Re-Estimation for Binary Outcomes

Why Binary Outcomes Need Their Own Re-Estimation Calculation

For a binary endpoint, each participant either experiences the event or does not. Interim event rates therefore determine both the estimated treatment effect and the estimated variability of that effect. In an unblinded re-estimation, both quantities can enter the conditional-power calculation: the estimated effect describes the signal projected forward, while the estimated variance describes the uncertainty in the remaining data.

The calculation below concerns a two-arm trial with equal allocation and a fixed final analysis. Define a positive treatment effect to mean that the treatment arm has a lower event probability than control. This direction is convenient when the event is undesirable; for a beneficial event, reverse the contrast or state the direction explicitly. The method is an approximation based on binomial sampling and a normal approximation to the final test statistic.

For the one-sided superiority test, the hypotheses are \(H_0:\delta\leq 0\) and \(H_a:\delta>0\), with significance level \(\alpha\). Under independent Bernoulli outcomes within and between arms, the variance of a difference in event rates based on \(n\) participants per arm is

Replacing the unknown probabilities with their interim estimates gives the plug-in variance component

The arm-specific terms matter: under the alternative, the two event probabilities need not be equal. The pooled variance sometimes used for a null-based score test answers a different question and should not be substituted automatically for this plug-in variance in a power calculation.

Conditional Power With Interim Binary Data

Suppose the final analysis will include \(N\) participants per arm, where \(N>n_I\). The interim data are fixed when calculating conditional power. The remaining \(N-n_I\) participants per arm are assumed to have event probabilities represented by the interim estimates. This is the plug-in assumption: \(\hat p_C\) and \(\hat p_T\) are treated as estimates of the probabilities governing future outcomes.

Let \(D_I=\hat p_C-\hat p_T\) be the observed interim difference, and let \(D_R\) denote the difference between the two event rates among the remaining participants. The final risk-difference estimator is the weighted average of interim and remaining differences. Conditional on the interim data, its approximate mean and variance under the plug-in model are

The variance expression follows because the remaining-arm difference has variance approximately \(\hat v_I/(N-n_I)\), and it receives weight \((N-n_I)/N\) in the final estimator. For a simple plug-in Wald calculation, use \(\hat v_I\) in the final standard error as well, so that the final statistic is standardized by \(\sqrt{\hat v_I/N}\). Its conditional standard deviation is then approximately \(\sqrt{(N-n_I)/N}\).

More generally, if the final standard error uses a specified variance component \(v_{\text{plan}}\), the conditional mean and standard deviation of the final statistic are approximated by

With \(v_{\text{plan}}=\hat v_I\) and \(\hat\delta_I=D_I\), these simplify to \(\mu_Z(N)=D_I\sqrt{N/\hat v_I}\) and \(\sigma_Z(N)=\sqrt{(N-n_I)/N}\). For a one-sided final Wald test, the approximate conditional power is

where \(z_{1-\alpha}\) is the \(1-\alpha\) quantile of the standard normal distribution and \(\Phi\) is its cumulative distribution function. This is conditional power given the interim observations under the plug-in model; it is not the probability that the null hypothesis is true.

Applying a Promising-Zone Re-Estimation Rule

A promising-zone rule uses conditional power to distinguish interim results that are too weak to justify an increase, results in a prespecified intermediate range that may justify an increase, and results sufficiently strong to continue without an increase. The cutoffs and any maximum sample size must be specified in advance. Conditional power alone does not dictate a unique decision rule.

For example, a rule might leave the planned size unchanged when conditional power is below 0.55, seek a target conditional power of 0.80 when conditional power lies from 0.55 up to but not including 0.80, and make no increase above that range. A cap \(N_{\max}\) prevents the revised size from exceeding the maximum allowed. The target is pursued by evaluating \(CP(N)\) at candidate final sizes using the same interim estimates and increasing \(N\) only within the prespecified constraints.

Justification and Assumptions

The conditional-power formula follows directly from the future-data model and the final rejection boundary. Given the interim observations, the remaining outcomes are independent Bernoulli variables under the assumed model. For \(N-n_I\) future participants in each arm, the variance of their difference in event rates is approximately \(\hat v_I/(N-n_I)\). Multiplying that future difference by its weight in the final estimate gives

After standardizing by the plug-in final standard error \(\sqrt{\hat v_I/N}\), the variance of the final statistic is \((N-n_I)/N\), giving the stated conditional standard deviation. If this statistic is approximately normal, rejection occurs when it exceeds \(z_{1-\alpha}\). Standardizing that boundary by the conditional standard deviation yields the normal-tail expression for \(CP(N)\).

This justification relies on several conditions: equal allocation at interim and final, independent participants, stable event probabilities across enrollment, no unaccounted clustering or repeated observations, and a sufficiently accurate normal approximation. It also treats the interim rates as the best estimates of the future rates. Departures such as changing standard of care, differential dropout, delayed event ascertainment, or a change in participant mix can make that projection unreliable.

Diagnostics, Alternatives, and Limitations

Before using the result, inspect the event counts and rates in both arms, not only the estimated difference. A rate of zero or one gives a zero plug-in variance contribution; with sparse data, the normal approximation and the resulting conditional-power estimate may be poor even if the formula returns a number. Check whether expected event counts are adequate for the intended approximation, whether the arm-specific rates are based on comparable follow-up, and whether the observed difference is plausible in light of clinical and operational knowledge. An unusually large interim effect can be driven by sampling noise, so projecting it unchanged may yield an optimistic re-estimation.

The unblinded plug-in approach differs from a blinded variance re-estimation, which may update nuisance variability without using the treatment contrast to revise the assumed effect. It also differs from a rule based directly on the interim \(Z\)-statistic: the binary plug-in calculation explicitly represents event rates and their arm-specific variance contributions. These approaches answer different design questions and should not be treated as interchangeable.

A critical limitation is that changing sample size based on unblinded interim effect estimates can affect the null distribution of a naive final test. A plug-in conditional-power calculation does not by itself establish control of the type I error rate. The adaptation and final inferential procedure therefore need to be prespecified and evaluated together, with suitable error-rate control for the design. The method is also sensitive to the assumed future event rates, can be unstable with sparse interim events, and does not account automatically for multiplicity, nonadherence, missing outcomes, or other deviations from the stated model.

Common Mistake