Changing The Planned Look Schedule

Why the Planned Look Schedule Matters

A classical group sequential boundary is calibrated for a particular sequence of analysis times, usually expressed as information fractions. Those times determine how strongly the statistics at different looks are correlated under the null hypothesis. Consequently, the probability of crossing one or more boundaries depends on both the cutoff values and the schedule. If an analysis is moved or added but the original boundary values are retained, the original overall type I error calibration does not automatically carry over.

Here, a look means a planned interim or final analysis. The overall null crossing probability is the probability, assuming \(H_0\), that the efficacy statistic crosses its boundary at any planned look. This is the design-level probability relevant to the type I error rate; it is not the nominal p-value threshold at one particular look.

Null Model and Dependence Across Looks

To make the schedule effect precise, consider a one-sided efficacy design with \(K\) planned looks. Let \(t_i\) denote the information fraction at look \(i\), with \(0<t_1<\cdots<t_K=1\). Under the canonical null model, the standardized statistics \(Z_i\) have a joint multivariate normal distribution with mean zero, variance one, and \(\operatorname{Corr}(Z_i,Z_j)=\sqrt{t_i/t_j}\) for \(i<j\). Equivalently, the information increments are independent under \(H_0\). This model is standard for a normal-theory design with independent increments and known variance, and is also used as an approximation for many score statistics when the information scale is appropriate.

The calculation assumes that the information fractions describe the actual analysis times, the null distribution is correctly specified (or adequately approximated), and the efficacy boundaries are specified before examining the interim results. It also assumes the schedule is fixed for the calculation. If the timing itself is selected in response to accumulating data, a fixed-schedule probability calculation may not describe the resulting procedure.

For two looks at \(t_1\) and \(t_2=1\), the correlation is \(\rho=\sqrt{t_1}\). If the same upper cutoff \(c\) is used at both looks, the crossing event is \(Z_1\ge c\) or \(Z_2\ge c\). Inclusion-exclusion gives

where \(P_0\) denotes probability under \(H_0\), \(\Phi\) is the standard normal cumulative distribution function, and \(Q(c,c;\rho)\) is the joint upper-tail probability for two standard normal variables with correlation \(\rho\). The marginal terms do not change when the look moves, but the joint-tail overlap does. Less overlap means a larger probability of crossing at least once.

Moving a Look Changes the Overlap

Suppose the first look is moved earlier while the final look remains at information fraction 1.00. The first statistic then has a different correlation with the final statistic. Keeping the same cutoff changes the joint-tail subtraction in the crossing-probability formula. In particular, moving the first look from 0.50 to 0.25 reduces the correlation from \(\sqrt{0.50}=0.7071\) to \(\sqrt{0.25}=0.5000\). The two statistics overlap less in their upper-tail events, so the union of those events becomes more probable.

Adding a Look Can Only Preserve or Increase Crossing Probability

The effect of adding a look has a general justification that does not depend on a particular boundary family. Let \(A\) be the event of crossing at one or more of the original looks, and let \(B\) be the event of crossing at a newly added look. With the same boundaries retained at all original looks, the new crossing event is \(A\cup B\). Since \(A\) is a subset of \(A\cup B\),

The increase is strict when the new look has positive probability of crossing while all the original looks do not. Under a nondegenerate joint normal model with information fractions strictly between zero and one, there are sample paths that cross at the new look and stay below the original boundaries. Those paths have positive probability. Thus, with a genuine added look and unchanged boundaries, the overall crossing probability is generally larger. The inequality does not say how large the increase is; that requires the joint null distribution for the full schedule.

How to Handle a Schedule Change

For a fixed, planned schedule change, the direct remedy is to calculate or numerically evaluate the null crossing probability for the actual information fractions and boundary values. If the design target remains \(\alpha\), select or recalibrate the boundary values using that new joint distribution. A common cutoff is only one possible illustration; for a shaped classical boundary, the values may differ across looks, but the same principle applies: evaluate the probability of crossing any of the scheduled boundaries jointly under the null.

The main alternatives are therefore practical design choices rather than interchangeable calculations:

• Keep the original schedule. If the original boundary is to be used without recalculation, preserve the information fractions for which it was calibrated.
• Change the schedule and recalibrate. Use the changed information fractions and the chosen boundary form to evaluate the overall null crossing probability, then adjust the boundary to the intended error-rate target.
• Use a procedure designed for timing flexibility. If timing may change, choose a framework whose calibration accounts for that flexibility, rather than assuming a fixed-schedule classical calibration remains valid. The method still requires its own assumptions and implementation rules.

These options are not equivalent to retaining the old cutoff and simply reporting its original \(\alpha\). The original value describes the original joint crossing event only.

Diagnostics, Limitations, and Common Mistakes

Before applying a classical boundary after a change, verify the information fractions, not merely the calendar dates. Information is determined by the amount of statistical information accumulated, which may not grow linearly with elapsed time. Check also that the statistic’s null variance and dependence structure are appropriate, that every planned look is included, and that the cutoff being applied at each look matches the boundary used in the calculation. A change in sample size, event rate, allocation, or analysis method can alter the information schedule or the statistic’s null behavior.

A common mistake is to reason, “The cutoff at each look is unchanged, so the type I error is unchanged.” This confuses a marginal tail probability with the probability of crossing at least once. The correction is to calculate the joint crossing event for the actual schedule. Another mistake is to treat a nominal p-value threshold at each analysis as an overall error-rate guarantee. A nominal threshold describes one look’s marginal test; it does not by itself account for repeated opportunities to cross.

The calculations above also have limits. A canonical normal calculation can be inaccurate if the statistic is poorly approximated by the model, information fractions are mismeasured, or the look schedule is selected using interim data in a way not represented in the calculation. Two-sided boundaries require accounting for both upper and lower crossing events; the one-sided formulas here should not be reused unchanged. Finally, numerical recalibration is only as reliable as the specified model, schedule, and boundary values.