Choosing A Wang-Tsiatis Shape

Why the Shape Choice Matters

A group sequential efficacy design can allow a study to stop early when the accumulating evidence is sufficiently strong. The Wang–Tsiatis family describes a range of boundary shapes, from the relatively flat Pocock shape to the more stringent-early-look O’Brien–Fleming shape. Choosing a shape is therefore a way of expressing how demanding the design should be at interim analyses relative to the final analysis.

The shape parameter does not, by itself, specify a complete test. A design also needs a planned information schedule, a testing direction, a target type I error rate, and a boundary constant calibrated to that schedule. The comparisons below first isolate the effect of shape using a common final threshold; the practical design step is to recalibrate the constant for each candidate.

For this one-sided rule, the null hypothesis concerns no effect in the prespecified direction, and the alternative concerns an effect in that direction. The planned looks and their information fractions must be fixed for the calibration being used. Under the null, the standardized statistics are assumed to follow the canonical joint normal model: each \(Z_k\) has mean 0 and variance 1, and for \(i<j\), their correlation is \(\sqrt{t_i/t_j}\). This model follows from independent increments of information in the usual group sequential setting. If the analysis uses a different valid joint model, the calibration must use that model instead.

Reading the Shape From Early and Final Thresholds

Because the final information fraction is \(t_K=1\), the final threshold is \(b_K=c\), whatever the value of \(\Delta\). At an earlier look \(t_k<1\), the threshold is higher than the final threshold when \(\Delta>0\). Dividing the two thresholds removes the common constant:

This ratio is the cleanest way to compare shapes before calibration. For any fixed early information fraction, it increases as \(\Delta\) increases. Thus, for a common final threshold, larger \(\Delta\) makes an early efficacy stop harder to achieve relative to stopping at the final look. At \(\Delta=0\), all thresholds equal \(c\). At \(\Delta=0.5\), an early threshold is multiplied by \(1/\sqrt{t_k}\).

Worked Comparisons

Calibration and a Practical Selection Process

For each candidate \(\Delta\), the boundary constant is selected using the joint null distribution and the planned information fractions. In the one-sided case, calibration sets the probability of at least one efficacy crossing under the null to the target \(\alpha\):

This probability accounts for the dependence among repeated looks. It is not obtained by assigning the full \(\alpha\) independently to every look. The earlier material on familywise type I error and the Wang–Tsiatis boundary family provides the relevant background; for choosing a shape, the operational point is that the calibration is repeated for every candidate value of \(\Delta\).

A flat Pocock-like profile places less relative penalty on an early crossing. An O’Brien–Fleming-like profile demands substantially stronger evidence early and places more of its opportunity for rejection near the final analysis. Intermediate \(\Delta\) values provide intermediate relative profiles, but after calibration, the absolute thresholds must be read from the resulting design rather than inferred from the shape label alone.

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to check the calculated ratio at every planned interim look. For \(\Delta>0\), it should exceed 1 at every \(t_k<1\), and it should approach 1 as \(t_k\) approaches 1. If a calculation gives an early-to-final ratio below 1, check the sign in the exponent: the boundary is \(c/t_k^\Delta\), not \(ct_k^\Delta\). Also verify that the final look is assigned \(t_K=1\); otherwise its threshold will not reduce to \(c\).

The main alternative is not another formula for the same fixed shape, but a different boundary emphasis. The Pocock and O’Brien–Fleming boundaries are the endpoint shapes within the conventional Wang–Tsiatis range. A candidate between them is useful when the desired early-to-final contrast lies between those endpoints. A flat boundary is more permissive early relative to the final look, while a strongly increasing early threshold is more conservative at interim analyses. The right comparison depends on the study’s priorities; the ratio alone does not rank designs on every operating characteristic.

Several limitations matter in practice. First, the ratio comparison holds the final threshold fixed, while actual calibrated candidates generally have different values of \(c\). Second, the shape does not encode a change to the information schedule: moving a look changes \(t_k^{-\Delta}\) and requires recalibration. Third, an efficacy-boundary shape does not specify futility stopping rules or other adaptations. Finally, a design calibrated under one null distribution or analysis plan may not retain its intended error rate if the actual analysis departs from those assumptions. Such departures require an appropriate design-specific evaluation.

Common Mistakes

• Choosing \(\Delta\) and treating \(c\) as universal. The constant that achieves a target \(\alpha\) depends on the shape and planned schedule. Calibrate each candidate separately.
• Calling a lower early threshold “more conservative.” With the same final threshold, a lower early cutoff is easier to cross and thus allows earlier efficacy stopping more readily. Error-rate comparisons still require calibration.
• Comparing absolute thresholds while changing both \(c\) and \(\Delta\) without explanation. Use \(t_k^{-\Delta}\) to isolate shape, or state that the values being compared are fully calibrated design thresholds.
• Assuming an intermediate shape guarantees an intermediate operating characteristic. The relative threshold profile is intermediate, but power, stopping probabilities, and other operating characteristics depend on the effect and full design.