Why Competing Events Change Absolute Risk
An absolute genetic risk estimate answers a probability question over a specified time horizon: for a person with a particular genetic score and clinical profile, what is the probability of the event by that time? The answer can depend on what happens before the event of interest. For example, death from another cause prevents a person from subsequently experiencing a first diagnosis of the disease being predicted. That death is not simply missing follow-up; it changes which outcomes remain possible.
A competing event is an event that precludes the target event from occurring as the first event under study. In genetic risk prediction, the competing event may itself be associated with age, clinical characteristics, or the genetic score. The cumulative incidence of the target event must therefore account for the chance that a competing event occurs first.
Here \(x\) can include a polygenic score and clinical predictors. The event definition and time origin must be clear: for example, the question might concern first diagnosis of a disease before death from another cause, within ten years of assessment. A probability of a first event is not the same as the probability of ever experiencing the disease under a hypothetical world in which other causes of death could not occur.
Cause-Specific Hazards and Cumulative Incidence
For each cause \(k\), the cause-specific hazard \(\lambda_k(t\mid x)\) describes the instantaneous rate of that event at time \(t\), among individuals with predictors \(x\) who have experienced no event yet. Formally, it is the limit of the conditional probability of cause \(k\) in a short interval divided by that interval's length. Someone who has already experienced any first event is no longer in the event-free risk set.
The factor \(S(u-\mid x)\) is essential: a person can experience cause \(k\) at time \(u\) only if they have remained free of every event up to just before \(u\). Multiplying this event-free probability by the cause-specific hazard gives the instantaneous contribution to the target event's cumulative probability. Summing those contributions over time gives the cumulative incidence.
These expressions require the hazards for all first-event causes, not just the target cause. They do not require independent latent times to different causes. The cause-specific hazards describe the observed first-event process among those still event-free. In practice, a genetic risk model needs appropriate cause-specific hazards, predictors, and a defined target population to produce absolute risks.
A Short Derivation
Consider a short interval from \(t\) to \(t+dt\). Conditional on being event-free just before \(t\), the probability of cause \(k\) in this interval is approximately \(\lambda_k(t\mid x)dt\), for a sufficiently small \(dt\). The unconditional probability of a cause-\(k\) event in that interval is therefore approximately the probability of still being event-free multiplied by that conditional probability:
Thus the rate of accumulation of \(F_k(t\mid x)\) is \(S(t-\mid x)\lambda_k(t\mid x)\). Integrating from time zero to \(t\) gives the cumulative incidence formula. To obtain the corresponding event-free survival expression, note that the total instantaneous rate of any first event is the sum of the cause-specific hazards. Therefore \(S\) obeys the differential equation below, with \(S(0\mid x)=1\):
Dividing by \(S(t\mid x)\) and integrating both sides from zero to \(t\) gives \(\log S(t\mid x)=-\int_0^t\sum_j\lambda_j(u\mid x)\,du\). Exponentiating yields the stated expression for event-free survival. This derivation also shows why every competing cause affects the target cumulative incidence: each contributes to the total rate at which event-free survival declines.
Worked Example: Constant Hazards Over Ten Years
Suppose, hypothetically, that a genetic-risk model assigns a person a constant target-disease cause-specific hazard of \(0.020\) per year and a competing-death hazard of \(0.030\) per year. Assume these hazards are constant over the ten-year horizon and that these are the only first-event causes. Calculate the ten-year target-disease cumulative incidence, then compare it with a calculation that ignores competing death.
The total hazard is \(0.020+0.030=0.050\) per year. Event-free survival at ten years is therefore \(S(10)=\exp(-0.050\times10)=\exp(-0.5)=0.6065\). With constant hazards, integrating the target cause's contribution gives:
The model's ten-year probability of experiencing the target disease first is therefore about 15.74%. The probability of experiencing competing death first is \(0.030/0.050\times(1-e^{-0.5})=0.2361\), or about 23.61%. The remaining 60.65% are event-free at ten years; these probabilities sum to 100%, subject to rounding.
If competing death were removed from the risk calculation, the target-only calculation would be \(1-e^{-0.020\times10}=0.1813\), or 18.13%. This is not the observed-world probability of target disease before competing death. It is a different quantity based on the target hazard without allowing competing events to reduce the time spent event-free.
Piecewise Hazards and Changing Risk Over Time
Cause-specific hazards need not be constant. They may change with age or time since assessment, and their relative sizes may change too. The cumulative incidence is still obtained by accumulating the target hazard among those who remain event-free. For piecewise-constant hazards, calculate each interval's event-free survival and add the target-event contributions across intervals.
Worked Example: A Target Hazard That Rises After Year Five
Consider a hypothetical ten-year prediction. During years 0–5, the target-disease hazard is \(0.010\) per year and the competing-event hazard is \(0.020\) per year. During years 5–10, the corresponding hazards are \(0.030\) and \(0.020\) per year. Assume hazards are constant within each interval. Calculate the target cumulative incidence by year 10.
In the first five years, the total hazard is \(0.030\) per year. The target-event contribution in that interval is:
The probability of remaining event-free through year 5 is \(S(5)=e^{-0.15}=0.8607\). In years 5–10, the total hazard is \(0.030+0.020=0.050\) per year. The target-event contribution during this second interval, conditional on survival to year 5 and then weighted by that survival probability, is:
Adding the two contributions gives \(F_1(10)=0.04643+0.1142=0.1607\). Thus the hypothetical ten-year probability of the target disease occurring first is about 16.07%. By contrast, ignoring the competing cause gives a target-only risk of \(1-e^{-(0.010\times5+0.030\times5)}=1-e^{-0.2}=0.1813\), or 18.13%. The rising target hazard increases later disease risk, but competing events still reduce the probability that a person remains able to experience the target event.
Estimating Cumulative Incidence From Event Counts
The same logic applies to observed event data. At each event time, the Aalen–Johansen estimator updates event-free survival using all causes of first event, then adds the target-event share of the risk set weighted by survival just before that time. In the simple case of distinct observed event times, let \(Y(u)\) be the number at risk immediately before time \(u\), and \(d_k(u)\) the number of cause-\(k\) events at \(u\). The estimated survival just after \(u\) is multiplied by \(1-\sum_j d_j(u)/Y(u)\), and the estimated cause-\(k\) cumulative incidence increases by the survival just before \(u\) times \(d_k(u)/Y(u)\). Tied event times use the same increment based on all events at that time.
Worked Example: A Small Event Table
In a hypothetical cohort of 100 people in a genetic-score group, 5 have the target diagnosis first at year 1 and 10 have a competing event first at year 1. The remaining 85 are event-free and at risk at year 2. At year 2, 6 experience the target diagnosis first and 8 experience a competing event first. Assume there is no loss to follow-up before year 2. Estimate the target cumulative incidence through year 2.
At year 1, the target contribution is \(1\times(5/100)=0.0500\), so the estimated cumulative incidence is 5.00%. The event-free proportion immediately after year 1 is \(1-(5+10)/100=0.85\). At year 2, the target contribution is \(0.85\times(6/85)=0.0600\). Therefore:
The estimated probability of the target diagnosis first by year 2 is 11.00%. The competing events reduce the event-free risk set; they are not counted as if those people could still later have the target diagnosis during the study.
Why Treating Competing Events as Censoring Gives a Different Answer
A common alternative is to censor people when a competing event occurs and use \(1-\widehat{S}(t)\) from a Kaplan–Meier calculation for the target event. In the small example, this calculation treats the 10 competing events at year 1 as censored. It then uses 85 people at risk at year 2, of whom 6 have the target event. The estimated target-only survival is:
Consequently, \(1-\widehat{S}_{\mathrm{KM}}(2)=0.1171\), or 11.71%, compared with the cumulative incidence estimate of 11.00%. The Kaplan–Meier calculation removes competing events from later risk sets but does not reduce estimated survival at the time those competing events occur. It therefore estimates a target-only, or net-risk, quantity based on the target cause-specific hazard, rather than the real-world probability of the target event before a competing event.
| Quantity | How competing events are handled | Risk question answered |
|---|---|---|
| Cumulative incidence | Counted as first events that remove people from the event-free risk set | Probability of the target event occurring first by the horizon |
| One minus Kaplan–Meier survival with competing events censored | Removed from subsequent risk sets, without being counted as events | Target-only or net risk based on the target cause-specific hazard |
Censoring competing events can be useful when the intended question is explicitly a hypothetical target-only risk, but it does not estimate the observed-world cumulative incidence merely by ignoring the competing cause. Interpreting net risk as the risk that would occur if competing events were eliminated also requires additional causal assumptions; the calculation alone does not establish that interpretation.
Assumptions and a Common Pitfall
A competing-risk absolute-risk estimate depends on a clearly specified first-event outcome, a time origin and horizon, and cause-specific hazards that represent the target population and predictors of interest. Estimation also assumes that loss to follow-up is handled appropriately: for example, conditional on modeled predictors, censoring due to incomplete observation must not be informative about future event risk. Misclassified causes, incomplete follow-up, or hazards that do not transport to the target population can distort the resulting risk.
Common mistake: “The target-event hazard is 2% per year, so the ten-year risk is \(1-e^{-0.02\times10}=18.13\%\), even though some people die of other causes.” This calculation omits the competing-event hazard from event-free survival. It is a target-only risk calculation, not necessarily the probability of target disease before competing death. For the latter, calculate \(F_1(t\mid x)=\int_0^t S(u-\mid x)\lambda_1(u\mid x)\,du\), with \(S\) incorporating the hazards of every competing first event.
Check Your Understanding
Answer each question by identifying the risk quantity and showing the relevant reasoning.
- In your own words, define the cumulative incidence of a target event by time \(t\) in the presence of competing events.
- Why does the formula for target cumulative incidence include event-free survival based on the hazards of all causes?
- Suppose the target hazard is \(0.01\) per year and the competing-event hazard is \(0.04\) per year, both constant for five years. Write the formula for target cumulative incidence over this horizon.
- In the small event-table example, why does the year-2 target contribution use \(0.85\) rather than \(0.95\) as the event-free weight?
- What risk quantity does one minus Kaplan–Meier survival estimate when competing events are censored, and why is that quantity not generally the observed-world cumulative incidence?