Two Different Features of a Sequential Design
A two-sided group sequential design has upper and lower efficacy limits at each interim analysis. It is tempting to describe any difference between those limits as a difference in tail allocation. That conflates two distinct features. Tail allocation concerns how much of the null rejection probability occurs through the upper rather than the lower stopping event. Boundary shape concerns the pattern of the limits across analyses, including how they change over time and whether the lower sequence mirrors the upper one.
The features are related, but they are not interchangeable. In particular, with a centered symmetric null distribution, mirrored limits necessarily give equal upper- and lower-tail crossing probabilities. Thus, under that model, unequal directional error requires non-mirrored limits. Conversely, non-mirrored limits do not by themselves specify how unequal the directional error probabilities are: those probabilities depend on the joint distribution across looks and must be calculated for the full stopping rule.
For example, an upper crossing at look \(i\) requires \(Z_i\geq U_i\) and no efficacy crossing at an earlier look. A lower crossing requires \(Z_i\leq L_i\) and no earlier efficacy crossing. A statistic that crosses an upper limit at an earlier look cannot later contribute to the lower first-crossing probability: the trial has already stopped. This is why the directional probabilities are properties of the entire stopping rule, not merely of the limits at the final look.
Assumptions and the Symmetry Constraint
Use the standard canonical null model for this discussion: under \(H_0\), the vector of test statistics is multivariate normal with mean zero, and its joint distribution is unchanged when every component is multiplied by \(-1\). For the usual information-fraction formulation, \(Z=(Z_1,\ldots,Z_K)\sim N_K(0,\Sigma)\), with \(\operatorname{Corr}(Z_i,Z_j)=\sqrt{t_i/t_j}\) for \(i<j\), where \(t_i\) is the information fraction at look \(i\). The symmetry argument requires the centered, sign-invariant joint null distribution; it does not require independent statistics across looks.
The limits are called mirrored when \(L_i=-U_i\) at every look. Under the stated null model, reflecting the entire observed path to its negative maps every upper first crossing to a lower first crossing. It preserves probability because the joint null distribution is sign-invariant. Therefore the upper and lower first-crossing probabilities must be equal.
This result is a constraint, not a calibration shortcut. If the total null rejection probability is \(0.05\), a mirrored design has \(0.025\) in each direction under this model. It cannot be calibrated to \(0.03\) in the upper direction and \(0.02\) in the lower direction merely by choosing different mirrored limits at different looks. A mirrored sequence can have many possible shapes, but every such shape retains equal directional error under the symmetric null.
Worked Example: Unequal Allocation at One Look
Worked Example: Unequal Allocation at One Look
Suppose a hypothetical one-look test uses a standard normal statistic under \(H_0\), and the total two-sided significance level is \(0.05\). The design assigns \(0.03\) to upper rejection and \(0.02\) to lower rejection. Find the limits.
At one look, the upper event is \(Z\geq U\), so \(P(Z\geq U\mid H_0)=0.03\). Hence \(U\) is the 97th percentile of the standard normal distribution:
The lower event is \(Z\leq L\), so \(P(Z\leq L\mid H_0)=0.02\). Therefore
The limits are not mirrored: \(-U\approx -1.881\), which is not \(L\). The allocated probabilities add to \(0.03+0.02=0.05\), as required. For comparison, an equal split at the same total level would use \(U\approx1.960\) and \(L\approx-1.960\). This one-look calculation makes the connection between limits and tail allocation transparent; with multiple looks, the same quantile calculation at each look does not in general give the directional first-crossing probabilities.
Changing Shape Without Changing Tail Allocation
A boundary’s shape can be described by its sequence of limits over information time. For a mirrored design, one convenient description is a positive profile \(g_1,\ldots,g_K\) and a scale \(c\), giving \(U_i=cg_i\) and \(L_i=-cg_i\). The profile describes the relative pattern across looks; the scale controls the overall distance of the limits from zero. For a target total error \(\alpha\), the scale must be calibrated using the joint null distribution and the complete stopping rule.
Changing the profile changes the shape. To preserve the same total error, the scale will generally need to be recalibrated. Because the limits remain mirrored, the two directional error probabilities remain equal; the shape change does not change the equal allocation. In contrast, keeping a profile fixed while using different upper and lower scales creates non-mirrored limits. The resulting directional error probabilities depend on the full joint crossing calculation, so the scales are not themselves the probabilities.
Worked Example: Changing Shape While Retaining Equal Allocation
Consider a hypothetical two-look design with information fractions \(t_1=0.50\) and \(t_2=1\). Under the canonical null model, the correlation between looks is \(\sqrt{0.50/1} \approx 0.7071\). Compare two mirrored boundary profiles for a target total Type I error of \(0.05\):
The first profile has an interim-to-final limit ratio of \(1.20/1.00=1.20\); the second has a ratio of \(1.50/1.00=1.50\). Thus profile B has a more stringent interim limit relative to its final limit. For each profile, use limits \(U_i=cg_i\) and \(L_i=-cg_i\). Choose \(c\) separately for each profile so that the probability of any efficacy crossing is \(0.05\).
For profile A, the calibration equation is
For profile B, replace \(1.20c_A\) by \(1.50c_B\) and solve the corresponding equation for \(c_B\). These equations use the bivariate normal distribution with correlation \(0.7071\); the cutoffs require numerical calibration. In both designs, reflection symmetry gives \(\alpha_U=\alpha_L=0.025\), even though the boundary shapes differ. The example isolates the distinction: changing the time profile while recalibrating can preserve the tail allocation.
Changing Allocation While Holding a Profile in View
A useful way to specify non-mirrored boundaries is to start with a common positive time profile \(g_i\) and use separate positive scales:
If \(c_U=c_L\), the limits are mirrored and the directional errors are equal under the symmetric null. If \(c_U\neq c_L\), they are not mirrored; unequal directional error is then possible. The actual values of \(\alpha_U\) and \(\alpha_L\) must still be computed from the joint first-crossing events. The profile alone does not determine the allocation, and unequal scales do not imply a particular numerical allocation without that calculation.
Worked Example: Same Time Profile, Different Upper and Lower Limits
For illustration, take the two-look profile \(g=(1.25,1.00)\), so the interim limit is 25% farther from zero than the final limit on either side when the design is mirrored. Now specify \(c_U=1.90\) and \(c_L=2.10\). The upper limits are
and the lower limits are
Both sides retain the same relative time profile: \(2.375/1.900=1.25\) and \(2.625/2.100=1.25\). But the limits are not mirrors, since \(-U_1=-2.375\neq-2.625=L_1\). Under the centered canonical null model, the upper and lower crossing probabilities need not be equal. These limits illustrate a geometric change, not a calibrated error allocation: to claim particular values of \(\alpha_U\) and \(\alpha_L\), one must evaluate the two first-crossing probabilities using the joint null distribution and, if necessary, adjust the scales.
Calibration, Diagnostics, and Alternatives
For a two-look design, for example, the upper first-crossing probability is
with an analogous expression for \(\alpha_L\). The continuation condition at the first look is essential. In designs with more looks, each term for a later crossing includes the requirement that all earlier looks remained in the continuation region. These probabilities are evaluated under the joint null distribution, which accounts for dependence across looks.
A practical diagnostic is to check both the boundary geometry and the calibrated crossing probabilities. First verify whether \(L_i=-U_i\) at every look; if so, equal directional errors should follow under a centered sign-symmetric null. Then verify that the overall error is the sum of the two disjoint first-crossing probabilities and that it matches the target. A result reporting unequal directional errors for mirrored limits under the stated symmetric null signals an implementation, coding, or interpretation problem. Conversely, non-mirrored limits alone do not establish that a specified unequal allocation has been achieved.
The main alternatives answer different design questions. Mirrored limits are appropriate when equal directional allocation is intended under a symmetric null. Non-mirrored limits can be used when the two directions need different operating characteristics, but require separate calibration of the complete stopping rule. Different mirrored shapes can also be compared while retaining equal directional allocation, provided each is calibrated to the desired total error. Per-look tail cutoffs or a simple split of \(\alpha\) do not replace this joint calculation: they generally fail to account for repeated opportunities to cross and dependence between looks.
Common Mistake and Key Takeaway
A common mistake is to say, “I want more null error in the upper tail, so I will keep mirrored boundaries but use a different sequence of limits over time.” The reasoning confuses changing the time profile with changing the directional allocation. Under the centered symmetric null, every mirrored design has equal upper and lower first-crossing probabilities, regardless of its shape. To obtain unequal directional error under that model, the limits must be non-mirrored and calibrated using the joint null distribution.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Under a centered sign-symmetric null, a three-look design has \(L_i=-U_i\) at all three looks. Can its upper and lower first-crossing probabilities be \(0.03\) and \(0.02\)? Explain.
- A one-look test has upper-tail error \(0.01\) and lower-tail error \(0.04\). What are the upper and lower limits on the standard normal scale, to three significant figures?
- Two mirrored designs have different interim-to-final limit ratios, but each is calibrated to total Type I error \(0.05\). What are their directional errors under the stated symmetric null?
- Why is it not generally valid to add the upper-tail probabilities calculated separately at each look to obtain \(\alpha_U\)?
Solutions. (1) No. Reflection maps upper first crossings to lower first crossings, so the probabilities must be equal; their sum cannot be \(0.05\) with that unequal split. (2) \(U=\Phi^{-1}(0.99)\approx2.33\), and \(L=\Phi^{-1}(0.04)\approx-1.75\). The probabilities sum to \(0.05\), and the limits are non-mirrored. (3) Each has \(\alpha_U=\alpha_L=0.025\). The different shapes do not break sign symmetry. (4) A path may cross an upper limit only if it has not already crossed either efficacy limit at an earlier look. In addition, the statistics across looks are dependent under the canonical model. The first-crossing probabilities must include those continuation conditions and use the joint null distribution.