Continuation Wedges In Two-Sided Monitoring

Reading the Central Continuation Wedge

In two-sided group sequential monitoring, an efficacy boundary at each analysis look marks a statistic value extreme enough to stop monitoring and declare efficacy in one direction. The upper boundary corresponds to evidence in the positive direction; the lower boundary corresponds to evidence in the negative direction. Values between them do not cross either efficacy boundary. Across successive looks, those intervals form the central continuation wedge.

The picture is useful, but it can be misleading if interpreted only one look at a time. A path that is currently between the boundaries may already have crossed a boundary at an earlier look. The path’s history therefore matters: to remain eligible for continued monitoring, it must have stayed between the limits at every preceding look as well as at the current one.

Definition: Let \(Z_k\) be the standardized test statistic at scheduled look \(k\), with lower and upper efficacy boundaries \(L_k\) and \(U_k\), where \(L_k<U_k\). At look \(k\), the central continuation interval is \(L_k<Z_k<U_k\). A path continues through look \(k\) only if it is inside this interval at look \(k\) and was inside the continuation interval at every earlier look. A value at or beyond either boundary stops monitoring for efficacy in the corresponding direction.

The boundary pair may be displayed against analysis look or against information accrued. Connecting the lower limits and connecting the upper limits outlines the central band. The word “wedge” describes this band’s appearance; it does not require straight sides, equal widths, or boundaries that move in only one direction. The interval can widen or narrow across looks, depending on the design.

The statistic and boundaries must be on the same scale. For example, if the plan specifies boundaries for standardized \(Z\)-statistics, compare \(Z_k\) with \(L_k\) and \(U_k\), not an unstandardized score with those limits. In the examples below, the limits are hypothetical, fixed before monitoring, and are used only to illustrate path classification.

Path-Wise Continuation

A single point inside the current interval does not by itself establish continuation. Define \(C_k\) to mean that monitoring has continued through look \(k\). For an efficacy-only stopping rule, \(C_k\) requires both continuation through look \(k-1\) and a statistic at look \(k\) strictly between the current boundaries.

C_k=C_{k-1}\cap\{L_k<Z_k<U_k\},\qquad C_k=\bigcap_{j=1}^{k}\{L_j<Z_j<U_j\},

The second expression follows by applying the first expression repeatedly: continuation through look \(k\) requires continuation through look \(k-1\), which itself requires continuation through all earlier looks. Thus the full continuation region in the sequence of statistics is the intersection of the between-boundary conditions across all looks reached so far.

This is the key distinction between the current continuation interval and the path-wise continuation region. At look \(k\), the former is just \(L_k<Z_k<U_k\). The latter additionally requires that no earlier statistic crossed either efficacy boundary. If the design has only efficacy boundaries, being in the central interval at an interim look means monitoring continues. At the final planned look, being inside the interval means neither efficacy boundary was crossed; monitoring ends because the planned analyses are complete.

Justification: The stopping rule ends monitoring as soon as \(Z_j\leq L_j\) or \(Z_j\geq U_j\) at any look \(j\). Therefore, a path is still active at look \(k\) exactly when none of those stopping conditions occurred for \(j=1,\ldots,k\). Negating each stopping condition gives \(L_j<Z_j<U_j\), and requiring this at every look gives the intersection above.

Worked Path Classifications

Worked Example: A Path That Stays in the Central Band

Consider a hypothetical three-look design with lower and upper efficacy limits \((-2.6,2.6)\), \((-2.2,2.2)\), and \((-2.0,2.0)\). Suppose the standardized statistics are obtained from scores \(S_k\) and information values \(I_k\) using \(Z_k=S_k/\sqrt{I_k}\). The observed values are as follows:

At look 1, \(S_1=1.8\) and \(I_1=1.00\), so \(Z_1=1.8/\sqrt{1.00}=1.800\). Since \(-2.6<1.800<2.6\), the path continues.

At look 2, \(S_2=2.1\) and \(I_2=2.25\), so \(Z_2=2.1/\sqrt{2.25}=2.1/1.5=1.400\). Since \(-2.2<1.400<2.2\), and the path continued at look 1, it continues at look 2.

At look 3, \(S_3=2.4\) and \(I_3=4.00\), so \(Z_3=2.4/\sqrt{4.00}=1.200\). This is inside \((-2.0,2.0)\). The path has not crossed either efficacy boundary at any look. Because look 3 is the final planned analysis, monitoring ends without an efficacy boundary crossing. Being inside the final interval does not mean that monitoring continues to another planned look.

Worked Example: Why the Current Look Is Not Enough

For a second hypothetical design, use boundaries \((-2.5,2.5)\) at look 1 and \((-2.2,2.2)\) at look 2. Compare two possible statistic paths that both have \(Z_2=1.9\).

Path A: \(Z_1=0.8\), followed by \(Z_2=1.9\). At look 1, \(-2.5<0.8<2.5\), so monitoring continues. At look 2, \(-2.2<1.9<2.2\), so this path remains in the continuation region.

Path B: \(Z_1=2.6\), followed, hypothetically, by \(Z_2=1.9\). At look 1, \(2.6\geq2.5\), so the upper efficacy boundary is crossed and monitoring stops. The later value \(1.9\) is inside the look-2 interval, but it cannot make this path eligible to continue: under the stopping rule, the look-2 statistic would not be collected as a monitoring decision for this path.

The identical current value does not imply the same monitoring status. Path A continues through look 2; Path B stopped at look 1. A table or diagram that records only \(Z_2\) would conceal this difference.

Worked Example: Crossing a Lower Boundary as the Band Narrows

Consider a hypothetical design with symmetric limits \((-2.8,2.8)\), \((-2.3,2.3)\), and \((-1.9,1.9)\). A statistic path has scores \(S_1=-1.4\), \(S_2=-3.0\), and \(S_3=-4.2\), with information values \(I_1=1.00\), \(I_2=2.25\), and \(I_3=4.00\).

At look 1, \(Z_1=-1.4/\sqrt{1.00}=-1.400\), which is between \(-2.8\) and \(2.8\). At look 2, \(Z_2=-3.0/\sqrt{2.25}=-3.0/1.5=-2.000\), which is between \(-2.3\) and \(2.3\). The path therefore continues through look 2.

At look 3, \(Z_3=-4.2/\sqrt{4.00}=-2.100\). Since \(-2.100\leq-1.9\), the statistic crosses the lower efficacy boundary. Monitoring stops at look 3 for evidence in the negative direction. The statistic was inside the first two continuation intervals, but it is outside the third.

This example also shows why the band should be read using the limits scheduled for each look. A value can be inside a wider earlier interval and beyond a narrower later limit. The observed score becoming more negative is not, by itself, the rule; the decision follows from comparing the standardized statistic with the applicable boundary.

Checking and Interpreting a Boundary Diagram

A boundary diagram summarizes the allowed range at each scheduled look, but it does not display the complete history of any particular path unless the path itself is plotted or its prior status is recorded. For reliable interpretation, use the following sequence:

Identify the scale and look.
Confirm which statistic is being monitored, which analysis look is shown, and the corresponding lower and upper efficacy limits.

Compare with both limits.
If \(Z_k\leq L_k\), the lower limit is crossed. If \(Z_k\geq U_k\), the upper limit is crossed. Otherwise the current value is between the limits.

Check the path history.
If the value is between the current limits, verify that no earlier look crossed either boundary. Only then is the path still eligible to continue.

Account for the final look.
At an interim look, an eligible path continues monitoring. At the final planned look, the decision is complete even if the statistic remains inside the interval.

In practice, several diagnostics help catch errors. First, check that \(L_k<U_k\) at every look; otherwise the stated central interval is empty or invalid. Second, verify that the statistic and boundaries use the same sign convention and scale. Third, trace each path in order, marking the first look at which a boundary is crossed. A path should never be labelled as continuing after that point. Finally, check that the boundary values have been assigned to the correct looks. A correctly drawn band with look labels shifted by one can still lead to incorrect decisions.

The limits need not move monotonically for the path rule to apply. For example, a later interval may be wider than an earlier one. The continuation condition remains the same: the statistic must be between the applicable limits at every look reached. Do not infer the decision from the visual direction of the boundary lines alone.

What the Wedge Does—and Does Not—Tell You

The central wedge gives a compact description of the statistic values that do not trigger an efficacy stop at each scheduled look. It is especially useful for checking whether a path is currently inside the boundaries and for communicating the monitoring plan. A path plot can show how a statistic evolves relative to both directional limits; a look-by-look table makes the same decision rule explicit in a form that is easy to audit.

Neither representation replaces the other in every setting. A plot makes the changing width and shape of the band visible, but can obscure the exact numerical comparison or the status of a path that crossed earlier. A table is precise about each value and decision, but may make the overall boundary shape less apparent. The logical rule \(L_k<Z_k<U_k\) at every look reached is the standard that both displays should implement.

A terminal-only comparison is not equivalent to sequential monitoring. Checking only whether the final statistic is between the final limits misses an earlier crossing; checking only the final statistic against a conventional fixed-sample cutoff also does not reconstruct the planned sequential stopping rule. Likewise, the geometry of a wedge alone does not establish that a design has a specified Type I error rate or power. Those properties depend on the joint distribution of the statistics across looks and on how the boundaries were chosen. The wedge describes the decision region; it is not by itself a calibration argument.

Common Mistakes

Using only the current statistic. “\(Z_k\) is inside the limits, so monitoring continues” is incomplete. The correction is to confirm that every earlier look was also inside its limits.
Reversing the directional interpretation. The lower boundary is crossed by a sufficiently small statistic, and the upper boundary by a sufficiently large statistic. Use the stated sign convention rather than judging direction from a plot’s orientation.
Assuming a boundary must be crossed exactly. Under the rule used here, values equal to or beyond a limit stop monitoring: \(Z_k\leq L_k\) or \(Z_k\geq U_k\). Continuation requires strict inequalities.
Treating “inside at the final look” as ongoing monitoring. At the final planned analysis, monitoring ends whether or not an efficacy boundary is crossed. An inside value means no efficacy boundary crossing, not an extra analysis.
Assuming every wedge is symmetric or monotone. Symmetry and boundary shape are properties of a particular design, not requirements of the continuation rule. Compare the statistic with the actual limits at each look.

Key takeaway: At each look, the interval between the lower and upper efficacy limits is the local continuation interval. A statistic path remains eligible only if it has stayed inside every such interval up to that look. The final look ends monitoring even when the path remains inside the central band.

Check Your Understanding

Answer each question using the path-wise continuation rule. The solutions follow the questions.

At look 1, the limits are \((-2.4,2.4)\), and \(Z_1=2.4\). Does the path continue?
At two successive looks, the limits are \((-2.6,2.6)\) and \((-2.1,2.1)\). A path has \(Z_1=-0.5\) and \(Z_2=-2.0\). Is it still eligible to continue after look 2?
In the same design as Question 2, another path has \(Z_1=-2.7\) and a hypothetical later value \(Z_2=0.2\). Is the path eligible to continue at look 2? Explain.
Why does a statistic inside the final look’s interval not imply that another planned analysis will occur?
What additional information is needed before a boundary diagram can establish a design’s Type I error rate?

Solutions. (1) No. Since \(Z_1=2.4\geq U_1=2.4\), the upper boundary is crossed; equality counts as crossing under the stated rule. (2) Yes. At look 1, \(-2.6<-0.5<2.6\); at look 2, \(-2.1<-2.0<2.1\). The path is inside at both looks and is eligible to continue if look 2 is an interim analysis. (3) No. At look 1, \(-2.7\leq-2.6\), so the lower boundary is crossed and monitoring stops. The later value cannot restore continuation. (4) The final look is the last scheduled analysis, so monitoring ends there whether or not an efficacy boundary is crossed. (5) The joint distribution of the test statistics across looks and the procedure used to choose the boundaries are needed; the band’s geometry alone does not determine the probability of a Type I error.