Accrual Changes the Calendar, Not the Spending Scale
An information-based group sequential design specifies interim analyses at particular fractions of the planned final information. Those fractions do not, by themselves, tell the trial team which calendar dates to use. Participants may accrue faster or slower than expected, outcomes may take time to become available, and event rates may change. As a result, the same information fraction can be reached earlier or later in calendar time.
The key separation is between when information is available and how much alpha has been spent at that information fraction. An alpha-spending curve is indexed by information fraction, not by the proportion of calendar time that has elapsed. If the information fractions at planned looks stay the same, changing accrual changes their calendar timing, but does not change the curve's value at those fractions.
For example, a design might plan looks at \(t_1=0.50\) and \(t_2=0.75\). The corresponding spending targets are \(A(0.50)\) and \(A(0.75)\). Those targets are not replaced by \(A(c/C)\), where \(C\) is a planned trial duration, unless calendar time happens to be the design's chosen indexing scale. Boundary calibration still uses the planned information fractions; the calendar schedule tells the team when those fractions are expected or observed to be reached.
From Accrual to Available Information
A simple approximation assumes that each accrued participant immediately contributes a fixed amount of information. Let \(r(s)\) be the accrual rate at calendar time \(s\), measured in participants per month, and let \(J\) be the information contributed per participant. Under this approximation, cumulative information by time \(c\) is \(I(c)=J\int_0^c r(s)\,ds\). If \(r(s)=r\) is constant, then \(I(c)=Jrc\).
That approximation can be useful for planning, but it is not universal. In a trial with delayed outcomes, a participant enrolled today may contribute little or no usable information today. A more general representation is
where \(b(c,s)\) is the information available at time \(c\) from a participant enrolled at time \(s\). The planned calendar time \(c_k\) for look \(k\) solves \(I(c_k)=t_k I^*\). If \(I(c)\) is strictly increasing over the relevant range, this equation identifies a unique time. If information arrives in jumps, such as when event counts increase by one, the operational look is generally the first time the target information is reached or exceeded.
This mapping depends on the assumptions used to predict information. A fixed information contribution per participant is reasonable only when participant-level information is sufficiently similar and available on the assumed schedule. In time-to-event settings, event counts may be used as an information proxy, but the relationship between events and statistical information depends on the analysis model and design. Expected event accrual predicts a calendar date; the realized event count determines when the target is actually attained.
Worked Calendar-Time Calculations
Worked Example: Constant Early and Late Accrual
Consider a hypothetical trial with planned final information equivalent to 400 participants, with one equal information unit per participant. The two interim looks are at \(t_1=0.50\) and \(t_2=0.75\), so they require 200 and 300 participants, respectively. Assume all outcomes are immediately available and accrual is constant. Compare an early-accrual plan of 40 participants per month with a late-accrual plan of 20 participants per month.
Under the early-accrual plan, the first look is expected after \(200/40=5\) months and the second after \(300/40=7.5\) months. The final information is expected after \(400/40=10\) months.
Under the late-accrual plan, the same targets are expected after \(200/20=10\) months and \(300/20=15\) months, with final information after \(400/20=20\) months. The information fractions remain 0.50 and 0.75 in both plans. Thus, the spending targets remain \(A(0.50)\) and \(A(0.75)\); only the expected calendar dates change.
Worked Example: A Change in Accrual Rate
Suppose a hypothetical study needs 300 participants for its final information, with 150 participants corresponding to \(t_1=0.50\) and 240 corresponding to \(t_2=0.80\). Outcomes are immediately available. Accrual is 30 participants per month for the first 4 months and then 15 participants per month.
After 4 months, \(30\times4=120\) participants have accrued. The first look requires 30 more, which at 15 participants per month takes \(30/15=2\) months. It is therefore expected at month \(4+2=6\).
The second look requires 240 participants. After reaching 150 at month 6, another \(240-150=90\) participants are needed. At 15 per month, that takes \(90/15=6\) months, so the second look is expected at month 12. Final information requires 60 further participants after the second look, taking \(60/15=4\) months; the final analysis is expected at month 16.
The calculations use \(I(c)\) proportional to accrued participants, as assumed. The spending curve is evaluated at 0.50 and 0.80, not at \(6/16\) and \(12/16\). In particular, \(12/16=0.75\) does not make the second look a 0.75-information look; it is still the planned 0.80 look.
Worked Example: Delayed Information Availability
Suppose a hypothetical trial accrues 20 participants per month and requires 200 participants' worth of information for its planned interim look. Assume, for this simplified example, that each participant contributes one information unit exactly 2 months after enrollment and contributes none before then. There is no loss to follow-up.
By calendar month \(c\), the participants enrolled by month \(c-2\) have reached the assumed information time. For \(c\geq2\), the available information is therefore \(I(c)=20(c-2)\) participant units. The look occurs when \(20(c-2)=200\), so \(c-2=10\) and \(c=12\) months.
Ignoring the delay would give \(200/20=10\) months, underestimating the look date by 2 months. The spending target is determined by the design's information fraction at the look, not by whether the team used a delayed-outcome model in its calendar forecast. In a real design, the assumed delay distribution and the definition of analysis information would need to be justified rather than treated as a fixed 2-month rule.
Why the Information Fraction Index Is Preserved
The invariance of the spending target follows directly from how the design is specified. Suppose the planned look thresholds are \(t_1,\ldots,t_K\), and \(A(t)\) is the prespecified cumulative spending curve. The target at look \(k\) is \(A(t_k)\). Accrual determines the calendar time \(c_k\) at which the information process reaches \(t_k I^*\); it does not enter the definition of \(A(t_k)\).
This is a justification of the indexing, not a claim that any calendar-time prediction is exact. If actual information at an intended calendar visit is less than \(t_k I^*\), the analysis is not at the planned information fraction merely because the scheduled date has arrived. Conversely, if the information target has been reached early, the trial may need to conduct the planned look earlier, subject to its protocol and operational rules. Procedures for calibrating boundaries at irregular or changed information times determine the statistical boundary values; the present point is that accrual governs the mapping between information and calendar time.
For a one-sided design, \(A(t_k)\) is the cumulative type I error spending target through look \(k\), under the null hypothesis and the specified design. For a two-sided design, the convention for total versus per-tail spending must be consistent throughout the design. In either case, the increment \(A(t_k)-A(t_{k-1})\) is a difference in cumulative spending targets; it is not generally the conditional probability of crossing at look \(k\), given no earlier crossing.
Diagnostics, Alternatives, and Limitations
A calendar forecast should be checked against the assumptions that link enrollment to information. Important diagnostics include comparing observed accrual with the assumed accrual curve, checking the proportion of enrolled participants with outcomes available, and comparing observed information or event counts with their forecast. A growing gap between accrued sample size and analyzable information is a warning that a participant-count model may be overstating progress.
- Check the scale. Confirm that the planned look is expressed in information units or information fraction, not mistakenly in months or a fraction of trial duration.
- Check outcome availability. Delayed assessments, incomplete follow-up, and data-cleaning lag can make accrued participants differ substantially from participants contributing information.
- Check event predictions. In event-driven designs, compare projected with realized event accumulation. Changes in event rates can shift look dates even when enrollment is on schedule.
- Check the final information target. If the total planned information or analysis assumptions change, the denominator \(I^*\) may change; this is more than a simple calendar shift.
- Check operational rules. Data cutoffs and review schedules can mean a statistical information target is reached between formal analyses. The protocol should specify how such timing is handled.
An alternative is to schedule looks at fixed calendar dates. This can simplify operations, but the information fraction at each date may vary with accrual and follow-up. The design must then address the actual information times rather than assume the dates correspond to the original fractions. Event-driven timing instead schedules analyses at event targets, which can be useful when events are the main information source, but the calendar date remains uncertain. Information-based timing is the organizing principle that links these approaches: a calendar date or event count is a way of locating information, not a replacement for defining it.
The main limitation is that a forecasted accrual curve is not a guarantee. Recruitment can be seasonal, sites can open late, and outcome availability can vary across participants. Moreover, not all analyses have information exactly proportional to the number enrolled or the number of events. If the information model is misspecified, predicted look dates can be wrong even though the spending curve is correctly defined. A calendar forecast should therefore be reported as a planning estimate, while actual analysis timing should use the information measure prespecified for the design.
Check Your Understanding
Answer each question using information fractions, not elapsed calendar time alone.
- A trial plans a look at \(t=0.60\). If slower accrual delays the look by 3 months, what happens to its target cumulative spending?
- A trial needs 240 participant-equivalents of information. Accrual is 24 per month, with immediate information. When is the look at \(t=0.75\) expected?
- Why might a look not occur when the required number of participants has been enrolled?
- A look occurs at month 8 of a planned 16-month trial. Is it necessarily a 0.50-information look?
Solutions. (1) The target remains \(A(0.60)\), assuming the design's information fraction and spending curve are unchanged; only the calendar date shifts. (2) The target information is \(0.75\times240=180\) participant-equivalents, so the expected time is \(180/24=7.5\) months. (3) Outcomes may not yet be observed or usable, or the design's information measure may depend on events or other data features rather than enrollment alone. (4) No. It is a 0.50-information look only if \(I(8)/I^*=0.50\); elapsed calendar time alone does not establish that fraction.