Blinded Re-Estimation For Binary Outcomes

Why Re-Estimate a Binary-Outcome Sample Size?

For a binary endpoint, the variance depends on the event probabilities in the treatment groups. Those probabilities may be uncertain at the design stage, and an incorrect planning value can lead to a trial that is larger or smaller than intended. A blinded sample-size re-estimation uses the overall event rate observed so far to update a variance input without using treatment labels. The target treatment difference and the decision rule for updating the sample size remain prespecified.

This approach does not reveal the separate treatment-group event rates: the pooled rate alone cannot identify them. It can, however, inform the overall frequency of events and hence a variance estimate. The key qualification is that pooled variability also reflects any separation between the groups. The re-estimation rule therefore needs to be understood as a particular variance approximation, not as a way to recover both group-specific risks from blinded data.

Model and Sample-Size Formula

Definition: Let \(Y_{ij}\) indicate whether participant \(j\) in group \(i\) experiences the event, with \(Y_{ij}=1\) for an event and \(Y_{ij}=0\) otherwise. Assume independent outcomes, with \(Y_{ij}\sim\operatorname{Bernoulli}(p_i)\) for group \(i\), where \(i=0,1\). The treatment effect is the risk difference \(\delta=p_1-p_0\). In an equal-allocation design, \(n\) denotes the number randomized to each group, so the total sample size is \(2n\).

The basic calculation below uses a two-sided test at significance level \(\alpha\), target power \(1-\beta\) for a prespecified risk difference \(\delta^*\ne 0\), equal allocation, and a normal approximation. It assumes independent participants, complete binary outcomes, and event probabilities that are adequately represented by the planning values. It uses the variance under the specified alternative; other sample-size conventions, such as a score-test calculation using a null variance, must be used consistently rather than mixed into this formula.

With treatment-group event probabilities \(p_0\) and \(p_1\), the estimated difference in sample proportions has variance approximately \(\{p_0(1-p_0)+p_1(1-p_1)\}/n\). Equating the target difference to the sum of the normal critical values for the test and power gives the familiar approximation:

n \approx \frac{(z_{1-\alpha/2}+z_{1-\beta})^2 \left[p_0(1-p_0)+p_1(1-p_1)\right]}{(\delta^*)^2}.

Here \(z_u\) is the \(u\)-quantile of the standard normal distribution. The formula returns a per-group sample size; round up to a whole participant. In practice, a protocol may also specify a minimum and maximum sample size and account separately for loss to follow-up. Those rules should be set in advance.

What the Blinded Pooled Rate Estimates

Suppose an interim group of \(m\) participants per arm has completed the binary endpoint. With treatment labels masked, let \(K\) be the total number of events among the \(2m\) participants. The pooled event-rate estimate is \(\hat q=K/(2m)\). Under equal allocation, its population counterpart is \(q=(p_0+p_1)/2\). A common blinded approximation treats \(q(1-q)\) as the Bernoulli variance for each arm, replacing the sum of group-specific variances in the sample-size formula by \(2\hat q(1-\hat q)\). Thus the updated per-group target is

n_{\text{blind}} \approx \frac{(z_{1-\alpha/2}+z_{1-\beta})^2\,2\hat q(1-\hat q)} {(\delta^*)^2}.

The target effect \(\delta^*\) is the effect the design is intended to detect; it is not estimated from the blinded pooled data. The calculation estimates a nuisance variance input, not the treatment effect.

Why Treatment Separation Changes the Estimate

The pooled rate is a mixture of two group-specific Bernoulli distributions. Its variance includes both within-group outcome variability and variation between the two group means. The following identity shows exactly how these components relate:

\begin{aligned} q(1-q) &= \frac{p_0(1-p_0)+p_1(1-p_1)}{2} + \frac{(p_1-p_0)^2}{4},\\ 2q(1-q) &=p_0(1-p_0)+p_1(1-p_1)+\frac{\delta^2}{2}. \end{aligned}

To verify the first line, substitute \(q=(p_0+p_1)/2\) into \(q(1-q)\), expand, and collect terms. The difference between the pooled plug-in variance and the sum of the two within-group variances is therefore \(\delta^2/2\), which is nonnegative and is zero only when \(p_0=p_1\). If the actual treatment separation is appreciable, using \(2q(1-q)\) in place of the within-group variance sum tends to increase the calculated sample size, all else equal. This is a property of the variance approximation; it does not mean that the pooled rate reveals the size or direction of the treatment effect.

In a finite interim sample, \(\hat q(1-\hat q)\) is a plug-in estimate. Its sampling variability can be material when \(m\) is small or the event is rare. The population identity explains the effect of separation, but it does not remove uncertainty in \(\hat q\).

Worked Example: Planning With Group-Specific Event Rates

Consider a hypothetical equal-allocation trial with planning event rates \(p_0=0.20\) and \(p_1=0.30\), a target risk difference of \(0.10\), two-sided \(\alpha=0.05\), and power \(0.80\). Assume independent outcomes, complete ascertainment, and use the normal approximation under the specified alternative.

The relevant quantiles are \(z_{0.975}=1.960\) and \(z_{0.80}=0.8416\). Therefore, \((1.960+0.8416)^2=7.849\), and the sum of the group variances is \(0.20(0.80)+0.30(0.70)=0.1600+0.2100=0.3700\).

Substituting into the formula gives \(n=7.849(0.3700)/(0.10)^2=290.4\), so the design requires 291 participants per group, or 582 in total, before any allowance for incomplete outcomes.

The planning pooled rate is \(q=(0.20+0.30)/2=0.25\). The blinded variance approximation would use \(2q(1-q)=2(0.25)(0.75)=0.3750\), slightly above the group-specific variance sum of \(0.3700\). This difference is \(\delta^2/2=(0.10)^2/2=0.0050\), as the identity predicts. With this approximation, \(n=7.849(0.3750)/(0.10)^2=294.3\), or 295 per group. The assumptions for the normal formula are design assumptions here; in a real protocol, their adequacy would need to be assessed for the planned event rates and sample size.

Applying the Update at an Interim

A practical rule uses the pooled interim event rate to calculate a new target for the total number per group. It then compares that target with the number already enrolled and applies any prespecified floor or cap. The rule should specify the endpoint status required for inclusion, the allocation ratio, the target effect and power, and how rounding and incomplete outcomes are handled. The example below assumes all interim participants have complete outcomes and that the design uses equal allocation.

Compute the blinded rate.
Pool the endpoint outcomes across both groups and calculate \(\hat q=K/(2m)\), without using treatment assignments.

Calculate the updated target.
Substitute \(\hat q\) into the blinded formula, using the prespecified \(\alpha\), power, and target difference \(\delta^*\), then round up to obtain \(n_{\text{blind}}\) per group.

Apply the enrollment rule.
Compare the target with the number already enrolled per group. Apply the protocol’s prespecified floor and cap, and do not reduce enrollment below the number already randomized.

Translate the target into additional enrollment.
If the final target is \(n_{\text{final}}\) per group and \(m\) participants per group are already enrolled, the planned additional enrollment is \(n_{\text{final}}-m\) per group.

Worked Example: Updating From a Pooled Interim Rate

In a hypothetical trial, 100 participants per group have complete outcomes at the interim. Across the 200 participants, 64 events are observed. The design targets a risk difference of \(0.10\), power \(0.80\), and two-sided \(\alpha=0.05\), using the same normal approximation as above. Assume the protocol has no binding cap below the calculated target and the event data meet the planned endpoint definition.

The blinded pooled rate is \(\hat q=64/200=0.3200\). Its variance estimate is \(\hat q(1-\hat q)=0.3200(0.6800)=0.2176\), so the variance term for the difference is \(2(0.2176)=0.4352\).

Using the same critical-value sum, the updated target is \(n_{\text{blind}}=7.849(0.4352)/(0.10)^2=341.6\), which rounds up to 342 participants per group. Since 100 per group are already enrolled, the planned additional enrollment is \(342-100=242\) per group. The final target is 684 participants in total.

The pooled rate supports a variance update under the stated rule; it does not show whether one group had more events or estimate the treatment difference. The calculation also does not address loss to follow-up: any inflation for incomplete outcomes would need to follow a separately prespecified rule.

Worked Example: Comparing Separation With Pooled Variability

Suppose, hypothetically, the underlying event rates are \(p_0=0.20\) and \(p_1=0.40\), so \(q=0.30\) and the actual separation is \(\delta=0.20\). The trial still uses a prespecified target difference \(\delta^*=0.10\), with two-sided \(\alpha=0.05\) and power \(0.80\). Assume the same equal-allocation, independent-outcome normal approximation.

The pooled variance term is \(2q(1-q)=2(0.30)(0.70)=0.4200\). The group-specific variance sum is \(0.20(0.80)+0.40(0.60)=0.1600+0.2400=0.4000\). Their difference is \(0.0200=\delta^2/2=(0.20)^2/2\).

Using the pooled term with the target effect gives \(n_{\text{blind}}=7.849(0.4200)/(0.10)^2=329.7\), or 330 per group. If the group-specific variances were known and used instead, the same target effect would give \(n=7.849(0.4000)/(0.10)^2=314.0\), or 314 per group. In this example, the blinded approximation adds 16 participants per group relative to that variance calculation. This comparison isolates the variance terms; it does not claim that blinded data can establish the actual rates or treatment separation.

Diagnostics, Alternatives, and Limitations

A re-estimation rule should be checked both statistically and operationally. The pooled rate is a noisy estimate, and the resulting sample size can change sharply when the event rate is near zero or one. Before using the rule, examine whether the interim sample is large enough for a stable rate estimate, confirm that endpoint definitions and follow-up are consistent across participants, and verify that the pooled count includes only eligible outcomes. A calculated target outside the prespecified range is a reason to apply the protocol’s stated cap or floor—not to improvise a different rule after seeing the data.

The main alternatives have different information and assumptions:

Approach	What informs the variance	Main consideration
Fixed sample size	Planning event rates only	Simple and prespecified, but sensitive to planning-rate error.
Blinded pooled-rate update	Overall event rate, with treatment labels masked	Does not estimate group-specific rates; pooled variability includes a component from treatment separation.
Unblinded group-specific update	Separate event rates by treatment group	Uses more direct variance information, but requires access to treatment-specific interim outcomes and different design safeguards.

The pooled-rate formula is most directly suited to equal allocation and a simple two-group binary endpoint. Unequal allocation requires a variance expression that reflects the allocation ratio; repeated, clustered, stratified, or time-dependent outcomes require methods that represent their dependence or analysis structure. Rare events, small interim samples, substantial missingness, or changing event rates over calendar time can also make a simple pooled plug-in estimate unreliable. These cases call for design-specific methods and sensitivity calculations rather than mechanical use of the formula here.

Common Mistake

Common mistake: “The pooled event rate is 0.30, so both groups must have event rates near 0.30; I can treat the pooled variance as the exact variance of the treatment difference.” The pooled rate is only the average \((p_0+p_1)/2\), and it cannot identify either group rate. Moreover, \(2q(1-q)\) exceeds the sum of the within-group variances by \(\delta^2/2\). Use it as the prespecified blinded variance approximation, recognize how separation affects it, and do not interpret it as a treatment-effect estimate.

Key takeaway: A blinded binary-outcome re-estimation substitutes the pooled event rate into a prespecified variance-based sample-size rule while retaining the planned effect size. Under equal allocation, pooled variability includes a nonnegative contribution from between-group separation, so the update can be conservative relative to a calculation using the true group-specific variances.

Check Your Understanding

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

With \(p_0=0.10\) and \(p_1=0.30\), calculate \(q\), the sum of the group-specific variances, and the blinded pooled variance term \(2q(1-q)\).
A blinded interim includes 150 participants per group and 72 total events. Calculate \(\hat q\). What information about the separate group event rates does this calculation provide?
For \(\alpha=0.05\), power \(0.80\), \(\hat q=0.25\), and target difference \(\delta^*=0.10\), calculate the approximate blinded target per group and round up.
State two assumptions or conditions that should be checked before applying the equal-allocation pooled-rate formula.

Solutions. (1) \(q=(0.10+0.30)/2=0.20\). The variance sum is \(0.10(0.90)+0.30(0.70)=0.0900+0.2100=0.3000\). The pooled term is \(2(0.20)(0.80)=0.3200\), larger by \(0.0200=\delta^2/2\), where \(\delta=0.20\). (2) \(\hat q=72/(2\cdot150)=72/300=0.2400\). It estimates the pooled event rate, not either separate rate or their difference. (3) Using \((1.960+0.8416)^2=7.849\), \(n=7.849[2(0.25)(0.75)]/(0.10)^2=7.849(0.3750)/0.0100=294.3\), so round up to 295 per group. (4) Examples include equal allocation, independent binary outcomes, consistent endpoint ascertainment, and a sample size large enough for the normal approximation to be reasonable.