Futility Boundaries When An Interim Look Is Omitted

Why an Omitted Look Changes the Calculation

A futility boundary is calibrated for a particular sequence of analyses. If one planned interim analysis is omitted, its statistic is not compared with a boundary and no futility decision is made there. Consequently, its scheduled beta increment is not automatically counted as having been spent. At the next conducted look, both the cumulative target and the increment since the last conducted look must be identified using the actual sequence of looks.

This distinction matters because a boundary is not just a cutoff attached to a calendar date. It is chosen so that, under a specified alternative, the probability of stopping for futility by that look reaches a planned cumulative target. The probability depends on which earlier analyses were actually conducted and on whether the process survived their boundaries. Skipping a look changes that event and can therefore require recalculating later cutoffs.

Definition: Let \(t_j\) be the information fraction at the \(j\)th conducted analysis, with \(0<t_1<\cdots<t_J\leq 1\). Let \(B(t)\) be the prespecified cumulative beta-spending function under the design alternative. The cumulative target at a conducted look \(t_j\) is \(B(t_j)\); the incremental target since the previous conducted look is \(B(t_j)-B(t_{j-1})\), where \(B(t_0)=0\). A skipped look is not a conducted look and does not itself generate an observed futility crossing.

Here, beta spending refers to the planned cumulative probability of a futility crossing under the chosen alternative. The boundary calculation must use that same alternative throughout. For a lower-boundary calculation, assume the canonical joint-normal model: under the alternative indexed by \(\delta\), \(Z(t)\) has mean \(\delta\sqrt{t}\), variance 1, and \(\operatorname{Cov}(Z(s),Z(t))=\sqrt{s/t}\) for \(s\leq t\). Thus, for two conducted looks \(s<t\),

Z(t)\mid Z(s)=z \sim N\!\left(\sqrt{\frac{s}{t}}\,z+\delta\sqrt{t}\left(1-\frac{s}{t}\right),\ 1-\frac{s}{t}\right).

The conditional mean has two parts: the earlier observed statistic contributes \(\sqrt{s/t}\,z\), and the alternative contributes the remaining drift. The conditional variance is \(1-s/t\). These expressions rely on the canonical model, correctly specified information fractions, and a fixed design alternative for spending calculations. If the monitoring statistic does not follow this model, the resulting boundary probabilities need not match the targets.

Recalibrating at the Conducted Looks

Suppose a planned look at \(u\) is skipped between conducted looks \(s\) and \(t\). The actual next increment is \(B(t)-B(s)\), not \(B(t)-B(u)\). The skipped look contributes no first-crossing event because no decision was made there. For lower boundaries \(l_1,\ldots,l_j\), the calibration condition at conducted look \(t_j\) is

P_\delta\!\left(Z(t_1)\geq l_1,\ldots,Z(t_{j-1})\geq l_{j-1},\ Z(t_j)<l_j\right) = B(t_j)-B(t_{j-1}).

The conditions \(Z(t_k)\geq l_k\) mean that the path did not cross a previous lower boundary. For the first conducted look, there are no earlier survival conditions, so the boundary can be found from a marginal normal probability. At later looks, the probability is generally a joint probability over the surviving paths, not the marginal probability \(P_\delta(Z(t_j)<l_j)\).

Key result: After a look is omitted, recalibrate using the actual sequence of conducted looks. Preserve the specified cumulative spending targets at the actual information fractions, and calculate each increment relative to the previous conducted look. Find later boundaries from the probability of a first crossing among paths that survived all earlier conducted boundaries.

Worked Recalculation Examples

Worked Example: The First Conducted Look Follows a Skipped Look

Suppose three looks were planned at information fractions \(0.25\), \(0.50\), and \(1.00\), but the \(0.25\) look is omitted. Use a linear spending function \(B(t)=0.20t\), so the total planned beta is \(0.20\), and use \(\delta=1\). Find the lower boundary at the first conducted look, \(t=0.50\).

Because no earlier analysis was conducted, the cumulative target at \(t=0.50\) is \(B(0.50)=0.20(0.50)=0.10\). The skipped look's nominal target \(B(0.25)=0.05\) is not an observed crossing and is not subtracted from the target. Under the model, \(Z(0.50)\sim N(\sqrt{0.50},1)\), so \(l\) satisfies

P_\delta\{Z(0.50)<l\}=0.10,\qquad l=\sqrt{0.50}+\Phi^{-1}(0.10) =0.7071-1.2816=-0.5745.

Thus the first conducted boundary is approximately \(-0.5745\). At the final look, the cumulative target is \(B(1)=0.20\), so the increment after the conducted \(t=0.50\) look is \(0.20-0.10=0.10\). The final boundary must be calibrated to a further first-crossing probability of \(0.10\) among paths that did not cross at \(t=0.50\); it cannot be obtained by simply assigning the skipped look's \(0.05\) to the final increment.

Worked Example: A Previously Conducted Look and a Skipped Middle Look

Now suppose the look at \(t=0.25\) was conducted, the look at \(t=0.50\) was skipped, and the next analysis is at \(t=1\). Keep \(B(t)=0.20t\) and \(\delta=1\). The first boundary is \(l_1=0.5+\Phi^{-1}(0.05)\approx-1.1449\), giving cumulative spending \(B(0.25)=0.05\). What is the target increment at the final conducted look?

The final cumulative target is \(B(1)=0.20\). The last conducted look was at \(0.25\), where the target was \(0.05\). Therefore the final increment is

$$ B(1)-B(0.25)=0.20-0.05=0.15. $$

The final boundary \(l_3\) must satisfy the first-crossing equation

P_\delta\{Z(0.25)\geq 0.5+\Phi^{-1}(0.05),\ Z(1)<l_3\}=0.15.

This equation is the recalibration: the skipped \(0.50\) statistic is absent, and the probability is evaluated among paths that survived the one earlier conducted boundary. In general, \(l_3\) is found by evaluating the joint normal probability as a function of \(l_3\) and solving numerically. Using \(B(1)-B(0.50)=0.10\) would incorrectly treat the skipped look as the last conducted analysis.

Worked Example: Checking a Conditional Probability After a Skip

A conditional check can help explain why an old cutoff may no longer deliver an intended conditional probability. Suppose \(t_1=0.25\), \(t_3=1\), \(\delta=1\), and the observed value is \(Z(0.25)=1\). The planned look at \(0.50\) is omitted. Calculate the conditional probability of crossing an old final cutoff of 0, and the cutoff giving a conditional crossing probability of \(0.10\).

The conditional mean and variance are

E_\delta\{Z(1)\mid Z(0.25)=1\} =\sqrt{0.25}(1)+1(1-0.25)=0.5+0.75=1.25, \qquad \operatorname{Var}_\delta\{Z(1)\mid Z(0.25)\}=1-0.25=0.75.

Therefore the conditional standard deviation is \(\sqrt{0.75}=0.8660\). With cutoff 0, the conditional probability is

P_\delta\{Z(1)<0\mid Z(0.25)=1\} =\Phi\!\left(\frac{0-1.25}{\sqrt{0.75}}\right) =\Phi(-1.4434)\approx 0.0745.

A conditional cutoff for probability \(0.10\) instead solves

\frac{l-1.25}{\sqrt{0.75}}=\Phi^{-1}(0.10)=-1.2816, \qquad l=1.25+\sqrt{0.75}(-1.2816)\approx 0.1401.

This is a conditional calculation for the specified history \(Z(0.25)=1\), not the unconditional design boundary from the previous example. It shows why one must distinguish a conditional predictive probability from the spending equation, which averages over all surviving histories.

Why the Boundary Equation Has This Form

At each conducted look, the first-crossing events are mutually exclusive: a path that crossed a lower boundary earlier is no longer eligible to make a first crossing later. Consequently, cumulative futility probability through look \(j\) is the sum of the first-crossing probabilities at the conducted looks through \(j\). The derivation is simply a partition of the event of having crossed by look \(j\).

\begin{aligned} P_\delta(\text{futility by }t_j) &=\sum_{k=1}^{j} P_\delta\{Z(t_1)\geq l_1,\ldots,Z(t_{k-1})\geq l_{k-1},\ Z(t_k)<l_k\}\\ &=\sum_{k=1}^{j}\{B(t_k)-B(t_{k-1})\}\\ &=B(t_j)-B(t_0)=B(t_j). \end{aligned}

The second equality is the calibration condition at each conducted look. The last equality follows by cancellation of consecutive cumulative targets and \(B(t_0)=0\). A skipped look does not add a term to this sum: there is no first-crossing event there. This argument also shows why recalibrating only the next increment while leaving all other boundaries untouched may fail. Changing the conducted sequence changes which paths remain eligible at later looks, so later first-crossing probabilities can change as well.

If an efficacy boundary can also stop the trial, the survivor event must include not having stopped for efficacy at earlier looks. The equation above is for lower-boundary calibration without that competing stopping event; in a design with both stopping rules, calibrate the futility probability under the actual joint stopping rule and the specified alternative.

Diagnostics, Alternatives, and Limitations

After a look is omitted, check the schedule and calculations before implementing a threshold. The following checks often identify errors:

List actual conducted looks. Record their information fractions in order. Do not leave the skipped time in the recursive survival conditions.
Check cumulative targets. Confirm that the target at a conducted time is \(B(t)\), and that its increment is the difference from the previous conducted time's target.
Verify monotonicity and feasibility. A nondecreasing spending function should not produce negative increments. A proposed increment must also be attainable given the probability mass among surviving paths; if not, the requested boundary equation has no valid solution.
Check the model inputs. Confirm the alternative \(\delta\), information fractions, and conditional normal transition are the ones specified for the design. Using a different alternative changes the calibrated beta probabilities.
Recompute attained spending. Evaluate the first-crossing probabilities under the actual schedule and sum them. The resulting cumulative value should agree with \(B(t_j)\), subject to numerical tolerance and any boundary rounding.

The main alternative is to retain the original boundaries and report the operating characteristics of the modified schedule rather than recalibrate. That can be appropriate when the protocol specifies fixed cutoffs or when updating is not permissible, but the original cumulative spending targets are no longer guaranteed. Recalibration instead targets the planned spending function at the conducted times; it may require numerical integration or simulation when there are several earlier boundaries. These are different decisions: one preserves the original cutoffs, while the other adapts cutoffs to the new schedule.

Recalibration also has limits. It does not restore an analysis that was never performed, recover information lost by delay, or ensure that a realized trial spends exactly its planned probability. Spending is a design probability under the assumed model and alternative. Operational changes to the timing, data, or analysis statistic can make the canonical model inadequate, and rounded boundaries can slightly alter attained spending. If an efficacy rule is present, its stopping behavior must be included rather than ignored.

Common Mistake

Pitfall: “The \(0.50\) look was skipped, so its scheduled beta increment has already been used. At \(t=1\), subtract \(B(0.50)\) to get the remaining target.” This is wrong when no futility decision was made at \(0.50\). The correction is to evaluate cumulative spending at the actual conducted look and subtract the target at the previous conducted look. A skipped analysis has no observed crossing event; it is not equivalent to a conducted analysis with a boundary that happened not to be crossed.

A related error is to use the marginal probability \(P_\delta\{Z(t_j)<l_j\}\) at a later look. That probability includes paths already stopped for futility. The design boundary concerns first crossings, so the calculation must restrict to paths that survived earlier conducted looks.

Check Your Understanding

Answer each question, then compare with the solutions below.

A look at \(t=0.30\) is skipped. The previous conducted look was \(t=0.20\), and the next conducted look is \(t=0.70\). Write the incremental beta target at \(0.70\).
For \(B(t)=0.18t\), a trial's first conducted look is at \(t=0.40\). What is its cumulative target and its increment from the start?
Why is \(P_\delta\{Z(t_3)<l_3\}\) generally not the correct calibration probability after an earlier futility look?
Under the canonical model, take \(s=0.25\), \(t=1\), \(\delta=1\), and \(Z(s)=1\). Give the conditional mean, variance, and probability of crossing cutoff 0.

Solutions. (1) The increment is \(B(0.70)-B(0.20)\); the skipped \(0.30\) look is not the previous conducted look. (2) The cumulative target is \(B(0.40)=0.18(0.40)=0.072\), which is also the increment from the start because there was no earlier conducted look. (3) The marginal probability includes paths that crossed an earlier boundary and would already have stopped. Use the joint first-crossing event restricted to survival at earlier conducted looks. (4) The conditional mean is \(\sqrt{0.25}(1)+1(1-0.25)=1.25\), the variance is \(1-0.25=0.75\), and the crossing probability is \(\Phi\{(0-1.25)/\sqrt{0.75}\}=\Phi(-1.4434)\approx0.0745\).