Expected Sample Size Tradeoffs Across Boundary Families

Why Expected Information Use Depends on the Effect

A group sequential efficacy design has a planned maximum information level, but it may stop before reaching that limit if an interim statistic crosses its efficacy boundary. The planned maximum is therefore not necessarily the average information used across repeated trials. That average depends on how often a boundary is crossed early, and those crossing probabilities depend on both the boundary shape and the true treatment effect.

Pocock, O’Brien-Fleming, and Wang-Tsiatis boundaries allocate stringency differently across the interim looks. Broadly, Pocock boundaries are less stringent early than O’Brien-Fleming boundaries, while Wang-Tsiatis boundaries provide a family of shapes between and around these familiar patterns. These geometric descriptions do not, by themselves, tell us which design has the smallest expected sample size. We must also specify the look schedule and the effect scenario under which expected use is being evaluated.

This expected value is scenario-specific. It is not an intrinsic, fixed property of a boundary family: the same design can have different expected information use under the null hypothesis and under different nonzero effects.

Assumptions and the Expected-Information Formula

Consider a one-sided efficacy design with \(K\) prespecified looks, information fractions \(0<t_1<\cdots<t_K=1\), and upper boundaries \(c_1,\ldots,c_K\). At look \(k\), the standardized statistic is \(Z_k\), and efficacy stopping occurs at the first look satisfying \(Z_k\geq c_k\). Assume there is no separate futility stopping rule and that a trial not stopped earlier continues to the next planned look.

For the usual canonical group sequential model, the joint statistics are approximately multivariate normal. Under the null hypothesis, \(E(Z_k)=0\); under a fixed standardized effect \(\delta\), \(E(Z_k)=\delta\sqrt{I_k}\). Their dependence matters: for looks \(j\leq k\), the canonical covariance is \(\operatorname{Cov}(Z_j,Z_k)=\sqrt{I_j/I_k}\). Boundary-crossing and look-reach probabilities must therefore be calculated from the joint distribution, not by treating the interim statistics as independent.

Write \(R_k=P(\tau\geq k)\), the probability that a trial reaches look \(k\). By definition, \(R_1=1\). A trial uses information up to fraction \(t_k\) exactly when it reaches that look, so the increments in its information use can be counted one at a time.

Taking expectations gives the key identity:

Here \(\mathbf{1}(\tau\geq k)\) equals 1 when the trial reaches look \(k\) and 0 otherwise. The formula weights each increment of planned information by the probability that the trial actually uses that increment. It applies to unequal as well as equal information spacing.

Worked Calculations

What Changes Across Effect Scenarios?

Under an effect that makes positive \(Z_k\) values more likely, crossing an efficacy boundary tends to become more likely, and stopping can occur earlier. If one boundary is easier to cross at an early look, it may therefore produce lower expected information use under that effect. The effect on expected use is accumulated over all looks: reaching an early look is nearly certain, but avoiding an early crossing is what determines whether later information increments are used.

Under the null hypothesis, the probability of an efficacy crossing across the design is controlled by the design’s overall type I error rate. That control does not make expected information use the same across boundary families. Different calibrated shapes can distribute null crossing opportunities differently over time, and hence can have different probabilities of reaching later looks. Also, reaching the final look is not the same as not rejecting: a trial may reach that look and cross its boundary there.

The shape comparison is therefore about the timing of opportunities to stop, not a claim that one family always saves more information. Earlier, less stringent boundaries can create more early stopping when the effect is sufficiently strong, but they can also change operating characteristics at later looks. The Wang-Tsiatis family provides a shape parameter that changes this pattern; a particular Wang-Tsiatis design should be compared using its actual calibrated boundaries and scenario-specific reach probabilities.

Diagnostics, Alternatives, and Limitations

Before interpreting an expected sample size, check that the calculation matches the design being evaluated:

• Use reach probabilities, not crossing probabilities alone. \(R_k\) is the probability of reaching look \(k\), which means no earlier efficacy crossing. The probability of crossing at look \(k\) is a different quantity.
• Use the correct information increments. For unequal information spacing, weight each reach probability by \(t_k-t_{k-1}\), not by an assumed equal share.
• Keep the scenario explicit. Reach probabilities under \(H_0\) need not describe expected use under a clinically relevant alternative.
• Respect dependence across looks. Calculating each look as though it were an independent test generally gives incorrect probabilities of reaching later looks.
• Confirm the stopping rule. Adding a futility rule, changing the look schedule, or changing the boundary changes the stopping time and requires corresponding reach probabilities.

The main alternative summary is the maximum information or maximum sample size: it describes the planned cap, not the average use. Power and type I error answer different questions again. A boundary comparison should report expected information under clearly named scenarios alongside the operating characteristics relevant to the design; expected sample size alone does not summarize the consequences of a stopping rule.

The conversion \(E(n)=n_{\max}E(t_\tau)\) assumes sample size is proportional to information, as with a fixed allocation and stable information per participant. With unequal accrual, changing allocation, variable event rates, or information that does not grow proportionally with enrollment, expected sample size should be calculated using the actual mapping from information to enrollment. The expected value also does not describe the spread of possible stopping times: two designs can have similar means but different probabilities of stopping at each look.

Common Mistake

A common error is to say, “O’Brien-Fleming has very strict early boundaries, so it must have the largest expected sample size,” or to infer average savings from the boundary picture alone. The first statement may describe a tendency under a particular effect, but it is not a calculation. Expected use depends on the joint probabilities of reaching each look, the information increments, and the effect scenario. Calculate \(E(t_\tau)\) from those inputs before comparing designs.