Simulating Accrual and Interim Information Fractions

Why Stage Timing Needs an Accrual Model

A MAMS design specifies when interim analyses are intended to occur, but the intended stage does not by itself determine a calendar date. Participants may arrive at a variable rate, and an enrolled participant may not yet have an outcome available. A simulation therefore needs to represent both when participants accrue and how much usable information is available at each planned interim.

These are related but distinct quantities. A recruitment target is a count of participants; an information fraction is a ratio of statistical information to the information planned at the final analysis. The two coincide only under particular assumptions, such as balanced allocation, equal outcome availability, and a suitable fixed-variance model. A calendar-time target is different again: reaching a date does not guarantee that either the planned count or the planned information has been reached.

Specify Accrual Before Specifying Interim Timing

One standard representation is a nonhomogeneous Poisson process for recruitment. Let \(N(t)\) be the number of participants who have enrolled by calendar time \(t\), and let \(\lambda(t)\) be the accrual rate, measured in participants per unit time. The model assumes independent counts in non-overlapping time intervals, with \(N(t_2)-N(t_1)\sim\operatorname{Poisson}(\int_{t_1}^{t_2}\lambda(u)\,du)\) for \(t_2>t_1\). If \(\lambda(t)=\lambda\) is constant, interarrival times are independent exponential random variables with rate \(\lambda\).

This model represents recruitment variability, but it does not automatically represent allocation, withdrawal, or outcome follow-up. A simulation can generate an arrival time for each participant and then assign the participant to an arm according to the protocol’s allocation probabilities. It should also represent, when relevant, the delay between enrollment and an outcome becoming usable. The assumptions should match the intended trial scenario: for example, a constant rate is a poor representation if recruitment is expected to increase after a site opens or decrease during a holiday period.

Alternatively, accrual may be represented by a prespecified schedule of counts by time interval. This can be useful when the recruitment plan is known in blocks, but a deterministic schedule suppresses random variation in the times at which targets are reached. Whichever representation is used, the simulation should distinguish the planned accrual pattern from the realized pattern in an individual replicate.

Map Recruitment to Information

To translate a recruitment target into an information target, specify the outcome model and the contrast. For independent normally distributed outcomes with known common variance \(\sigma^2\), the sample mean in an arm with \(n\) participants has variance \(\sigma^2/n\). For a treatment-control difference, the variance is the sum of the two arm-mean variances. The information for that contrast, defined as the reciprocal of its variance, is therefore

where \(n_t\) and \(n_c\) are the numbers of participants with usable outcomes in the treatment and control arms. This formula assumes independent participant outcomes, a common known variance, and no additional variance contribution from clustering or other design features. If these assumptions do not hold, the information calculation must reflect the model actually used for the analysis.

With equal numbers \(n\) in the two arms, \(I=n/(2\sigma^2)\), so doubling \(n\) doubles information. But this proportionality need not hold when the allocation ratio or the balance of available outcomes differs. In a multi-arm design, each treatment-control contrast has its own information. Shared control observations can induce dependence between contrasts; that dependence does not make the information for each individual contrast equal to the total number enrolled.

Representing an Information-Fraction Target

A design may specify target fractions \(f_1,\ldots,f_{K-1}\), with \(0<f_1<\cdots<f_{K-1}<1\), relative to the planned final information. In a simulation, these targets need a concrete operational rule. For example, under the known-variance normal model, one can calculate arm-specific sample-size targets that attain the desired information for each contrast, then determine when the required outcomes are available. With shared control, the control count contributes to every contrast, so targets for different contrasts should be checked against the same evolving control count.

Discrete participants mean a target may not be attainable exactly. If information increases in jumps as outcomes become available, a rule such as “analyze at the first time the planned target is met or exceeded” will generally produce an achieved fraction at least as large as the target. A rule based on a fixed recruitment count may instead stop below or above the intended fraction, especially when allocation is random or some outcomes are missing or delayed. The simulation should preserve the rule specified for the design and record the achieved information, rather than silently treating the target and the achieved value as identical.

A Practical Simulation Sequence

A useful separation is to define accrual, stage targets, and information calculations as distinct components. This makes it easier to identify whether a discrepancy arises from the recruitment model, the allocation pattern, or outcome availability.

Diagnostics, Alternatives, and Limitations

Several checks can reveal a timing model that is not doing what was intended. First, inspect the distribution of time to each recruitment target across replicates; a constant-rate model should show random variation, not a fixed date. Second, check arm-specific counts at each stage, particularly when allocation is randomized. Third, verify that information fractions are calculated using outcome-available counts and the correct final-information denominator. Within a replicate, information for a given contrast should not decrease as additional usable outcomes accumulate under the fixed-variance model. A decrease may indicate a coding error, a changing variance assumption, or a different definition of the target.

The main alternatives answer different design questions:

A count target is often simpler to implement and explain, whereas an information target more directly represents statistical precision. A calendar trigger may be necessary when decisions must occur by specified dates, but it should not be described as an information-fraction rule unless the information is checked. The right choice depends on the protocol, not on which rule is easiest to simulate.

These representations also have limits. A Poisson accrual process assumes a particular pattern of recruitment variability and may not capture site-level pauses, seasonal effects, or dependence between recruitment and participant characteristics. A fixed known-variance formula is not an exact information calculation for every outcome type or analysis model. If information depends on estimated nuisance parameters, the realized information can itself be variable. The simulation should state these assumptions and should not interpret a planned information fraction as a guarantee that every replicate reaches precisely that amount.

Common Mistake