Why Enrollment Dates Matter
A planned analysis date does not give every participant the same opportunity to experience an event. Someone enrolled early may have been followed for months, while someone enrolled shortly before the analysis date has had little time at risk. Projecting events from the total number enrolled alone ignores this difference and can distort the expected event count.
The key calculation is participant-specific: determine how much follow-up each participant can contribute by the proposed cutoff, then use the event probability appropriate to that duration. This tutorial focuses on that calendar-to-follow-up calculation and on summing the resulting expected event contributions. It takes the event-probability model as specified by the design; deriving those probabilities from survival models is a separate step.
If participant \(i\) has not enrolled by \(T\), then \(e_i>T\), so \(u_i=0\): that participant contributes no event probability to the projection at this cutoff. If participants share the same event-probability curve, write \(q_i(u)=q(u)\). If the curve differs by treatment group or another relevant characteristic, use the appropriate \(q_i\) for each participant.
Assumptions and the Expected-Count Calculation
This projection requires a clearly defined event, an enrollment date for each participant (or each participant’s cohort), a common proposed cutoff, and an event-probability model that applies to the participants being projected. The follow-up clock must start at the same point used by that model, ordinarily enrollment or randomization. The probability \(q_i(u)\) must refer to the event of interest occurring within \(u\) units of follow-up—not to an event by a calendar date measured from some different origin.
The basic setup assumes participants are observed for their planned follow-up through \(T\), and that the specified event probabilities account for the relevant study conditions. If participants may leave observation early, or if other causes prevent the event of interest, a projection must handle those features through an appropriate event-probability model. Simply assigning everyone their planned follow-up would not account for unmodeled early loss.
The expected count follows from a simple indicator argument. Let \(I_i(T)\) equal 1 if participant \(i\) has experienced the event by \(T\), and 0 otherwise. The total number of events is \(D(T)=\sum_i I_i(T)\). Given the participant’s follow-up, the mean of this indicator is its event probability, \(E[I_i(T)]=q_i(u_i)\). Therefore:
The middle equality uses linearity of expectation. It does not require participants’ event indicators to be independent. Independence would matter for calculating the variance of the total or a prediction interval, not for this expected-count formula. The result is an average over repeated realizations under the assumed model; it is not a promise that the observed event count will equal the projection.
Worked Example: Six Individual Enrollment Dates
Suppose a hypothetical trial proposes an analysis cutoff on day 180. Six participants have enrollment dates on days 0, 30, 60, 90, 120, and 150. For illustration, the design’s event-probability model gives the following probabilities by each listed duration of follow-up.
| Enrollment day | Follow-up by day 180 | Specified event probability |
|---|---|---|
| 0 | 180 days | 0.30 |
| 30 | 150 days | 0.26 |
| 60 | 120 days | 0.22 |
| 90 | 90 days | 0.17 |
| 120 | 60 days | 0.12 |
| 150 | 30 days | 0.06 |
For each row, planned follow-up is \(u_i=T-e_i\). For example, the participant enrolled on day 90 has \(180-90=90\) days of follow-up, so their event contribution to the expectation is 0.17.
Adding all six probabilities gives:
The projected expected event count is 1.13 events by day 180. A fractional expectation is valid: it describes a model average, while an observed event count must be a whole number. The calculation uses the supplied probabilities; the enrollment dates alone cannot determine the event count without an event-probability model.
Working With Enrollment Cohorts
When many participants enroll on the same date, calculating one row per person is unnecessary. If \(n_j\) participants share enrollment date \(e_j\) and the same applicable event-probability curve, their combined expected contribution is \(n_j q(T-e_j)\). Summing across enrollment cohorts gives the same result as summing across participants. This is an exact regrouping when everyone in a cohort has the same follow-up and event probability.
Worked Example: Three Enrollment Cohorts
Consider a hypothetical study with a planned cutoff at month 12. The enrollment plan is 20 participants at month 0, 15 at month 6, and 10 at month 10. The prespecified event-probability model gives probabilities of 0.28, 0.16, and 0.06 for 12, 6, and 2 months of follow-up, respectively.
The cohort enrolled at month 0 has \(12-0=12\) months of follow-up, so its expected contribution is \(20(0.28)=5.60\) events. For the month-6 cohort, follow-up is \(12-6=6\) months, giving \(15(0.16)=2.40\). The month-10 cohort has \(12-10=2\) months, giving \(10(0.06)=0.60\).
The projected expected count is 8.60 events. The cohort calculation is equivalent to applying the appropriate probability to each participant in each cohort, but it is easier to audit when enrollment dates occur in groups.
A cohort table is only as precise as its time grouping. If enrollment spans a month but the projection treats everyone as enrolling on the first day, it assigns some participants too much follow-up. Grouping by finer calendar intervals—or using the actual dates when available—reduces this approximation. When participants within a nominal cohort have different event probabilities for another reason, they must be separated into groups for which the relevant probabilities are the same.
Different Event Probabilities Across Groups
The calendar calculation is unchanged when event probabilities differ across treatment groups: calculate each participant’s \(u_i\), then apply the probability appropriate to that participant’s group and follow-up. A pooled curve is not appropriate if it does not represent the groups’ event processes. These calculations project event totals under the specified assumptions; they are not, by themselves, a comparison of treatment effects.
Worked Example: Applying Arm-Specific Probabilities
In a hypothetical trial with a cutoff at month 12, 30 control participants enroll in three cohorts: 10 at month 0, 10 at month 6, and 10 at month 9. The corresponding follow-up times are 12, 6, and 3 months. The specified control-group event probabilities at those times are 0.24, 0.14, and 0.07. In the treatment group, 10 participants enroll at each of months 0, 3, and 6, giving follow-up times of 12, 9, and 6 months; the corresponding treatment-group probabilities are 0.18, 0.15, and 0.11.
For the control group, the projected expected event count is:
For the treatment group, apply the treatment-group probabilities to its own enrollment cohorts:
The combined projected expected count is \(4.50+4.40=8.90\) events. The separate calculations preserve both differences in enrollment timing and differences in the specified event probabilities. The result does not establish that the groups will have equal observed event counts, nor does it quantify uncertainty in either projected count.
Diagnostics, Shortcuts, and Limitations
A reliable projection can be checked in a few direct ways. Verify that all enrollment dates and the cutoff use the same time scale; confirm that no participant is assigned negative follow-up; and check that later enrollment never produces more follow-up at a fixed cutoff. Also verify that each probability corresponds to the calculated follow-up duration and the correct group. A spreadsheet or script should be tested on a small hand-calculated example before being used for a large enrollment roster.
Several shortcuts can fail:
- Using the final sample size alone. This treats a participant enrolled just before the cutoff as if they had the same event opportunity as someone enrolled much earlier.
- Giving everyone the average follow-up and applying one probability. In general, \(q\) of the average follow-up is not the average of the individual probabilities. The shortcut is exact only under restrictive conditions, such as a probability function that is linear over the follow-up range, or identical follow-up for everyone.
- Multiplying total person-time by an event rate without checking the model. This can be a useful approximation under suitable assumptions, but it is not automatically identical to summing probabilities of first events. Use the event-probability approach when the design supplies probabilities by follow-up time.
- Reading an expectation as a guaranteed count. An expected count of 8.6 does not mean that 8 or 9 events must occur; realized counts vary around their model-based expectation.
The individual-date method is direct when a roster or detailed enrollment schedule is available. Cohort grouping is a more compact equivalent when participants within each cohort share follow-up and event probabilities. An aggregate person-time calculation may be simpler, but it relies on a compatible rate-based approximation. The choice should preserve the features that matter for the event process rather than merely reduce calculation effort.
Finally, the projection depends on the enrollment schedule and on the event probabilities being appropriate. If actual enrollment differs from plan, or the probability model does not represent the study population or treatment groups, the projected count may be inaccurate. This calculation produces a model-based mean, not a prediction interval or a guarantee of when a target will be reached.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A cutoff is at month 10. Two participants enroll at months 2 and 8. What are their planned follow-up times?
- Three participants have event probabilities of 0.10, 0.25, and 0.40 by their respective follow-up times. What is the projected expected event count?
- Why does summing participant-specific probabilities not require independent event outcomes?
- When is it valid to replace individual calculations with a cohort calculation?
- A planner applies one event probability at the average follow-up time. State one reason this can differ from the participant-specific projection.
Solutions. (1) The follow-up times are \(10-2=8\) months and \(10-8=2\) months. (2) The expected count is \(0.10+0.25+0.40=0.75\) events. (3) Linearity of expectation gives \(E[\sum_i I_i]=\sum_i E[I_i]\) whether or not the indicators are independent. (4) Group participants when they share enrollment timing and the same applicable event probability; multiply that probability by the cohort size, then sum across cohorts. (5) In general, applying a nonlinear probability function to average follow-up does not equal averaging the probabilities calculated at individual follow-up times.