An Unplanned Timing Worked Example

Why an Unplanned Look Changes the Boundary

A group sequential boundary is calibrated for both the amount of type I error spent at each look and the dependence between the test statistics at those looks. If an analysis takes place at an information fraction different from the planned one, that dependence changes. The old boundary at the later look may therefore no longer produce the intended cumulative type I error.

This tutorial works through a two-look example in which the second analysis occurs earlier than planned. The goal is to retain a specified cumulative one-sided type I error target by recalibrating the second efficacy boundary at the actual information fraction. The example assumes the protocol assigns a cumulative alpha target of 0.025 by the terminal analysis; changing the timing does not, in this example, change that target. In a design where a spending function specifies a different cumulative target at the actual time, that function’s target must be used instead.

The canonical model assumes a prespecified test statistic with approximately normal increments under \(H_0\), information that correctly reflects the variance of the accumulating data, and a common information-time scale across analyses. The actual timing must also be treated as fixed or otherwise independent of the interim test results for this simple calibration to apply. There is no futility stopping in the example, and efficacy is assessed only in the positive direction. A two-sided test or a design with additional stopping rules requires the corresponding crossing probabilities to be calibrated.

The correlation formula follows directly from independent increments. If \(t_2>t_1\), write \(B(t_2)=B(t_1)+[B(t_2)-B(t_1)]\), where the bracketed increment is independent of \(B(t_1)\). Since \(\operatorname{Cov}\{B(t_1),B(t_2)\}=t_1\), standardizing gives:

Moving the second look earlier, while holding \(t_1\) fixed, increases the correlation. That change affects the chance of crossing the second boundary among trials that did not already cross the first one. Consequently, recalibration must account for both the marginal probability of crossing at the second look and the overlap with first-look crossings.

Worked Recalibration With an Earlier-Than-Planned Analysis

Suppose the first boundary is \(b_1=2.326\), chosen so that \(P\{Z(t_1)\geq b_1\}\) is approximately 0.0100. The desired cumulative probability of an efficacy crossing by the second, terminal look is 0.0250. Thus, the second look must contribute approximately \(0.0250-0.0100=0.0150\) in probability of crossing at look 2 without having crossed at look 1.

Let \(b_2\) denote the recalibrated second boundary and let \(\rho=\sqrt{2/3}\). The second-look-only crossing probability is the probability of \(Z(t_2)\geq b_2\) while \(Z(t_1)<b_1\). Conditioning on \(Z(t_1)=z\), the canonical bivariate normal model gives \(Z(t_2)\mid Z(t_1)=z\sim N(\rho z,1-\rho^2)\). Therefore, the required probability is:

where \(\phi\) and \(\Phi\) are the standard normal density and cumulative distribution functions. The boundary is found by setting this integral equal to 0.0150. Numerical integration and one-dimensional root-finding give \(b_2\approx2.096\). The cumulative crossing probability is then approximately \(0.0100+0.0150=0.0250\).

Why Reusing the Old Boundary Can Miss the Target

A boundary calibrated for one information fraction cannot automatically be carried over to another. The following calculation illustrates how even a nearby value can leave the cumulative type I error slightly below its target. It also shows why checking the cumulative probability is more informative than inspecting the boundary alone.

A Practical Recalibration and Verification Sequence

This approach is consistent with the earlier material on recursive boundary calibration and cumulative type I error verification: later crossing probabilities are conditional on not having crossed earlier, and the cumulative probability is the sum of disjoint first-crossing events. For more than two looks, the same principle requires accounting for all earlier noncrossing conditions in the joint distribution; a two-look calculation should not be treated as a substitute for that larger calibration.

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to calculate the cumulative crossing probability after solving for the boundary, using an evaluation that is separate from the root-finding step when possible. Check that the first-look probability matches its assigned value, the later increment is nonnegative, and the cumulative probability matches the stated target within the chosen numerical tolerance. A computed boundary that moves in an unexpected direction, or a crossing probability that changes materially when numerical settings are refined, calls for investigation.

The main alternatives are to retain the original boundary, to recalibrate the boundary at the actual information time, or to redesign the schedule and spending targets. Retaining the original boundary is simplest but does not guarantee the intended cumulative alpha. Recalibration preserves the specified target in the stated model and setup. Redesign may be preferable when the timing change is substantial or when the original spending plan defines a different target at the actual time. These approaches are not interchangeable: the target itself must be stated before a boundary can be judged.

The calculation has important limits. It assumes a valid canonical joint-normal approximation, correctly measured information, and an analysis time that is not selected based on the interim efficacy statistic. If the look is triggered by emerging results, conditioning only on the realized time and applying this calculation can fail to account for the selection mechanism. A two-sided efficacy rule also requires both tails to be included in the crossing calculation. Finally, a nominally correct boundary does not guarantee correct error control if the test statistic, variance model, or information-time estimates are materially misspecified.

Common Mistakes

• Using the planned correlation. Reusing \(\sqrt{t_1/t_{2,\text{planned}}}\) after the look moves ignores the changed dependence. Recalculate it from the actual information fractions.
• Adding marginal tail probabilities. The probability of crossing at either look is not generally the sum of the two marginal crossing probabilities, because paths can cross both boundaries. Use disjoint first-crossing probabilities or subtract the joint crossing probability.
• Claiming error control from a boundary value alone. A boundary is meaningful only with its information time, earlier boundaries, and cumulative target. Verify the resulting cumulative crossing probability.
• Changing timing and silently changing the target. Decide whether the design preserves a specified cumulative alpha budget or follows a spending function evaluated at the actual time. State which rule is being applied.