Blinded Sample Size Re-Estimation from Pooled Variability

Why Re-Estimate Sample Size While Blinded?

Sample-size planning for bioequivalence depends strongly on variability. A study planned with an underestimated within-subject standard deviation may have less power than intended; one planned with an overestimated value may enroll more participants than necessary. Blinded sample-size re-estimation addresses this uncertainty by using accumulating variability data to update enrollment while keeping treatment identities concealed and withholding the estimated treatment difference from the decision process.

The method does not replace an initial design calculation. It adds a prespecified interim calculation: use a blinded estimate of the variance relevant to the primary endpoint, recalculate the required number of evaluable participants under the original design assumptions, and apply a stated minimum and maximum. The update is about variability, not whether the treatments appear equivalent.

Which Variance Is Needed?

For a two-period crossover study, suppose the primary endpoint is analyzed on the log scale. Let \(Y_{iT}\) and \(Y_{iR}\) be participant \(i\)'s log endpoint under test and reference, and let \(D_i=Y_{iT}-Y_{iR}\) be the within-participant contrast. The sample-size calculation requires the standard deviation of these contrasts, denoted here by \(\sigma_D\). It is not the standard deviation of all individual log measurements pooled together.

A common model for a two-period, two-sequence crossover has fixed effects for sequence, period, and anonymously coded treatment, a participant effect nested within sequence, and a residual error. If residual errors in the two periods are independent with common variance \(\sigma_e^2\), the participant effect cancels from the within-participant contrast, and \(D_i\) has residual variance \(2\sigma_e^2\). Thus \(\sigma_D=\sqrt{2\sigma_e^2}\). A blinded analyst can estimate the residual variance from the prespecified model while the treatment labels remain masked, and provide the variance estimate without releasing a treatment contrast.

These relationships assume the model is appropriate: the endpoint is analyzed on the specified scale; participants are independent; residual variance is sufficiently stable across sequences and periods; and the residual errors have the structure used in the calculation. If the design or variance structure differs, the estimator must match that design rather than borrowing this conversion automatically.

How the Enrollment Update Works

Let \(n\) denote the number of evaluable participants, \(\alpha\) the one-sided significance level for each equivalence test, and \(1-\beta\) the target power. Let \(L\) and \(U\) be the lower and upper equivalence limits on the log scale, and let \(\theta_0\) be the assumed true mean contrast for planning. A normal-approximation calculation illustrates the dependence of sample size on variance:

Here \(z_q\) is the \(q\) quantile of the standard normal distribution. The expression uses the distance from the assumed true contrast to the nearer equivalence boundary; it is a planning approximation, not a substitute for an exact power calculation where that is required. With the margins, \(\alpha\), target power, and \(\theta_0\) held fixed, it shows why the required sample size scales with \(\sigma_D^2\). In practice, the planned calculation should use the same power method before and after re-estimation.

Derivation of the Variance-Scaling Rule

The scaling rule follows directly from the normal-approximation formula when all quantities except \(\sigma_D\) are fixed. Write \(c=(z_{1-\alpha}+z_{1-\beta})/\min(U-\theta_0,\theta_0-L)\). Then \(n\approx c^2\sigma_D^2\). Applying the same expression to the old and new standard deviations gives

The constant \(c^2\) cancels. This derivation explains both the convenience and the limitation of the shortcut: it is valid for the stated approximation with unchanged design inputs. Small-sample power calculations, integer sequence balance, or a changed target power can prevent this ratio from giving the exact revised sample size. In those settings, recompute power using the original method rather than assuming the scaling identity is exact.

Keeping the Update Blinded

Blinding is a data-handling procedure, not a property guaranteed by the word “pooled.” A defensible process keeps treatment identities masked, uses a prespecified model that accounts for the design’s treatment and period structure, and returns only the variance information needed for the size decision. The people deciding whether to expand enrollment should not receive treatment-specific summaries or the estimated treatment contrast.

The decision rule also needs clear operational details. Specify when the interim data are available, how many participants or complete profiles are required, how the variance estimator is calculated, the target power and other inputs that remain fixed, how rounding and sequence balance are handled, and the minimum and maximum enrollment. A rule such as “increase if the interim result looks unfavorable” is not blinded variance re-estimation: it is an undefined response to accumulating results.

Diagnostics, Alternatives, and Limitations

Before using an interim estimate, check whether it is numerically and scientifically credible. Review the number of complete profiles contributing to it, model convergence, residual variation by sequence and period, influential observations, protocol deviations, and whether the endpoint transformation and analysis population match the plan. A striking estimate based on very few participants may reflect ordinary sampling noise; a variance estimate is itself uncertain. Data quality problems should be investigated rather than treated automatically as evidence that enrollment must increase.

A blinded update is not the only option. Fixed enrollment avoids interim estimation and its operational complexity, but cannot respond to a meaningful mismatch between planned and observed variability. Unblinded re-estimation may use more direct information about treatment-specific effects, but it reveals comparative information to those conducting the update and requires additional safeguards. Blinded re-estimation preserves concealment of treatment identities and the comparative result, but its validity still depends on the estimator, model, and prespecified decision rule; “blinded” does not by itself guarantee correct statistical behavior.

This procedure also has limits. It cannot repair a misspecified endpoint, biased measurements, incorrect equivalence limits, or a design with inadequate power for reasons other than variance. A hard maximum may leave the study below its target power if variability is higher than anticipated. A cap can be necessary for feasibility, but its consequences should be evaluated during planning. Nor should the updated estimate be interpreted as evidence for or against bioequivalence: the interim calculation concerns enrollment, while the final equivalence conclusion follows from the prespecified final analysis.

Common Mistake

• Wrong reasoning: “The pooled data have a larger standard deviation, so the treatments must be different; I will use the observed difference to decide how many participants to add.”
• Correction: A blinded sample-size update uses a prespecified estimate of the variance relevant to the design and does not use the treatment contrast to make the enrollment decision. Pooled raw variability can include mean differences, so use a design-appropriate estimator and keep comparative summaries concealed.
• Wrong reasoning: “The standard deviation increased by 20%, so the sample size also increases by 20%.”
• Correction: Under the variance-scaled approximation, sample size is proportional to the square of the standard deviation. A 20% increase gives a factor of \(1.20^2=1.44\), or approximately a 44% increase before rounding and applying limits.