Benefit-Risk Estimands for Competing Outcomes

Why the estimand must match the outcome structure

A treatment may reduce a beneficial event, increase a harmful event, or affect the chance of experiencing either. Those outcomes do not necessarily share a scale or occur independently. Some are mutually exclusive first events; others can recur or occur in the same person. A useful benefit–risk comparison therefore begins by defining which population quantity is being compared, rather than by combining outcomes into a single score.

Here, an estimand is a precisely defined population-level treatment contrast. It specifies the treatment strategies being compared, the population, the outcome variable and time frame, and the summary measure. It is not itself a numerical estimate: the estimate is calculated from observed data using an analysis appropriate to that estimand. The focus here is on defining contrasts that retain the meaning of distinct benefits and harms.

This definition uses potential outcomes to describe the target of inference; it does not imply that both treatment outcomes can be observed for the same person. In a randomized comparison, randomization supports estimating the strategy contrast, provided the assigned strategies and outcomes are defined consistently, outcome data are sufficiently complete or missingness is handled with justified assumptions, and each strategy has a positive chance of assignment in the target population. If follow-up stops at treatment discontinuation but the estimand concerns outcomes regardless of discontinuation, that loss of outcome information must be addressed rather than silently changing the target.

Fixed-horizon risks with competing event types

Suppose the outcome is the first event after treatment begins, and its type \(J\) may be a benefit event, a harm event, or another event such as death. For a specified event type \(k\), define the cumulative incidence by time \(\tau\) as the probability of experiencing that event type by \(\tau\), before any competing first event. This is a population probability, not merely the event rate among people still event-free.

A direct treatment contrast is the risk difference \(F_k^1(\tau)-F_k^0(\tau)\), in percentage points or probability units. A negative value means a lower probability of that event type under strategy 1. A risk ratio \(F_k^1(\tau)/F_k^0(\tau)\) is another possible relative contrast when the comparator risk is nonzero. The choice changes the question: the risk difference describes an absolute population difference, while the ratio describes a relative one.

When event types are mutually exclusive first events, their cumulative incidences are components of the overall first-event distribution. An event of another type prevents the focal first event from occurring later in that first-event classification. Consequently, the probability of a focal event depends both on its occurrence process and on the probability of remaining free of all competing first events.

Here \(S^a(u-)\) is the probability of having experienced no first event just before time \(u\), and \(\Lambda_k^a\) is the cumulative cause-specific hazard for event type \(k\). The formula follows by adding, over time, the probability of still being event-free just before \(u\) multiplied by the instantaneous probability of a type-\(k\) event in the next instant, conditional on being event-free. Thus, even if the cause-specific hazard for a benefit event is unchanged, an increase in a competing fatal event can reduce the cumulative incidence of that benefit event.

Time-to-event contrasts answer different questions

For a time-to-event outcome, the event and time origin must be explicit. If \(T^a\) is time to the event of interest under strategy \(a\), survival at time \(\tau\), \(S^a(\tau)=P(T^a>\tau)\), is the probability of remaining event-free beyond that time. A fixed-time survival difference, \(S^1(\tau)-S^0(\tau)\), compares event-free probabilities at that horizon. For a composite endpoint such as first hospitalization or death, it compares the probability of avoiding either component; it does not reveal which component accounts for the difference.

Another absolute contrast is the difference in restricted mean event-free time through \(\tau\). It compares the expected time alive and event-free accumulated over the interval from zero to \(\tau\), without requiring proportional hazards. It is the area between the two survival curves, with units of time.

A cause-specific hazard contrast, such as a hazard ratio, instead compares instantaneous event rates among people who remain event-free just before each time. It is conditional on remaining at risk and is not a direct comparison of population probabilities. A hazard ratio does not generally equal a risk ratio, and a hazard ratio alone does not state the absolute number of events avoided by a fixed horizon. Proportional-hazards summaries additionally rely on a proportionality assumption for a constant hazard ratio; survival differences and restricted mean contrasts do not require that assumption.

Recurrent events require a count or rate estimand

When an outcome can occur repeatedly, a first-event estimand discards later episodes. Let \(N^a(\tau)\) be the number of events per person by time \(\tau\) under strategy \(a\). The difference in mean cumulative counts, \(E[N^1(\tau)]-E[N^0(\tau)]\), compares average event burden per person over a common horizon. A recurrent-event rate instead relates the total number of events to the total time at risk. Define it as the population mean count divided by the population mean exposure time; a rate ratio compares those rates.

Here \(L^a(\tau)\) is exposure time through \(\tau\), measured in person-time. A rate difference has units such as events per person-year; a rate ratio is dimensionless. This population rate is a ratio of totals in expectation, not necessarily the average of each participant’s individual event rate. The distinction matters when exposure times differ.

Choosing and checking the estimand

Each contrast supports a different treatment comparison. Fixed-horizon cumulative incidence compares the probability of a particular first-event type by a clinically specified time. Survival probability compares event-free status at one time, while restricted mean survival time compares accumulated event-free time up to that horizon. Recurrent-event mean counts compare average total burden through a horizon; recurrent-event rates compare event frequency per unit of observed exposure time. None of these summaries is automatically a complete benefit–risk judgment. They preserve separate outcome meanings so that later assessment can consider them without confusing event probabilities, time, and rates.

Before estimating a contrast, check that the event is operationally defined, the time origin and horizon are clinically meaningful, and the treatment strategies correspond to the intended comparison. For competing first events, confirm that categories are mutually exclusive and define what happens when event times tie or cannot be classified. For recurrent events, specify whether episodes are counted, how a recurrence is distinguished from continuation of one episode, and when time at risk ends.

Censoring also matters. A person lost before the horizon has an unknown later outcome, not automatically “no event.” Time-to-event and competing-risk estimates require appropriate handling of censoring; in ordinary analyses, this commonly entails that censoring is independent of the event process, at least conditional on variables included in the analysis. If that assumption is implausible, use a justified adjustment and sensitivity analyses. Check follow-up completeness, event adjudication, risk-set sizes, and whether estimated curves or rates are driven by a small number of participants or late observations.

A frequent mistake is to treat a competing death as ordinary censoring and then interpret one minus a Kaplan–Meier estimate as the real-world probability of the nonfatal event. That calculation targets a different quantity: it behaves as if those who die could remain under observation for the event of interest, under assumptions that are not warranted simply by labeling death as censoring. For the probability of a particular first-event type in the presence of competing events, use its cumulative incidence. Cause-specific hazards can still be useful for describing event processes, but they do not replace the population risk contrast.

Another pitfall is to compare treatment groups only on a rate ratio when exposure time is affected by a terminal event. A lower rate per observed person-year can coexist with fewer days alive and at risk. Report the denominator definition and interpret the rate alongside an appropriate fixed-horizon outcome. Conversely, a first-event analysis can miss an important treatment effect on repeated episodes after the first event. The estimand should follow the clinical question, not just the easiest available model.