Competing-Risk Outcomes in External Comparisons

Why competing events change the question

In a time-to-event comparison, some participants may experience an event that prevents the event of interest from occurring later. Death before disease progression is a familiar example: after death, progression can no longer be observed. Such an event is a competing event. Treating it as ordinary loss to follow-up changes the quantity being estimated and can distort an external comparison.

The central quantity here is the probability of experiencing a specified event type by a fixed time, before experiencing any competing event. In an external-control analysis, the treatment and control risks must refer to the same event definition, time origin, horizon, and target population. These conditions are particularly important when trial and routine-care records classify events differently.

This is a probability on the original patient scale, not a hazard. A cause-specific hazard describes the instantaneous event rate among people still event-free. The cumulative incidence combines the cause-specific hazards for all event types: a high competing-event rate leaves fewer people available to experience the event of interest. The earlier material on time-to-event analysis with external controls provides the broader setting for comparing follow-up outcomes; here the focus is on the competing-event estimand and its implementation.

From event probabilities to cumulative incidence

For clarity, first divide follow-up into intervals. In interval \(r\), let \(h_{kr}\) be the conditional probability of event type \(k\) among people who are event-free at the start of that interval. The probabilities for all event types, together with remaining event-free, must sum to one. Define \(S_{r-1}\) as the probability of still being event-free at the start of interval \(r\). Then the contribution of interval \(r\) to the risk of event type \(k\) is \(S_{r-1}h_{kr}\).

The last expression adds the chance of event \(k\) in each interval, weighted by the chance of reaching that interval without any event. This is the discrete-time form of the cumulative-incidence calculation. In continuous time, the corresponding expression is \(F_k(t)=\int_0^t S(u-)\,d\Lambda_k(u)\), where \(S(u-)\) is the probability of remaining event-free just before time \(u\), and \(\Lambda_k\) is the cumulative cause-specific hazard for type \(k\). With observed right-censored data, the Aalen–Johansen estimator uses estimated event hazards and event-free survival to estimate these quantities.

Defining a comparable external-control contrast

Let \(A=1\) denote trial treatment and \(A=0\) the external-control strategy. Let \(X\) be baseline covariates, and let \(F_k(t\mid A=a,X=x)\) denote the event-\(k\) cumulative incidence under strategy \(a\), conditional on covariates \(x\). For a target covariate distribution \(G\), a standardized risk is the average of the conditional cumulative incidences over that distribution:

The risk difference \(\Delta_k(t;G)\) is measured in probability units (or percentage points) at the specified horizon. It contrasts the two strategies in the same target population, rather than comparing an unadjusted trial risk with an unadjusted external-cohort risk. A risk ratio may also be reported when the control risk is nonzero, but it answers a different scale-specific question.

For a causal interpretation, the usual requirements for an adjusted external comparison apply: the strategies and outcome must be well-defined; measured baseline covariates must suffice to control treatment-selection differences; both strategies must be represented across covariate patterns in the target population; and the observed outcome under a received strategy must correspond to the potential outcome under that strategy. Follow-up and censoring must also be handled so that censoring does not selectively hide event outcomes after conditioning on the variables in the model. These assumptions do not become true merely because estimates are Bayesian.

Bayesian estimation and uncertainty

A Bayesian analysis can represent uncertainty in the event processes and in the standardized contrast. One approach models the interval-specific probabilities of each event type among people still event-free, conditional on treatment and covariates. For every posterior draw, those probabilities are converted to cumulative incidences using the recursion above, then averaged over the same target covariate distribution for both strategies. Subtracting the two standardized risks gives a posterior draw of the treatment contrast.

The posterior mean or median summarizes the contrast, while a credible interval summarizes posterior uncertainty under the specified likelihood, priors, and model. For example, a 95% credible interval is an interval containing 95% of the posterior probability for the contrast. It is not a guarantee that selection bias, outcome misclassification, or model misspecification has been accounted for. Posterior draws should preserve dependence among event probabilities and among contrasts that share model parameters; separately combining marginal interval endpoints does not generally produce a valid interval for a risk difference.

Harmonization, alternatives, and diagnostics

Before fitting a model, specify the event of interest, every competing event, and the rule for ambiguous or unknown causes. Apply the same clinical definition in the trial and external data. Also align the time origin, eligibility criteria, follow-up horizon, and event-ascertainment procedures. For instance, a trial may adjudicate disease-specific death while routine-care records only identify all-cause death. If the external source cannot distinguish the relevant cause, it cannot directly estimate the same cause-specific cumulative incidence. Relabeling an unknown cause as “no event” does not solve that problem.

The main alternatives answer different questions. A cause-specific hazard comparison concerns event rates among people still free of every event; it is useful for describing the event process but is not itself a cumulative-incidence risk contrast. A subdistribution-hazard model parameterizes the cumulative-incidence function differently and can be useful for regression, but its hazard is not an ordinary event rate among an event-free risk set. Neither hazard-scale result should be substituted for the standardized risk at a chosen horizon when that is the target estimand.

Common warning signs include a fitted risk outside the range from 0 to 1, standardized risks that fail to account for all competing event types and event-free status, large changes after modest changes to event coding, and sparse covariate strata with little or no representation in one source. Check that estimated cause-specific event probabilities are valid, that calculated event-type cumulative incidences and event-free probability sum to one, and that posterior predictive or fitted event counts are not plainly inconsistent with the observed follow-up. These checks can reveal computational or model problems, but they do not establish exchangeability or eliminate differential ascertainment.

A useful final report names the event of interest and competing events, gives the target population and time horizon, reports the standardized risks and contrast on a clinically interpretable scale, and describes posterior uncertainty. It should also state which outcome and adjustment assumptions are most consequential. Where event definitions cannot be reconciled or covariate support is inadequate, a precise posterior distribution can still describe the fitted model while providing weak evidence about the intended clinical contrast.