Sample-Size Updates For Risk-Difference Trials

Why Update a Risk-Difference Design?

For a binary outcome, the sample size needed to detect a specified risk difference depends on the event rates in both arms. Before enrollment, those rates are often uncertain. A blinded internal-pilot update can use the overall event rate observed so far to revise the enrollment target without using treatment labels. The absolute effect used for planning remains prespecified: the pooled rate informs the nuisance-rate estimate, while the planned effect determines how that rate is apportioned between arms.

This tutorial considers a two-arm, individually randomized trial with equal allocation and a binary endpoint. The outcome is recorded for every participant, and participants' outcomes are independent. The calculation is a normal approximation for a two-sided comparison of risks. It is a planning method, not a substitute for checking the design's operating characteristics.

The update assumes that the planned risk difference is still a suitable design effect and that the interim pooled rate is informative about the event-rate level in the trial. For a given pooled rate \(q\) and planned difference \(\delta\), the implied arm risks are

Both implied risks must lie between zero and one. If they do not, the pooled rate and planned difference are incompatible; the calculation cannot be used as written. The blinded pooled rate is not itself an estimate of either arm's risk.

Sample-Size Formula and Its Justification

Suppose the trial tests \(H_0:p_C=p_T\) against a two-sided alternative at significance level \(\alpha\), and is planned to have power \(1-\beta\) for the risk difference \(\delta\). With \(n\) participants per arm, the usual large-sample test has a null standard error based on the common risk. At the planning value \(q\), the null variance of the difference in sample proportions is approximately \(2q(1-q)/n\). Under the alternative, its variance is approximately \(\{p_C(1-p_C)+p_T(1-p_T)\}/n\).

A normal-approximation design chooses \(n\) so that the planned difference is large enough to exceed the two-sided critical value under the null and the desired normal quantile under the alternative. This gives the commonly used formula below. Here \(z_a\) is the \(a\) quantile of the standard normal distribution.

For the blinded update, substitute \(\hat q\) for \(q\), then reconstruct the arm risks using the prespecified \(\delta\). The first variance term is the null variance; the second is the alternative variance. Keeping these roles distinct matters: using the same variance in both terms is a different approximation.

The variance relationship also explains why simply treating the pooled rate as though both arms had that rate under the alternative is not quite right. Substituting \(q+\delta/2\) and \(q-\delta/2\) gives

Thus the pooled-rate null variance coefficient \(2q(1-q)\) exceeds the alternative variance coefficient by \(\delta^2/2\). The formula uses the former for the null critical threshold and the latter for power under the planned alternative.

Because \(n\) is a number per arm, round it up to a whole participant per arm. With equal allocation, the total target is twice that rounded number. If participants already enrolled in the internal pilot are included in the final analysis and meet the trial's outcome requirements, the update revises the total target rather than adding the full target again.

Recalculating Enrollment at an Interim Point

At an interim point, the blinded event rate is calculated across both arms. For example, with equal allocation and complete binary outcomes, 52 events among 160 participants gives \(\hat q=52/160\). The same prespecified effect, \(\alpha\), and target power are then used to obtain a revised total sample size. The observed pooled rate changes the estimated nuisance event rate; it does not reveal which arm had the higher risk.

An internal-pilot update should be specified before the interim data are examined. That plan should state the event-rate calculation, the assumed effect, the sample-size formula, and how the interim participants count toward the final target. The update is a blinded nuisance-parameter calculation, not an interim comparison of treatment groups.

Diagnostics, Alternatives, and Limitations

A blinded update is most useful when uncertainty in the event-rate level is an important source of uncertainty in the original sample-size calculation. It cannot establish that the planned effect is correct. Nor does a single pooled rate show whether treatment groups differ in risk. The following checks help identify when the calculation needs careful review:

• Check the implied risks. Confirm that \(\hat q+\delta/2\) and \(\hat q-\delta/2\) are both between zero and one. Invalid values indicate incompatible inputs, not a reason to silently truncate a risk.
• Check the interim information. A pooled rate based on few participants or few events can be imprecise. Consider whether the interim sample is large enough to provide a useful nuisance-rate estimate, and examine how reasonable changes in \(\hat q\) would affect the target.
• Check the design assumptions. Unequal allocation, clustering, repeated outcomes, missing outcomes, or dependence between participants can invalidate this equal-allocation, independent-outcome formula unless separately accounted for.
• Check whether the approximation is adequate. When event rates are near zero or one, or expected event counts are small, a normal approximation may be poor. A more suitable design calculation may be needed; do not treat this formula as reliable merely because it returns a number.

The main alternative is to retain the original fixed sample size, which avoids changing enrollment in response to an interim nuisance estimate but may leave the study under- or over-sized if the event rate differs from planning assumptions. A blinded pooled-rate update can adapt the target to the observed overall rate while keeping the effect assumption fixed. A calculation that uses the pooled rate as the alternative risk in both arms is simpler, but it fails to represent the prespecified difference in the alternative variance. An unblinded recalculation uses arm-specific information and raises different design and operating-characteristic questions; it is not the procedure derived here.

Finally, plugging in an estimated rate does not automatically establish exact power or preserve every error-rate property of an adaptive trial. The formula is an approximate conditional planning calculation. Its suitability depends on the update rule, timing, endpoint availability, and analysis plan. The update should therefore be prespecified and evaluated as part of the full design, rather than improvised after seeing interim results.