Estimation Under Asymmetric Boundaries

Why Asymmetric Boundaries Change Estimation

A stopping rule does more than determine when a trial ends. It also determines which estimates are observed at each stage. If the upper and lower efficacy boundaries differ in magnitude, or if futility rules treat the two tails differently, the set of estimates selected for an early stop is not balanced around the true effect. The resulting stage-specific estimate can therefore have a different conditional bias in each stopping region.

The central issue is selection. For example, an upper efficacy stop selects unusually large positive interim statistics, so estimates reported at that stop tend to exceed the true effect. A lower efficacy stop selects unusually small statistics and tends to produce estimates below the true effect. Unequal boundaries change both how often each kind of stop occurs and how extreme the selected estimates are. A single bias summary can hide that structure.

The calculations below use the canonical two-look normal model with known variance. Let \(I_1<I_2\) be the information at the first and second looks. The cumulative standardized statistics satisfy \(Z_k=\sqrt{I_k}\hat{\theta}_k\), and under a fixed effect \(\theta\), \(Z_k\sim N(\theta\sqrt{I_k},1)\). The joint model assumes that information increases as planned, the effect is constant across looks, and the increments contain independent information. In particular, the statistics are correlated because they use overlapping data.

This model is an approximation in many applications, especially when the standard error is estimated or the outcome model is not normal. Such settings require additional care; the present derivations treat the stated canonical model as given.

Representing Unequal Boundaries

At the first look, a general two-sided rule can have a lower efficacy boundary \(L_1\) and an upper efficacy boundary \(U_1\), with \(L_1<U_1\). Efficacy stopping occurs when \(Z_1\le L_1\) or \(Z_1\ge U_1\). A futility rule may assign a central interval \((F_{L1},F_{U1})\) to futility, with \(L_1<F_{L1}<F_{U1}<U_1\). The remaining regions are continuation regions. This is one way to express a rule that continues only when the interim result is promising enough in either direction but has not crossed an efficacy boundary.

Actual designs may instead use a one-sided futility rule, different futility definitions for the two tails, or a nonbinding futility recommendation. The calculations do not require one particular rule: define the event corresponding to each reported stopping stage, then condition on that event. The introductory material on binding and nonbinding futility distinguishes whether a futility rule is enforced or advisory; here the focus is how the resulting selection regions affect estimates.

For any stage-one region \(A\), the estimate is \(\hat{\theta}_1=Z_1/\sqrt{I_1}\). Consequently,

Thus, boundary-specific bias is determined by the mean of a normal statistic truncated to the relevant region. An upper crossing and a lower crossing generally have different probabilities and different conditional means, particularly when the boundaries are asymmetric relative to the model mean \(\theta\sqrt{I_1}\).

For the upper-tail result, standardize \(X=Z-\mu\sim N(0,1)\) and set \(a=c-\mu\). The conditional expectation is \(\mu+E(X\mid X\ge a)\). Since \(\phi'(x)=-x\phi(x)\),

For the lower tail, integrating over \((-\infty,a]\) gives \(\int_{-\infty}^a x\phi(x)\,dx=-\phi(a)\), yielding the stated expression. These formulas show the direction of the selection effect: the upper-tail conditional mean exceeds \(\mu\), while the lower-tail conditional mean falls below \(\mu\).

Selection at the Second Look

If the trial continues, the second-look estimate is selected by both the first-look continuation event and the second-look stopping rule. The conditional distribution given \(Z_1=z\) provides a direct way to understand the second part of this selection. Write its conditional mean as \(m(z)\) and variance as \(v\). If an upper efficacy stop at look two requires \(Z_2\ge U_2\), then, conditional on \(Z_1=z\) and that crossing,

The conditional mean estimate is this expectation divided by \(\sqrt{I_2}\), and its conditional bias is obtained by subtracting \(\theta\). To obtain the average among all trials that reach and stop at look two, one must also average over the distribution of \(Z_1\) conditional on both continuation and the relevant look-two stopping event. For an upper stop, this weights each continued value \(z\) by \(P(Z_2\ge U_2\mid Z_1=z)\); analogous event-specific weighting applies to other stopping events. The first-look boundaries therefore affect second-look estimation even though the final crossing is assessed using \(Z_2\).

How Asymmetry Changes the Reported Stage-Specific Bias

A design’s boundaries determine the probability of each stopping event as well as the estimate distribution within that event. For a stage-one upper efficacy region, the crossing probability is \(1-\Phi(U_1-\theta\sqrt{I_1})\); for a lower region it is \(\Phi(L_1-\theta\sqrt{I_1})\). If \(L_1\ne -U_1\), these probabilities need not match even when the two sides are treated as a nominally two-sided test. A shifted model mean can make their imbalance still larger.

The mean estimate conditional on stopping for efficacy on either side combines the two tail means with weights proportional to their crossing probabilities. It is not generally the average of the upper and lower conditional means. Nor does it describe either tail well when one tail is rare. A clear report therefore distinguishes, where relevant, the upper efficacy stop, lower efficacy stop, futility stop, and continuation-to-final-look groups.

Assumptions, Diagnostics, and Alternatives

The derivations rely on a correctly specified canonical joint normal model, known information levels, a fixed effect across looks, and stopping regions that are defined in terms of the stated statistics. Before interpreting a calculated bias, verify that the efficacy and futility regions are disjoint, that each boundary is expressed on the same statistic scale, and that the information values correspond to the planned analysis. For estimated standard errors, non-normal statistics, or changing effects, the conditional law may not have the form used here.

Useful diagnostics include tabulating the simulated or calculated frequency of each stopping region under several plausible values of \(\theta\), then comparing the conditional mean estimate with \(\theta\) separately in each region. Check that upper-tail conditional estimates move upward and lower-tail estimates move downward under the normal model. Also check that pooled summaries are weighted by stopping probabilities, rather than formed by giving each boundary equal weight. Large discrepancies between a model-based calculation and simulation can indicate incorrect boundary coding, an incorrect covariance, or a mismatch between the assumed and implemented stopping rule.

The ordinary maximum-likelihood estimate at the observed look is a simple descriptive alternative, but it does not adjust for the event that selected the look or the boundary crossing. Conditional-likelihood or stopping-rule-based estimators target selection more explicitly, as covered in the earlier material on likelihood and bias-adjusted estimation. Those approaches answer different estimation questions and do not make the boundary geometry irrelevant: asymmetric boundaries still determine the selection event being modeled. No single stage-specific conditional bias characterizes all possible stopping events.