Event Timing in Longitudinal Estimands

Why Event Timing Changes a Longitudinal Estimand

A longitudinal outcome is measured more than once, but an intercurrent event may occur between scheduled visits. The timing matters: a score recorded after rescue medication, treatment discontinuation, or another event can answer a different question from a score recorded before it. Even when everyone has measurements at the same scheduled visits, the target changes if some measurements are included only when they occur before an event.

The central task is to specify the target before deciding how to use the observed records. State when follow-up begins, how the event time is defined, which outcome window is relevant, and whether measurements before, at, and after the event enter the summary. The event-handling strategy determines the meaning of those rules; the introductory material on intercurrent-event strategies establishes the main strategy choices.

This notation describes the target, not necessarily what can be directly observed. For a treatment-policy strategy, for example, outcomes after the event remain relevant to the target. Under a while-on-treatment strategy, a rule might include measurements strictly before the event and exclude measurements at or after it. Under a hypothetical strategy, the target concerns outcomes under the specified scenario in which the event does not occur. The strategy alone does not specify a complete longitudinal summary: the measurement window and its inclusion rule must also be stated.

For an outcome recorded at scheduled visit times \(v_0,\ldots,v_K\), a summary can be defined using a window \(W\) and an inclusion rule \(q_g(v_j,T_i)\). The rule equals 1 when a visit contributes and 0 otherwise. A participant-level average over eligible visits, when at least one eligible visit exists, can be written as follows:

This formula is one possible summary, not a universal definition. A target might instead use one fixed visit, a change from baseline, a weighted average, or a time-to-event outcome. If the denominator is zero for some participant, the summary as written is undefined for that participant; the estimand must specify an appropriate alternative rather than silently dropping that person.

Specify Time, Window, and Inclusion Before Summarizing

A usable specification should make the following choices explicit.

• Time origin and units. For example, measure elapsed days from randomization, not from the first post-randomization visit. Use the same convention in the event and outcome timelines.
• Event definition and timing convention. Say which event counts and whether \(T_i\) is its onset, occurrence, or recorded date. If a visit only reveals that the event happened since the last visit, its time is interval-censored: the exact date is not known from those visit records alone.
• Outcome window. A fixed calendar window, such as days 28–56 after randomization, differs from a window anchored to the event, such as days 0–14 after event onset.
• Boundary rule. State whether a measurement exactly at \(T_i\) is included. “Before the event” commonly means \(t<T_i\), but a measurement taken at the event may warrant its own convention.
• Summary and weighting. Specify whether the target is a single visit, a participant-level average, or another summary. If participants contribute different numbers of visits, decide whether the target weights participants equally or weights each recorded measurement equally.

These details are scientifically meaningful. A fixed day-84 outcome asks about status at a common time after randomization. A measurement taken 14 days after each participant’s event asks about status at a common time relative to that event. Those measurements may occur on different calendar days and do not estimate the same quantity.

Event-Dependent Windows and Participant Weighting

When the inclusion rule depends on \(T_i\), participants can contribute different numbers of measurements. If the scientific target is the average of participants’ own eligible-visit averages, each participant’s average should be calculated first, then averaged across the target population. Pooling all eligible visits instead gives more weight to participants with more visits. The two summaries are generally unequal.

This example assumes the event dates are known accurately enough to apply the strict \(t<T_i\) rule, and that the listed scores are available at the scheduled visits. If the exact event date is uncertain, the rule may not determine whether a measurement contributes. The measurement-inclusion rule does not resolve missing scores either: a scheduled value that belongs to the target but is unobserved remains a missing outcome, not a value that can be discarded by redefining the window.

Justification: Why Different Windows Produce Different Targets

The distinction between a fixed-time and an event-aligned summary follows directly from their definitions. Let the fixed-time summary be the outcome at day 84, \(Y_i^{a,g}(84)\). Let an event-aligned summary instead be the outcome 14 days after the event, \(Y_i^{a,g}(T_i^{a,g}+14)\), when that time is defined under the chosen strategy. They are equal only under additional conditions, such as \(T_i^{a,g}=70\) for every participant, or a trajectory that happens to have the same value at both times.

Consequently, changing from a common calendar-time endpoint to an event-relative endpoint changes the time at which the outcome is evaluated, not just the formatting of the data. It may also change which participants have a defined event-relative measurement and which outcomes are available. The estimand must make that target explicit before a contrast is interpreted.

A related distinction arises with averaging. If participant \(i\) has \(m_i\) eligible measurements \(Y_{ij}\), the average of participant-specific means is \(\frac{1}{n}\sum_{i=1}^{n}\frac{1}{m_i}\sum_{j=1}^{m_i}Y_{ij}\). The pooled-measurement average is \(\frac{\sum_i\sum_jY_{ij}}{\sum_i m_i}\). In the latter, participant \(i\)'s total weight is proportional to \(m_i\); in the former each participant has weight \(1/n\). Unless the number of eligible measurements and their values align in special ways, the summaries differ. This algebra is why the weighting rule belongs in the estimand specification.

Choosing and Checking the Timing Rule

Different choices can be appropriate for different scientific questions. A treatment-policy target may include outcomes at a fixed visit regardless of an event. A while-on-treatment target may use measurements before an event, with a precise boundary convention. A hypothetical target may refer to outcomes in a scenario where an event does not occur. A composite strategy can make the event itself part of a combined outcome. These approaches are not interchangeable ways of handling the same endpoint: they can target different questions.

Before analysis, check the protocol and data for inconsistencies that can alter these rules. Confirm that the event’s time origin and recorded date use compatible units; compare scheduled visits with the event timeline; identify event dates known only within intervals; and verify the exact boundary rule for a visit on the event date. Check whether the outcome window actually contains eligible visits for every participant in the target population. If not, the specification needs to say how the summary is defined in those cases.

The main failure modes are using a vague phrase such as “before discontinuation,” silently replacing an event-aligned window with the nearest scheduled visit, or treating every recorded outcome as part of the target regardless of timing. Another is pooling eligible measurements while describing the result as an equally weighted participant average. Each can produce a numerical result, but the result may answer a different question from the one intended.