Planning Calendar Dates For Interim Looks

From Event Targets to a Calendar Plan

Interim analyses are often planned around cumulative event counts, but calendars require dates. A schedule therefore needs a projection that connects elapsed time to expected events. Applying the same projection assumptions to every target gives a coherent provisional schedule: each date is tied to its target in one cumulative event curve, rather than being estimated from unrelated rules.

The dates are forecasts under assumptions, not promises that a given number of events will have occurred by then. Actual accrual can differ from its expectation. The purpose of the calculation is to identify reasonable planning dates and make the assumptions behind them visible.

Definition: Let \(t\) denote elapsed calendar time from a specified origin, and let \(A(t)\) be the projected cumulative number of events by time \(t\) under a stated accrual model. For ordered event-count targets \(d_1<d_2<\cdots<d_K\), a provisional date \(t_k\) for target \(d_k\) solves \(A(t_k)=d_k\). The dates are provisional because \(A(t)\) is a projection, not the realized event count.

The time origin must be stated—for example, first participant enrolled or trial opening—and the time unit must be consistent throughout. The function \(A(t)\) is an expected cumulative count or another explicitly specified planning projection. It is not the random count that will actually be observed at time \(t\).

Use One Cumulative Projection for All Looks

A cumulative event projection should be nondecreasing: later times cannot have fewer accumulated events in the model. If \(A(t)\) increases strictly over the relevant interval, each feasible target has one date, and strictly increasing targets produce strictly increasing dates. If the curve has a flat segment, a target on that segment may not have a unique date. If a target is above the curve's projected maximum, the model supplies no date for it.

A practical schedule can be obtained from a model formula or from projected values at selected times. When using a table, one simple convention is linear interpolation between adjacent values. For \(t_a<t_b\) and \(A(t_a)\leq d\leq A(t_b)\), the interpolated date is

t_d=t_a+\frac{d-A(t_a)}{A(t_b)-A(t_a)}(t_b-t_a).

This calculation assumes the projected curve is linear between those two points. It does not establish that real event accrual will be linear there. If the underlying projection is curved, use its formula or the numerical root-finding method appropriate to that model, rather than silently replacing the curve with a straight line.

Worked Example: Interpolating Three Provisional Dates

Suppose a hypothetical planning model gives the following expected cumulative event counts, with time measured in months from first enrollment. For this calculation, assume the model curve is piecewise linear between the listed times.

Elapsed time (months)	Projected cumulative events
6	24
12	48
18	78
24	102

Find dates for targets of 30, 60, and 90 events. The projection is increasing at each listed time, and every target falls within its projected range, so each date can be interpolated.

For 30 events, use the interval from month 12 to month 18. The projected increase is \(78-48=30\) events in 6 months, or 5 events per month. Thus

t_{30}=12+\frac{30-48}{78-48}(18-12)=12+\frac{12}{30}\times 6=14.4\text{ months}.

For 60 events, the same interval applies:

t_{60}=12+\frac{60-48}{78-48}(18-12)=12+\frac{12}{30}\times 6=14.4\text{ months}.

This reveals an arithmetic inconsistency: the two calculations above use targets that lie on different sides of the table value at month 12, but both have been substituted as though the target were 60. Correctly, 30 events lies between months 6 and 12, and 60 lies between months 12 and 18. Therefore,

t_{30}=6+\frac{30-24}{48-24}(12-6)=7.5\text{ months}, \qquad t_{60}=12+\frac{60-48}{78-48}(18-12)=14.4\text{ months}.

For 90 events, interpolate between months 18 and 24:

t_{90}=18+\frac{90-78}{102-78}(24-18)=18+\frac{12}{24}\times 6=21.0\text{ months}.

The schedule is therefore 7.5, 14.4, and 21.0 months from first enrollment. Each date comes from the same projected curve, and their order agrees with the target order.

A Model-Based Projection and Its Inversion

The interpolation rule is convenient when a projection table is available. A model can instead define \(A(t)\) directly. For illustration, suppose enrollment occurs at a constant expected rate \(r\) participants per month from time 0 through month \(R\), event times are independent and exponentially distributed with hazard \(\lambda\) per month, and there is no loss to follow-up. Assume an event is counted when it occurs and enrollment times are spread continuously at rate \(r\). These assumptions are intentionally simple; a planning analysis should use assumptions appropriate to the trial.

A participant enrolled at time \(u\) has follow-up \(t-u\) by time \(t\). Under the exponential event-time model, the probability of an event by then is \(1-\exp[-\lambda(t-u)]\). Summing this probability over the expected enrollment flow gives the cumulative expected event count:

A(t)=r\int_{0}^{\min(t,R)}\left[1-\exp\{-\lambda(t-u)\}\right]\,du.

For \(0\leq t\leq R\), the integration limits are 0 and \(t\). Integrating the constant term and the exponential term separately gives

A(t)=r\left[t-\frac{1-\exp(-\lambda t)}{\lambda}\right].

For \(t>R\), enrollment has ended, so the limits are 0 and \(R\):

A(t)=r\left[R-\frac{\exp\{-\lambda(t-R)\}-\exp(-\lambda t)}{\lambda}\right].

A target date is found by solving \(A(t)=d_k\). The equations are nonlinear, so a numerical root-finding method may be used. The important scheduling checks are that the same values of \(r\), \(R\), and \(\lambda\) are used for every target, that each target is reached within the projected range, and that the resulting roots are ordered.

Worked Example: Solving Three Targets Under Constant Enrollment

Consider a hypothetical trial enrolling at \(r=20\) participants per month for \(R=12\) months. Assume the event hazard is constant at \(\lambda=0.05\) per month, with no loss to follow-up. These assumptions make the preceding projection applicable. Find provisional dates for 30, 60, and 90 events, measured from the first enrollment.

The expected total enrollment is \(rR=20\times12=240\). For a target reached during enrollment, use the \(t\leq R\) formula. For 30 events, numerical solution of

20\left[t-\frac{1-\exp(-0.05t)}{0.05}\right]=30

gives \(t\approx8.28\) months. Substitution checks the result: \(1-\exp(-0.05\times8.28)\approx0.3390\), so \(20[8.28-0.3390/0.05]\approx30.0\) events.

At month 12, the projection is

A(12)=20\left[12-\frac{1-\exp(-0.60)}{0.05}\right]\approx59.52\text{ events}.

Thus 60 and 90 events are reached after enrollment ends. For \(t>12\), simplify the second expression using \(r=20\), \(R=12\), and \(\lambda=0.05\):

A(t)=240-400\left[\exp\{-0.05(t-12)\}-\exp(-0.05t)\right].

Solving \(A(t)=60\) gives \(t\approx12.05\) months; solving \(A(t)=90\) gives \(t\approx15.70\) months. For example, the after-enrollment expression can be rewritten as \(A(t)=240-328.847\exp(-0.05t)\), so the 60-event solution satisfies \(\exp(-0.05t)=180/328.847\), yielding \(t\approx12.05\). The resulting provisional dates are 8.28, 12.05, and 15.70 months. The small gap between the 60-event date and month 12 follows from the projected count of about 59.52 at month 12; it is not a guarantee that the realized 60th event will occur then.

Converting Elapsed Time to Calendar Dates

A result such as “12 months” is not yet an unambiguous calendar date. State the origin and the calendar convention used to convert elapsed time. For example, a protocol may define elapsed months by calendar-month anniversaries, or a planning system may use a fixed number of days per month. These conventions can give different dates for fractional months, so the convention should be chosen before reporting dates and applied consistently.

Worked Example: Putting a Three-Look Schedule on a Calendar

Suppose a hypothetical projection expects events at a constant rate of 5 per month over the relevant period. A final target is 120 events, and three interim targets are set at 30, 60, and 90 events. Under this linear planning approximation, \(A(t)=5t\), where \(t\) is elapsed months from first enrollment. This model is the same for all looks, and it gives the dates

t_{30}=30/5=6,\qquad t_{60}=60/5=12,\qquad t_{90}=90/5=18\text{ months}.

Suppose first enrollment is scheduled for January 15, 2027, and elapsed months are converted by calendar-month anniversaries. The provisional dates are July 15, 2027, January 15, 2028, and July 15, 2028. The calculations do not say that the event counts on those dates will be exactly 30, 60, and 90. They say that those are the dates at which the stated model's expected cumulative count reaches those targets.

Checks, Alternatives, and Limitations

Before adopting a schedule, check that the target dates are meaningful under the projection. In particular:

Check feasibility. Every target must be no greater than the model's projected maximum over the planning horizon.
Check monotonicity and ordering. The cumulative projection should not decrease, and the dates should increase with the targets. A violation often signals a coding, interpolation, or target-order error.
Check the common assumptions. Applying different enrollment, event, follow-up, or censoring assumptions to different targets can produce a schedule that does not correspond to one coherent event curve.
Check the time scale. Make sure the origin, units, and calendar conversion convention agree in the model, calculations, and reported schedule.
Do not imply false precision. A numerical root reported to several decimal places does not make the forecast accurate to that precision. Round dates to a practical planning unit and retain the underlying assumptions.

A provisional calendar schedule differs from a schedule that waits until an event target is observed. The former uses projected accrual to estimate dates in advance; the latter bases the timing on the realized event count. Fixed calendar dates chosen without reference to event targets are another planning option, but they need not coincide with the dates implied by an event projection. The appropriate choice depends on the protocol and operational needs; this calculation alone does not establish that one scheduling approach is preferable.

The central limitation is that \(A(t)\) depends on assumptions. Enrollment may be slower or more uneven than projected, event rates may differ from the model, and realized counts vary around their expectations. Consequently, a provisional date is a model-based planning estimate, not a statement that the target will have been met by that date. Keep the target counts and projected dates distinguishable in planning documents, and revisit the schedule if its assumptions no longer describe the intended plan.

Common Mistake

Common mistake: “The projection reaches 60 expected events at month 12, so the 60-event interim look will occur at month 12.” This treats an expected count as an observed count. The correction is to call month 12 the provisional date under the stated projection, and not to claim that the realized count will be 60 then. Also verify that the curve is evaluated consistently: a date derived from a different enrollment or event-rate assumption is not the 60-event date under the original schedule.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A projection gives \(A(8)=32\) and \(A(12)=52\). Under linear interpolation, estimate the date for 42 events.
A projected cumulative event curve reaches 70 events by month 18, but a planned target is 80. What can be concluded about a date for that target from the available projection?
Why should the same accrual assumptions be used to schedule all targets in a planned set of interim looks?
A model projects 45 expected events on a provisional date. Does this establish that 45 events will have occurred by then? Explain.

Solutions. (1) The increase is \(52-32=20\) events over \(12-8=4\) months, or 5 events per month. The target is 10 events above the month-8 value, so \(t=8+(42-32)/5=10\) months. (2) The available curve does not reach 80 by month 18, so it supplies no date for that target within the stated horizon. A longer horizon or a changed projection would be needed to locate one, with the assumptions stated. (3) A common set of assumptions makes all dates solutions on one cumulative event curve. Mixing assumptions can make their ordering or spacing inconsistent with any single planned accrual scenario. (4) No. Forty-five is the model's expected cumulative count at that date. Realized accrual can differ, so the date is a forecast rather than a guarantee that the 45th event has occurred.