Documenting A Prespecified Re-Estimation Algorithm

From a Design Intention to an Executable Rule

A plan to “re-estimate the sample size using the interim variance” is not yet an algorithm. It leaves unresolved which participants count as available, when the data are frozen, how the variance is calculated, how the estimate becomes a sample size, and what happens if the estimate cannot be calculated. Different answers can produce different trial sizes from the same trial data.

A prespecified re-estimation algorithm closes those gaps before interim results are examined. It should make the mapping from permitted data to action sufficiently explicit that an independent reviewer can reproduce the decision. The rule concerns the planned sample size; it does not authorize changing the primary endpoint, treatment contrast, or analysis question.

The specification should be written before the interim data are accessed by anyone responsible for the adaptation. It should also identify who can access the data and who makes, verifies, and records the decision. These controls do not replace the statistical rule; they help ensure the rule is followed as written.

State the Model and Calculation

The worked specifications below use a hypothetical two-arm trial with equal allocation and a continuous endpoint. The planned comparison is a two-sided test of a fixed mean difference \(\delta=5\) points at \(\alpha=0.05\), with target power \(90\%\). Assume independent participant outcomes, normally distributed within each arm with a common variance \(\sigma^2\), and a normal-approximation sample-size calculation. These assumptions motivate the calculation; they are not guaranteed by writing them into a protocol.

For \(n\) evaluable participants per arm, the variance of the difference in sample means is \(2\sigma^2/n\). The approximate sample size that achieves the specified power is therefore:

Here, \(n_{\mathrm{eval}}\) is the required number of evaluable participants per arm; \(z_q\) is the \(q\)-quantile of the standard normal distribution; and the ceiling function rounds upward to the next integer. With \(\alpha=0.05\), \(z_{0.975}=1.960\); with 90% power, \(z_{0.90}=1.282\). Thus the squared quantile sum is approximately \(10.51\). If the anticipated proportion with an evaluable endpoint is \(q=0.90\), the corresponding randomized target per arm before applying bounds is \(\lceil n_{\mathrm{eval}}/q\rceil\).

The variance estimate used at interim must also be defined exactly. For \(n_A\) and \(n_B\) evaluable participants in arms A and B, let \(s_A^2\) and \(s_B^2\) be their usual sample variances. Under the common-variance model, the pooled within-arm estimate is:

This calculation uses within-arm variation, not the difference between the arm means. A protocol must still specify the variance estimator, data cutoff, and eligibility rules: “use the pooled variance” alone is not sufficiently detailed.

Worked Example: Specify and Apply the Main Rule

Make Thresholds, Bounds, and Fallbacks Unambiguous

An auditable rule states both the threshold and the action on either side of it. For example, “at least 30 evaluable outcomes in each arm” means that 30 qualifies and 29 does not. It should also state what happens if the variance estimate is non-finite, a required field is missing, or the data query remains unresolved at the cutoff. Do not leave those cases to a discretionary judgment made after seeing the interim data.

For a concrete rule, suppose the protocol uses the planning standard deviation \(\sigma_0=15\) points as the fallback if either arm has fewer than 30 evaluable outcomes, or if the prespecified variance calculation cannot be completed. It then applies the same sample-size formula, attrition adjustment, and bounds. This fallback is a design choice, not a universal recommendation; its adequacy should be assessed for the particular trial.

The order of operations also matters. One precise sequence is: calculate the unrounded evaluable target; round it upward; divide by \(q\) and round upward to obtain the randomized target; apply the minimum and maximum to that randomized target; then subtract participants already randomized per arm to determine additional enrollment. If current enrollment already exceeds the bounded target, the protocol must say whether to stop enrollment at the next feasible point rather than attempt to remove participants.

What the Written Specification Must Let a Reviewer Reproduce

A useful specification is operational, not merely statistical. It should let a reviewer reconstruct the decision from a dated data extract, the locked analysis code or calculation record, and the protocol’s stated constants. At minimum, record:

• Timing: the trigger for review, outcome window, data cutoff, and any rule for a delayed review.
• Inputs: endpoint definition, evaluability rules, analysis population, arm counts, and the source and units of each data field.
• Calculation: the variance estimator, target contrast, type I error level, power, quantiles or method, attrition assumption, and rounding order.
• Decision rules: the exact threshold for using interim data, fallback action, lower and upper bounds, and treatment of a target below current enrollment.
• Governance and record: who receives which data, who verifies the calculation, the final decision, date, code or worksheet version, and any deviation and its resolution.

These details also support diagnostics. Check that arm counts match the cutoff, variances are in squared endpoint units, the denominator of the pooled variance is correct, and the computed target follows the documented rounding and bounding order. Compare the entered values with the source extract and have a second qualified person independently reproduce the decision. Investigate unexpected values or inconsistencies using prespecified procedures; do not silently remove observations, change the estimator, or choose a more convenient fallback.

A fixed sample-size design avoids interim re-estimation and its operational data-access requirements, but cannot respond to an unanticipated nuisance variance in the same way. A re-estimation design can use a prespecified nuisance-parameter update, but requires a clear rule and evaluation of its statistical behavior. Blinded and unblinded inputs are not interchangeable by default: the chosen data access and estimator must be specified and justified. A written algorithm by itself does not establish that the adaptive design controls type I error or attains its intended power; those properties require design-specific assessment, including simulation when appropriate.

Common Documentation Failures

• “Re-estimate as needed.” This permits discretion after the data are seen. Replace it with a trigger, calculation, thresholds, and actions for every specified case.
• Unclear data availability. “Use interim data” does not define how late, missing, or unverified outcomes are handled. State the cutoff, evaluability rules, minimum counts, and fallback.
• Bounds without an order. Rounding, attrition adjustment, and truncation can yield different targets if applied in different orders. Write the sequence explicitly.
• Confusing a cap with a successful target. A maximum can constrain the calculated sample size. Report the capped decision as such rather than implying the uncapped target was reached.
• Changing the question while updating the size. A variance-based update does not authorize changing the endpoint or target contrast. Keep the primary question fixed and document any permitted adaptation separately.