Estimands For Recurrent-Event Studies

Why the Estimand Comes Before the Model

A study of recurrent events can summarize the same observations in several valid ways. It might ask how quickly a second event occurs among people who have already had a first, how many events the average person experiences by one year, or how frequently events occur per unit of observed time at risk. These are different questions, so their answers need not agree. Choosing a model before specifying which quantity is of interest can produce a precise estimate of the wrong scientific target.

An estimand specifies the quantity a study aims to learn about a target population. For recurrent events, state at least the event being counted, the population or treatment group, the time scale and horizon, whether the summary is conditional or marginal, and how death or other terminal events affect follow-up. The earlier material on recurrent-event data and event order distinguishes the first, second, and subsequent events; here the focus is what population quantity those event orders can define.

The phrase “conditional on the stated history” matters. A hazard for the third event concerns people who have had two events and are still under observation and at risk for a third. It is not a hazard among everyone who entered the study. Further, if the hazard conditions on a person's observed history, it is a conditional quantity; an average rate or event count across the target population is marginal. The target and conditioning set are part of the estimand, not optional details.

Event-Order-Specific Hazards

For a given event order \(k\), the hazard at time \(t\) concerns the next event of that order among people who have not yet experienced it. In a continuous-time formulation, given relevant history \(\mathcal{H}_i(t-)\), its definition is

The time origin must be stated: \(t\) might be time since study entry or time since the previous event. The hazard also depends on which information is included in \(\mathcal{H}_i(t-)\). An event-specific hazard is not itself the probability of having that event by time \(t\), and it does not directly report the average number of events in the population. Its units are inverse time, such as events per person-year in a hazard-rate interpretation.

For a research question about progression through event orders, one may compare the hazards of the first and second events, or ask how covariates relate to the hazard of a particular order. This targets event timing within the corresponding risk set. It does not automatically answer how a treatment changes the total burden of events: people who have an earlier event, remain alive, and continue under observation make up a selected group for later event orders.

Mean Event Counts and Marginal Event Rates

The mean event count at horizon \(\tau\), \(E\{N_i(\tau)\}\), is the expected number of qualifying events per person by that time in the target population. It is a population-level summary, not a hazard and not necessarily the number experienced by a “typical” individual. It can be non-integer: a mean of 1.4 events per person does not mean that anyone experiences 1.4 events. It means the population average count is 1.4 over the stated horizon.

A marginal event rate instead puts expected event count over expected time at risk. If \(U_i\) is the time individual \(i\) contributes at risk under the specified observation regime, a population rate can be written

Its units are events per unit of person-time. With common, complete follow-up to \(\tau\), the corresponding rate is \(E\{N_i(\tau)\}/\tau\). With variable follow-up, the mean count per person and the event rate per person-time are distinct summaries: the former uses a fixed horizon if one is specified, while the latter accounts for the amount of time contributed. In data, a ratio of total events to total observed person-time is a descriptive rate; interpreting it as a target-population rate requires appropriate sampling and censoring assumptions.

The connection between event order and the mean count can be shown directly. For any nonnegative integer count, write it as a sum of indicators that the count reaches each event order. Taking expectations gives

The first equality holds because a person with \(m\) events by \(\tau\) contributes one to each of the first \(m\) indicators and zero thereafter. Taking expectations term by term for nonnegative indicators yields the second equality. This identity shows why a mean count reflects the accumulation of event orders, whereas a hazard for order \(k\) is conditional on being at risk for that particular event.

Match the Estimand to the Research Question

Before analysis, state the question in a form that makes the target quantity explicit. A useful checklist is: who is the target population; what event counts; what is the time origin and horizon; does the question concern a particular event order or all events; and is the summary conditional on history or averaged across the population? For a treatment comparison, also specify the treatment strategies and whether the comparison concerns each strategy's population-level outcomes.

These summaries can point in different directions without contradiction. A treatment could delay second events among people who reach the second-event risk set, yet have little effect on the overall mean count if many people never experience a first event. Conversely, an overall mean count could fall because fewer people remain alive and able to experience events, even if the event process among survivors is unchanged. The interpretation must follow the estimand, not a preferred verbal label such as “benefit.”

Assumptions, Diagnostics, and Limitations

The definitions alone do not guarantee that the estimand is identifiable from observed data. A fixed-horizon mean count requires adequate information about event histories through that horizon. If people are censored before \(\tau\), their unobserved later counts cannot simply be treated as zero. Estimation then requires assumptions or methods that address censoring, such as independent censoring conditional on relevant information or a justified censoring adjustment. The earlier material on censoring addresses these issues; the key point here is that the target quantity and the assumptions needed to recover it are separate.

A marginal rate based on observed person-time can be distorted if censoring is related to future event experience. A hazard for a later event order can also be hard to interpret when the risk set is small or selected by earlier events, death, or loss to follow-up. For either estimand, report the time origin, event definition, follow-up rules, and risk population. Examine event counts by order and follow-up, the amount of person-time, and the number remaining at risk; sparse higher-order events make estimates unstable and narrow-looking model-based summaries potentially misleading.

Death needs explicit treatment because it prevents future recurrent events. If death ends event observation, a mean count under the observed treatment strategy includes both event occurrence and the amount of time people remain alive to experience events. That may be exactly the desired treatment-policy summary, but it is not the same as an event rate in a hypothetical world where everyone remained alive. A rate calculated among observed person-time likewise does not automatically isolate an effect on event propensity independent of survival.

The main alternatives answer different questions. An event-specific hazard focuses on timing and conditioning for one event order. A mean count focuses on total population burden by a fixed time. A marginal rate standardizes event accumulation by expected time at risk. None is universally best, and none can be substituted for another without changing the question. These definitions also do not specify a regression model or estimation procedure; those require additional choices about time scale, risk sets, and dependence among events.

Common Pitfall