Net Risk and Observed Cumulative Incidence

Two Different Risks for the Same Event

When an event of interest can be precluded by another event, two questions that sound similar have different answers. The observed cumulative incidence asks how often the event of interest actually occurs by time \(t\), while allowing competing events to occur. Net risk asks what the risk would be by time \(t\) in a hypothetical setting where competing events are eliminated. The second quantity is therefore not simply another name for the first, nor is it automatically identified by observed data.

Let \(k\) denote the event of interest and \(c\) a competing event. For treatment level \(a\), let \(T^a\) be the time to the first event and \(J^a\) its cause. The observed cumulative incidence for cause \(k\) is \(P(T^a\le t,J^a=k)\). Define \(T_k^{a,\mathrm{elim}}\) as the counterfactual time to event \(k\) under treatment \(a\) if the competing event were eliminated. Net risk is \(P(T_k^{a,\mathrm{elim}}\le t)\). This definition concerns a hypothetical intervention and requires assumptions before it can be given a causal interpretation from observed data.

How the Risks Are Related

For a clear comparison, first consider one treatment group and continuous event-time distributions. Let \(\lambda_k(u)\) and \(\lambda_c(u)\) be the cause-specific hazards among individuals still free of either event just before time \(u\). Their cumulative hazards are \(\Lambda_j(t)=\int_0^t\lambda_j(u)\,du\), for \(j=k,c\). The probability of remaining free of both events through time \(u\) is \(S(u)=\exp\{-\Lambda_k(u)-\Lambda_c(u)\}\). Thus the observed cumulative incidence is the rate of cause \(k\) weighted by the probability of still being at risk:

If elimination of the competing event leaves the target-event hazard equal to \(\lambda_k(u)\), the corresponding net risk is

The equality on the right follows by integrating the derivative of \(1-\exp\{-\Lambda_k(u)\}\). Comparing the integrands gives a direct justification for the usual ordering and for its equality condition.

The integrand is nonnegative at every time. Therefore net risk is at least as large as observed cumulative incidence under these hazard definitions. Equality holds exactly when the integral is zero. For example, equality holds if there is no competing-event hazard before \(t\), or if \(\lambda_k(u)=0\) almost everywhere on the set where \(\Lambda_c(u)>0\). Merely having no simultaneous overlap between the two cause-specific hazards is not sufficient: a competing event occurring earlier can reduce the number at risk when target events occur later.

When Does Net Risk Have a Causal Interpretation?

The formulas above are mathematical relationships between hazards. Their causal interpretation depends on what it means to eliminate the competing event and whether the target-event process under that intervention can be learned from the observed setting. In particular, a cause-specific hazard is conditional on being free of all event types up to that time. People who remain event-free can differ from those who would have experienced a competing event. Consequently, the observed cause-specific hazard need not equal the hazard of the target event in the hypothetical population where competing events are prevented.

One sufficient, but strong and generally untestable, condition is conditional independence of the latent target and competing event times given treatment and measured baseline covariates \(L\). Under this condition, at each \(L\), the target-event hazard among people who have not yet had either event agrees with the hazard of their target-event time in the absence of the competing event. A further structural assumption is needed: the intervention that eliminates the competing event does not itself change the target-event time. Under these assumptions, the conditional net risk can be identified from the target cause-specific hazard, then averaged over the covariate distribution of interest.

For a causal contrast between treatment levels, additional standard requirements include consistency (the observed outcome agrees with the relevant counterfactual outcome under the treatment received), exchangeability of treatment given measured covariates, positivity of treatment assignment at covariate levels being standardized over, and appropriate handling of loss to follow-up. The elimination intervention must also be sufficiently well specified: different ways of preventing a competing event could have different effects on the target event. These requirements are separate from the algebra that relates net risk and cumulative incidence.

Standardization Across Baseline Covariates

When net risks are identified conditional on baseline covariates, a population-level net risk is obtained by averaging the conditional risks over the covariate distribution of interest. This standardization prevents a comparison from silently changing the target population between treatment groups. It does not repair a failure of the causal assumptions needed to identify the conditional risks.

Diagnostics, Alternatives, and Limitations

The observed cumulative incidence is the appropriate quantity when the question concerns the probability of actually experiencing the target event by a stated time under the natural competing-risk process. Net risk addresses a different hypothetical question. A cause-specific hazard model can describe the instantaneous target-event rate among those still event-free, while a Fine-Gray model targets the observed cumulative-incidence process through its subdistribution hazard. Neither model, by itself, makes net risk a causal effect of eliminating competing events.

A common estimation approach censors competing events when estimating a net risk from the target cause-specific hazard. That calculation can estimate a net-risk quantity under suitable assumptions, but censoring a competing event in software does not make the event independent or make the hypothetical intervention well defined. Compare the resulting estimate with the observed cumulative incidence, check whether the assumed target hazard is plausible in the scientific setting, and examine whether results are sensitive to the covariates used to account for dependence between latent event times. These checks can expose instability or implausible modeling choices; they cannot verify untestable elimination assumptions.

A frequent mistake is to say, “The complement of the observed cumulative incidence is the probability of avoiding the target event, so its complement is the net risk.” This is wrong: \(1-F_k(t)\) includes people who had a competing event as well as people who remained event-free. Net risk instead uses the hypothetical target-event process under elimination. Another mistake is to assume that because a competing event is coded as censored, the resulting Kaplan–Meier complement automatically has a causal interpretation. The correction is to name the estimand first, state the assumptions connecting it to observed hazards, and report observed cumulative incidence separately when that is the real-world probability of interest.