Piecewise Enrollment Rates In Event Projections

Why Piecewise Enrollment Rates Matter

Event-driven trial plans often assume that recruitment proceeds at a constant rate. In practice, enrollment can accelerate after additional sites open, slow during holidays, or change when a recruitment strategy is revised. If the plan uses one average rate for the entire recruitment period, it can misstate how many participants have enrolled by a given calendar date. That matters for event projections because participants recruited at different times may contribute differently to the total number of events observed by a cutoff.

The task here is to represent recruitment with a rate for each calendar period, calculate the expected enrollment contributed by each period, and add the corresponding expected event contributions. The event yield for each period is treated as a specified planning input. Calculating yields from survival probabilities, or modeling each participant's follow-up time in detail, requires additional modeling beyond this calculation.

Definition: A piecewise-constant enrollment-rate schedule specifies consecutive calendar periods and a constant recruitment rate $r_j$ within each period. If period $j$ runs from $a_j$ to $b_j$, then $r_j$ has units of participants per unit of calendar time. For a projection cutoff $T$, the period's active duration is $L_j(T)=\max\{0,\min(T,b_j)-a_j\}$, and its expected enrollment is $E[M_j(T)]=r_jL_j(T)$. Here, $M_j(T)$ denotes the actual, potentially random, number enrolled in that period by $T$.

The boundaries $a_j$ and $b_j$ must use the same calendar-time scale as $T$, and the periods should not overlap or leave gaps unless recruitment is intentionally zero in a gap. The formula also handles a cutoff inside a period: only the time from the period's start through $T$ contributes. A fractional expected enrollment is legitimate; it is an average over the recruitment process, not a claim that a fraction of a participant enrolled.

From Recruitment Periods to Expected Events

For event projections, let $D_j(T)$ be the number of observed events by cutoff $T$ among participants enrolled during period $j$. The actual participant count is $M_j(T)$, not its expectation. If $I_{ij}(T)$ is an indicator equal to 1 when participant $i$ from period $j$ has an observed event by $T$, and 0 otherwise, then the actual event total is

D_j(T)=\sum_{i=1}^{M_j(T)} I_{ij}(T).

To get a simple period-by-period projection, suppose that within period $j$, each enrolled participant has the same expected event indicator by $T$, denoted $q_j(T)$. In other words, conditional on enrollment in that period, $E[I_{ij}(T)]=q_j(T)$. This is a planning assumption: it treats the expected event yield as common for enrollees in that period. It is not automatically valid if participants enrolled at different times within the period have materially different expected yields.

Under that assumption, the expected event contribution is the expected enrollment multiplied by the period's expected event yield. One justification is to condition first on the actual enrollment count. Given $M_j(T)=m$, linearity of expectation gives $E[D_j(T)\mid M_j(T)=m]=m q_j(T)$. Taking expectation over the enrollment count then gives the result below. This reasoning does not replace the actual count by its expectation inside a sum; it explicitly averages over the random count.

E[D_j(T)\mid M_j(T)]=M_j(T)q_j(T), \qquad E[D_j(T)]=E[M_j(T)]q_j(T) =r_jL_j(T)q_j(T).

The equality for the unconditional expectation holds when the per-participant expected yield $q_j(T)$ is common within the period and remains applicable conditional on the enrollment count. A deterministic planning calculation can also use $r_jL_j(T)$ as the expected cohort size and multiply by a stated period yield; that calculation is an expected-value approximation, not a count of actual events. If the expected yield varies substantially within a period, split the period into shorter intervals with more nearly common yields, or use a model that explicitly averages yield over enrollment times.

Key result: For a schedule with $J$ periods and compatible period-specific yields, the projected expected event total is the sum of the expected contributions: $$ E[D(T)]=\sum_{j=1}^{J}r_jL_j(T)q_j(T). $$ This sum combines expected values; it does not assert that the period-specific actual enrollment or event counts are fixed.

Worked Projections

Worked Example: Enrollment When the Cutoff Falls Inside a Period

Suppose a hypothetical trial recruits at 12 participants per month from month 0 to month 4, then at 20 participants per month from month 4 to month 9. Find the expected enrollment by month 7.

The first period is complete, so its duration is $4-0=4$ months. Its expected enrollment is $12\times4=48$ participants. The cutoff falls three months into the second period, so the active duration there is $7-4=3$ months and the expected enrollment is $20\times3=60$ participants.

E[M(7)]=(12\ \text{participants/month})(4\ \text{months}) +(20\ \text{participants/month})(3\ \text{months}) =48+60=108\ \text{participants}.

Thus the expected enrollment by month 7 is 108 participants. This is an expectation, not a guarantee that exactly 108 people will have enrolled. The second period contributes only three months of recruitment; using its full five-month duration would incorrectly project enrollment through month 9.

Worked Example: Combining Period-Specific Event Yields

Suppose a hypothetical projection has expected enrollment of 48 participants in an early recruitment period and 60 in a later period by the cutoff. Planning inputs specify expected event yields of 0.30 and 0.18 events per enrolled participant, respectively. Assume each yield is common to the participants in its period. Calculate the expected event total.

The early period contributes $48\times0.30=14.4$ expected events. The later period contributes $60\times0.18=10.8$ expected events. Add the two contributions:

E[D(T)]=(48)(0.30)+(60)(0.18)=14.4+10.8=25.2\ \text{events}.

The projected total is 25.2 expected events. This need not be an integer because it is an expectation. The later period has more expected enrollees but a smaller event yield per enrollee, so it contributes fewer expected events than the earlier period in this projection. The yields are supplied planning inputs here; this calculation does not derive them from a survival model.

Worked Example: A Full Piecewise Schedule With a Partial Final Period

Consider a hypothetical schedule with three periods: months 0–3 at 10 participants per month, months 3–6 at 16 participants per month, and months 6–10 at 12 participants per month. The event projection cutoff is month 8. Suppose the specified expected event yields by month 8 are 0.40, 0.28, and 0.12 events per enrollee for the three periods, respectively, and assume a common yield within each period.

The active durations through month 8 are 3, 3, and 2 months. Expected enrollment is therefore $10\times3=30$, $16\times3=48$, and $12\times2=24$ participants in the three periods. Their projected event contributions are $30\times0.40=12.0$, $48\times0.28=13.44$, and $24\times0.12=2.88$ events.

E[D(8)]=(10)(3)(0.40)+(16)(3)(0.28)+(12)(2)(0.12) =12.0+13.44+2.88=28.32\ \text{events}.

The projected total is 28.32 expected events by month 8. The final rate applies for only two months, not for the full four-month period ending at month 10. The calculation assumes that each listed yield is appropriate for the participants enrolled in its corresponding period by this cutoff.

Why the Calculation Works—and When It Needs Refinement

The enrollment part follows from rate multiplied by exposure time: a rate of $r_j$ participants per month over $L_j(T)$ months gives expected enrollment $r_jL_j(T)$. The event part follows from expected yield per enrollee. Under the common-yield assumption, conditional on an actual count $m$, the expected sum of $m$ event indicators is $mq_j(T)$; averaging over the actual count replaces $m$ by $E[M_j(T)]$, yielding $r_jL_j(T)q_j(T)$. Adding across periods is valid because the total event count is the sum of the period-specific event counts, and expectation is additive.

Independence between period-specific event counts is not required to add their expectations. However, the multiplication within a period needs a valid yield for that period. If event yield changes meaningfully across enrollment times within the same period, a single $q_j(T)$ may be inadequate unless it is defined as an enrollment-weighted expected yield compatible with that cohort's expected event total. In practice, dividing a period at a rate change does not necessarily resolve changes in yield; the analyst should also consider whether the yield assumption is reasonable across that interval.

Set the calendar intervals.
Write each start and end time in a common unit, and identify the projection cutoff $T$.

Find active durations.
For each interval, use only its overlap with the calendar time through $T$; a period beginning at or after $T$ contributes zero.

Calculate expected enrollment.
Multiply each period's rate by its active duration, checking that participants per time multiplied by time gives participants.

Apply the appropriate yields.
Multiply each period's expected enrollment by its stated, compatible expected event yield, then add the contributions.

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to check both the schedule and the arithmetic before interpreting the result. Confirm that rates are attached to the intended periods, that adjacent intervals meet at the correct boundaries, and that no period is counted twice. Recalculate the active duration when $T$ lies inside an interval. Check units at every step: participants per month times months gives participants, and participants times expected events per participant gives events. If the projected event total exceeds total projected enrollment, at least one yield or calculation is inconsistent with a yield bounded between 0 and 1 when each participant can contribute at most one event.

A constant-rate approach is a reasonable alternative when recruitment is expected to be approximately stable, or when a single average rate is an adequate planning summary. It is simpler, but can hide important changes in the timing and size of recruitment cohorts. A piecewise schedule is more transparent when rates change at known dates. More detailed approaches model enrollment timing and participant-level event probabilities directly; they can be appropriate when yields vary substantially within periods, but require additional assumptions and inputs.

The piecewise calculation is only as credible as its inputs. Rates may be uncertain, planned site openings may be delayed, and observed enrollment can depart from its expected trajectory. Likewise, the expected event yields are not guaranteed outcomes. This calculation gives an expected total, not a prediction interval, a probability of reaching a target by a date, or a guarantee about the realized number of events. Those questions require methods that explicitly represent uncertainty. When updated information shows that a rate or yield has changed, revise the schedule rather than forcing the new data into an obsolete average.

Common Mistakes

Using an expected enrollment as a sum limit. An expected count can be fractional and is not the actual number of enrolled participants. Write actual event totals using the actual count; then derive or state the expected-value calculation under its assumptions.
Applying the last period's rate for its full planned duration. If the cutoff occurs partway through that period, multiply by only the elapsed portion.
Adding rates instead of expected counts. Rates from different periods cannot be added to obtain total enrollment; multiply each rate by its active duration first.
Using one event yield without checking its scope. A yield should match the enrollment group and cutoff to which it is applied. If yield varies within a period, refine the grouping or use a more detailed model.
Reporting an expected event total as an actual count. A value such as 28.32 is an expected number of events, not a claim that a fraction of an event will be observed.

Key takeaway: For each recruitment period, calculate its active duration through the cutoff, multiply by its enrollment rate to obtain expected enrollment, and multiply by a compatible period-specific event yield. Add these expected contributions only after checking the assumptions behind each yield.

Check Your Understanding

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

A hypothetical trial recruits at 8 participants per month from month 0 to month 5 and 14 participants per month from month 5 to month 8. What is expected enrollment by month 6?
For the two periods in Question 1, suppose the expected event yields by month 6 are 0.25 and 0.10. What is the expected event total, assuming each yield is common within its period?
Why is it not generally correct to use the expected enrollment $E[M_j(T)]$ as the upper limit in a sum over actual participant event indicators?
A projection cutoff is month 4, but a recruitment period runs from month 3 to month 7. What duration of that period contributes to the projection?

Solutions. (1) The first period contributes $8\times5=40$ expected participants. The second contributes $14\times1=14$, so expected enrollment is $40+14=54$ participants. (2) The expected events are $(40)(0.25)+(14)(0.10)=10+1.4=11.4$ events. (3) The actual count $M_j(T)$ is the number of participants in the realized data and must index the actual event indicators. Its expectation may be fractional and is not that realized count. Under a common conditional yield, first condition on the actual count and then average: $E[D_j(T)]=E[M_j(T)]q_j(T)$. (4) Only month 3 to month 4 contributes, so the active duration is one month.