Participant Burden and Cost Trade-Offs

Why Participant Burden and Cost Belong in the Design Comparison

A promising-zone rule can increase the final sample size when the interim results are encouraging but not strong enough to make the original design look reliably successful. That option may improve power, but it also exposes more participants to the trial and can raise its cost. Comparing rules therefore requires more than listing their maximum sample sizes: the chance of reaching each sample size matters.

This tutorial compares rules through their expected number of participants and their expected trial cost, alongside power. The expected value summarizes average burden over repeated trials under specified assumptions; it is not a promise about the enrollment of any one trial. A rule with a moderate expected sample size can still have a high maximum, and a rule with a large maximum may reach it only rarely.

A Model for Comparing Rules

For clarity, consider a two-arm trial with a continuous endpoint, equal allocation, independent normally distributed outcomes, and a known common standard deviation \(\sigma\). Let \(N\) be the total number of participants at the final analysis, so each arm has \(N/2\) participants. Under these assumptions, the estimated treatment difference has standard error \(2\sigma/\sqrt{N}\), and the information is \(I=N/(4\sigma^2)\). The same comparisons can be made with other endpoints and allocation ratios, but their variance and cost models must be specified accordingly.

Assume one interim look after \(n_1\) participants have provided the planned interim information, and that the rule specifies what to do in each interim region. A simple rule might stop for futility below a lower cutoff, continue to a base sample size in a high region, and increase enrollment to a maximum in a promising region. For each true-effect scenario, the probabilities of these regions can be calculated under the assumed interim model. Earlier material on conditional power supplies one way to describe the chance of a final rejection given interim data; here the emphasis is how those chances translate into expected burden and cost.

A useful power calculation partitions the interim outcomes into regions. Let \(q_j\) be the probability of rejecting \(H_0\), averaged over interim data in region \(R_j\), conditional on being in that region, the specified true effect, and the sample size chosen there. If the rule stops without rejection in a region, set \(q_j=0\). Then the unconditional power under that scenario is

This weighted average is important: a large conditional power in a region contributes little to overall power if that region is rarely reached. Conversely, increasing the sample size in a frequently reached promising region can have a noticeable effect on both expected enrollment and power.

Worked Expected-Burden Comparison

From Enrollment to Expected Cost

Expected cost can be calculated by applying the cost model to the same interim branches. A basic model assigns a variable cost \(c\) to each enrolled participant and a fixed cost \(F\) to conducting the trial. If extra enrollment requires a separate expansion cost \(K\), incurred only when expansion occurs, then

This is a planning model, not a universal accounting formula. It assumes that the specified costs apply to the relevant participants and that the expansion cost is incurred whenever the rule expands. A real budget may need to distinguish costs by treatment arm, follow-up duration, site, or calendar time. Participant burden can also include more than enrollment, such as additional visits or procedures; those should be represented explicitly if they differ across branches.

Weighing a Larger Maximum Against Power

A design team may consider raising the maximum sample size while leaving the interim categories unchanged. The effect on expected enrollment is especially direct: if enrollment in the promising region rises from \(N_a\) to \(N_b\), with \(N_b>N_a\), and that region has probability \(p\), then

The power change is not obtained by simply multiplying this enrollment increase by a constant. It depends on the rejection probability at each chosen sample size and on how often the trial enters each region. The comparison should therefore report both the expected burden and the resulting power under clearly stated true-effect scenarios.

How to Make a Fair Comparison

A transparent comparison keeps the assumptions and scenarios visible. At minimum, report the interim-region probabilities, the enrollment selected in each region, expected enrollment, maximum enrollment, the cost components, and power under each relevant true-effect scenario. If a design has different follow-up or procedures after expansion, include those costs and burdens rather than treating all participants as interchangeable.

These calculations complement, rather than replace, other design criteria. A fixed-sample design has a simpler enrollment commitment and predictable maximum, but it does not adapt enrollment to interim results. A promising-zone rule can reduce enrollment when results are weak and increase it when more information may be useful, at the cost of variable enrollment and a larger possible maximum. A rule that always increases to the maximum may be easier to budget conservatively, but may expose more participants than necessary in scenarios where the interim results do not justify expansion.

Expected enrollment is useful for planning average resource use, but it can obscure the experience of participants and sites in the branch with the largest enrollment. For budgeting, the maximum may matter more than the mean if funds must be available before the interim decision. For participant protection, the number assigned to additional follow-up or procedures may matter more than total enrollment alone. Report the measure that answers the decision at hand, and do not substitute one for another.

Diagnostics, Limitations, and Common Pitfalls

The comparison is only as credible as its inputs. Check that interim-region probabilities sum to 1, that every branch’s final enrollment is at least the number already enrolled, and that the expected enrollment lies between the smallest and largest possible totals. Check that costs are applied to the participants and branches that actually incur them. Power summaries must correspond to the same rule, true-effect scenario, and final analysis used for the comparison.

A particularly common mistake is to compare rules by their maximum sample sizes alone: “Rule A can enroll 600 and Rule B only 400, so Rule A costs 50% more.” This ignores the probability of reaching the larger sample size and any early stopping. The correction is to calculate \(E(N)\) from the branch probabilities and then separately report the maximum. The reverse error is also possible: citing a modest expected enrollment while hiding a high maximum can understate the resources needed to guarantee that the trial can finish.

Another limitation is that expected values average over repeated trials under a specified model. They do not guarantee a particular power or budget outcome in one realized trial. Interim probabilities can change substantially across true-effect scenarios, and costs may change over time or with recruitment difficulty. A single scenario is therefore not enough to characterize the design. These calculations also do not establish that the adaptive testing procedure controls its intended error rates; that requires a separate assessment of the test procedure.