Reading the Inner Wedge
A two-sided group sequential design has scheduled looks at accumulating information. At each look, the monitoring statistic can be compared with a lower and an upper efficacy limit. In a boundary plot, the region between those limits is the inner wedge: a path inside it has not crossed either efficacy limit at that look. The plot is useful only when its axes, limits, and stopping convention are read together.
The geometric interpretation is distinct from the calibration of a design. A plot can show where a particular statistic path lies, but the picture alone does not show the probability of that path, establish the type I error rate, or prove that the limits are appropriate. Here the task is to interpret the region and the decisions it represents.
The sign of \(Z_j\) has a prespecified meaning. For example, positive values might favor the experimental treatment and negative values might favor the control. The upper limit then represents evidence in the positive direction and the lower limit evidence in the negative direction. “Lower” and “upper” describe locations on the statistic axis; they do not by themselves mean “less important” and “more important.”
For the usual canonical group sequential model, information accrues at prespecified looks \(0<t_1<\cdots<t_K=1\). A standardized statistic at look \(j\) is formed from the cumulative score or estimate and its standard error. Under the null, a common reference model has \(Z_j\sim N(0,1)\), with independent increments on the information scale. In that model, for \(i<j\), the correlation is \(\operatorname{Corr}(Z_i,Z_j)=\sqrt{t_i/t_j}\). These assumptions support probability calculations for the design. The geometric rule itself—whether an observed value is between two specified limits—is a comparison and does not require normality. In application, the statistic, look schedule, limits, and equality convention must be specified in advance for the plot to represent the planned rule.
From a Point on the Plot to a Decision
At a particular look, plot the observed \(Z_j\) at the corresponding information value \(t_j\). A point strictly between \(L_j\) and \(U_j\) is inside the inner wedge, so neither efficacy limit has been reached at that look. A point on or beyond either limit is outside the continuation region and calls for stopping for efficacy. The decision is based on the first look at which a limit is reached, not on whether a later hypothetical value would return to the wedge.
Because the limits can change from look to look, the wedge is generally bounded by two curves or step functions rather than by two horizontal lines. The relevant comparison at look \(j\) is always with \(L_j\) and \(U_j\), not with a neighboring look’s limits. A rising path can remain in the wedge at one look and cross the upper limit at the next; similarly, a falling path can cross the lower limit.
Worked Example: A Path Reaches the Upper Limit
Consider a hypothetical four-look design with limits \((-3.00,3.00)\), \((-2.50,2.50)\), \((-2.20,2.20)\), and \((-2.00,2.00)\) at looks 1 through 4. The pairs list \((L_j,U_j)\). Suppose the observed effect estimates and standard errors, in the same effect units, are respectively \((0.12,0.20)\), \((0.20,0.18)\), \((0.36,0.16)\), and \((0.42,0.20)\). For this example, calculate \(Z_j=\widehat{\theta}_j/SE_j\), where \(\widehat{\theta}_j\) is the estimate at look \(j\).
At look 1, \(Z_1=0.12/0.20=0.600\), which lies between \(-3.00\) and \(3.00\). At look 2, \(Z_2=0.20/0.18=1.111\), which lies between \(-2.50\) and \(2.50\). At look 3, \(Z_3=0.36/0.16=2.250\), which is at least \(U_3=2.20\). The path therefore reaches the upper efficacy limit at look 3 and stops for efficacy there. The look-4 values are not used to reverse that decision.
Worked Example: A Path Reaches the Lower Limit
In a hypothetical three-look design, the lower and upper limits are \((-3.00,3.00)\), \((-2.50,2.50)\), and \((-2.10,2.10)\). Suppose the effect estimates and standard errors are \((-0.10,0.20)\), \((-0.54,0.20)\), and \((-0.70,0.25)\). Use \(Z_j=\widehat{\theta}_j/SE_j\) to locate the path.
At look 1, \(Z_1=-0.10/0.20=-0.500\), and \(-3.00<-0.500<3.00\), so the point is inside the wedge. At look 2, \(Z_2=-0.54/0.20=-2.700\), and \(-2.700\leq L_2=-2.50\). The lower efficacy limit is reached at look 2, so monitoring stops for efficacy in the negative direction. The look-3 estimate would give \(Z_3=-0.70/0.25=-2.800\), but it is not a later decision point for this trial: after stopping at look 2, that scheduled look is not reached.
What Staying Inside the Wedge Does—and Does Not—Guarantee
The statement “the path is inside the wedge” describes the efficacy decision at one look. It does not automatically mean that the trial continues. A protocol may also stop for safety, futility, loss of feasibility, or another prespecified reason. These rules are separate from the two efficacy limits and must be considered when deciding whether a later look is actually reached.
The precise path statement can be derived directly from the efficacy-only stopping rule. Suppose the efficacy limits are the only stopping rules, and let \(\tau\) be the first look at which either limit is reached. If no such limit is reached before look \(m\), then the trial reaches look \(m\). Conversely, if it reaches look \(m\), it cannot have reached either efficacy limit at any earlier look. Thus:
The left side includes the possibility that a limit is reached at look \(m\); it says the trial was not stopped by an efficacy limit before that look. The strict inequalities on the right reflect the stated convention that equality with a limit is a stopping event. If another stopping rule applies, the right side must also require that no such rule stops the trial before look \(m\). Being inside the efficacy wedge at all earlier looks is then necessary for avoiding an efficacy stop, but is not sufficient to establish that the next look occurs.
Worked Example: Inside the Efficacy Wedge but Stopped for Safety
Suppose a hypothetical trial has an efficacy interval \((-3.00,3.00)\) at look 1. The estimated effect is \(0.08\) units and its standard error is \(0.10\) units, giving \(Z_1=0.08/0.10=0.800\). Since \(-3.00<0.800<3.00\), the statistic is inside the inner wedge at look 1.
Now suppose the protocol also specifies a separate safety stopping rule, and the observed safety data meet that rule at look 1. The trial stops for safety, despite not reaching either efficacy limit. It does not reach look 2. Therefore, the condition \(L_1<Z_1<U_1\) alone does not imply that look 2 occurs when other stopping rules are in force. The correct interpretation is that efficacy monitoring would continue from look 1 if no other applicable rule stopped the trial.
How to Read and Check a Boundary Plot
A reliable interpretation follows the path in chronological order and checks the limits at each scheduled look. A quick visual impression can be misleading when the axes are reversed, the boundary values are labeled for a different statistic, or the plot connects discrete looks with lines that suggest decisions between scheduled analyses. Unless the design includes continuous monitoring, decisions are made at the scheduled looks; a curve between them is a visual guide, not an extra set of analysis times.
Confirm which axis gives information or look number and which gives the signed statistic. Verify that the plotted path uses the same statistic and scale as the listed limits.
Use the lower and upper values for that specific look. Confirm that \(L_j<U_j\); otherwise there is no nonempty inner interval as defined here.
Compare \(Z_j\) with both limits. It is in the efficacy continuation region only if \(L_j<Z_j<U_j\); equality counts as reaching a limit under this convention.
If the point is inside, check for other prespecified stopping rules before concluding that the next look will occur. If a limit has been reached, record the first such look and direction.
These checks also reveal common plot failures. If a displayed point is called “continuing” even though it equals a limit, the decision convention has been overlooked. If the path is compared with one constant pair of limits despite boundaries changing by look, the wrong region has been used. If the conclusion says the trial must proceed because \(Z_j\) lies inside the efficacy wedge, it has ignored any additional stopping rules. Finally, a path description is not a probability statement: observing a statistic inside the wedge does not itself quantify how likely that path was under \(H_0\) or under an alternative.
Why the Inner Wedge Matters
The inner wedge gives a compact visual summary of the decision not to stop for efficacy at a particular look. It makes the two-sided nature of monitoring visible: a sufficiently positive statistic may stop in one direction, while a sufficiently negative statistic may stop in the other. A one-sided plot, or a plot showing only one limit, can conceal the fact that evidence in either direction has a planned efficacy boundary.
Compared with a fixed-sample decision rule, a group sequential rule checks for efficacy more than once and uses look-specific limits. It is therefore not correct to apply a single fixed-sample critical value at every look and regard repeated crossing opportunities as if there were only one analysis. The boundary plot displays the planned decisions, while error-rate calibration requires the joint behavior of the statistics across looks. That calibration is not supplied by the wedge geometry alone.
The wedge also has limits as a communication tool. It does not show the treatment effect in original units unless the statistic is mapped back to that scale, and it does not display uncertainty intervals unless those are added separately. Its central message is narrower: at a scheduled look, a value strictly between the two efficacy limits has not triggered an efficacy stop. That message should be reported separately from whether the protocol’s other rules allow the trial to continue.
Check Your Understanding
Work through the questions, then compare your reasoning with the solutions.
- At a look with limits \(L_j=-2.40\) and \(U_j=2.40\), classify \(Z_j=-2.40\), \(Z_j=0.70\), and \(Z_j=2.41\) under the stated equality convention.
- A trial has only efficacy limits. At look 1, \(Z_1=1.10\) lies between \(-3.00\) and \(3.00\). Does this establish that look 2 is reached?
- In a three-look plan, the limits are \((-3.00,3.00)\), \((-2.50,2.50)\), and \((-2.10,2.10)\). The path is \(Z_1=0.20\), \(Z_2=-2.00\), and \(Z_3=2.15\). At which look and in which direction does efficacy monitoring stop?
- Explain why a statistic inside the efficacy wedge at look 1 may still be followed by no look 2.
Solutions. 1. \(Z_j=-2.40\) equals the lower limit and stops at the lower efficacy boundary; \(0.70\) is strictly between the limits and is inside the wedge; \(2.41>2.40\) crosses the upper limit and stops there. 2. Yes, under the stated assumption that efficacy limits are the only stopping rules: the statistic has not reached either limit at look 1, so the efficacy rule allows look 2. 3. At look 1, \(0.20\) is inside \((-3.00,3.00)\); at look 2, \(-2.00\) is inside \((-2.50,2.50)\); at look 3, \(2.15\geq2.10\), so the upper efficacy limit is reached at look 3. 4. A separate rule, such as a prespecified safety stop, may stop the trial even though neither efficacy limit has been reached. The inner wedge describes the efficacy decision only.