Accrual And Censoring Scenarios For Trial Calibration

Why Accrual and Censoring Belong in Calibration

In a time-to-event trial, the planned sample size does not by itself determine how much information the analysis will contain. A participant enrolled early may be observed for much longer than one enrolled near the study’s end. If the trial closes on a fixed calendar date, those follow-up differences affect how many events are observed, how precisely treatment effects are estimated, and potentially whether a decision rule is crossed.

Calibration should therefore represent not only event-time distributions, but also when participants enter and when observation ends. The earlier material on time-to-event outcomes in operating-characteristic simulations supplies the general simulation framework. Here the focus is how accrual timing and administrative censoring change the data generated within that framework.

Definition: Let \(U_i\) be participant \(i\)'s enrollment time, measured from trial opening, and let \(\tau\) be the fixed calendar time at which the trial closes. Under administrative censoring at closure, that participant's potential follow-up is \(C_i=\tau-U_i\). If the event time from enrollment is \(T_i\), the observed time is \(Y_i=\min(T_i,C_i)\), and the event indicator is \(\delta_i=1\) when \(T_i\le C_i\), and \(0\) otherwise.

This setup assumes \(\tau\ge U_i\) for all enrolled participants. Administrative censoring means observation ends because the planned study-end date is reached, not because the participant experiences the event. In the basic model here, event times are independent across participants, and censoring is determined by the enrollment schedule and calendar close rather than by a participant's unobserved event time. Other censoring mechanisms require their own assumptions and simulation scenarios.

Event Probabilities Under a Fixed Study-End Date

Suppose enrollment is uniform over the interval from trial opening to month \(A\), with closure at month \(\tau\), where \(\tau\ge A\). Then \(U_i\) is uniform on \([0,A]\), and follow-up is \(\tau-U_i\). For a constant event hazard \(\lambda\), assume \(T_i\) is independent of enrollment time \(U_i\) and has an exponential distribution, so \(P(T_i>t)=e^{-\lambda t}\). Conditional on enrollment time \(u\), the probability of an event before closure is \(1-e^{-\lambda(\tau-u)}\). Averaging over enrollment times gives the marginal probability of an observed event.

q(\lambda,A,\tau) =\frac{1}{A}\int_0^A\left(1-e^{-\lambda(\tau-u)}\right)\,du =1-e^{-\lambda(\tau-A)}\frac{1-e^{-\lambda A}}{\lambda A}.

The final expression follows by integrating the probability of remaining event-free to closure. In particular,

P(T_i>C_i) =\frac{1}{A}\int_0^A e^{-\lambda(\tau-u)}\,du =e^{-\lambda(\tau-A)}\frac{1-e^{-\lambda A}}{\lambda A}, \qquad q=1-P(T_i>C_i).

If \(n\) participants enroll independently according to this schedule, the expected event count is \(nq\). If enrollment times are fixed rather than independently sampled, follow-up times differ across participants; the expected count is still the sum of their individual event probabilities, but the event count need not follow a binomial distribution with a common probability. Either way, expected counts are not guarantees: simulated trial replicates will vary.

Worked Example: Expected Events With Six Months of Accrual

Consider a hypothetical equal-allocation trial with 100 participants in each arm. Enrollment is uniform over the first 6 months, and the study closes at month 18. Assume constant hazards of \(0.08\) events per person-year in control and \(0.06\) events per person-year in treatment, with all times converted to years. Thus \(A=0.5\) years and \(\tau=1.5\) years. The formula can be used with these time units because each hazard is per year.

For the treatment arm, the hazard times the accrual and follow-up durations are \(\lambda A=0.06(0.5)=0.03\) and \(\lambda(\tau-A)=0.06(1)=0.06\). Therefore,

q_T=1-e^{-0.06}\frac{1-e^{-0.03}}{0.03} \approx 1-(0.941765)(0.985149) \approx 0.07222.

The expected treatment event count is \(100(0.07222)=7.222\) events. For control,

q_C=1-e^{-0.08}\frac{1-e^{-0.04}}{0.04} \approx 1-(0.923116)(0.980264) \approx 0.09510.

The expected control event count is \(100(0.09510)=9.510\) events. These values are expectations under the stated exponential-hazard and uniform-accrual assumptions, not observed counts. The low counts result from measuring time in years: the hypothetical study ends after only 1.5 years, with hazards of 0.06 and 0.08 per year.

How Accrual Duration Changes Follow-Up

With a fixed closure date, extending the accrual period shifts more participants toward the end of the study. They consequently have less potential follow-up. This usually reduces the event probability per participant, although the effect on total events in a real design also depends on whether the total sample size, accrual rate, or closure date changes. A simulation must specify which of these design features is held fixed; changing several at once makes the source of a difference harder to interpret.

Worked Example: Longer Accrual With the Same Sample Size

Suppose, hypothetically, that 100 participants in an arm have exponential event times with hazard \(0.06\) per year, and the study still closes at month 18 (\(\tau=1.5\) years). Compare uniform accrual over 6 months (\(A=0.5\) years) with uniform accrual over 12 months (\(A=1\) year). For the 12-month accrual scenario, the average event probability is

q=1-e^{-0.06(1.5-1)}\frac{1-e^{-0.06(1)}}{0.06(1)} =1-e^{-0.03}\frac{1-e^{-0.06}}{0.06} \approx 1-(0.970446)(0.970591) \approx 0.05809.

The expected count is \(100(0.05809)=5.809\) events. Under 6-month accrual with the same closure and hazard, the calculation is

q=1-e^{-0.06(1.5-0.5)}\frac{1-e^{-0.06(0.5)}}{0.06(0.5)} =1-e^{-0.06}\frac{1-e^{-0.03}}{0.03} \approx 0.07222,

or \(7.222\) expected events among 100 participants. The longer accrual schedule yields fewer expected events per participant because it leaves less follow-up before closure. This comparison holds the sample size, hazard, and closure date fixed; it does not claim that longer accrual always reduces the total number of events in designs where enrollment continues to add participants.

Administrative Censoring and Trial Decisions

Administrative censoring is not a nuisance to be removed from a simulation: it is part of the planned observation process. If a protocol closes when a target number of events is reached, that is a different design from one that closes on a fixed calendar date. Likewise, a fixed minimum follow-up period for every participant implies a different closure rule from uniform accrual followed by a common calendar close. Simulate the rule that the trial will actually use.

A useful calibration strategy keeps the analysis and decision rule fixed while changing the accrual or censoring scenario. For example, the analysis might declare success when the posterior probability that the treatment hazard is lower than the control hazard exceeds 0.95. The same model, prior, and cutoff should be applied in every replicate and scenario. Otherwise, differences in decision rates could reflect a changed decision procedure rather than changed follow-up.

The next example uses an explicitly approximate information calculation to illustrate why event counts can matter for such a decision. It is not an exact posterior calculation for a survival model. Under a balanced-arm proportional-hazards approximation, the standard error of an estimated log hazard ratio is about \(2/\sqrt{D}\), where \(D\) is the total number of events. For illustration, combine that approximate normal likelihood with a \(N(0,1)\) prior on the log hazard ratio.

Worked Example: Event Information Near a Decision Cutoff

Imagine two hypothetical simulated scenarios with balanced allocation and estimated log hazard ratio \(\hat\beta=-0.30\). Use a normal likelihood with standard error \(SE=2/\sqrt{D}\), and prior \(\beta\sim N(0,1)\). The posterior variance is \(V=1/(1+1/SE^2)\), and the posterior mean is \(m=V\hat\beta/SE^2\). Under this approximation, \(P(\beta<0\mid\text{data})=\Phi(-m/\sqrt{V})\), where \(\Phi\) is the standard normal cumulative distribution function.

For \(D=129\) events, \(SE=2/\sqrt{129}\approx0.1761\), so \(1/SE^2=32.25\) and \(V=1/33.25\approx0.03008\). The posterior mean is \(m=0.03008(32.25)(-0.30)\approx-0.2910\), and the posterior standard deviation is \(\sqrt{V}\approx0.1734\). Thus,

P(\beta<0\mid\text{data}) \approx \Phi\left(\frac{0.2910}{0.1734}\right) =\Phi(1.678) \approx 0.953.

This crosses a 0.95 decision cutoff. For \(D=111\), \(SE=2/\sqrt{111}\approx0.1898\), \(1/SE^2=27.75\), \(V=1/28.75\approx0.03478\), and \(m=0.03478(27.75)(-0.30)\approx-0.2896\). The posterior standard deviation is about \(0.1865\), giving \(\Phi(0.2896/0.1865)=\Phi(1.553)\approx0.940\), below the cutoff. Thus, with this same estimated effect, the approximate decision changes as information decreases. In an actual calibration, simulate event times and censoring, then apply the prespecified Bayesian survival analysis to every replicate; this calculation is only an illustration of the information mechanism.

Simulation Workflow, Diagnostics, and Limits

A trial-level simulation should generate enrollment times, event times, and censoring times in a way that matches the design. It should retain the observed time and event indicator for each participant, fit the planned analysis to those data, and record the decision. Repeating this process estimates the operating characteristic of interest, such as the probability of a successful decision under a specified treatment effect and accrual schedule.

Specify the calendar design.
Set the accrual distribution, accrual duration, closure rule, sample size or enrollment rate, and any required follow-up period.

Generate participant times.
Draw enrollment times and event times from the scenario's specified distributions. Apply the planned censoring rule to obtain \(Y_i\) and \(\delta_i\).

Analyze and record the decision.
Fit the same prespecified model and apply the same posterior decision rule in each replicate. Record event counts as well as decisions.

Check the generated data.
Compare simulated follow-up summaries and event counts with design-based expectations, and inspect whether censoring or follow-up patterns differ by arm as intended.

Diagnostics should include enrollment-time distributions, follow-up summaries, the fraction administratively censored, and event counts by arm. Under the simple exponential model with uniform accrual, simulated mean event counts should be near the corresponding expectations \(nq\) over many replicates, allowing for Monte Carlo variation. Large discrepancies can indicate incorrect time units, a mistaken closure date, or an implementation error. Also verify that no participant has follow-up beyond the administrative close and that events after that person's censoring time are not counted.

The exponential model assumes a constant hazard within each arm. If the intended scenario has changing hazards, the exponential event-probability formula above is not appropriate; use the specified survival distribution and simulate its event times directly. Uniform accrual is also a design assumption, not a universal description of enrollment. Actual recruitment may ramp up, pause, or differ by site. Scenarios should reflect plausible schedules and clearly state what is held constant when alternatives are compared.

A fixed-calendar-close design is often compared with a fixed-follow-up design. Under fixed follow-up, each participant is observed for a prespecified duration after enrollment, so closure generally occurs later for later entrants. Under fixed calendar closure, later entrants have shorter follow-up. Neither is inherently preferable for calibration; each represents a different design. The key is not to interpret a change in events or decisions as a treatment effect change when it arose from a changed observation schedule.

Common Mistakes

Using the study duration as every participant's follow-up. With staggered enrollment and a calendar close, follow-up is \(\tau-U_i\), not \(\tau\) for everyone.
Confusing expected and realized events. The value \(nq\) is an expectation under the model; any one simulated trial can have fewer or more events.
Changing accrual and sample size without saying so. A longer accrual period at fixed sample size differs from a longer accrual period at a fixed recruitment rate. State the design comparison precisely.
Treating administrative censoring as an event. A participant event indicator is zero when the participant remains event-free through their censoring time.
Assuming a change in event count determines the decision. Decisions also depend on event times, allocation, the analysis model, prior, effect estimate, and cutoff. Estimate decision rates by applying the full prespecified analysis in simulated replicates.

Key takeaway: Accrual timing determines how much follow-up participants can contribute under a calendar close. Simulate that timing and censoring explicitly, check event counts against design-based expectations, and assess decisions using the unchanged planned analysis.

Check Your Understanding

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

A trial closes at month 24. A participant enrolls at month 9. What is the participant's administrative follow-up time?
For exponential event times with hazard \(\lambda\), uniform accrual over \([0,A]\), and closure at \(\tau\ge A\), write the expression for the probability of an observed event.
At fixed sample size and closure date, what happens to follow-up for later entrants when accrual is extended?
Why should a calibration simulation record censoring indicators as well as observed times?
In a simulation comparison, why should the analysis model and posterior cutoff remain fixed across accrual scenarios?

Solutions. (1) Follow-up is \(24-9=15\) months, provided the participant has no earlier event. (2) The probability is \(q=1-e^{-\lambda(\tau-A)}(1-e^{-\lambda A})/(\lambda A)\). (3) More participants enroll closer to closure, so their potential follow-up is shorter; the effect on total events also depends on which design features are held fixed. (4) The indicator distinguishes an observed event from a participant censored while event-free; both are needed to represent the time-to-event data correctly. (5) Holding the analysis and cutoff fixed makes changes in decision rates interpretable as consequences of the changed accrual and censoring scenario, rather than changes in the decision procedure.