Joint Null Error Across Two Tails

Why the Two Tails Must Be Considered Together

In a two-sided group-sequential design, the null hypothesis can be rejected at an interim analysis because the statistic crosses either an upper efficacy boundary or a lower efficacy boundary. The overall type I error is therefore the probability, under the null, of stopping for efficacy in either direction. It is not enough to calculate a tail probability at one look: the monitoring rule allows several opportunities to cross, and the statistics from those looks are dependent.

The key is to describe the rejection event as a union of first-crossing events. These events partition the ways the design can first reject the null. Their probabilities can then be added, provided each event includes the requirement that no earlier efficacy boundary was crossed. The joint null distribution across looks is essential for evaluating those probabilities.

Assume efficacy stopping is the rejection rule: the trial rejects \(H_0\) if a first crossing occurs at any look. Any futility rule that stops monitoring without rejecting must also be reflected in the continuation conditions; here the formulas focus on the efficacy boundaries. If \(H_0\) is true, the design's two-sided type I error is

The equality follows because the first-crossing events are mutually exclusive: a path can have only one first efficacy crossing. This does not mean that all upper-tail and lower-tail exceedance events at all looks are independent. Rather, the events in the sum are defined to be disjoint, and their probabilities are calculated using the joint distribution of the sequential statistics.

Joint Null Model and First-Crossing Probabilities

For the standard canonical group-sequential null model, assume the statistic vector is jointly normal, centered under \(H_0\), and standardized to variance one at every look. Let the information at look \(i\) be \(t_i\), where \(0<t_1<\cdots<t_K=1\). With independent increments and the usual information-based standardization, the covariance between looks \(i\) and \(j\) is \(\sqrt{t_i/t_j}\) when \(i<j\). Thus

This model is an assumption, not a consequence of having multiple looks. The statistic and information scale must be appropriate, and the joint normal approximation and independent-increment structure must be reasonable for the design. If the actual analysis has a different joint null distribution, the same first-crossing definitions apply, but probabilities must be evaluated under that distribution instead.

Under the canonical model, a first-crossing probability is a multivariate normal probability over a region with continuation constraints at all earlier looks and a boundary crossing at the current look. For example, the probability of an upper first crossing at look \(i\) is an integral over \(z_1,\ldots,z_{i-1}\) restricted to their continuation intervals, and \(z_i\ge u_i\). Later statistics do not enter that event: the design has already stopped for efficacy.

A Two-Look Derivation

For two looks, write \(\rho=\sqrt{t_1/t_2}\). Under the canonical null model, \((Z_1,Z_2)\) is bivariate normal with correlation \(\rho\). The first-look upper and lower probabilities are simply \(P(Z_1\ge u_1\mid H_0)\) and \(P(Z_1\le l_1\mid H_0)\). The second-look upper first-crossing probability is

where \(\phi_2(\cdot,\cdot;\rho)\) is the standard bivariate normal density with correlation \(\rho\). The lower first-crossing probability integrates over \(z_1\in(l_1,u_1)\) and \(z_2\le l_2\). The bounds on \(z_1\) are crucial: they encode that neither boundary was crossed at the first look.

A useful equivalent calculation conditions on the second-look statistic. Since \(Z_1\mid Z_2=z\) is normal with mean \(\rho z\) and variance \(1-\rho^2\), the upper probability can be written as a one-dimensional integral:

This form shows directly how the first-look continuation interval and the dependence between looks affect the second-look crossing probability. The integrand is a marginal density at look 2 multiplied by the conditional probability that look 1 remained inside its continuation interval.

Worked Examples

Symmetry, Diagnostics, and Alternatives

If the null distribution is invariant under changing every \(Z_i\) to \(-Z_i\), and the boundaries are symmetric at every look (\(l_i=-u_i\)), the transformation swaps upper and lower first-crossing events without changing their probabilities. Consequently,

This is a result of joint sign symmetry and symmetric limits, not a general property of every two-sided design. If either the limits or the null distribution are asymmetric, calculate the two directional contributions separately. Also distinguish the total error across both directions from the error in one direction; those quantities coincide only under additional symmetry conditions.

Several checks help detect a faulty calculation. First, verify that each event imposes continuation at all earlier looks. Second, confirm that the reported first-crossing probabilities are nonnegative and that their sum lies between zero and one. Third, for symmetric limits under a sign-symmetric null model, upper and lower contributions should agree up to numerical error. A material discrepancy can indicate a coding, boundary-sign, or integration problem. Finally, compare the computed total with the design's target error: being close to the target is a calibration check, not proof that the model or boundary specification is correct.

The first-crossing sum is usually the clearest approach because it represents the actual stopping rule. An equivalent approach is to compute one minus the probability of remaining in the continuation region at every look. That can be efficient, but the complement must represent paths that have not crossed either boundary at any earlier look. A calculation based only on the marginal tail probability at each look is not an equivalent shortcut: it ignores the pathwise stopping rule and the dependence among looks. The introductory material on two-sided group-sequential decision regions provides the stopping-rule context; the essential calculation here is the joint null probability of each disjoint first-passage region.

Common Mistakes and Limitations

• Using a marginal tail as a first-crossing probability. \(P(Z_i\ge u_i\mid H_0)\) includes paths that may already have crossed at an earlier look. Restrict the earlier statistics to their continuation intervals.
• Adding overlapping events. Upper and lower exceedances defined without a first-crossing condition can overlap across different looks as descriptions of a path. Define the events by the first efficacy crossing before summing.
• Assuming equal directional error without checking symmetry. Equal contributions require both symmetric limits and a sign-symmetric joint null distribution.
• Treating numerical precision as model validity. Accurate integration cannot repair an incorrect covariance model, an inappropriate normal approximation, or incorrectly specified boundaries.

The result is specific to the stated null model and rejection rule. In practice, the joint distribution may only be approximately canonical, and discrete data or nuisance-parameter estimation can affect calibration. If the assumptions are doubtful, the null rejection probability should be checked using a method appropriate to the actual design and analysis. The event decomposition remains useful, but its probabilities must match that setting.