Why Competing Events Change the Meaning of Risk
A drug exposure may affect several possible first events, or it may alter one event while another event prevents it from occurring. For example, a safety analysis might distinguish a particular adverse event from death due to another cause. Once a participant experiences one of these first events, the other can no longer be observed as that participant’s first event during follow-up. These are competing risks.
A cause-specific hazard describes the instantaneous rate of one event among people who have not yet experienced any event. A cumulative incidence function (CIF), by contrast, gives the probability of having experienced that specific event by a stated time, before any competing event occurs. The distinction matters in exposure-response work: a change in one cause-specific hazard can change the observed probability of other causes, even if their own hazards do not change.
From Cause-Specific Hazards to Event Probabilities
Write \(\lambda_k(t\mid C)\) for the cause-specific hazard of cause \(k\), and \(S(t\mid C)\) for the probability of remaining free of every event through time \(t\). If each cause has a well-defined hazard and events are mutually exclusive first events, the instantaneous hazard of any event is the sum of the cause-specific hazards. Consequently, survival without any event and the CIF for cause \(k\) are:
The integrand in the CIF has two parts. \(S(u\mid C)\) is the probability of still being at risk of every first event just before \(u\); \(\lambda_k(u\mid C)\) is the instantaneous rate of cause \(k\) in that event-free group. Multiplying them gives the instantaneous probability contribution to cause \(k\), which is accumulated over time.
The result follows directly from the event-free probability. Over a short interval of length \(dt\), the probability of any event among those still event-free is approximately \(\sum_j\lambda_j(t\mid C)dt\). Thus the rate of change in event-free probability is:
Solving this differential equation with \(S(0\mid C)=1\) gives the exponential expression for \(S\). Over the same short interval, the probability contribution from cause \(k\) is approximately \(S(t\mid C)\lambda_k(t\mid C)dt\). Summing these contributions over time gives the integral for \(F_k\). This argument assumes continuous-time hazards; if event times are tied or recorded only in broad intervals, the likelihood or probability calculation must respect that observation structure.
Worked Example: Constant Hazards for Two Competing Events
Consider a hypothetical setting with a constant cause-1 hazard of \(0.08\) per year and a constant cause-2 hazard of \(0.32\) per year. Calculate the probability of each event as the first event within three years.
The total hazard is \(0.08+0.32=0.40\) per year, so the probability of remaining event-free to year 3 is:
With constant hazards, \(F_k(t)=\{\lambda_k/(\lambda_1+\lambda_2)\}\{1-e^{-(\lambda_1+\lambda_2)t}\}\). Substituting the cause-1 values:
Thus the three-year cumulative incidence of cause 1 is approximately \(13.98\%\). For cause 2:
The probability of either event by year 3 is \(F_1(3)+F_2(3)\approx0.6988\), which equals \(1-S(3)\). The remaining probability, approximately \(0.3012\), is the probability of no event by that time. These probabilities add to 1, as they should.
An Exposure Effect on One Cause Can Change Both CIFs
Suppose exposure changes \(\lambda_1\) but leaves \(\lambda_2\) unchanged. Because the hazards are added when calculating \(S(t\mid C)\), the exposure-related change in \(\lambda_1\) changes the number of people who remain event-free and therefore still able to experience cause 2. It follows that the CIF for cause 2 can change even when its cause-specific hazard is constant across exposure levels. This is a probability consequence of competing events, not evidence that the cause-2 hazard itself changed.
Worked Example: Exposure Raises One Hazard but Lowers the Other Cause’s CIF
Use the constant hazards from the previous example as the unexposed condition: \(\lambda_1=0.08\) per year and \(\lambda_2=0.32\) per year. In a hypothetical exposed condition, suppose \(\lambda_1\) doubles to \(0.16\) per year, while \(\lambda_2\) remains \(0.32\) per year. Compare the three-year CIFs.
For the unexposed condition, the total hazard is \(0.40\) per year. The previous calculation gives \(F_1(3)\approx0.1398\) and \(F_2(3)\approx0.5590\).
For the exposed condition, the total hazard is \(0.16+0.32=0.48\) per year. The event-free probability at year 3 is \(e^{-(0.48)(3)}=e^{-1.44}\approx0.2369\). Therefore:
and:
The cause-1 CIF rises from about \(13.98\%\) to \(25.44\%\), an increase of about \(11.46\) percentage points. Although the cause-2 hazard is unchanged at \(0.32\) per year, its CIF falls from about \(55.90\%\) to \(50.87\%\), a decrease of about \(5.03\) percentage points. More cause-1 events occur earlier, leaving fewer people available to experience cause 2 as their first event. The total event probability increases from about \(69.88\%\) to \(76.31\%\), consistent with the decline in event-free probability.
Time-Varying Hazards and Exposure Histories
In PK/PD applications, exposure may vary over time with concentration, dosing, or a delayed drug effect. The same CIF formula applies: use the relevant time-varying cause-specific hazards inside the overall survival and CIF integrals. If the hazards are piecewise constant, calculate each interval’s event contribution using the event-free probability at the start of that interval. That probability reflects all competing hazards accumulated before the interval.
Worked Example: Piecewise-Constant Cause-Specific Hazards
Suppose a hypothetical exposure-related cause-1 hazard is \(0.10\) per year during year 1 and \(0.20\) per year during year 2. The competing cause-2 hazard is \(0.10\) per year throughout. Calculate the cause-1 CIF by the end of year 2.
During year 1, the total hazard is \(0.20\) per year. The event-free probability at year 1 is \(e^{-0.20}\approx0.8187\), and the cause-1 CIF accumulated during that year is:
During year 2, the total hazard is \(0.20+0.10=0.30\) per year. Conditional on being event-free at the start of that year, the cause-1 event probability during the year is \((0.20/0.30)(1-e^{-0.30})\approx0.1728\). Weighting by the probability of reaching year 2 event-free gives the interval’s contribution to the overall CIF:
Adding the two interval contributions gives \(F_1(2)\approx0.09063+0.1415=0.2321\), or about \(23.21\%\). It would be incorrect to calculate the second-year contribution as though everyone remained at risk until year 1: only about \(81.87\%\) are event-free at that point.
Estimation, Assumptions, and Diagnostics
In a cause-specific hazard model, a competing event ends a person’s time at risk for the cause being modelled. For example, when estimating the cause-1 hazard, a cause-2 event removes that person from the cause-1 risk set at the time of the competing event. This convention estimates the observed cause-specific hazard among those still event-free; it does not mean the competing event is ordinary loss to follow-up for the purpose of predicting the CIF.
To estimate a CIF from observed event times, a standard nonparametric approach is the Aalen–Johansen estimator. At each event time, it updates estimated event-free survival using events of all causes, then accumulates the estimated cause-specific event contribution. In schematic form, the increment for cause \(k\) is the estimated event-free probability just before an event time multiplied by the cause-\(k\) events divided by the number at risk. The estimate therefore accounts for competing events rather than treating them as if they were independent censoring.
A model-based CIF prediction requires more than a well-fitting hazard for the target cause. It requires the cause-specific hazards for all relevant competing causes over the prediction period, along with an appropriate exposure history. Important conditions and checks include:
- Event definition: First-event causes must be defined clearly and recorded consistently. Ambiguous or misclassified causes can distort both hazards and CIFs.
- Censoring: Loss to follow-up should be independent of future event risk, conditional on variables included in the analysis, or handled using a method that accounts for its informativeness. Administrative study end is commonly treated as censoring.
- Exposure and model specification: The exposure history, time scale, baseline hazard, and exposure-effect form must be appropriate for each cause. Inspect whether predicted hazards are plausible and compare predicted CIFs with observed event proportions over time.
- Support and precision: At late times, few individuals may remain event-free. Estimates then become imprecise, especially for rare causes or exposure patterns with little data.
These assumptions do not make an exposure coefficient automatically causal. In an observational exposure-response analysis, confounding, treatment changes driven by health status, and exposure measurement error can undermine causal interpretation. A predicted CIF is a model-based probability under the specified covariates and exposure history; a causal interpretation requires an appropriate causal design and assumptions in addition to the competing-risks calculations.
Choosing the Quantity to Report
| Quantity | Question answered | Main caution |
|---|---|---|
| Cause-specific hazard | Among individuals still free of all events, how rapidly is a particular cause occurring now? | It is an instantaneous rate, not a probability of the event by a chosen time. |
| Cumulative incidence | What is the probability that this cause is the first event by time \(t\)? | It depends on the hazards of all competing causes. |
| Subdistribution hazard | How does a covariate relate to a hazard constructed to model the CIF directly? | Its risk-set construction differs from the cause-specific hazard and does not have the same instantaneous-rate interpretation among event-free people. |
Cause-specific hazard models are useful for describing how exposure relates to the rate of each event type among those still at risk. CIFs are generally the direct quantity for communicating absolute probabilities of specific event types by a clinically relevant time. Subdistribution hazard models are an alternative regression approach for CIF-related questions, but their coefficients should not be read as cause-specific hazard ratios. No single summary answers every scientific question; state the estimand and time horizon explicitly.
Common Mistakes and Limitations
- Wrong reasoning: “Cause 2’s hazard is unchanged, so its three-year probability must also be unchanged.” Correction: The CIF uses overall event-free survival. If exposure raises the cause-1 hazard, fewer people remain able to experience cause 2, so its CIF can fall.
- Wrong reasoning: “I can treat competing events as ordinary censoring and use one minus the cause-specific survival curve as the event probability.” Correction: That calculation removes competing causes from the probability accounting. Use the cause-specific hazards for all causes in the CIF formula, or an appropriate competing-risks estimator.
- Wrong reasoning: “A larger cause-specific hazard proves that exposure causes more events.” Correction: A hazard association alone does not establish causation; confounding and exposure assignment must also be considered.
Competing-risks methods address mutually exclusive first events. They do not by themselves describe a sequence of repeated events after the first event; that requires a framework that represents recurrent events and their dependence. Also, the CIF is tied to the event definitions, follow-up period, and population being studied. A different competing event or a different time horizon can change the probability being reported.
Check Your Understanding
Answer each question using the distinction between cause-specific rates and event probabilities.
- For two constant hazards, \(\lambda_1=0.05\) per year and \(\lambda_2=0.15\) per year, what is the CIF for cause 1 by two years?
- If an exposure changes \(\lambda_1\) but leaves \(\lambda_2\) unchanged, must the CIF for cause 2 remain unchanged? Explain.
- Why is a competing event not treated as ordinary independent censoring when estimating a CIF?
- What does a cause-specific hazard describe, and how does that differ from a CIF?
Solutions. (1) The total hazard is \(0.20\) per year, so \(F_1(2)=(0.05/0.20)(1-e^{-(0.20)(2)})=0.25(1-e^{-0.40})\approx0.08242\), or \(8.242\%\). (2) No. A change in \(\lambda_1\) changes overall event-free survival and therefore can change the probability of remaining available to experience cause 2, even with \(\lambda_2\) fixed. (3) A competing event prevents the target event from being the first event later; treating it as ordinary censoring would act as though the competing event did not affect the target-event probability. (4) A cause-specific hazard is the instantaneous rate for one cause among people still free of every event. A CIF is the probability that the cause is the first event by a specified time and incorporates all competing hazards.