Why Calendar-Time Definitions Matter
In a platform or umbrella trial, treatment arms can become available at different times and can stop recruiting at different times. A single platform may therefore contain several arm-specific recruitment windows within a longer operating period. Those windows describe when eligible participants may be enrolled to an arm; they do not, by themselves, define how long an enrolled participant is followed.
A clear calendar-time representation prevents a basic classification error: treating the date an arm stops recruiting as if it were the date that all participants in that arm stop being observed. It also makes it possible to state consistently which platform period contains a given enrollment date. These are descriptive definitions of trial operations, not treatment-effect conclusions.
Specify the Time Scale and Boundaries
Before listing dates, specify the time origin and unit. For example, time might be recorded in weeks since platform activation, or as calendar dates with a stated time zone and precision. The origin need not be the same as the date every arm opens. What matters is that the same convention is applied to all arms and platform periods.
A convenient convention for recruitment windows is the half-open interval \([a_k,b_k)\). It includes the entry boundary \(a_k\), but excludes the exit boundary \(b_k\). Thus recruitment is permitted at \(t=a_k\), while \(t=b_k\) is not included. The convention prevents a time exactly on a shared boundary from belonging to two adjoining intervals. If a protocol uses a different rule, such as a specified date through the end of that day, that rule should be stated and represented consistently.
A platform’s calendar periods can be represented similarly. If the periods are intended to partition a specified operating horizon, write them as \([c_0,c_1),[c_1,c_2),\ldots,[c_{J-1},c_J)\), with strictly increasing boundaries. Each time in the horizon then belongs to exactly one period. The periods are platform-wide labels; an arm can be open during all, some, or none of a particular period.
An arm’s recruitment windows are often recorded in ordered, nonoverlapping form because that is easy to inspect. But overlapping listed intervals do not automatically make the availability set ambiguous or make one interval redundant: their union is well-defined. For example, \([1,4)\cup[3,6)=[1,6)\). If an overlap reflects two conflicting operational records—for example, one says the arm was open and another says it was paused—the intended operational dates need clarification. The mathematical union alone cannot resolve conflicting records.
Recruitment Windows Are Not Follow-Up Windows
Let \(E_i\) be participant \(i\)'s enrollment time. The arm’s availability determines whether that enrollment time falls within its recruitment schedule. Follow-up is a separate participant-level quantity. Let \(L_i\) denote the participant’s observed follow-up duration under the study’s follow-up rules; the corresponding follow-up endpoint can be written \(E_i+L_i\). That endpoint may be determined by an event, censoring, withdrawal, a planned assessment limit, or an administrative data cutoff, depending on the study and endpoint.
Consequently, if arm \(k\) exits recruitment at \(b_k\), a participant enrolled shortly before \(b_k\) may still be followed after \(b_k\). Whether that participant has complete or incomplete follow-up depends on the endpoint-specific follow-up plan and observed history, not just on the arm’s recruitment window. A useful record keeps at least the arm, enrollment time, arm availability dates, platform period, follow-up duration or endpoint, and the rule used to determine follow-up.
Worked Example: An Arm Closes While Follow-Up Continues
Suppose a hypothetical arm recruits during weeks \([0,20)\), measured from platform activation. The follow-up plan aims for 24 weeks after each participant’s enrollment, provided no earlier endpoint-specific event or censoring rule applies. Participant A enrolls at week 19.5. Is the enrollment within the recruitment window, and when would 24 weeks of follow-up end?
The enrollment is within the window because \(0\leq19.5<20\). Under the stated target follow-up, the planned endpoint is
Thus the arm’s recruitment exit at week 20 does not itself end Participant A’s follow-up at week 20. The planned 24-week endpoint is week 43.5, unless an earlier event, censoring rule, or other specified follow-up rule applies. By contrast, a participant enrolling exactly at week 20 is outside \([0,20)\) under the half-open convention.
Worked Examples With Calendar Periods
Worked Example: Intermittent Availability and Overlapping Records
In a hypothetical platform schedule, recruitment to an arm is permitted during weeks \([2,7)\) and \([10,15)\). A data extract also contains two overlapping records, \([1,4)\) and \([3,6)\), for a different arm. Find the availability set represented by the overlapping records, and classify enrollment times 6.5, 8, 10, and 15 for the intermittent arm.
For the overlapping records, the union covers every time from 1 up to, but not including, 6:
Neither listed interval is redundant; each extends the union over part of the range. The union is unambiguous as a set. If the records are intended to describe one arm’s permitted recruitment, \([1,6)\) is a concise equivalent representation. If they encode conflicting operational states, those states must be clarified separately.
For the intermittent arm, 6.5 is included in \([2,7)\), so recruitment is permitted. Week 8 lies in the gap between windows, so recruitment is not permitted then. Week 10 is included because the second window includes its entry boundary. Week 15 is excluded because the second window excludes its exit boundary. Therefore the four classifications are: permitted, not permitted, permitted, and not permitted, respectively.
Worked Example: Assigning Enrollment to a Platform Period
A hypothetical platform defines three calendar periods in months since activation: \([0,6)\), \([6,12)\), and \([12,18)\). Arm A is available during \([0,12)\), Arm B during \([4,18)\), and Arm C during \([10,18)\). A participant is enrolled to Arm B at month 11.5 and has nine months of observed follow-up under the specified follow-up rule. Identify the platform period and the follow-up endpoint, and state whether Arm B’s recruitment exit alone ends follow-up.
Since \(6\leq11.5<12\), the enrollment belongs to the second platform period, \([6,12)\). Arm B is available at month 11.5 because \(4\leq11.5<18\). Adding the observed follow-up duration gives
The observed follow-up endpoint is month 20.5. Arm B’s recruitment exit at month 18 does not, by itself, end that follow-up. The endpoint lies beyond the stated platform-period horizon ending at month 18; this illustrates why the calendar periods used to label recruitment need not cover all subsequent participant follow-up. Arm C’s window begins at month 10, but that does not change the period assignment for this Arm B enrollment.
Deriving the Boundary Rules
The interval convention has two practical consequences. First, adjacent windows \([a,b)\) and \([b,c)\) meet without a gap or overlap: times below \(b\) belong to the first, and time \(b\) belongs to the second. Second, if platform periods have strictly increasing boundaries and cover the horizon, each enrollment time within the horizon receives exactly one period label.
To see the unique-label result, take any time \(t\) in \([c_0,c_J)\). Because the boundaries increase from \(c_0\) to \(c_J\), there is a period whose left boundary is no greater than \(t\), while its right boundary is greater than \(t\). The half-open rule assigns \(t\) to that period, including when \(t\) equals a boundary: it is excluded from the period ending there and included in the period starting there. Strictly increasing boundaries ensure that no two periods can both contain \(t\). This conclusion depends on the periods actually covering the stated horizon; if a gap is intentional, a time in that gap has no period label unless an additional rule is supplied.
How to Check a Calendar-Time Specification
A schedule is easier to audit when dates are shown on a common time scale, and the interval rules are written before classifying enrollments. A timeline or a table can expose staggered entries, exits, gaps, and overlaps. The representation should also distinguish planned dates from actual operational dates if both are being recorded.
- Check the origin and units. Do not mix, for example, weeks from platform activation with calendar months without an explicit conversion.
- Check boundary precision. If a date is only known to the nearest day, do not imply that the record resolves an exact time within that day.
- Check for unexplained gaps. A gap may mean recruitment was not permitted, or it may be missing documentation; the schedule should say which.
- Check overlaps in context. Merge overlapping intervals when they simply describe the union of permitted times. Clarify operational conflicts rather than assuming the union resolves them.
- Check that follow-up has its own rule. An arm exit date is not a substitute for a participant-specific event, censoring, or follow-up endpoint.
- Check period coverage. If every enrollment must receive a period label, the periods must cover the relevant calendar-time horizon without gaps.
These checks diagnose problems in the time representation; they do not establish that participants enrolled in different periods are comparable or that any treatment contrast is unbiased. Those questions require additional design and analysis assumptions.
Choosing a Representation and Its Limits
| Representation | What it describes | Useful when | What it does not establish |
|---|---|---|---|
| One platform-wide recruitment interval | A broad span of platform recruitment activity | All arms share the same recruitment dates or a platform overview is needed | Which arms were actually available at each time |
| Arm-specific interval or interval union | Permitted recruitment times for each arm | Arms enter, exit, pause, or resume at different times | Participant follow-up duration or comparability across enrollment times |
| Platform calendar-period partition | A common period label for each time in a stated horizon | Operational periods need consistent labels across arms | That every arm recruited in every period, or that period labels explain outcome differences |
| Participant-level follow-up record | Observed time after an individual’s enrollment | Follow-up continues independently of recruitment closure | When recruitment to the arm was permitted |
A single platform-wide window is simpler but can conceal arm-specific entry and exit dates. Arm-specific intervals preserve those operational details but require consistent date definitions and careful recording of pauses. Calendar periods provide a shared time partition, but they are not a substitute for arm availability records. Participant follow-up must still be represented separately. None of these representations alone addresses changing outcome distributions, confounding, or causal interpretation.
Common Mistakes
- Equating arm exit with the end of follow-up. Recruitment may close while enrolled participants remain under observation. Use participant-level follow-up rules to define their endpoints.
- Leaving boundary inclusion implicit. Saying “open until week 20” is insufficient if enrollment exactly at week 20 is possible. State the endpoint convention.
- Treating any overlap as an error. Overlapping intervals have a well-defined union. Merge them when they represent one availability set, and clarify the intended operational dates when the records conflict.
- Assuming a period label implies arm availability. A platform period can include times when a particular arm is closed. Check the arm’s own availability set.
- Assuming a clean timeline solves an analysis problem. Precise dates improve classification, but do not alone justify comparisons across calendar time.
Check Your Understanding
Answer each question using the half-open interval convention unless the question states otherwise. Solutions follow the questions.
- An arm is available during \([3,9)\). Is an enrollment at time 3 eligible by date, at time 9, and at time 8.99?
- Two recorded windows are \([2,5)\) and \([4,7)\). What is their union, and does the overlap alone make the availability set ambiguous?
- Platform periods are \([0,4)\), \([4,8)\), and \([8,12)\). Which period contains an enrollment at time 8?
- An arm exits recruitment at month 15. A participant enrolled at month 14.5 has 6 months of observed follow-up. What is the follow-up endpoint, and does the arm exit itself end follow-up?
Solutions. (1) Time 3 is included, time 9 is excluded, and time 8.99 is included because each satisfies \(3\leq t<9\), except \(t=9\). (2) The union is \([2,7)\). The overlap does not make the union ambiguous; if the records imply conflicting operational states, clarify those states separately. (3) Time 8 belongs to \([8,12)\), because the interval includes its left boundary and the preceding period excludes its right boundary. (4) The endpoint is \(14.5+6=20.5\) months. Recruitment exit at month 15 does not, by itself, end this participant’s follow-up.