Deriving Symmetric Boundaries From Sign Invariance

Why Derive the Mirror Condition?

In a two-sided group sequential design, the efficacy rule may stop a trial for evidence in either direction. A natural design choice is to use an upper boundary for evidence of a positive effect and a lower boundary that is its negative. That choice is not merely a visual convention: under a zero-effect null, it follows from a symmetry of the joint distribution of the statistics across all planned looks. The joint qualification matters because stopping depends on an entire path, not on one statistic considered in isolation.

The earlier material on mirror symmetry under a zero-effect null motivates this property. Here the argument is made directly: reverse the signs of every statistic in a path, and show how this transformation maps one directional first-crossing event to the other. The result is conditional on the null model and on the form of the stopping rule; sign symmetry alone does not force every possible design to have mirrored boundaries.

Joint Null Model and Sign Invariance

Suppose a study has \(K\) planned analyses. Let \(Z_i\) be the standardized test statistic at look \(i\), where \(i=1,\ldots,K\). Under the null hypothesis \(H_0\) of zero effect, assume the vector \(\mathbf{Z}=(Z_1,\ldots,Z_K)\) has a centrally symmetric joint distribution:

\mathbf{Z}\mid H_0 \ \overset{d}{=}\ -\mathbf{Z}\mid H_0 .

This equality in distribution means that for any set of possible paths \(A\), \(P(\mathbf{Z}\in A\mid H_0)=P(-\mathbf{Z}\in A\mid H_0)\). A standard setting is the canonical joint-normal model: the statistics have a joint multivariate normal distribution with mean vector zero and covariance matrix determined by the information at each look. For example, with information fractions \(0<t_1<\cdots<t_K=1\), the canonical covariance is

\operatorname{Cov}(Z_i,Z_j\mid H_0) = \sqrt{\frac{t_i}{t_j}}, \qquad i\leq j,

with the other entries supplied by symmetry of the covariance matrix. The covariance formula is one way to specify dependence across looks; the derivation below needs only central symmetry of the full joint null distribution.

This model is appropriate when the standardized statistics have a zero-mean joint normal null distribution, as under the usual independent-increment assumptions for the canonical model. In applications where that distribution is an approximation, the symmetry result is correspondingly approximate. The argument also presumes that the statistic is oriented consistently: positive values indicate the positive-effect direction at every look.

Definition: A sequential null distribution is sign-invariant if the full statistic vector has the same null distribution after every component is multiplied by \(-1\). For two-sided efficacy boundaries \(U_i\) and \(L_i\), the mirrored-boundary condition is \(L_i=-U_i\) at every look \(i\).

First-Crossing Events and the Derivation

Let \(U_i\) be the upper efficacy boundary and \(L_i\) the lower efficacy boundary at look \(i\). Assume that the study stops for upper efficacy when \(Z_i\geq U_i\), and stops for lower efficacy when \(Z_i\leq L_i\). If neither condition holds, it continues. For clarity, this derivation considers efficacy stopping only; any additional stopping rule must also respect sign reversal for the same argument to apply to the full design.

Define \(E_+\) as the event that the first efficacy crossing is at the upper boundary, and \(E_-\) as the event that the first efficacy crossing is at the lower boundary. Their path descriptions are

E_+ = \bigcup_{i=1}^{K} \left\{ L_j<Z_j<U_j\ \text{for every }j<i,\quad Z_i\geq U_i \right\},

E_- = \bigcup_{i=1}^{K} \left\{ L_j<Z_j<U_j\ \text{for every }j<i,\quad Z_i\leq L_i \right\}.

The events are disjoint: a path cannot have its first efficacy crossing in both directions. Under mirrored boundaries, \(L_i=-U_i\), consider a path in \(E_+\) and reverse all its signs, setting \(W_i=-Z_i\). If it stayed between the boundaries before look \(i\), then

-U_j<Z_j<U_j \quad\Longrightarrow\quad -U_j<W_j<U_j,

and if it crossed the upper boundary at look \(i\), then

Z_i\geq U_i \quad\Longrightarrow\quad W_i\leq -U_i=L_i.

Thus sign reversal maps every upper first-crossing path into a lower first-crossing path, and maps every lower first-crossing path back into an upper one. Since the joint null distribution is sign-invariant, the two mapped events have equal probability. Therefore,

P(E_+\mid H_0)=P(E_-\mid H_0).

If the design’s total two-sided type I error is \(\alpha\), and rejection occurs on either efficacy crossing, then the disjointness of \(E_+\) and \(E_-\) gives

P(E_+\cup E_-\mid H_0)=\alpha, \qquad P(E_+\mid H_0)=P(E_-\mid H_0)=\frac{\alpha}{2}.

This conclusion concerns the whole sequential crossing event. It does not follow by treating the looks as independent or by assigning a separate \(\alpha\) to each look. Dependence across looks is already part of the joint distribution whose sign invariance is being used.

Worked Example: Mirroring a Two-Look Boundary Vector

Consider a hypothetical two-look design under a sign-invariant zero-effect null. Suppose the planned upper limits are \(U_1=2.50\) and \(U_2=2.10\). Find the mirrored lower limits and verify the path mapping for a possible statistic vector \((Z_1,Z_2)=(2.60,1.00)\).

The mirror condition is \(L_i=-U_i\). Hence

L_1=-2.50,\qquad L_2=-2.10.

At look 1, the given path has \(Z_1=2.60\geq 2.50\), so it crosses the upper boundary at the first look. Reversing its signs gives \((-2.60,-1.00)\). The reversed path has \(-2.60\leq -2.50\), so it crosses the lower boundary at the first look. The second-look value does not affect the stopping decision because the trial has already stopped at look 1.

This example verifies the event mapping for one path; the equality of directional probabilities comes from applying that mapping to every path and using the joint null distribution’s sign invariance. The particular boundary values alone do not establish that the total type I error is a chosen value: the design still needs calibration to its target \(\alpha\).

What Symmetry Does—and Does Not—Determine

Sign invariance supplies a clean way to impose equal directional error when a two-sided design is intended to treat positive and negative effects alike. With mirrored boundaries, a valid calibration of one directional crossing probability also determines the other: it is the same probability. The total null rejection probability is then twice either directional probability.

The converse requires care. Equal directional crossing probabilities do not, by themselves, prove that every look-specific boundary satisfies \(L_i=-U_i\). Aggregate probabilities can be equal for reasons other than a coordinate-by-coordinate mirror, and more than one boundary vector can yield the same total error. The mirror condition is a structural design choice justified by sign invariance when the decision rule is meant to be directionally symmetric; it is not a uniqueness theorem for all calibrated designs.

Likewise, sign invariance does not imply symmetry if the stopping rule itself distinguishes directions. Direction-specific futility rules, different clinical thresholds, or unequal error budgets can break the correspondence between the two rejection events. If such features are present, a designer must analyze the actual joint decision regions rather than infer equal directional errors from the null distribution alone.

Worked Example: Detecting an Asymmetric Rule

Suppose a hypothetical two-look design has upper limits \((2.40,2.00)\) and lower limits \((-2.00,-1.70)\). Is the design mirrored? What boundary values arise when the original design is transformed by reversing all statistic signs?

At look 1, the mirror of the upper limit \(2.40\) would be \(-2.40\), not \(-2.00\); at look 2, the mirror of \(2.00\) would be \(-2.00\), not \(-1.70\). The design is not mirrored.

Under sign reversal, the original upper limits become the negatives of the lower limits, and the original lower limits become the negatives of the upper limits. The sign-transformed design therefore has upper limits

U'_1=-L_1=2.00,\qquad U'_2=-L_2=1.70,

and lower limits

L'_1=-U_1=-2.40,\qquad L'_2=-U_2=-2.00.

Sign invariance equates an upper-crossing probability in the original design with the corresponding lower-crossing probability in this transformed design. It does not equate the upper and lower probabilities within the original design, because the transformed limits differ from the original limits. Thus one cannot claim equal tail error merely because the null vector is sign-invariant.

Worked Example: Checking the Joint Model and an Early-Look Tail

Suppose a hypothetical canonical design has two analyses at information fractions \(t_1=0.40\) and \(t_2=1.00\). Its null covariance is

\operatorname{Corr}(Z_1,Z_2\mid H_0) = \sqrt{\frac{0.40}{1.00}} = \sqrt{0.40} \approx 0.6325.

The assumed joint null model is a mean-zero bivariate normal distribution with unit variances and this correlation. Its density is centrally symmetric, so the sign-invariance assumption holds. Now set \(U_1=2.60\), \(U_2=2.00\), and use their negative mirrors as the lower limits. At the first look, \(Z_1\sim N(0,1)\), so the upper first-look crossing probability is

P(Z_1\geq 2.60\mid H_0) = 1-\Phi(2.60) \approx 0.004661,

where \(\Phi\) is the standard normal cumulative distribution function. By symmetry, the lower first-look crossing probability is also \(0.004661\). The combined probability of stopping at the first look for efficacy is therefore \(2(0.004661)=0.009322\). This is only the first-look contribution; the final-look first-crossing probability must also account for the joint distribution and continuation at look 1. Sign invariance guarantees equality of the two complete directional probabilities, but it does not make the looks independent or supply the final-look probability from a marginal tail alone.

Diagnostics, Alternatives, and Limitations

Before relying on the mirror argument, check both the statistical model and the decision rule. A useful diagnostic sequence is:

Check the null center. The joint null distribution must be centered at zero on the chosen statistic scale. A nonzero null margin requires a different coordinate system and cannot be handled by this zero-centered argument as written.
Check joint sign symmetry. A zero mean at each look is not sufficient by itself; the full joint distribution must be invariant under simultaneous sign reversal. Joint normality with mean zero is a standard sufficient condition.
Check every look’s boundaries. Verify \(L_i=-U_i\) for each \(i\), not just at the final analysis.
Check the rest of the rule. Any additional stopping or rejection condition must also map to itself under sign reversal. Otherwise the efficacy-boundary argument may not describe the full design.
Check the calibration target. Equality of directional probabilities follows from symmetry; it does not prove that the total rejection probability equals the target \(\alpha\).

A direct numerical calculation of the joint first-crossing probabilities is an alternative when the model or boundaries are asymmetric. Simulation can also estimate operating characteristics, provided it simulates the joint path under the specified null model and applies the complete stopping rule. A simpler single-look normal calculation cannot replace a sequential calculation, because it omits the probability of reaching later looks and the dependence among statistics.

The central limitation is that symmetry is a null-distribution and rule property, not a guarantee about power under nonzero effects. Under an alternative, the joint distribution is generally shifted, so positive and negative crossing probabilities need not be equal. Nor does the mirror condition establish clinical equivalence of the consequences of stopping in the two directions. It formalizes statistical symmetry under the stated null assumptions.

Key takeaway: If the joint null statistic vector is invariant under simultaneous sign reversal and the complete efficacy rule is also symmetric, then mirrored limits \(L_i=-U_i\) pair every upper first-crossing path with a lower one. The directional null error probabilities are equal, but the total error still requires calibration.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

Under what distributional property does simultaneous sign reversal preserve probabilities of statistic paths?
A design has \(U_1=2.30\) and \(U_2=1.95\). What lower limits satisfy the mirror condition?
Why does \(P(E_+\mid H_0)=P(E_-\mid H_0)\) not, by itself, show that the total type I error is \(0.05\)?
For upper limits \((2.2,1.9)\) and lower limits \((-2.2,-1.6)\), is the design mirrored at both looks?
Does a zero mean for each \(Z_i\), without any further joint-distribution assumption, establish sign invariance?

Solutions. (1) The full joint null distribution must be centrally symmetric: \(\mathbf{Z}\overset{d}{=}-\mathbf{Z}\). (2) The lower limits are \((-2.30,-1.95)\). (3) Equality of the two disjoint directional probabilities does not specify their sum; the total must be calibrated to the target, and equals twice either direction only after that total is known. (4) No. The first pair is mirrored, but the second lower limit should be \(-1.9\), not \(-1.6\). (5) No. Marginal zero means do not alone establish the required symmetry of the joint path distribution.