Re-Estimation For Equivalence Trials

Why Re-Estimation Is Different for Equivalence

In an equivalence trial, the goal is not to show that a treatment differs from control, nor merely to rule out a difference in one direction. The goal is to show that the treatment difference is sufficiently close to zero: specifically, that it lies within prespecified lower and upper equivalence margins. Because success requires two one-sided tests to pass, the sample size must support both conclusions. Uncertainty in the outcome variance matters directly: if the variance is larger than planned, the confidence interval is wider and equivalence is harder to establish.

This tutorial develops a variance-based re-estimation rule for a continuous outcome with equal allocation. The calculations use a normal approximation and a common variance. They are planning approximations, not a substitute for checking the final test and the full adaptive design. The earlier material on sample size re-estimation for noninferiority trials uses a one-sided margin; equivalence instead requires both one-sided conditions below.

Assume independent participants, a continuous outcome, equal numbers \(n\) in the two groups, and normally distributed outcomes with a common variance \(\sigma^2\). For the planning calculation, treat \(\sigma\) as known or use a large-sample normal approximation. The estimated difference \(D\) then has approximate distribution \(D\sim N(\theta,2\sigma^2/n)\), with \(SE=\sigma\sqrt{2/n}\). At one-sided significance level \(\alpha\), both tests pass exactly when

where \(z_q\) is the \(q\)-quantile of the standard normal distribution. If the lower endpoint is not below the upper endpoint, this acceptance interval is empty and equivalence cannot be declared. The corresponding two-sided confidence interval has confidence level \(1-2\alpha\); for \(\alpha=0.05\), it is a 90% interval.

Power and the Sample-Size Rule

Under a true difference \(\theta\), the probability that both tests pass under this approximation is the probability that \(D\) falls inside the acceptance interval. Subtracting the normal probabilities below its lower endpoint and above its upper endpoint gives

where \(\Phi\) is the standard normal cumulative distribution function. The formula makes the two-sided requirement explicit. For planning at the centered alternative \(\theta=0\), symmetry reduces power to

To ensure this power is at least \(1-\beta\), it is sufficient to require \(\Delta/SE-z_{1-\alpha}\geq z_{1-\beta/2}\). This allocates no more than \(\beta/2\) probability to failure on either side. Substituting \(SE=\sigma\sqrt{2/n}\) and rearranging gives the familiar conservative planning rule:

The resulting \(n\) is the number of participants per group, rounded up to an integer. For the centered alternative, the rule gives the exact continuous-\(n\) threshold for the centered normal-approximation power equation; rounding up gives the smallest integer \(n\) meeting the target. It is not an exact solution for a noncentered alternative or for a different final testing procedure. If the anticipated true difference is not zero, use the general power expression and solve for \(n\) numerically. Power generally declines as the anticipated difference moves toward either equivalence margin.

Updating the Size From an Interim Variance

Suppose an interim review estimates the common outcome variance without examining the treatment difference. Let \(s_p^2\) denote the pooled within-group variance estimate. Replacing \(\sigma^2\) in the planning rule by \(s_p^2\) gives a plug-in target total per-group size:

where \(n_1\) is the number already enrolled per group and \(\lceil\cdot\rceil\) rounds upward. The update is a total sample size per group, not the number of additional participants. If a prespecified minimum or maximum is used, apply it according to the protocol, while ensuring the final total cannot be less than the number already enrolled.

Why Variance Uncertainty Changes the Interpretation

A plug-in update does not make the nuisance parameter known. Under the normal, common-variance model, if the interim estimate pools \(n_1\) independent observations per group, its degrees of freedom are \(\nu=2n_1-2\), and

The second expression is the coefficient of variation of the variance estimate. Since the unrounded re-estimated size is proportional to \(s_p^2\), its sampling variability is substantial when the interim sample is small. A realized estimate below \(\sigma^2\) tends to produce a smaller trial than the true variance would require; an estimate above it tends to produce a larger one. Thus, achieving the target power conditional on the observed estimate is not the same as guaranteeing that target over repeated interim samples.

The following example quantifies this point using the same hypothetical design. It treats the variance estimate and the resulting size as random across repeated internal pilots, before any floor, cap, or rounding is applied.

Assumptions, Diagnostics, and Safeguards

The rule is most defensible when the interim estimate reflects the variance relevant to the final endpoint, the groups have a common variance, allocation is balanced, and participants contribute independent outcomes. The normal approximation also becomes less reliable with very small samples or markedly nonnormal outcomes. If the final analysis uses a pooled \(t\) statistic rather than a normal statistic, planning should use an appropriate \(t\)-based calculation or simulation, particularly when sample sizes are small. The normal formula is a transparent approximation, not an exact guarantee for a finite-sample \(t\) procedure.

Before applying an update, check that the interim estimate is based on the prespecified variance estimator and that data handling, endpoint definition, and follow-up are consistent with the final analysis. Review group-specific spreads for a conspicuous variance imbalance, and inspect outliers or data errors that could dominate \(s_p^2\). Such diagnostics do not license an informal change of method: if the common-variance assumption is implausible, a preplanned alternative calculation is needed. A floor and cap can control operational consequences but may leave the final design below its target power or require more enrollment than is feasible.

A blinded, variance-only update is different from changing sample size in response to an observed treatment difference. Blinding helps keep the decision from being driven by apparent efficacy, but the adaptation rule and final testing procedure still need to be prespecified and assessed as a design. In particular, do not infer that any adaptive update automatically preserves the nominal type I error. The rule should specify the interim timing, variance estimator, target, rounding, minimum and maximum size, and how the final test uses the accrued data. A fixed sample size avoids interim-estimate variability but may be inefficient if the planned variance is wrong. A plug-in update responds to observed nuisance information, while a more comprehensive predictive or simulation-based assessment can incorporate uncertainty in that information and the details of the final procedure.