Prespecifying A Re-Estimation Algorithm Without Unblinding

Why the Rule Must Be Operational

A blinded sample-size re-estimation is not fully specified by saying, “review the variance at interim and adjust enrollment if needed.” That description leaves open which data are reviewed, who sees them, how the nuisance quantity is estimated, when the calculation occurs, and what happens if the result falls outside the permitted enrollment range. Each choice can affect the final sample size and the integrity of the treatment mask.

The goal is to define a rule that can be executed without revealing treatment-specific outcomes or the treatment contrast. The rule should update planning for nuisance quantities, such as a pooled event rate or an outcome variance, while keeping the target effect, significance level, and power fixed. A blinded calculation does not itself guarantee appropriate error rates: the full procedure, including estimation, timing, limits, and rounding, must be evaluated under its intended design assumptions.

Definition: A prespecified blinded re-estimation algorithm is a written, reproducible rule that identifies the masked data available at a planned review, specifies the estimator and sample-size formula to apply, and sets the resulting enrollment action and its limits. The algorithm must not use a treatment-specific estimate or disclose the treatment contrast to those conducting the review.

What the Protocol Must Fix in Advance

The introductory material on blinded re-estimation distinguishes nuisance-parameter updating from using an interim treatment contrast to guide sample size. Here the emphasis is on making that distinction operational. A protocol or statistical analysis plan should answer each of the following questions before trial data are reviewed.

Access and masking: State which variables are available, whether treatment codes remain inaccessible, and which roles may receive the pooled summaries. Specify how reports, analysis files, and communications are protected from revealing arm-specific results.
Analysis population and data status: Define who contributes to the estimator and how incomplete, duplicate, out-of-window, or otherwise unusable records are handled. State whether the estimator uses observed outcomes only or a prespecified missing-data method.
Estimator: Give its formula, including any adjustment, boundary rule, or site handling. If a target treatment effect is used to translate a pooled nuisance estimate into planning values, identify that target and keep it distinct from an interim effect estimate.
Timing: Set the review trigger and the data cut-off, such as a fixed number of participants with the required endpoint assessment. Specify what happens if data arrive late or the target is not met on schedule.
Recalculation and action: Name the sample-size formula and its inputs. Define how the result is rounded, whether the cap applies to total enrollment or enrollment per group, and how attrition is allowed for.
Documentation: Record the data cut-off, estimator inputs, calculation, checks, decision, and authorized recipients. Keep the rationale and outputs sufficient for independent reproduction without circulating treatment-specific information.

These are not administrative details added after the statistical work. They define the procedure whose performance should be assessed. For example, an estimator calculated among participants with observed outcomes can behave differently from one calculated using every randomized participant, particularly when outcome availability varies.

A Binary-Outcome Recalculation

Consider a two-group trial with independent binary outcomes, equal allocation, and a prespecified target absolute risk difference $\delta^*$. Let $\bar p$ denote the pooled event proportion at the masked review. Under the planning convention $p_1=\bar p+\delta^*/2$ and $p_0=\bar p-\delta^*/2$, the equal-group normal-approximation sample size for a two-sided test is

n_{\text{raw}} = \frac{(z_{1-\alpha/2}+z_{1-\beta})^2 \left[2\bar p(1-\bar p)-(\delta^*)^2/2\right]} {(\delta^*)^2},

where $n_{\text{raw}}$ is the required number per group before integer rounding, $\alpha$ is the two-sided significance level, and $1-\beta$ is the target power. This formula assumes the stated binary-outcome model and the planning rates implied by $\bar p$ and $\delta^*$; it is not a universal formula for other outcome types or designs.

Justification: For independent Bernoulli outcomes, the variance contribution to the variance of a difference in group proportions is proportional to $p_1(1-p_1)+p_0(1-p_0)$. Substituting $p_1=\bar p+\delta^*/2$ and $p_0=\bar p-\delta^*/2$ gives $$ p_1(1-p_1)+p_0(1-p_0) =2\bar p(1-\bar p)-(\delta^*)^2/2. $$ The usual normal-approximation planning equation divides the squared sum of the critical-value and power quantiles, multiplied by this variance term, by the squared target difference. That yields the displayed sample-size formula.

Worked Example: Prespecifying a Pooled-Rate Update

Suppose a hypothetical equal-allocation trial plans for a two-sided $\alpha=0.05$ test, 90% power, and a target risk difference of $\delta^*=0.10$. The rule sets one masked review after 480 participants have the endpoint recorded. The analyst receives only the pooled number of events and the eligible denominator; treatment codes and group-specific event counts are unavailable. At the review, 184 of the 480 participants have an event.

The masked estimator is the pooled event proportion:

\bar p=\frac{184}{480}=0.3833.

Using $z_{0.975}=1.960$ and $z_{0.90}=1.282$, the squared quantile sum is approximately $(1.960+1.282)^2=10.511$. The variance term, calculated from the unrounded proportion, is

2\left(\frac{184}{480}\right)\left(1-\frac{184}{480}\right)-\frac{0.10^2}{2} =0.4678\ \text{(approximately)}.

Therefore,

n_{\text{raw}} = \frac{10.511(0.4678)}{0.10^2} \approx 491.7\ \text{participants per group}.

If the prespecified rule rounds upward to a whole participant per group, the target is 492 per group, or 984 total. This is an enrollment target, not an instruction to inspect or report the observed risk difference. Before adopting it, the protocol must still apply any prespecified floor, cap, and attrition adjustment. The calculation also relies on independent binary outcomes, equal allocation, and feasible implied planning rates: here $p_0=0.3833-0.05=0.3333$ and $p_1=0.3833+0.05=0.4333$, both within zero and one.

Bounds, Timing, and Attrition Are Part of the Rule

The raw recalculation is not yet an operational decision. A complete rule determines the target after upward rounding, then applies the stated enrollment limits in a fixed order. It should say whether a floor or cap refers to participants per group or to the total number randomized. If a cap forces enrollment below the recalculated target, the resulting power may be below the nominal target; a cap must not be described as preserving the original power automatically.

Timing must also be unambiguous. A review triggered by “about halfway through” is open to interpretation. A trigger such as “when 480 participants have the primary endpoint recorded” identifies an analysis population and a data condition. A separate rule should handle delayed outcomes or an unattained trigger. The review should not be moved earlier or later because interim results seem favorable or unfavorable.

If the sample-size formula targets participants with observed endpoints but the trial enrolls participants before their endpoints are available, an attrition adjustment may be needed. Under a simple prespecified assumption that a fraction $r$ of randomized participants will lack the required endpoint, a target of $n$ evaluable participants per group translates to

n_{\text{randomized per group}} = \left\lceil \frac{n}{1-r}\right\rceil.

This adjustment assumes the planning attrition fraction is appropriate for the relevant participants and does not itself account for more complex missing-data mechanisms. The chosen value of $r$ and any alternative missing-data procedure must be stated in advance.

Worked Example: Applying a Cap to a Variance Update

Suppose a hypothetical continuous-outcome trial uses equal group sizes, a two-sided $\alpha=0.05$, 90% power, and a target difference of 5 outcome units. The masked pooled variance estimate is $s^2=64$ units squared. Under the independent, equal-variance normal planning model, the raw sample size per group is

n_{\text{raw}} = \frac{2(1.960+1.282)^2s^2}{5^2} = \frac{2(10.511)(64)}{25} \approx 53.82.

Rounding upward gives 54 per group. Suppose the protocol specifies a maximum of 52 per group. The final rule therefore sets the target at 52 per group, 104 total. Because the cap is below the recalculated requirement, the design no longer achieves the nominal 90% target under the planning model. The report should say that plainly rather than treating the cap as a harmless rounding choice.

The calculation is appropriate only if the outcome variance estimator is based on the prespecified masked data and the normal-model assumptions used for planning are reasonable. A variance estimate calculated separately by treatment group would not satisfy the stated masking rule.

Worked Example: Combining a Masked Rate Update With Attrition

In another hypothetical binary-outcome trial, a review is triggered after 400 participants are randomized. The prespecified analysis includes participants with an observed endpoint by the data cut-off. Of 400 randomized participants, 360 have an endpoint recorded, with 126 events in the pooled masked data. The planning target is $\delta^*=0.12$, with two-sided $\alpha=0.05$ and 90% power. Assume the protocol specifies a 10% attrition allowance and a maximum of 740 randomized participants total.

First calculate the pooled rate among the defined analysis population:

\bar p=\frac{126}{360}=0.3500.

The implied planning rates are $p_0=0.3500-0.0600=0.2900$ and $p_1=0.3500+0.0600=0.4100$, so both are feasible. The variance term is

2(0.3500)(0.6500)-\frac{0.12^2}{2} =0.4550-0.0072 =0.4478.

Thus the evaluable target per group is

n_{\text{raw}} = \frac{(1.960+1.282)^2(0.4478)}{0.12^2} = \frac{10.511(0.4478)}{0.0144} \approx 326.9,

which rounds upward to 327 evaluable participants per group. With a 10% attrition allowance, the randomized target is $\lceil327/0.90\rceil=364$ per group, or 728 total. This is below the 740-participant cap, so the cap does not change the target. The pooled calculation uses 360 observed endpoints, not all 400 randomized participants; the distinction follows the stated analysis-population rule. The attrition adjustment relies on the protocol’s 10% planning assumption and should not be mistaken for a guarantee that exactly 10% will be missing.

Masking Checks and Failure Modes

A useful algorithm includes checks that can be performed without exposing the treatment contrast. Before accepting a recalculation, confirm that the data cut-off and denominator match the protocol, the estimator used only authorized masked fields, the formula inputs are recorded, and the result is within the estimator’s domain. For the binary example, the implied planning rates must lie between zero and one. For a variance-based rule, a zero, implausibly small, or non-finite estimate should trigger the prespecified fallback rather than an improvised calculation.

Operational signs that the process is going wrong include a report containing separate group event counts, a data file with decodable treatment assignments, a changed endpoint denominator, an unplanned review date, or a sample-size calculation that cannot be reproduced from the recorded inputs. These are not solved by merely calling the review “blinded.” The access pathway and the calculation need to be checked.

Using a treatment-specific estimate: Group-specific event rates, variances, or treatment differences can disclose information the rule is meant to withhold. Use only the specified masked summary.
Changing the estimator after seeing the result: Switching denominators, exclusions, or adjustment methods in response to the estimate makes the procedure data-dependent beyond the written rule. Use the prespecified method or invoke a prespecified fallback.
Confusing a target effect with an observed effect: The target difference in the planning formula is fixed by design. Replacing it with an interim estimate changes the procedure and may disclose treatment information.
Reporting a capped target as if it met the power goal: A binding upper limit may prevent the sample size required by the recalculation. Report that consequence accurately.

How This Rule Differs From Other Approaches

A fixed sample-size design does not recalculate enrollment from interim nuisance estimates. A blinded re-estimation rule updates a prespecified nuisance input while withholding the treatment contrast. An approach that uses unblinded treatment information is a different design and requires its own explicit rules and operating-characteristic assessment. The distinction is about the information used and the decision rule, not simply whether an interim meeting occurs.

Approach	Interim information used	Operational implication
Fixed sample size	No interim estimate for recalculation	Enrollment remains at the original target, subject to other protocol provisions.
Blinded nuisance re-estimation	Prespecified masked nuisance summaries	Update the sample size using a locked estimator and formula, without using the treatment contrast.
Unblinded adaptation	Treatment-specific interim information	Requires a distinct, explicitly specified procedure; it is not the blinded rule described here.

Blinded re-estimation can still have limitations. The pooled nuisance estimate may be imprecise, sensitive to the chosen analysis population, or affected by departures from the planning model. A single interim estimate cannot remove uncertainty about future data. The method also does not replace evaluation of the full design’s type I error, power, and expected enrollment under plausible scenarios.

Turning the Plan Into a Reproducible Procedure

A concise algorithm is most useful when each step has an observable input and a defined action. The following outline illustrates the order of operations; the study-specific choices must be fixed before the review.

Lock the review.
At the prespecified trigger and data cut-off, identify the authorized masked records and analysis population.

Calculate the nuisance estimate.
Apply the stated estimator and any predefined data-quality or feasibility checks; do not calculate treatment-specific summaries.

Recalculate enrollment.
Use the prespecified design formula, target effect, error rate, power, and nuisance estimate. Apply rounding and attrition rules in the stated order.

Apply limits and record the decision.
Apply the floor and cap, document whether either binds, and communicate only the authorized enrollment decision.

The record should allow a reviewer to reconstruct the result from the masked inputs and identify any deviation. It should also distinguish a scheduled calculation from the final enrollment decision and state who may see each output. Evaluating the rule before trial conduct should include its actual trigger, estimator, rounding, and bounds, not an idealized version that omits operational details.

Key takeaway: Prespecification means fixing the information, estimator, timing, formula, and enrollment action—not merely naming a blinded interim review. A reproducible rule protects the mask, makes deviations detectable, and permits the complete procedure to be evaluated before use.

Check Your Understanding

Answer each question using the stated rule, then compare your reasoning with the solutions.

A masked review reports pooled events and the pooled denominator, but the analyst also has access to treatment-coded event counts. Is the process adequately masked? Explain.
In a two-sided binary-outcome update, $\bar p=0.40$ and $\delta^*=0.10$. Calculate the variance term $2\bar p(1-\bar p)-(\delta^*)^2/2$, and check whether the implied planning rates are feasible.
A continuous-outcome update gives 71 participants per group after upward rounding, but the protocol caps enrollment at 68 per group. What target is used, and what must be stated about power?
A trial needs 250 evaluable participants per group and prespecifies 20% attrition. Calculate the randomized target per group using the simple attrition adjustment.

Solutions. (1) No. Access to treatment-coded counts can reveal group-specific outcomes; the access procedure must prevent or control that information, not merely report a pooled summary. (2) The variance term is $2(0.40)(0.60)-0.10^2/2=0.48-0.005=0.475$. The implied rates are $0.40-0.05=0.35$ and $0.40+0.05=0.45$, both feasible. (3) The final target is 68 per group. Since the cap is below the recalculated requirement, the nominal power target may not be attained and this limitation must be reported. (4) The randomized target is $\lceil250/(1-0.20)\rceil=\lceil312.5\rceil=313$ per group.