Expected Sample Size Under Transition Rules

Planning Around More Than One Trial Path

At a seamless phase boundary, a trial may stop, continue with one treatment, or continue with another. Each decision leads to a different amount of future enrollment and may take a different amount of calendar time. Planning with only the largest possible sample size can overstate typical resource needs; planning with a single “average” pathway can conceal how much the actual trial may vary.

The expected enrollment and expected duration summarize the design across its possible pathways. They are useful for budgeting and operational planning, but they do not guarantee that a realized trial will match either expectation. The calculation begins by listing the possible transition outcomes, assigning a probability to each, and specifying the enrollment and timing that each outcome entails.

Here, enrollment means participants randomized into the trial. If a protocol also tracks screened patients, screen failures, or participants needed to replace those with unusable outcomes, those quantities require their own assumptions and calculations. The number enrolled in stage 1 is counted once on every pathway. Stage 2 enrollment is zero on a stop pathway.

Expected Enrollment From Transition Probabilities

Let \(N\) denote total enrollment. If pathway \(j\) occurs, the total is \(N_j=n_1+n_{2j}\). The expected total is therefore

The second equality uses \(\sum_j\pi_j=1\): stage 1 is enrolled regardless of the later decision. This expression also works when some pathways stop, provided their \(n_{2j}\) is set to zero. It requires no assumption that the probabilities of the transition outcomes are independent of the stage 1 data; the probabilities describe the design's chance of taking each route.

When treatment arms have different allocations, calculate \(n_{2j}\) as the number of unique participants randomized on pathway \(j\). A shared control group is not counted again for each active-treatment comparison. If the stage 2 total is specified through arm sizes, add the distinct arm allocations once, applying the intended randomization ratio and any planned control sharing.

Expected Calendar Duration

Enrollment and duration are related, but duration is not always total enrollment divided by one accrual rate. Recruitment may be constrained by the slowest arm, the active treatment may change between pathways, or a phase transition and outcome follow-up may add calendar time. Define a pathway-specific duration from a clear start point to a clear endpoint, such as time from first enrollment to availability of the required primary outcome data.

Let \(T_j\) be that duration if pathway \(j\) occurs. If the stage 1 period is common to every pathway and stage 2 follows it, one useful decomposition is \(T_j=T_1+d_j+T_{2j}+f_j\), where \(T_1\) is stage 1 accrual time, \(d_j\) is any transition delay, \(T_{2j}\) is stage 2 accrual time, and \(f_j\) is additional follow-up required after the relevant accrual period. Include a component only if it is not already included in another term. Then

For parallel arm recruitment with arm-specific enrollment targets \(n_{aj}\) and accrual capacities \(r_{aj}\) participants per month, a simple planning model for stage 2 is

This maximum applies when recruitment in all required arms proceeds concurrently, each rate is a stable capacity, and there are no other bottlenecks. If participants are recruited through one trial-wide pool at a stable rate \(r_j\), a different appropriate model may be \(T_{2j}=n_{2j}/r_j\). The operational assumptions determine which calculation is defensible.

Derivation and Checks

The expectation formulas follow from summing over the possible pathways. To see this directly for enrollment, define \(I_j\) to equal 1 if pathway \(j\) occurs and 0 otherwise. Exactly one pathway occurs, so \(N=n_1+\sum_j I_j n_{2j}\). Since \(E[I_j]=P(I_j=1)=\pi_j\), linearity of expectation gives

The same indicator argument gives \(T=\sum_j I_jT_j\), and therefore \(E[T]=\sum_j\pi_jT_j\). No independence between pathway indicators is required; indeed, because exactly one pathway occurs, they are mutually exclusive. These expectations do require a complete list of pathways and a duration or enrollment value specified for each one.

Assumptions, Diagnostics, and Alternatives

A planning calculation is only as sound as its inputs. Transition probabilities may come from the assumed operating characteristics of the design under a specified scenario. They should represent the probabilities of the design taking each action, not probabilities that a treatment is truly effective. If probabilities are estimated from a set of scenarios, identify the scenarios and weighting used; different assumptions can yield different expected resource needs.

Before relying on the result, check that:

• Every possible action at the boundary is represented, including stopping, and the probabilities sum to 1.
• Stage 1 enrollment is counted once, and each \(n_{2j}\) counts unique randomized participants on that pathway.
• Sample sizes reflect the planned randomization ratio and any shared control group.
• Accrual rates, arm bottlenecks, transition delays, follow-up, and the duration endpoint match the operational plan.
• The probabilities and sample sizes refer to the same design and decision rule.

For detailed budgeting, a single expectation may not be enough. Report the pathway-specific totals alongside the expected value, especially when a low-probability pathway has substantially higher enrollment or duration. A maximum or other upper planning value answers a different question from the expectation: it describes a more demanding scenario rather than the probability-weighted average. A fixed-sample-size plan is another useful comparator, but it does not represent the variable enrollment produced by transition rules.

Another diagnostic is to compare the calculated expectation with the possible pathway values. The expected enrollment must lie between the smallest and largest pathway enrollment, and expected duration must lie between the smallest and largest pathway duration when the pathway probabilities sum to one. A result outside those bounds usually signals a probability, arithmetic, or pathway-definition error. Large differences between pathways are not themselves errors; they are a reason to show the pathway breakdown rather than report only the average.