Two Different Questions About Risk
For a person who has not yet experienced an event, several mutually exclusive causes may lead to failure. The probability of experiencing a particular cause by a chosen horizon depends on what happens first: an earlier competing event prevents that person from experiencing the cause of interest as the first event. A different quantity asks what the risk of the cause of interest would be if competing causes were eliminated. These are not interchangeable questions.
The distinction matters when communicating absolute risk. A cumulative incidence is an observed-world probability. Net risk is calculated from a cause-specific hazard and is often given a hypothetical interpretation, but the data alone do not generally establish what would happen under an intervention that removes competing events.
At time \(u\), the cause-specific hazard \(\lambda_k(u)\) describes the instantaneous rate of cause \(k\) among people who are still event-free just before \(u\). Its cumulative hazard is \(\Lambda_k(t)=\int_0^t \lambda_k(u)\,du\). In a continuous-time model with hazards for all causes, the event-free survival probability is \(S(u)=\exp\{-\sum_j\Lambda_j(u)\}\). Consequently, the cumulative incidence for cause \(k\) is
The factor \(S(u)\) in the cumulative-incidence integral is crucial: a person can experience cause \(k\) at time \(u\) only if no cause has occurred before \(u\). In the net-risk calculation, survival is instead determined only by the hazard for cause \(k\). Thus, competing causes affect cumulative incidence directly, while they do not enter the mathematical definition of net risk.
Assumptions and Interpretation
These quantities are defined for a specified population, event definitions, and time origin. Causes must be classified consistently, and the event of interest must be distinguished from competing first events. Right censoring is different: it means that the event status or time becomes unobserved, not that a competing event occurred. Estimating either quantity from censored data requires appropriate handling of censoring; in particular, independent censoring is usually required, at least conditional on variables used in the analysis.
Cumulative incidence answers a descriptive prediction question: in the observed setting, what is the probability that cause \(k\) occurs first by \(t\)? Net risk is a transformation of the cause-specific hazard. Its hypothetical interpretation—risk if other causes were eliminated—requires more than calculating that transformation. It requires that eliminating competing causes would not itself change the cause-\(k\) hazard for people who remain at risk. If preventing one cause changes treatment, behavior, or the underlying disease process, that assumption may be implausible.
A related latent-time interpretation is also assumption-dependent. The observed data reveal the first event, not the time at which each person would have experienced every possible cause if all other causes could somehow be suppressed. Independence assumptions about latent failure times, or a sufficiently specified causal model, are needed to give the net quantity that stronger interpretation. Without such assumptions, the net risk is best described as a hazard-based hypothetical quantity, not as an identified probability under a real intervention.
Worked Example: Constant Hazards Over Five Years
Consider a hypothetical population with two constant cause-specific hazards: \(\lambda_1=0.08\) per year for the event of interest and \(\lambda_2=0.12\) per year for a competing event. Calculate the cause-1 cumulative incidence and net risk by five years. For the hypothetical net-risk interpretation, suppose eliminating cause 2 would leave the cause-1 hazard unchanged.
The total hazard is \(0.08+0.12=0.20\) per year, so the probability of remaining event-free to time \(u\) is \(\exp(-0.20u)\). Integrating the cause-1 hazard over those who remain event-free gives
The cause-1 net risk uses only its own cumulative hazard, \(0.08(5)=0.40\):
The observed-world probability of cause 1 by five years is about 25.3%; the hazard-based net risk is about 33.0%. The latter is not the observed probability of cause 1, nor does it show that an intervention preventing cause 2 would produce a 33.0% risk. That causal interpretation depends on the stated assumption that removing cause 2 leaves the cause-1 hazard unchanged.
Why Cumulative Incidence Is No Larger
When the same cause-specific hazard is used for both quantities, cumulative incidence cannot exceed net risk. In the continuous-time setting above, each cause has a nonnegative hazard, so \(\sum_j\Lambda_j(u)\geq\Lambda_k(u)\). Therefore
Multiplying both sides by the nonnegative \(\lambda_k(u)\) and integrating from 0 to \(t\) gives
For the equality, note that the derivative of \(1-e^{-\Lambda_k(u)}\) is \(e^{-\Lambda_k(u)}\lambda_k(u)\). Equality holds if no competing hazard removes anyone over the period in question; otherwise the inequality is generally strict. This is a mathematical comparison using the same cause-specific hazard, not evidence that eliminating competing events would leave that hazard unchanged.
Worked Example: Hazards That Change Over Time
Suppose, hypothetically, that from year 0 to year 2 the cause-1 hazard is \(0.05\) per year and the competing hazard is \(0.15\) per year. From year 2 to year 4, suppose the corresponding hazards are \(0.10\) and \(0.05\) per year. Calculate both cause-1 quantities at year 4.
During the first interval, the total hazard is \(0.20\) per year. The cause-1 incidence accumulated in that interval is
The event-free probability at year 2 is \(e^{-0.4}\approx0.6703\). During the second interval, the total hazard is \(0.15\) per year. The conditional cause-1 incidence over those two years is \((0.10/0.15)(1-e^{-0.15(2)})\approx0.1728\). Weighting this by the probability of reaching year 2 event-free gives an additional incidence of \(0.6703(0.1728)\approx0.1158\). Thus,
The cause-1 cumulative hazard through year 4 is \(0.05(2)+0.10(2)=0.30\). Therefore, \(R_1^{\mathrm{net}}(4)=1-e^{-0.30}\approx0.2592\). The net risk exceeds the cumulative incidence because the latter accounts for the competing hazard in both intervals. As before, interpreting 25.9% as a risk under elimination of the competing cause requires an assumption that the cause-1 hazard would remain as specified.
Estimation, Diagnostics, and Alternatives
The cumulative incidence is estimated with an estimator that accounts for the fact that competing events remove people from being able to experience the event of interest. In a nonparametric analysis, the Aalen–Johansen estimator combines the estimated event-free probability with the cause-specific event increments. Net risk can instead be calculated by estimating the cause-specific hazard and transforming its cumulative value using \(1-\exp\{-\widehat{\Lambda}_k(t)\}\). The earlier material on cumulative incidence functions develops their estimation and uncertainty; the key distinction here is the target each calculation represents.
A common shortcut is to treat competing events as censored and use one minus a Kaplan–Meier estimate for cause \(k\). This produces a net-type quantity based on the cause-specific hazard, not the observed-world cumulative incidence. Treating competing events as censored in that calculation does not make them ordinary independent losses to follow-up, and it does not by itself identify a causal risk under elimination of those events.
Useful checks include confirming that the horizon, population, and cause definitions are identical for the two reported quantities; verifying that the net risk is at least as large as the cumulative incidence when both are computed from the same cause-specific hazard; and checking that the estimated risks are in the interval from 0 to 1. If a reported cumulative incidence exceeds the corresponding net risk, investigate whether different models, populations, horizons, or hazard estimates were used, or whether there is a calculation error. A numerical ordering check is a diagnostic, not proof that either quantity is well estimated.
Worked Example: A Two-Time Discrete Calculation
In a hypothetical cohort of 100 event-free people at time 0, 10 experience cause 1 and 20 experience cause 2 at the first event time. The remaining 70 are event-free at the next time, when 7 experience cause 1 and 7 experience cause 2. Assume no censoring before those event times. Calculate the cause-1 cumulative incidence and the net-type risk formed from the cause-1 conditional event probabilities.
At the first time, the cause-1 probability is \(10/100=0.10\). At the second time, the cause-1 conditional probability among those still event-free is \(7/70=0.10\). For cumulative incidence, the second increment is weighted by the probability of remaining event-free after the first time, \(70/100=0.70\). Hence
For the corresponding net-type calculation, use the two cause-1 conditional probabilities without reducing the second interval's starting group by cause-2 events:
Thus the observed-world cause-1 probability is 17%, while the hazard-based net quantity is 19%. Calling 19% the probability under actual prevention of cause 2 would require the additional assumption that the cause-1 conditional probabilities would remain unchanged in that hypothetical setting.
Common Mistakes and Limits
- Calling net risk the observed probability. The observed-world probability is the cumulative incidence. Net risk removes competing hazards from its calculation and has a different interpretation.
- Assuming censoring competing events estimates cumulative incidence. A calculation that treats competing events as censored targets a net-type quantity, not the observed-world probability that cause \(k\) occurs first.
- Interpreting net risk as an intervention effect without qualification. The data do not generally show how the cause-\(k\) hazard would change if competing events were prevented. State the required invariance or causal assumptions.
- Comparing numbers from different settings. A valid comparison requires the same population, time origin, horizon, and cause definition, as well as compatible hazard estimates.
Neither quantity is universally preferable. Cumulative incidence is appropriate for prediction and counseling about what is likely to happen in the observed setting. Net risk may be useful for describing the cause-specific hazard on a scale without competing causes, or for a clearly specified hypothetical question. Its limitations are substantial when the intervention needed to eliminate competing events would alter the process generating the event of interest.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In one sentence, distinguish the event described by cumulative incidence from the hypothetical event described by net risk.
- A constant cause-\(k\) hazard is \(0.04\) per year for three years. Calculate net risk by year 3.
- Using the hazards in Question 2, suppose a competing hazard of \(0.06\) per year is also constant. Calculate the cumulative incidence by year 3 and explain why it is smaller than net risk.
- Does a net-risk estimate by itself identify the probability that would occur if a competing cause were prevented? State the key qualification.
Solutions. (1) Cumulative incidence is the observed-world probability that cause \(k\) occurs first by the horizon; net risk is derived from the cause-\(k\) hazard as if competing causes did not remove people. (2) \(1-e^{-0.04(3)}=1-e^{-0.12}\approx0.1131\), or 11.3% by three years. (3) The total hazard is \(0.10\) per year, so \(F_k(3)=(0.04/0.10)(1-e^{-0.10(3)})=0.4(1-e^{-0.3})\approx0.1037\), or 10.4%. Competing events reduce the probability of remaining event-free and therefore reduce observed cause-\(k\) incidence. (4) No. That causal interpretation requires assumptions, including that eliminating the competing cause would not change the cause-\(k\) hazard; a net-risk transformation alone does not establish that condition.