The Internal-Pilot Re-Estimation Workflow

Why an Internal-Pilot Workflow Needs Explicit Rules

An initial sample-size calculation depends on planning quantities, including the effect size worth detecting and nuisance parameters such as outcome variability. The effect size is usually a scientific design target; nuisance quantities may be less certain before data collection. A blinded internal-pilot re-estimation uses accumulating trial data to update a nuisance estimate and recalculate the total enrollment target, while keeping treatment labels masked.

The workflow matters as much as the arithmetic. The interim timing, data included, estimator, recalculation rule, enrollment floor, and any maximum enrollment should be specified before the interim data are examined. Otherwise, decisions can become influenced by interim treatment patterns or made selectively after seeing an inconvenient estimate. An internal pilot is part of the main trial: its participants remain in the final analysis, subject to the trial’s prespecified analysis rules.

Definition: An internal pilot is an initial portion of the main trial whose outcome data are used at a planned interim cut to update one or more nuisance parameters and, if specified, the total sample-size target. The participants are retained in the final trial. In a blinded re-estimation, the re-estimation team does not use treatment-specific outcome summaries or treatment labels to make the update.

The earlier material on nuisance parameters in fixed-effect sample-size formulas establishes why an updated variance or other nuisance quantity can change the required sample size. Here the emphasis is the sequence of design and operational decisions. The particular masked estimator is a separate issue; the workflow must identify it in advance without selecting an estimator after seeing the interim result.

Specify the Design Before the Interim Cut

A useful protocol separates the quantities that are fixed from those that may be updated. For a two-arm continuous-outcome illustration, let \(\delta\) be the prespecified difference worth detecting, let \(\alpha\) be the two-sided significance level, and let \(1-\beta\) be the target power. These are design targets, not quantities to be replaced by the observed interim treatment difference. Let \(\sigma\) denote the common within-arm standard deviation under the planning model. The internal-pilot estimate of that nuisance quantity is denoted \(\hat{\sigma}_{\mathrm{int}}\).

For independent normally distributed outcomes, equal allocation, a common variance, complete outcome data, and a normal-approximation fixed-effect calculation, the required number of analyzable participants per arm is

n_{\mathrm{arm}}= \left\lceil \frac{2\sigma^2\left(z_{1-\alpha/2}+z_{1-\beta}\right)^2}{\delta^2} \right\rceil , \qquad N_{\mathrm{an}}=2n_{\mathrm{arm}} .

Here \(z_q\) is the \(q\)-quantile of the standard normal distribution, and \(N_{\mathrm{an}}\) is the total analyzable sample size. This formula is an illustration of the workflow, not a universal formula for every endpoint or analysis. The model and assumptions must match the trial design. If some randomized participants may not provide an analyzable outcome, a prespecified completion fraction \(r\) can be used to convert the analyzable target into an enrollment target:

E_{\mathrm{req}}= \left\lceil \frac{N_{\mathrm{an}}}{r}\right\rceil .

The planned initial enrollment \(E_0\) should be based on the initial nuisance assumption and this same conversion rule. The protocol should also state the interim cut—for example, after a specified number of participants have been randomized and their outcomes are available—and define who may access the masked data. A cut taken too early may produce an imprecise estimate; one taken very late may leave little opportunity to recruit additional participants.

The Re-Estimation and Continuation Rule

At the interim cut, the prespecified masked procedure produces \(\hat{\sigma}_{\mathrm{int}}\). The sample-size calculation is repeated with this updated nuisance estimate while retaining the target effect, significance level, power, allocation ratio, and other fixed design assumptions. The result is a revised total target, not a number of additional participants to recruit.

Set the initial plan.
Specify the initial enrollment target, design targets, nuisance assumptions, and any allowance for unavailable outcomes.

Define the masked interim cut.
State when the cut occurs, which outcome records are included, and how treatment information remains concealed from those making the update.

Estimate nuisance quantities.
Apply the prespecified masked estimator to the interim data. Do not substitute an observed treatment effect for a nuisance estimate.

Recalculate the total target.
Insert the updated nuisance estimate into the prespecified sample-size rule, then convert to enrollment if outcome availability is below 100 percent.

Apply operational constraints.
Respect a prespecified maximum and round for the allocation scheme. Set the final target no lower than the number already randomized.

Continue and document.
Continue recruitment to the resulting total target, retain internal-pilot participants, and record the update without disclosing masked treatment comparisons.

In symbols, if \(E_{\mathrm{int}}\) participants have already been randomized, a basic continuation rule is \(E_{\mathrm{final}}=\max(E_{\mathrm{int}},E_{\mathrm{req}})\), subject to prespecified allocation rounding and any cap. If the trial requires equal allocation, the final target may need to be rounded upward to a feasible even number. A protocol may also impose a floor such as the original target; that is an additional policy choice, not a mathematical requirement of the internal-pilot method.

The floor is important because the revised calculation estimates a total. If the revised target is 120 and 150 participants have already been randomized, the instruction cannot be to “recruit 120 more,” nor can the final enrollment be reduced to 120. The trial has already enrolled 150; the continuation rule must not undo that fact.

Worked Example: Updating a Continuous-Outcome Target

Hypothetical design. A two-arm trial plans equal allocation, a two-sided \(\alpha=0.05\), 80% power, and a target difference of \(\delta=5\) units. Assume independent normally distributed outcomes with a common within-arm standard deviation, and use the normal-approximation formula above. Initially, the planning standard deviation is 10 units. Assume 90% of randomized participants are expected to provide analyzable outcomes.

For these targets, \(z_{0.975}=1.96\) and \(z_{0.80}=0.8416\). The initial analyzable requirement per arm is

n_{\mathrm{arm},0} = \left\lceil \frac{2(10^2)(1.96+0.8416)^2}{5^2} \right\rceil = \left\lceil \frac{200(7.849)}{25} \right\rceil = \lceil 62.79\rceil =63 .

Thus \(N_{\mathrm{an},0}=126\). Allowing for 90% outcome availability gives \(E_0=\lceil126/0.90\rceil=\lceil140\rceil=140\) participants to enroll.

At the planned masked interim cut, suppose the prespecified nuisance procedure returns \(\hat{\sigma}_{\mathrm{int}}=12\) units. The design targets remain unchanged. The updated requirement is

n_{\mathrm{arm},1} = \left\lceil \frac{2(12^2)(1.96+0.8416)^2}{5^2} \right\rceil = \left\lceil \frac{288(7.849)}{25} \right\rceil = \lceil 90.42\rceil =91 .

The revised analyzable total is \(N_{\mathrm{an},1}=182\). Converting to enrollment gives \(\lceil182/0.90\rceil=\lceil202.222\ldots\rceil=203\). For equal allocation, round this upward to 204. If 80 participants have already been randomized at the cut, the continuation target is 204 total randomized participants, so recruitment continues for 124 more. It is not 204 additional participants.

This calculation relies on the stated normal-approximation model and on the illustrative standard deviation being the appropriate within-arm nuisance quantity. It does not establish that a particular masked estimator will estimate that quantity accurately; that estimator must be justified separately.

Worked Example: A Smaller Estimate Does Not Erase Enrollment

Hypothetical design. Use the same two-arm assumptions and targets as above, but suppose the interim nuisance estimate is \(\hat{\sigma}_{\mathrm{int}}=8\) units. The required analyzable number per arm is

n_{\mathrm{arm},1} = \left\lceil \frac{2(8^2)(1.96+0.8416)^2}{5^2} \right\rceil = \left\lceil \frac{128(7.849)}{25} \right\rceil = \lceil40.19\rceil =41 .

This gives \(N_{\mathrm{an},1}=82\). At 90% expected outcome availability, the enrollment target is \(\lceil82/0.90\rceil=\lceil91.111\ldots\rceil=92\), already even. Suppose 100 participants have been randomized by the interim cut. Then \(E_{\mathrm{final}}=\max(100,92)=100\). The trial does not reduce enrollment or remove participants; it continues with no further recruitment under this numerical rule, while all 100 remain in the planned final analysis.

This conclusion is conditional on the protocol allowing the target to fall below the initial enrollment target. If the protocol sets a minimum of 140 enrolled participants, for example, the final target would instead be at least 140. The rule must be specified in advance rather than chosen after the estimate is seen.

Worked Example: Converting an Analyzable Target to Enrollment

Hypothetical operational calculation. Suppose a prespecified calculation requires 175 analyzable participants in total, and the design assumes that 90% of randomized participants will provide the required outcome. The enrollment target is

E_{\mathrm{req}} = \left\lceil\frac{175}{0.90}\right\rceil = \left\lceil194.444\ldots\right\rceil =195 .

The ceiling is used because 194 enrolled participants would yield an expected \(194(0.90)=174.6\) analyzable participants, below 175; 195 yields \(195(0.90)=175.5\). If equal allocation is required, the operational target must be rounded upward to 196. If 200 participants have already been randomized, the continuation rule gives a final target of at least 200, not 196. The expected availability fraction is a planning assumption, not a guarantee that exactly 90% will provide outcomes.

Justification, Diagnostics, and Failure Modes

The enrollment floor follows directly from what a total sample-size target means. Let \(E_{\mathrm{req}}\) be the recalculated total and \(E_{\mathrm{int}}\) the number already randomized. Any feasible final enrollment must be at least \(E_{\mathrm{int}}\), since those participants have entered the trial and cannot be retrospectively un-enrolled. It must also meet the recalculated target if that target is feasible. Therefore the smallest total satisfying both requirements is their maximum:

E_{\mathrm{final}} \ge E_{\mathrm{int}} \quad\text{and}\quad E_{\mathrm{final}} \ge E_{\mathrm{req}} \quad\Longrightarrow\quad E_{\mathrm{final}}\ge\max(E_{\mathrm{int}},E_{\mathrm{req}}).

Operational rounding or a protocol floor can increase this minimum. A cap can make the recalculated target unattainable; in that case, report that the capped design may not meet its planned power under the updated nuisance estimate. Do not describe the cap as if it preserved the target automatically.

Diagnostics should check whether the update was implemented as planned, not just whether the final number looks plausible. Confirm that the cut was made at the specified time, the intended records were included, the estimator was applied consistently, the nuisance result was not derived from unmasked treatment comparisons, and rounding and enrollment accounting were correct. An unexpectedly large or small estimate should prompt a review for data errors, unusual outcome patterns, or model mismatch under prespecified procedures—not an improvised change to the effect target.

Blinding reduces the opportunity to tailor the target to an observed treatment difference, but it does not make every adaptive design automatically valid. In a group-sequential trial, changing the total sample size can affect the distribution of the final test statistic and the relationship to planned interim boundaries. The adaptation must be compatible with the trial’s prespecified inferential method and type I error control. Re-estimating a nuisance parameter alone is not proof that the original error rate is preserved.

Compared with a fixed sample-size plan, internal-pilot re-estimation can respond to uncertainty in nuisance quantities but adds operational complexity and may still be imprecise at an early cut. Compared with an unblinded effect-based update, a masked nuisance-based rule avoids using treatment-specific effect information for the sample-size decision, but masking does not eliminate estimation error or guarantee validity. A separate external pilot can inform planning before the main trial, but it does not update the target using data from participants already enrolled in that trial.

Common Mistakes and Limitations

Adding the recalculated total to the number already enrolled. A target of 204 means 204 total, not 204 additional participants.
Reducing the target below those already randomized. Apply the enrollment floor, and retain those participants in the trial.
Updating the effect target based on the interim treatment difference. In this workflow, the effect size is a prespecified design target; the update concerns nuisance information.
Confusing analyzable sample size with enrollment. Apply the prespecified outcome-availability adjustment and then allocation rounding.
Assuming masking alone preserves type I error. Re-estimation must be evaluated within the planned group-sequential inferential design.

The method also has limits: an interim nuisance estimate may be unstable, missing outcomes may differ from the planning assumption, and the assumed model may not fit the eventual data. Re-estimation cannot repair a poorly chosen effect target, an inappropriate outcome model, or inadequate operational planning. The update is a structured response to nuisance uncertainty, not a guarantee of power or a substitute for design review.

Check Your Understanding

Answer each question, then compare with the brief solutions.

A re-estimated total enrollment target is 160, but 172 participants have already been randomized. What is the minimum final enrollment under the basic continuation rule?
A design needs 120 analyzable participants and assumes an outcome-availability fraction of 0.80. What enrollment target is required before any allocation rounding?
Why should the target effect generally remain fixed in a blinded nuisance-based update?
Does using masked interim data alone prove that the group-sequential type I error rate is preserved? Explain.

Solutions. (1) At least 172, because the final enrollment cannot be below the number already randomized: \(\max(160,172)=172\). (2) \(E_{\mathrm{req}}=\lceil120/0.80\rceil=\lceil150\rceil=150\). (3) The target effect represents the difference the study is designed to detect; replacing it with the observed interim difference would make the update effect-driven and could make the decision depend on interim treatment patterns. (4) No. Masking limits access to treatment-specific information, but changing sample size can affect sequential-test operating characteristics; the adaptation must be compatible with the prespecified method for controlling type I error.