Why Event-Driven Timing Needs Participant-Level Follow-Up
In an event-driven trial, the primary analysis is planned after a specified number of endpoint events have accrued. That target does not determine a calendar date by itself. Participants enter at different times, have different opportunities to experience the event before analysis, and may leave observation before an event occurs. A useful simulation therefore represents when each participant enrolls, when the event occurs, and when observation ends.
This tutorial uses event time measured from each participant’s enrollment and calendar time measured from the start of recruitment. The distinction matters: an event at 10 months after enrollment is not necessarily an event at month 10 of the trial. The examples use hypothetical trial inputs, not results from an actual study.
Generating Enrollment, Events, and Censoring
A simulation begins by specifying an enrollment process. A simple planning assumption is that \(N\) participants enroll uniformly over a recruitment period of length \(R\), so each enrollment time is drawn as \(S_i\sim\operatorname{Uniform}(0,R)\). This is an assumption about the rate of recruitment, not a consequence of randomization. If recruitment is expected to start slowly, accelerate, or stop and restart, use an enrollment distribution or schedule that represents that pattern instead.
For an exponential event-time model with constant hazard \(\lambda\) per month, \(T_i\sim\operatorname{Exponential}(\lambda)\), and the survival probability is \(P(T_i>t)=e^{-\lambda t}\) for \(t\geq0\). If \(U_i\) is drawn from a uniform distribution on \((0,1)\), inverse-transform sampling gives \(T_i=-\log(U_i)/\lambda\). A larger hazard produces shorter event times on average. The exponential assumption is convenient for transparent calculations, but it implies a constant hazard over time; it should not be used without justification when the anticipated event pattern changes over follow-up.
If withdrawal is represented by an independent exponential time \(C_i\) with rate \(\delta\), the observed event indicator is one only when the event occurs before both withdrawal and administrative censoring. Independence is a substantive assumption: if withdrawal is related to prognosis or treatment response, independent censoring may be implausible. The simulation should represent the assumed dependence when that dependence could affect event accrual.
Worked Example: Turning Random Draws Into Observed Follow-Up
Suppose a hypothetical control participant enrolls at month \(S_i=8\), has event hazard \(\lambda=0.10\) per month, and the trial analysis occurs at calendar month \(A=18\). First take the simulated uniform draw \(U_i=0.30\).
The event time from enrollment is \(T_i=-\log(0.30)/0.10=1.20397/0.10=12.0397\) months. The event would occur at calendar time \(8+12.0397=20.0397\) months, after the planned analysis. The available administrative follow-up is \(18-8=10\) months, so \(Y_i=10\) months and \(\Delta_i=0\): this participant is censored at the analysis.
For contrast, with another hypothetical draw \(U_i=0.60\), the generated event time is \(-\log(0.60)/0.10=5.1083\) months. The calendar event time is \(8+5.1083=13.1083\) months, before analysis, so this participant contributes an observed event and \(Y_i=5.1083\) months. The draws illustrate the calculation; a simulation would generate a new draw for each participant and replicate.
Expected Events Under Uniform Accrual
A useful check on a simulation is the expected event count at a fixed calendar analysis time. Assume \(N\) participants accrue uniformly over \([0,R]\), have independent exponential event times with a common hazard \(\lambda\), and have no loss to follow-up before administrative censoring. Suppose \(A\geq R\), so all participants have enrolled by analysis. A participant enrolled at time \(s\) has \(A-s\) months of follow-up and event probability \(1-e^{-\lambda(A-s)}\). Averaging over the uniform enrollment time gives the marginal probability of an event by analysis:
The first expression averages each possible participant follow-up duration. For the second, integrate the survival term:
Thus the expected number of events among \(N\) participants is \(Np(A)\). If participants are independent under these assumptions, their event indicators at the fixed time are Bernoulli variables with probability \(p(A)\), so the event count has a binomial distribution with mean \(Np(A)\) and variance \(Np(A)\{1-p(A)\}\). The mean is useful for planning and checking code; it is not a promise that a particular simulated trial will have that count.
Worked Example: Expected Events With Unequal Follow-Up
Consider a hypothetical treatment arm with \(N=100\), uniform recruitment over \(R=12\) months, event hazard \(\lambda=0.04\) per month, and analysis at \(A=18\) months. Everyone has enrolled by analysis, but follow-up ranges from 6 to 18 months.
Using the derived expression, \(p(18)=1-e^{-0.04(18)}\{e^{0.04(12)}-1\}/\{0.04(12)\}\). Substitution gives \(p(18)=1-e^{-0.72}(e^{0.48}-1)/0.48\approx0.375259\).
The expected number of events is \(100(0.375259)=37.5259\), or about 37.53 events. This is a fractional expected count, not a possible realized count. A single simulated arm will have an integer count that varies around this expectation.
Simulating an Event Target and Its Calendar Time
For an event-driven design, simulate the participant-level enrollment and event processes, then find the calendar time at which the event target is reached. If \(D\) is the target, sort the calendar event times \(S_i+T_i\) for events observed before any censoring or other applicable study limit. The \(D\)th such event gives the event-target time in that replicate, provided the target is reached. If fewer than \(D\) events occur before the maximum permitted study time, record that the target was not reached within the limit; do not silently treat the last observed time as if it met the target.
Draw or assign each participant's enrollment time using the planned recruitment scenario, then assign treatment according to the simulated randomization scheme.
Use the arm-specific event model and the specified censoring model. Preserve any dependence the design assumes rather than generating all times as independent by default.
Determine which events are observed before administrative censoring, withdrawal, or the maximum study duration. An event after observation ends does not count as an observed event.
Record the calendar time of the \(D\)th observed event, or record that the target was not reached. Across replicates, summarize the distribution of target times and the frequency of failure to reach the target.
Worked Example: Comparing Recruitment Scenarios for a 120-Event Target
Suppose a hypothetical two-arm trial plans to enroll 100 participants per arm, uniformly over 16 months, with exponential event hazards of \(0.10\) per month in control and \(0.07\) per month in treatment. There is no pre-analysis withdrawal in this calculation. We first estimate expected events at calendar month 19.8 using the fixed-time formula.
For control, \(p(19.8)=1-e^{-0.10(19.8)}(e^{0.10(16)}-1)/(0.10(16))\approx0.65888\). The expected control event count is \(100(0.65888)\approx65.888\).
For treatment, \(p(19.8)=1-e^{-0.07(19.8)}(e^{0.07(16)}-1)/(0.07(16))\approx0.538960\). The expected treatment event count is \(100(0.538960)\approx53.896\). The total expected count is therefore \(65.888+53.896=119.784\), about 119.78 events.
At month 20, the same calculations give expected counts of approximately 66.56 in control and 54.54 in treatment, or 121.10 events in total. Linear interpolation between these two expected counts places the time at which the expected total reaches 120 at approximately \(19.8+0.2(120-119.784)/(121.100-119.784)=19.833\) months, or about month 19.83.
This interpolation estimates when the expected count reaches 120; it does not give the event-target time for any particular trial. A simulation must separately estimate the distribution of the actual \(D\)th-event time, including the probability that it falls after a calendar cap. A different recruitment duration changes the amount of follow-up available at each calendar time, even if the total enrollment is unchanged.
Diagnostics, Alternatives, and Limitations
Check the simulated records against the data-generating rules before interpreting trial-level summaries. Useful invariants include: no event can occur before enrollment; observed follow-up cannot exceed the time from enrollment to administrative analysis; an event indicator is one only when the event time is no later than every applicable censoring time; and the cumulative event count cannot decrease over calendar time. These checks complement the earlier material on validating simulation code with invariants. Compare generated event-time and accrual distributions with their specified inputs, as in the earlier material on checking simulated data against empirical distributions.
Investigate a projected target time that seems implausible by checking the enrollment calendar, the arm-specific hazards, the time origin used for event generation, and whether censored events are being counted accidentally. A common coding error is to compare event time \(T_i\) directly with calendar analysis time \(A\), ignoring enrollment time \(S_i\). The correct comparison is \(T_i\leq A-S_i\), or equivalently \(S_i+T_i\leq A\), subject to earlier censoring. Another error is to use the expected event count as though it were the realized count; event-count variability means some replicates reach the target earlier and others later.
| Approach | What it provides | Main limitation |
|---|---|---|
| Fixed-time expectation | A quick analytic mean event count under simple assumptions; useful for checking a simulation. | Does not describe the full distribution of the event-target time or complex accrual and censoring. |
| Participant-level simulation | Replicate-specific event counts, target times, and failure to reach a target under explicitly specified scenarios. | Results depend on the input assumptions and require enough replicates for adequate Monte Carlo precision. |
| Fixed calendar follow-up | A design in which analysis timing is set by a calendar date or planned follow-up duration. | The event count is an outcome of the design and may vary around its expectation. |
The exponential model and uniform accrual formula are benchmarks, not universal models. If event hazards vary over time, enrollment is not uniform, or dropout is informative, the simple closed-form calculation may not apply. Simulate the processes that are plausible for the planned trial, examine alternative assumptions, and report uncertainty in the simulated summaries. Monte Carlo error in estimated probabilities and quantiles should also be assessed using the earlier material on quantifying Monte Carlo error in simulation results.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- A participant enrolls at month 6 and has an event 9 months after enrollment. At what calendar month does the event occur? If analysis is at month 14 and there is no earlier withdrawal, is it observed?
- Under uniform accrual over \(R\) months and exponential event hazard \(\lambda\), why is the event probability averaged over enrollment times rather than calculated using a single follow-up duration?
- At a fixed analysis time, the expected event count is 80. Does that mean the event-driven target time in every replicate is the analysis time? Explain.
- Name two simulation checks that could detect a mismatch between event times and censoring or enrollment times.
Solutions. (1) The calendar event time is \(6+9=15\) months. Only 8 months of administrative follow-up are available by month 14, so the event is not observed by that analysis. (2) Participants enroll at different calendar times and therefore have different follow-up durations \(A-S_i\); the marginal probability averages their event probabilities over the enrollment-time distribution. (3) No. An expected count of 80 is a mean at a fixed time, while replicate event counts vary; the target time must be found separately in each replicate. (4) For example, verify that no event precedes enrollment and that an event is counted only if it occurs before both administrative censoring and withdrawal.