Design Workshop For Inner Wedges And Tail Limits

From Directional Question to Sequential Rule

An asymmetric two-sided design is not just a pair of unequal numbers at each interim look. It is a decision rule whose two tails have explicit meanings, such as evidence of benefit in one direction and evidence of harm in the other. The design must state those meanings before data are examined, specify when the data will be assessed, and make clear what happens if neither boundary is reached.

Consider a hypothetical randomized trial in which a positive treatment effect means improvement on a chosen endpoint. Let \(\theta\) denote the treatment effect on that scale, so \(\theta>0\) is beneficial and \(\theta<0\) is harmful. A basic two-sided formulation is \(H_0:\theta=0\) against \(H_a:\theta\ne 0\). The null concerns no effect; the two possible directions under the alternative have different practical interpretations. If the scientific question instead uses a nonzero null margin, the boundary coordinates must be shifted consistently with that margin.

The upper crossing is evidence in the positive, beneficial direction under the stated coding; the lower crossing is evidence in the negative, harmful direction. Those labels are not properties of the words “upper” and “lower” alone. If the endpoint is recoded so that negative values represent benefit, the directional interpretations must be reversed as well.

Assumptions and First-Crossing Interpretation

For a standard design workshop, suppose there are \(K\) planned analyses at increasing information levels \(I_1<\cdots<I_K\). Use the usual canonical normal approximation: under \(H_0\), each \(Z_k\) is marginally standard normal, and for \(i<j\), \(\operatorname{Cov}(Z_i,Z_j)=\sqrt{I_i/I_j}\). This joint model describes the dependence created by accumulating data; the statistics at different looks are not independent. It is an approximation whose suitability depends on the study model, the statistic, and the information estimates.

The limits are prespecified and ordered at every look. A trial continues after look \(k\) only when its statistic lies inside that look’s wedge. The first crossing determines the efficacy direction and look; a statistic that crosses at an early look is not reclassified by later hypothetical values. If no efficacy limit is crossed, the study continues through its planned looks and ends at the maximum information, unless a separately specified stopping rule applies.

If the set in this expression is empty, there is no efficacy crossing at the planned looks. This convention separates efficacy stopping from other possible rules, such as a futility rule; those rules should not be silently added to the wedge.

The boundaries must also be calibrated to the design’s error goals. Under the global null, define the upper and lower first-crossing probabilities as \(P_0(\text{first crossing is upper})\) and \(P_0(\text{first crossing is lower})\). These events are mutually exclusive, so the overall type I error from efficacy crossings is their sum. A design can assign distinct tail budgets to these probabilities, but setting a nominal one-look cutoff at each time does not by itself control the cumulative error across looks.

Designing and Checking the Wedge

A practical design begins with the effect direction, not with a convenient pair of cutoffs. The following sequence makes the choices explicit.

The geometric picture is simple: at each look, the continuation wedge is the band between the lower and upper limits. Across looks, a sample path is a sequence of standardized statistics, one point per analysis. A path continues while every observed point is inside its corresponding band and stops at the first point outside it. This picture is a decision aid, not a substitute for calibrating the probabilities under the joint distribution.

Why Tail Budgets Require Joint Calibration

For a two-sided sequential design, the two directional error probabilities can be written using disjoint first-crossing events. Let \(A_U\) denote the event that the first crossing is upper and \(A_L\) the event that the first crossing is lower. Then \(A_U\cap A_L\) is empty, and

This identity justifies accounting for the two tails separately. It does not imply that the probability of an upper crossing is the sum of independent per-look probabilities: the \(Z_k\) values are correlated, and the first-crossing condition excludes paths that stopped earlier. Thus, tail budgets are targets for cumulative crossing events, not instructions to apply the same single-look cutoff repeatedly.

A simple conservative check follows from the union bound: the probability of any boundary crossing is no greater than the sum, over looks, of the probabilities of crossing either boundary at that look. This bound can be loose because it ignores the dependence and overlap among look-specific crossing events. A precise calibration uses the joint sequential distribution or an appropriate validated design procedure.

This distinction matters in asymmetric designs. A design may assign more type I error to the beneficial-direction tail than to the harmful-direction tail, or the reverse, based on the scientific question and consequences of the decisions. The selected limits must then be calibrated to those separate goals. Unequal clinical crossing thresholds and unequal statistical error budgets are related design considerations, but they are not interchangeable quantities.

Diagnostics, Alternatives, and Limitations

Before adopting a wedge, check the direction and the rule as carefully as the numbers. Useful diagnostics include:

• Sign check: Confirm that the statistic’s sign agrees with the endpoint coding and that the upper and lower limits carry the intended clinical meanings.
• Ordering check: Verify \(L_k<U_k\) at every look. If the limits meet or reverse, the stated continuation region is not a valid interval.
• Timing check: Confirm that information levels and looks match the planned analysis schedule. Unplanned looks can change error properties.
• Calibration check: Evaluate both cumulative directional crossing probabilities under \(H_0\), as well as power in scientifically relevant effect directions.
• Path check: Trace paths that approach, cross, or change sign near a boundary. This often exposes mistaken labels or an unintended continuation rule.

A common mistake is to treat each look as a separate one-time test and conclude that limits chosen for a single analysis automatically preserve the intended overall two-sided type I error. That reasoning ignores repeated opportunities to cross and the dependence among looks. The correction is to calibrate the complete first-crossing rule under the joint sequential model, with the upper and lower tail targets assessed cumulatively.

Another mistake is to call a path beneficial because its final hypothetical \(Z\) value is positive, even though it crossed the lower limit earlier. The correction is to apply the prespecified first-crossing rule: the direction and timing are determined at the earliest crossing. If no efficacy boundary is crossed, the result is not an efficacy stop; the design must state what happens at the final look.

Compared with equal upper and lower limits, an asymmetric wedge can reflect directional priorities, unequal tail error budgets, or distinct clinical consequences. A symmetric design is easier to describe and may be appropriate when the directional objectives are balanced. Neither symmetry nor asymmetry guarantees suitable error control: each rule requires calibration. A nonsequential fixed-sample test is another alternative, but it does not provide the same planned opportunities for early efficacy stopping. The choice depends on the scientific goals, error constraints, and operational requirements.

The canonical normal model and prespecified information times are assumptions, not universal facts. Small samples, model misspecification, changing variance, inaccurate information estimates, or departures from the planned monitoring schedule can weaken the justification for the proposed limits. A wedge describes efficacy decisions only; it does not by itself specify futility stopping, safety monitoring, missing-data handling, or adaptations to the design. Those issues require explicit rules and appropriate validation.