Why Accrual Timing Belongs in Design Assurance
A trial can reach the same planned sample size under very different recruitment histories. One may enroll steadily across sites; another may experience a pause, followed by rapid enrollment at a few sites. If an adaptive decision depends on information available at a calendar-time interim, these histories can produce different numbers of mature outcomes, different treatment-arm or site composition, and different probabilities of crossing the decision threshold. The final sample size alone does not describe those differences.
The relevant question is not only how many participants are recruited, but when they are recruited, where they enter, and when their outcomes become available. A design-assurance simulation should therefore represent accrual and follow-up on a calendar-time scale, then apply the planned interim rule to the data available at that time.
For a simple continuous endpoint, let \(\theta\) be the treatment-minus-control mean difference, and suppose outcomes in each arm have a common known variance \(\sigma^2\). If the interim data contain \(n_1\) treatment and \(n_0\) control outcomes, the difference in sample means has variance \(\sigma^2/n_1+\sigma^2/n_0\). Its information, defined here as the reciprocal variance, depends on the outcomes available and the arm counts—not merely on the total number recruited.
Representing Accrual and Outcome Availability
A simulation needs a rule for generating or specifying enrollment times. In a deterministic scenario, enrollment might occur at a fixed monthly rate. In a stochastic scenario, the number arriving in an interval can vary; for example, it may be generated from a Poisson process with a specified rate. In either case, site-specific rates or time-varying rates can represent differences among sites or periods.
The simulation must also represent the delay from enrollment to an analyzable interim outcome. With a fixed follow-up requirement of \(\tau\) months and interim time \(T\), a participant enrolled at time \(t_i\) contributes only if \(t_i+\tau\leq T\). If outcome collection has additional delays, those should be represented only when they are part of the design scenario. This tutorial focuses on recruitment and accrual timing, not on modeling missing outcomes or nonadherence.
Allocation must be represented as it will occur in the trial. A 1:1 randomization probability does not guarantee equal realized arm sizes at every interim. If a calculation assumes exact balance, the scenario must specify a mechanism that guarantees it at the interim being analyzed, such as completed blocks of size two. Otherwise, simulate the realized arm counts and use those counts in the information calculation.
Worked Example: Recruitment Rate and a Calendar-Time Interim
Consider two hypothetical accrual scenarios for a two-arm trial. The interim analysis is scheduled at the end of month 10, and each participant needs two months of follow-up before the endpoint is available. Outcomes have variance \(\sigma^2=100\) points\(^2\) in both arms. Enrollment is in completed two-person blocks, so the mature interim cohort is exactly balanced.
In the faster scenario, three participants enroll per month during months 1 through 8. At the end of month 10, those enrolled by the end of month 8 have completed follow-up: \(3\times8=24\) participants, or \(n_1=n_0=12\). The variance of the estimated treatment difference is
In the slower scenario, two participants enroll per month over the same eight months. Sixteen have mature outcomes by the interim, giving \(n_1=n_0=8\). Thus
The faster scenario has more interim information: \(1/16.6667=0.06000\ \text{points}^{-2}\), compared with \(1/25.00=0.04000\ \text{points}^{-2}\). This comparison concerns precision at the fixed calendar-time interim. It does not by itself say how the design’s final power or error rate changes; those depend on the full adaptive rule and the outcomes generated under each scenario.
How Pauses Change the Decision Timeline
A pause can affect a trial in at least two distinct ways. At a fixed calendar-time interim, it may reduce the number of participants with mature outcomes. At a fixed participant target, it may delay the date when that target is reached and therefore delay the interim or trial completion. These questions should not be collapsed into a single “recruitment rate” summary.
The basic information calculation follows directly from the variance of a difference in independent sample means. For arm \(a\), the sample mean has variance \(\sigma^2/n_a\). Independence between arms gives the variance of their difference as the sum of those variances. Taking the reciprocal yields information:
This derivation also shows why total enrollment alone is insufficient: two trials with the same total can have different \(n_1\) and \(n_0\), and hence different precision. More generally, when sites have different outcome variances or the planned analysis stratifies by site, the contributions to information depend on the numbers in each site and arm.
Worked Example: A Pause Before a Fixed-Date Interim
Suppose a hypothetical trial plans two enrollments per month, with a two-month follow-up requirement and an interim at the end of month 10. Under uninterrupted recruitment in months 1 through 8, 16 participants have mature outcomes at the interim. Allocation is exactly balanced in completed blocks, so \(n_1=n_0=8\). With outcome variance \(100\) points\(^2\) per arm, the standard error is
Now suppose recruitment stops during months 5 and 6, with two participants enrolling in each of months 1 through 4 and 7 through 8. Only \(4\times2=8\) participants are enrolled by the end of month 8 and mature by the interim. With exact balance, \(n_1=n_0=4\), giving
The pause has reduced interim information from \(1/25=0.04000\) to \(1/50=0.02000\ \text{points}^{-2}\). If instead the design waits for a fixed enrollment target before analyzing, the pause primarily shifts the calendar date at which the interim can occur. Which consequence matters depends on whether the protocol specifies a calendar-time trigger, an information or participant-count trigger, or a combination.
Site-Level Accrual and the Mix of Information
Site patterns can matter even when total enrollment and overall treatment allocation are unchanged. This is especially clear when the analysis allows different outcome variances by site. For a common treatment effect estimated from independent, site-specific treatment contrasts, inverse-variance weighting gives total information equal to the sum of the site-specific information contributions. Under the normal model, the contribution at site \(j\) is
This formula assumes independent site contrasts, a common treatment effect across sites, and the stated site-specific variances. It should not be used automatically if the planned analysis uses a different model, or if treatment effects vary by site.
Worked Example: The Same Total Accrual Across Two Sites
Compare two hypothetical interim cohorts, each with 40 participants and exact treatment balance within each site. Site A has outcome variance \(100\) points\(^2\); Site B has variance \(400\) points\(^2\). The analysis estimates a common treatment effect by combining independent site-specific contrasts using their inverse variances.
In Pattern 1, each arm has 15 participants at Site A and 5 at Site B. Site A’s contrast variance is \(100(1/15+1/15)=100(2/15)\), so its information is \(1/[100(2/15)]=0.07500\ \text{points}^{-2}\). Site B’s contrast variance is \(400(1/5+1/5)=400(2/5)\), so its information is \(1/[400(2/5)]=0.006250\ \text{points}^{-2}\). Therefore,
In Pattern 2, each arm has 5 participants at Site A and 15 at Site B. The site information values are \(1/[100(2/5)]=0.02500\) and \(1/[400(2/15)]=0.01875\ \text{points}^{-2}\). Thus,
Both patterns have the same total enrollment and are exactly balanced by arm. The first is more precise under the stated model because more participants accrue at the lower-variance site. This is not a general argument for preferential recruitment at any one site: feasibility, representativeness, and the actual analysis model matter, and the simulation should reflect the design’s intended site mix.
Building and Checking the Sensitivity Simulation
A useful analysis compares operationally plausible accrual scenarios while holding the design rule and outcome-generating assumptions fixed. Scenarios might include a slower initial rate, a temporary site pause, different site-specific rates, or a more uneven distribution of participants across sites. For each replicate, generate enrollment times and sites, assign treatment using the planned allocation procedure, determine which outcomes are available at each interim, and then apply the adaptive rule exactly as specified.
State rates by period and site, pause timing and duration, the enrollment target, and the outcome follow-up requirement. Identify which quantities are fixed and which vary across simulated trials.
For each simulated trial, generate or assign participant enrollment times and sites, then apply the planned randomization method. Preserve realized arm and site counts rather than replacing them with expected counts.
At each planned decision time, include only outcomes available under the scenario. Calculate the analysis statistic or information using those data, then execute the specified adaptation or stopping rule.
Estimate decision probabilities, final trial outcomes, and calendar-time quantities relevant to the design. Report Monte Carlo uncertainty for these estimates and compare scenarios on the same defined metrics.
The main alternative to explicit timeline simulation is to use an average accrual rate or a single expected interim sample size. That can be adequate for a rough planning calculation when timing variability and site composition are immaterial. It is not a reliable substitute when a calendar-time decision is sensitive to outcome maturity, when pauses are plausible, or when sites contribute different amounts of information. A fixed participant-count interim avoids some sensitivity to recruitment speed, but it does not eliminate effects of site mix, realized arm counts, or calendar duration.
Diagnostics should examine more than the average number enrolled. Plot or tabulate enrollment and mature-outcome counts over calendar time, by site and arm. Check that simulated pauses occur when specified, that rates and targets are implemented correctly, and that the interim analysis uses only eligible outcomes. Compare the distribution—not only the mean—of interim information and time to interim across scenarios. If a trigger is close to a decision threshold, small changes in information can alter the probability of reaching that threshold; inspect the distribution around the trigger rather than relying on a rounded average.
Common Mistake and Limitations
- Wrong reasoning: “The trial plans 40 participants with 1:1 randomization, so the interim has 20 per arm.” A 1:1 allocation probability does not guarantee exactly equal realized counts. Correction: specify an allocation scheme that guarantees balance for the interim cohort, or simulate realized arm counts and calculate information from those counts.
- Wrong reasoning: “A site pause does not matter because the final sample size is unchanged.” Correction: a pause can reduce information at a fixed calendar-time interim or delay a target-based interim; assess the trigger and timeline used by the design.
- Wrong reasoning: “The mean recruitment rate fully describes operational timing.” Correction: different patterns with the same mean rate can have different pauses, site mixes, and interim information distributions.
Simulation results are conditional on the accrual scenarios and outcome model specified. A small set of scenarios cannot establish performance under every possible operational history, and a simplified model may miss dependencies such as sites opening at different times. Conclusions should therefore identify the assumptions and scenarios examined. They should not treat an expected enrollment curve as a guarantee about any particular trial.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- A calendar-time interim has 10 mature outcomes per arm, with outcome variance \(64\) units\(^2\) in each arm. What are the variance and standard error of the estimated treatment difference?
- Why is a 1:1 allocation probability not enough to conclude that an interim cohort of 30 participants has exactly 15 participants per arm?
- A pause leaves the number of participants recruited by the interim unchanged but delays when all have completed follow-up. Which interim quantity can change, and why?
- Under the site-stratified model in the tutorial, what is the total information if two independent sites contribute \(0.030\) and \(0.020\) units\(^{-2}\)?
Solutions. (1) The variance is \(64/10+64/10=12.8\) units\(^2\), and the standard error is \(\sqrt{12.8}=3.578\) units. (2) Random allocation with probability one-half permits unequal realized counts; exact balance requires a specified balancing mechanism, otherwise the realized counts must be used. (3) The number of outcomes available—and thus interim information—can change because some recruited participants may not yet have completed the required follow-up. (4) Information adds across the independent site contrasts: \(0.030+0.020=0.050\) units\(^{-2}\).