
• which observations represent ordinary censoring;
• whether the target is a cause-specific hazard or an absolute event probability;
• whether Kaplan–Meier estimates censor competing events;
• whether cumulative incidence is reported when actual event probability is the target;
• whether hazard ratios are explicitly identified as cause-specific or subdistribution hazard ratios;
• whether absolute-risk predictions account for important competing events; and
• whether the interpretation matches the estimand actually produced by the analysis.

Bottom Line

The central problem in competing risks survival analysis is not that Kaplan–Meier estimation is intrinsically wrong. It is that Kaplan–Meier can answer a different probability question from the one researchers think they are answering.

Ordinary Kaplan–Meier methods appropriately handle censored survival observations under their intended survival framework (Rosner, 2016). But when a competing event prevents the event of interest, censoring that competing event and interpreting 1 − Ŝ(t) as the actual probability of the target event can be misleading (Harrell, 2015; Steyerberg, 2019).

No single competing-risk model answers every scientific question.

FAQs

What Is a Competing Risk in Survival Analysis?

A competing risk is an event that precludes subsequent occurrence or observation of the event of interest. Steyerberg uses death before a nonfatal outcome as a clear example: once the patient dies, that later outcome cannot occur (Steyerberg, 2019).

Why Can Kaplan–Meier Overestimate Risk With Competing Events?

When competing events are treated as ordinary censoring, the Kaplan–Meier calculation does not represent those events as outcomes that prevent the target event. Consequently, 1 − Kaplan–Meier does not generally equal the real-world cumulative incidence of the target event when meaningful competing risks are present (Harrell, 2015; Steyerberg, 2019).

What Is Cumulative Incidence in Competing Risks?

The cumulative incidence function describes the probability of experiencing a particular event type by time t while acknowledging that competing event types can occur. Harrell represents it through the target cause-specific hazard combined with overall event-free survival (Harrell, 2015).

What Is a Cause-Specific Hazard?

The cause-specific hazard is the instantaneous rate of a particular event type among individuals who remain alive/event-free and capable of experiencing that event at the specified time (Harrell, 2015).

What Is the Difference Between Cumulative Incidence and Kaplan–Meier Risk?

With meaningful competing events, cumulative incidence represents actual probability of the particular event under the observed competing-event process. An ordinary Kaplan–Meier complement that censors competing events targets a different quantity and should not automatically be interpreted as actual event probability (Harrell, 2015; Steyerberg, 2019).

What Is the Fine–Gray Model?

The Fine–Gray model is a competing-risk regression approach based on the subdistribution hazard. Steyerberg contrasts it with cause-specific modeling: competing events are censored for the cause-specific hazard model, whereas the Fine–Gray construction retains those experiencing competing events in its modified risk-set framework (Steyerberg, 2019).

Should I Use Cause-Specific Cox Regression or a Fine–Gray Model?

Neither is universally preferable. The choice depends on the scientific target. Cause-specific models describe associations with an event-specific hazard. Fine–Gray models operate on the subdistribution-hazard scale and are linked to cumulative incidence. The interpretation should follow the estimand rather than treating the two hazard ratios as interchangeable (Steyerberg, 2019).

Can I Still Use Kaplan–Meier When Competing Risks Exist?

A cause-specific Kaplan–Meier analysis can be used for an event-specific or actuarial perspective in which other events are censored. The problem arises when its complement is interpreted as the actual absolute probability of the target event despite meaningful competing risks (Steyerberg, 2019).

When Can Competing Risks Reasonably Be Ignored for Absolute-Risk Prediction?

Steyerberg states that censoring competing events for absolute-risk prediction is adequate when competing risks are rare. Competing risks become particularly relevant for long-term absolute-risk prediction and in populations where competing mortality is substantial, including older populations (Steyerberg, 2019).

References

Harrell, F. E., Jr. (2015). Regression modeling strategies: With applications to linear models, logistic and ordinal regression, and survival analysis (2nd ed.). Springer.

Rosner, B. (2016). Fundamentals of biostatistics (8th ed.). Cengage Learning.

Steyerberg, E. W. (2019). Clinical prediction models: A practical approach to development, validation, and updating (2nd ed.). Springer. https://doi.org/10.1007/978-3-030-16399-0