Binary Outcome Re-Estimation With Unequal Allocation

Why allocation and pooled rates matter

For a binary endpoint, the variance of a difference in event rates depends on both event probabilities and the number assigned to each arm. When allocation is unequal, applying an equal-allocation formula can produce the wrong target sample size. A blinded interim review adds another issue: the pooled event rate does not reveal either arm’s rate separately. A defensible update must state how that pooled rate is used, which effect is assumed, and how attrition is handled.

This tutorial considers a two-arm superiority design with a binary outcome observed for each participant. Let \(p_C\) be the event probability in the control arm and \(p_T\) the event probability in the treatment arm. Define the target risk difference as \(\Delta=p_T-p_C\), with its sign oriented so that the anticipated treatment effect is positive. Let \(r=n_T/n_C\) be the planned treatment-to-control allocation ratio. The sample-size target is first calculated for participants with an observed endpoint; anticipated attrition is applied afterward.

Variance and sample-size calculation

For independent participants with observed binary outcomes, the estimated risk difference is \(\hat p_T-\hat p_C\). Under the alternative rates used for planning, its variance is approximately \(p_T(1-p_T)/n_T+p_C(1-p_C)/n_C\). This uses the binomial variance \(p(1-p)\) for each arm and independence between arms. With \(n_T=rn_C\), the variance becomes \(A(r)/n_C\), where \(A(r)=p_C(1-p_C)+p_T(1-p_T)/r\).

The pooled interim rate supplies one nuisance-rate estimate, but a target difference is also needed to assign rates to the arms. Suppose the interim allocation ratio is \(r_0=m_T/m_C\), and the design uses the prespecified difference \(\Delta\) to reconstruct planning rates. The pooled rate estimates the weighted average \(q=(p_C+r_0p_T)/(1+r_0)\). Combining this with \(p_T-p_C=\Delta\) yields the rates below. The interim ratio is used for the reconstruction; the planned final ratio is used in the final variance.

The reconstructed rates must lie between zero and one. If they do not, the pooled rate and assumed difference are incompatible with this reconstruction. The design should have a prespecified response, such as examining plausible alternative assumptions or applying a justified constrained method; silently replacing an invalid rate by zero or one changes the method.

A short justification makes the role of unequal allocation explicit. Under the alternative, the estimated difference has mean \(\Delta\) and approximate standard deviation \(\sqrt{A(r)/n_C}\), so its standardized mean is \(\Delta\sqrt{n_C/A(r)}\). For a two-sided normal test, the critical value is \(z_{1-\alpha/2}\). The common planning approximation sets the standardized mean to the critical value plus \(z_{1-\beta}\), giving the stated expression after squaring. It is a planning approximation, not an exact identity for power: exact normal-approximation power also includes the probability in the opposite tail.

From evaluable outcomes to randomized enrollment

The preceding calculation targets participants whose endpoint is observed. If a proportion \(a\) is expected to have no usable endpoint, the approximate randomized target is the evaluable target divided by \(1-a\). With unequal or arm-specific attrition, each arm must have enough randomized participants to meet its evaluable target, while also respecting the planned allocation. Attrition assumptions concern endpoint availability; they do not alter the event-rate variance formula for participants whose outcomes are observed.

When a sample-size update is made after enrollment begins, the calculated target is ordinarily the desired final total, not an additional number to recruit on top of everyone already randomized. Enrollment planning must therefore account for participants already randomized and any participants whose outcomes are already observed. The protocol should specify the rounding rule, the minimum and maximum sample sizes, and how accrued participants count toward the final target.

Blinded updating, assumptions, and diagnostics

This procedure uses the pooled interim rate as a nuisance-rate input while keeping treatment assignments concealed. It does not estimate the treatment effect from the interim data: the assumed \(\Delta\) remains the design target. The reconstruction also assumes the pooled rate corresponds to the interim allocation weighting and that the target difference is a useful planning assumption for separating the two rates. If the interim allocation differs from the planned final allocation, use the actual interim weighting to reconstruct rates and the planned final ratio in \(A(r)\).

The calculation relies on independent participant outcomes, a well-defined binary endpoint, accurate event and denominator counts, and a normal approximation adequate for the planned test. It also presumes that attrition can be represented by the anticipated arm-specific fractions and that the target difference, significance level, allocation ratio, and power were specified appropriately. If event probabilities are near zero or one, or the arm sizes and expected event counts are small, the normal approximation can be poor; a method based on exact binomial power or simulation may be preferable.

Several checks help identify a failing update. Verify that pooled counts use participants with observed outcomes, not all randomized participants if some have missing endpoints. Confirm the actual interim allocation ratio and the planned final ratio separately. Check that both reconstructed rates are in \([0,1]\), that the variance coefficient is positive, and that the rounded enrollment meets both arm targets after attrition. Finally, compare the updated target with the prespecified floor and cap. An unexpectedly large change can indicate a changed event rate, an incompatible target difference, or a data-handling error; it should not be accepted without investigation.

Alternatives and common pitfalls

A simple blinded approximation sets both planning rates equal to the pooled rate. That can be convenient, but it ignores the assumed difference when calculating the alternative variance. The reconstruction used here retains the target difference while using the pooled rate to estimate the weighted average. Neither variance approximation is guaranteed to be conservative: their relative size depends on the pooled rate, target difference, and allocation ratio.

An unblinded approach could use separate interim arm rates to estimate the alternative variance directly. It uses more information about the observed treatment difference, but that difference may be highly variable at an interim analysis and can lead to unstable or biased sample-size changes. A fixed nuisance-rate update avoids basing the target directly on the interim effect estimate, but still depends on its pooled-rate and target-effect assumptions. Exact power calculations or simulation are alternatives when normal approximations are questionable. Any adaptive update should be prespecified, and the final inferential procedure must support the intended type I error control; a blinded update alone does not guarantee it.

Common mistake: “There were fewer events than expected, so I can plug the pooled rate into the equal-allocation formula and multiply the total by an attrition factor.” This reasoning can be wrong twice: unequal allocation changes the variance, and a pooled rate is not generally the rate in each arm. Instead, reconstruct planning rates using the pooled allocation-weighted rate and the prespecified target difference, calculate the variance at the planned ratio, then inflate arm-specific evaluable targets for anticipated attrition.