Common Errors In Inner-Wedge Interpretation

Reading the Regions Between Two Boundaries

A boundary plot can make the middle of the graph look like an active decision region. That impression is misleading: in a two-sided group-sequential rule, the region between the lower and upper limits is ordinarily the continuation wedge, not a rejection region. Misreading the wedge can reverse a trial decision, especially when the limits are asymmetric or the plotted statistic has an unfamiliar sign convention.

Consider a sequence of planned analyses indexed by \(k=1,\ldots,K\). At look \(k\), let \(Z_k\) be the test statistic on a scale where larger values favor the upper direction. Let \(L_k\) and \(U_k\) be the lower and upper boundaries, with \(L_k<U_k\). This tutorial concerns how to interpret those plotted regions; it does not determine how the boundaries were calibrated.

The labels “lower” and “upper” describe the position of a boundary on the \(Z\) scale, not the clinical meaning of a result by themselves. Under the stated orientation, crossing \(L_k\) is evidence in the lower direction, while crossing \(U_k\) is evidence in the upper direction. If a design instead defines its statistic so that smaller values favor benefit, the interpretation of benefit and harm must follow that convention; the geometry alone does not establish it.

The wedge is the open interval between the two boundaries. Its lower edge is \(L_k\), adjoining the lower-tail rejection region; its upper edge is \(U_k\), adjoining the upper-tail rejection region. A point inside the wedge is not “near rejection” in both directions, nor is it rejected merely because it lies between two lines. It has not crossed either line.

Why the Plot Gives Three Different Decisions

At a given look, the two boundaries divide the possible values of \(Z_k\) into three disjoint regions. Since \(L_k<U_k\), every real-valued statistic falls into exactly one of them:

The first and third regions are the two boundary-crossing rejection regions. The middle region is the continuation wedge. The inequalities matter: equality with a boundary counts as crossing under this convention. For a continuous statistic, an exact equality has probability zero under the usual models, but it is still useful to state the rule explicitly, particularly when recorded values are rounded.

Across several looks, the relevant rejection event is crossing either boundary at any planned analysis. If no boundary is crossed at an interim look, the observed statistic lies in that look’s wedge, and the boundary rule calls for continuing when another look is planned. This does not mean the statistic is “accepted as null.” It only records that neither rejection boundary has been crossed at that look.

This expression describes rejection by crossing a boundary. Other prespecified stopping or decision rules may also apply, and a final analysis may have a separately stated decision convention. Do not infer those additional rules from the wedge alone.

How to Read the Two Edges Without Reversing Them

On a plot with information time on the horizontal axis and \(Z_k\) on the vertical axis, the lower boundary is the lower curve and the upper boundary is the upper curve at each look. The region below the lower curve is lower-tail rejection; the region above the upper curve is upper-tail rejection. The area between the curves is the wedge. A visual check is to choose a value clearly below both curves, one clearly between them, and one clearly above both. The first should be lower-tail crossing, the second continuation at a nonfinal look, and the third upper-tail crossing.

Asymmetric boundaries need not be equally far from zero. For example, \(L_k=-1.90\) and \(U_k=2.60\) are not mirror images. That does not make the lower boundary an upper boundary, or turn the interval between them into a rejection band. The locations and labels of the two edges determine the classification; symmetry is not required.

The Final Look Is Not an Interim Continuation

The geometric regions at the final look are still defined by the same inequalities. What changes is the operational consequence of an interior statistic: there is no later planned look to continue to. At a final look, a statistic inside the wedge does not trigger boundary-crossing rejection, and the trial ends. It is incorrect to describe that outcome as “continue” merely because the statistic lies between the limits.

A Short Justification and a Reliable Reading Routine

The classification follows directly from ordering the limits. Because \(L_k<U_k\), a value cannot be both at or below \(L_k\) and at or above \(U_k\). If it is not in either rejection region, then \(Z_k>L_k\) and \(Z_k<U_k\), which is exactly the wedge condition. Conversely, any value satisfying \(L_k<Z_k<U_k\) is neither a lower nor an upper crossing. This establishes the partition without relying on visual intuition.

For a plotted boundary, a disciplined reading takes only a few steps:

Common Errors, Diagnostics, and Limits

The most important diagnostic is to verify that the plot is internally consistent: at each look, the lower boundary must be below the upper boundary, and a test value should be classified by its vertical position relative to both. If a figure shades the wedge, check its legend rather than assuming that shading means rejection. A plot can use shading for continuation, rejection, or merely a visual guide.

• Wrong reasoning: “The statistic is between the two limits, so it has crossed a boundary region and is rejected.” Correction: The open interval between the limits is the continuation wedge, not a rejection region.
• Wrong reasoning: “The lower edge is the upper-direction boundary because the test is two-sided.” Correction: Lower and upper refer to their positions on the chosen statistic scale. Check the declared orientation before attaching clinical meaning.
• Wrong reasoning: “An interior value at the last look means continue.” Correction: Continuation requires a later planned look. At the final look, an interior statistic does not trigger boundary-crossing rejection, and the trial ends.
• Wrong reasoning: “A value near the wedge’s center has a particular p-value or proves no effect.” Correction: Region membership alone supplies neither a p-value nor proof of the null. Those require the relevant statistical model and analysis.

The main alternative to a boundary plot is a written decision table or an explicit set of inequalities. These can make the rule easier to audit, especially when plotted curves are close together or labels are hard to read. Neither representation changes the decision rule. A plot is useful for seeing how limits vary over looks; a table is often clearer for classifying a particular observed statistic. In either case, the plot by itself does not establish error-rate calibration, clinical interpretation, or the value of a p-value.

Interpretations also depend on using the correct look and scale. Comparing \(Z_k\) with a boundary from a different look, or comparing an effect estimate with a boundary defined on the \(Z\) scale, is not a valid boundary check. The limits must be read at the same analysis and on the same statistic scale as the observed value. If the figure does not make these features clear, the decision should not be guessed from its shape.