Why Compare Boundary Shapes?
A group sequential design sets a threshold for the standardized test statistic at each planned analysis. Plotting those thresholds against information fraction makes a design’s geometry visible: some boundaries are relatively flat, while others start high and decline toward the final analysis. This helps describe when a design is comparatively permissive or stringent. The shape alone, however, does not establish that two designs have the same Type I error or power.
Here the focus is a one-sided upper efficacy boundary. Let \(t_j=I_j/I_{\max}\) be the information fraction at look \(j\), where \(I_j\) is the information available at that look and \(I_{\max}\) is the planned maximum information. Thus \(0<t_j\leq 1\), and the final planned look has \(t_j=1\). The plotted vertical coordinate is the critical value for the standardized statistic \(Z_j\), not an effect estimate in the original measurement units.
The usual geometric comparison assumes a correctly specified information schedule, a valid standardized statistic, and the planned analysis times. In the canonical group sequential model under \(H_0\), the standardized statistics have mean zero, variance one, and joint normal dependence with \(\operatorname{Corr}(Z_i,Z_j)=\sqrt{t_i/t_j}\) for \(t_i\leq t_j\). This dependence reflects the accumulation of information; the statistics at successive looks are not independent. A valid design also requires the boundary constants to be calibrated for the intended Type I error, sidedness, and look schedule.
The Three Geometries
The standard functional forms use a positive constant \(a\). For the Wang-Tsiatis family, \(\Delta\) is the shape parameter. The forms below describe the shape; in an actual design, the constants are chosen or calibrated for the design’s error-rate target and look schedule.
The Pocock boundary is horizontal: its critical value does not change with information fraction. The O’Brien-Fleming boundary is steep near the beginning of the information range and falls toward its final value. The Wang-Tsiatis family includes these two shapes as \(\Delta=0\) and \(\Delta=1/2\), respectively. For \(0<\Delta<1/2\), its curve declines between those endpoint shapes in the sense that, with a shared final value, it is above the flat boundary and below the O’Brien-Fleming boundary at every \(0<t<1\).
That last comparison is geometric, not an error-rate result. In a calibrated Pocock, O’Brien-Fleming, or Wang-Tsiatis design, the constants need not be equal. A design’s actual plotted curve depends both on its shape and on its calibrated constant.
Worked Example: Plotting the Three Shapes on a Common Scale
Suppose the goal is to compare shapes alone, so all three curves are assigned the same hypothetical final cutoff \(a=2.10\). Use \(\Delta=0.25\) for the Wang-Tsiatis curve. These values are illustrative coordinates, not cutoffs claimed to have been calibrated to a particular Type I error.
At \(t=0.25\), the Pocock value is \(2.10\). The O’Brien-Fleming value is \(2.10/\sqrt{0.25}=4.20\). The Wang-Tsiatis value is \(2.10(0.25)^{-0.25}=2.10\sqrt{2}=2.970\).
At \(t=0.50\), the corresponding values are \(2.10\), \(2.10/\sqrt{0.50}=2.970\), and \(2.10(0.50)^{-0.25}=2.10(1.1892)=2.497\).
At \(t=0.75\), they are \(2.10\), \(2.10/\sqrt{0.75}=2.425\), and \(2.10(0.75)^{-0.25}=2.10(1.0746)=2.257\). All three equal \(2.10\) at \(t=1\).
| Information fraction \(t\) | Pocock | O’Brien-Fleming | Wang-Tsiatis, \(\Delta=0.25\) |
|---|---|---|---|
| 0.25 | 2.100 | 4.200 | 2.970 |
| 0.50 | 2.100 | 2.970 | 2.497 |
| 0.75 | 2.100 | 2.425 | 2.257 |
| 1.00 | 2.100 | 2.100 | 2.100 |
Plot \(t\) on the horizontal axis and the critical value on the vertical axis. These coordinates show a flat Pocock curve, a sharply declining O’Brien-Fleming curve, and an intermediate Wang-Tsiatis curve. In this matched-final-value comparison, the O’Brien-Fleming threshold is highest at each interim fraction, and the Wang-Tsiatis threshold lies between it and Pocock’s.
What the Wang-Tsiatis Parameter Changes
A short calculation makes the role of \(\Delta\) precise. Taking logarithms of the Wang-Tsiatis boundary gives \(\log b_{\Delta}(t)=\log a_{\Delta}-\Delta\log t\). Therefore, its log-boundary changes by \(-\Delta\) for a one-unit increase in log information fraction. Equivalently, the proportional rate of change is:
This is the boundary’s elasticity with respect to information fraction. A larger \(\Delta\) means a greater proportional decline as information accumulates. Differentiating with respect to \(t\) also shows that, for \(\Delta>0\), the boundary slopes downward and is convex:
The formulas explain two features of the plot. First, every positive-\(\Delta\) curve declines as information increases. Second, the rate of decline is most pronounced at small \(t\), and the curve flattens as \(t\) approaches one. At \(\Delta=0\), both derivatives are zero and the curve is flat. At \(\Delta=1/2\), the Wang-Tsiatis form is the O’Brien-Fleming form.
Worked Example: Inferring a Wang-Tsiatis Shape
Suppose a hypothetical Wang-Tsiatis curve has final value \(b(1)=2.00\) and value \(b(0.25)=3.00\). Find its shape parameter and its boundary value at \(t=0.50\).
Since \(b(1)=a_{\Delta}(1)^{-\Delta}=a_{\Delta}\), the constant is \(a_{\Delta}=2.00\). At \(t=0.25\), the boundary equation gives \(3.00=2.00(0.25)^{-\Delta}=2.00(4)^{\Delta}\). Thus \(4^{\Delta}=1.50\), so
The value at \(t=0.50\) is therefore \(2.00(0.50)^{-0.2925}=2.00(2^{0.2925})\approx 2.450\). Because \(0.2925\) lies between zero and \(1/2\), this curve is geometrically between the flat Pocock and O’Brien-Fleming shapes when the final values are matched. This calculation identifies the curve’s shape; it does not determine its calibrated Type I error.
Reading a Boundary at Planned Looks
A curve is useful for visual comparison, but the group sequential rule is evaluated at the scheduled information fractions. For example, a design with looks at \(t=0.20\), \(0.50\), and \(1.00\) uses the boundary values at those three points. The curve between the points helps communicate the shape; it does not create additional analyses or decision opportunities.
Worked Example: Comparing Interim Stringency at a Scheduled Look
Use a shared hypothetical final cutoff of \(2.40\), and compare Pocock, O’Brien-Fleming, and Wang-Tsiatis with \(\Delta=0.30\) at \(t=0.20\) and \(t=0.50\). At \(t=0.20\), the thresholds are \(2.40\), \(2.40/\sqrt{0.20}=5.367\), and \(2.40(0.20)^{-0.30}=2.40(1.6207)=3.890\), respectively.
At \(t=0.50\), they are \(2.40\), \(2.40/\sqrt{0.50}=3.394\), and \(2.40(0.50)^{-0.30}=2.40(1.2311)=2.955\). At both interim fractions, the ordering is Pocock, then Wang-Tsiatis, then O’Brien-Fleming from lowest to highest threshold.
If \(Z=3.00\) at the \(t=0.50\) look, it exceeds the illustrative Pocock and Wang-Tsiatis thresholds but not the illustrative O’Brien-Fleming threshold. This comparison illustrates the geometric meaning of a higher or lower threshold at a particular look. It is not evidence that these three hypothetical curves control the same Type I error.
Comparisons, Diagnostics, and Limitations
| Feature | Pocock | O’Brien-Fleming | Wang-Tsiatis |
|---|---|---|---|
| Shape | Flat | High early, declining toward the final look | Decline controlled by \(\Delta\) |
| Parameter form | Constant boundary | Information exponent \(1/2\) | Information exponent \(\Delta\) |
| Matched final value | Same threshold at every look | Largest interim threshold for \(0<t<1\) | Between the two shapes when \(0<\Delta<1/2\) |
The main alternative to a shape-only comparison is to compare the actual calibrated boundaries for the intended design. That comparison accounts for the constants and the planned schedule of looks; it is necessary before drawing conclusions about operating characteristics. The boundary geometry can describe where each design places its thresholds, but by itself it does not calculate the probability of crossing a boundary under \(H_0\), the probability of stopping for efficacy under an alternative, or the expected sample size.
Several checks help prevent misleading interpretations:
- Check the final value. For these forms, \(b(1)=a\). If a plotted curve does not meet its stated final cutoff, check the information scale or formula.
- Check the horizontal scale. The formulas use information fraction, not necessarily the fraction of participants enrolled. Those fractions may differ when information accrues unevenly.
- Check the calibration claim. Equal constants make a clean geometric comparison, but do not imply equal overall Type I error. Calibration depends on the full planned design, including the looks.
- Check the decision locations. A continuous-looking curve does not authorize unplanned tests between scheduled looks.
These curves are also limited to the specified one-sided efficacy-boundary comparison. A different sidedness, testing rule, or analysis schedule requires its own boundary specification and calibration. Nor does a lower threshold at one look alone establish that one design is preferable overall: the geometry is one feature of a sequential design, not a complete summary of its operating characteristics.
Check Your Understanding
Answer each question using the boundary shapes and distinctions developed above.
- For a Wang-Tsiatis boundary with \(\Delta=0.20\) and final value \(a=2.50\), calculate the threshold at \(t=0.25\).
- With equal final values, which of the three shapes has the highest threshold at an interim fraction when \(0<\Delta<1/2\)?
- A plotted curve is horizontal and has final value \(2.30\). Which shape does it represent, and what is its value at \(t=0.40\)?
- Why does setting the same final cutoff for three curves not establish that the corresponding designs have the same Type I error?
Solutions. (1) \(2.50(0.25)^{-0.20}=2.50(4^{0.20})\approx 3.299\). (2) O’Brien-Fleming, because \(t^{-1/2}>t^{-\Delta}>1\) when \(0<t<1\) and \(0<\Delta<1/2\). (3) Pocock; its threshold is constant, so the value is \(2.30\) at \(t=0.40\). (4) Overall Type I error depends on the calibrated boundary constants and the planned look schedule, including dependence among the statistics. A shared final value specifies only one point on each curve.