Two Different Interim Questions
A time-to-event MAMS design has to answer two questions at an interim analysis. When will the data be reviewed? And which arms will continue? The first is a timing rule: it specifies when the interim look occurs. The second is a selection rule: it specifies how the available outcome data are used to retain or drop arms. Keeping these questions separate is essential when an early outcome is used for selection while the final endpoint is time to an event.
For example, a committee might review the trial when 60 participants have experienced the final endpoint, then retain arms with a sufficiently favourable early response rate. The total event count triggers the review; the early response rate informs selection. The event count is not, by itself, the selection outcome. As established in the earlier material on defining intermediate outcomes, an intermediate outcome is measured before the final outcome is fully available and may be used to guide arm selection.
The distinction does not mean that timing is always statistically unrelated to outcomes. A calendar date can be set independently of observed treatment results, whereas an event-count trigger necessarily depends on events having occurred. The design and analysis must reflect how the look was actually triggered. But neither trigger, on its own, defines the rule for comparing arms on an intermediate outcome.
Three Ways to Trigger a Look
A calendar-time trigger schedules review at a specified date or elapsed trial time. Its operational advantage is predictability: staff know when to prepare the data. Its statistical disadvantage is that the amount of information available then can vary. Differences in recruitment, event rates, follow-up, or censoring can leave the trial with more or fewer events than anticipated.
An information-fraction trigger schedules the review when the accumulated information reaches a stated proportion of the planned information. If \(I(t)\) is the information available at time \(t\), and \(I_{\mathrm{planned}}\) is the planned information at the analysis endpoint, the information fraction is \(I(t)/I_{\mathrm{planned}}\). This aims to place the look at a defined statistical maturity rather than at a fixed date. It requires a defensible definition and estimate of information; information is not always identical to the number of observed events.
An event-count trigger schedules review after a specified number of final-endpoint events has occurred. This is often convenient in time-to-event studies because event accumulation is related to the precision of treatment-effect estimates. Under simple assumptions, event count can serve as a useful proxy for information. Under unequal allocation, changing risk sets, non-proportional hazards, or other complications, the relationship need not be exact.
| Trigger | What is fixed | What may vary at the look |
|---|---|---|
| Calendar time | Date or elapsed time | Event count and statistical information |
| Information fraction | Target proportion of planned information | Calendar date and, depending on the model, event count |
| Event count | Target number of events | Calendar date and exact statistical information |
Information, Events, and a Short Derivation
The distinction between event count and information can be made concrete under a standard simplifying model. Suppose a time-to-event treatment effect is summarized by a log hazard ratio, hazards are proportional, allocation is balanced, and the event contributions are sufficiently regular. A common large-sample approximation is that the variance of the log hazard ratio is \(4/D\), where \(D\) is the total number of observed events. Since information is the reciprocal of variance, the approximate information is \(D/4\).
If the planned event total is \(D_{\mathrm{planned}}\), this approximation gives an information fraction of \(D/D_{\mathrm{planned}}\). This is a justification for using event count as a practical proxy in that setting, not a universal identity. For example, changing allocation ratios or assumptions about censoring can change how much information each event contributes. A protocol should therefore identify whether its trigger is an event count or a computed information measure, rather than treating the terms as interchangeable.
Worked Example: A Calendar Look With Variable Event Accumulation
Suppose a hypothetical trial schedules an interim review 12 months after recruitment starts. For illustration, assume 80 participants all enter at the start, each has a constant event hazard of 0.05 per month, and there is no loss to follow-up or competing event before the review. What event count should the team expect by month 12?
For one participant followed for 12 months under a constant hazard \(\lambda\), the probability of an event by month 12 is \(1-e^{-12\lambda}\). Substituting \(\lambda=0.05\) gives \(1-e^{-0.6}=1-0.5488=0.4512\).
The expected total number of events is the number of participants multiplied by this event probability:
Thus, about 36 events are expected, but the review is still triggered by the date, not by observing exactly 36 events. The actual count will vary because event occurrence is random. The calculation also depends on the stated assumptions; staggered entry, censoring, or a different hazard would change the expected count. This illustrates why a calendar-triggered review can occur at different information levels across trials or scenarios.
Keep the Trigger Separate From Selection
At any of these looks, the selection rule should separately state the intermediate outcome, the comparison or summary used, and the action taken. For instance, a design may review at a fixed event total and then compare the proportions of participants with an early response across active arms. The trigger is based on the total final-endpoint event count; the selection measure is early response. A design that instead retains arms with the most favourable interim event-based estimates has made event outcomes part of its selection rule, not merely its timing rule.
Worked Example: An Event Trigger and an Early-Response Selection Measure
In a hypothetical three-arm trial, an interim review is scheduled after 60 final-endpoint events across the trial. At that review, the observed early-response data are 24 responses among 40 participants in Arm A, 18 among 40 in Arm B, and 28 among 40 in Arm C. Suppose the prespecified selection rule retains arms with an observed response proportion of at least 0.60.
The event count of 60 determines when the data are reviewed. It does not tell the team which arm meets the early-response rule. Calculate each response proportion:
Arms A and C meet the stated threshold, so they are retained and Arm B is dropped under this hypothetical rule. The arithmetic applies the stated selection threshold; it does not establish that early response predicts final benefit or that the retained arms will have better final outcomes. Those are separate scientific and statistical questions. Nor does the fact that the 60 events triggered the review turn the event count into the selection criterion.
A trial may instead use an information-fraction trigger. In that case, the information calculation must be specified independently of the intermediate-outcome comparison. For example, the review could occur at 50% of planned information and then apply the same early-response threshold. The timing definition says when the data are sufficiently mature for review; the threshold says how those data guide arm retention.
Worked Example: Translating an Event Target to an Approximate Information Fraction
Suppose a hypothetical balanced trial plans for 240 final-endpoint events and uses the approximation \(\operatorname{Var}(\widehat{\log HR})\approx4/D\). The team is considering a review at 120 events. Under this model, find the approximate information fraction and compare the standard errors at the interim and planned endpoint.
The interim information fraction is \(120/240=0.500\), or 50%. At 120 events, the approximate standard error is \(\sqrt{4/120}=0.1826\) on the log-hazard-ratio scale. At 240 events, it is \(\sqrt{4/240}=0.1291\) on that scale.
The interim look has about half the planned information under the stated approximation, but its standard error is not half the final standard error: it is larger by a factor of \(\sqrt{2}\). The standard errors are for a log hazard ratio and are dimensionless on that scale. Because the approximation relies on balanced allocation and proportional hazards, a design with materially different conditions should use an appropriate information calculation rather than assume that 120 events always means 50%.
Assumptions, Diagnostics, and Alternatives
A sound specification begins by stating what is monitored and what is acted upon. For a calendar trigger, confirm the time origin and the date rule. For an information trigger, state how information is defined, estimated, and compared with the planned total. For an event trigger, define which events count, how adjudication and data delays are handled, and whether the target is an overall total or a count within specified groups. Then state the intermediate outcome and selection rule independently.
Before applying a planned rule, check whether its conditions are plausible. Useful diagnostics include recruitment and follow-up summaries, event accrual against projections, the number of participants with evaluable intermediate outcomes, and confirmation that the trigger was reached as specified. If an information fraction is used, check that the information calculation matches the analysis model and allocation pattern. If the look is event-triggered, check event definitions and reporting delays; otherwise, the operational event count may not match the design target.
The main alternatives have different operational trade-offs. Calendar timing is straightforward to schedule but can yield uneven statistical maturity. Event-count timing more directly standardizes the number of endpoint events, but the date is unpredictable and event accrual can depend on treatment and follow-up patterns. Information-fraction timing targets statistical maturity more directly, but relies on a sound information measure and may be harder to calculate operationally. None of these choices determines which intermediate outcome is best for selecting arms; that choice requires its own justification.
A further limitation is that a trigger can affect the data available at review. In particular, an event-triggered look occurs because events have accrued, so it should not casually be treated as if the analysis time had been fixed independently of observed outcomes. Sequential monitoring procedures and their error-control implications require methods appropriate to the actual design. Similarly, an intermediate outcome’s association with the final endpoint cannot be inferred merely because it is measured at the interim or used to select arms.
The practical value of this separation is clarity. It lets readers distinguish an operational delay from a treatment-selection decision, identify which data caused the review to occur, and understand which outcome actually determined the arms that continued. It also makes simulations and design sensitivity checks interpretable: changing the timing trigger is not the same intervention as changing the selection threshold.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A trial reviews at a fixed date and retains arms with an early-response proportion above a threshold. Name the timing rule and the selection rule.
- Under the approximation \(I(D)\approx D/4\), what is the information fraction at 90 events if the planned total is 180 events?
- Why can two trials reviewed at the same calendar time have different amounts of information?
- Is an overall event-count trigger necessarily an arm-selection criterion? Explain.
Solutions. (1) The fixed date is the calendar-time trigger; the early-response threshold is the intermediate-outcome selection rule. (2) The approximate information fraction is \(90/180=0.500\), or 50%. (3) Event accrual may differ because of recruitment, follow-up, event rates, censoring, or other features, so the same elapsed time need not produce the same event count or information. (4) No. It specifies when to review. It becomes part of selection only if the selection rule itself uses event counts or event-based comparisons to decide which arms continue.