Why Schedule Futility Probability?
A futility rule can stop a trial early when the interim evidence makes the desired benefit unlikely to be established. The design question is not only whether to include such a rule, but also how much probability of stopping for futility to allocate to each interim look under the effect the trial is designed to detect. A schedule makes this timing choice explicit: it describes how the target probability builds over the planned analyses.
This probability is evaluated under a specified design alternative, not under the null hypothesis. The schedule is a design target, rather than a set of boundaries: it does not by itself determine which observed test-statistic values trigger futility. The earlier material on futility stopping rules distinguishes binding and nonbinding policies; whichever policy is used must also be specified when describing the probability that a trial stops for futility.
Model and Definition
Consider a one-sided group sequential test in which positive values of \(Z_k\) favor the treatment at look \(k\). Let \(K\) be the number of planned analyses, \(I_k\) the information at look \(k\), and \(t_k=I_k/I_K\) the information fraction, with \(0<t_1<\cdots<t_K=1\). Let \(\delta^*>0\) be the design alternative on the scale of the treatment effect. Under the canonical group sequential model, the test statistics under this alternative satisfy
This model assumes that information is correctly specified, the standardized statistics have the stated joint normal distribution, and the planned analysis times and stopping rules are known. In the usual independent-increments formulation, information increases across looks and the additional information at each look contributes new increments. The model is an approximation in some applications; its adequacy should be checked for the trial’s design and analysis.
More precisely, let \(F_k\) denote the event that the trial has stopped for futility by look \(k\), under the specified design and stopping policy. The target at that look is \(B(t_k)\), and the corresponding operating characteristic is \(P_{\delta^*}(F_k)\), the probability of \(F_k\) under the design alternative. The spending function states the desired schedule for this probability; it is not itself a probability model or a boundary-calibration method.
Why the Targets Must Be Ordered
The schedule has a simple necessary property. If “stopped for futility by look \(k\)” includes all earlier looks, then \(F_k\subseteq F_{k+1}\): stopping by look \(k\) also means stopping by the next look. Probability respects event inclusion, so
Thus the target values at successive looks must not decrease. This is a constraint on a valid schedule, not a guarantee that a proposed set of boundaries will attain it. In particular, boundaries that produce a smaller probability at an early look than a later target require a coherent joint-probability calculation; checking each look in isolation is not enough to establish the complete design.
The endpoint also needs careful interpretation. A design may specify a target such as \(\beta=0.20\), the nominal type II error at the design alternative for a test with power \(1-\beta=0.80\). It is tempting to set \(B(1)=\beta\) automatically. But the probability of a futility stop under \(\delta^*\) is not generally identical to the overall type II error: the latter concerns failure to reject the null by the end, while the former concerns stopping for futility. Efficacy stopping, the final decision rule, and the binding or nonbinding convention can affect their relationship. State the intended endpoint probability explicitly in a design rather than relying on notation alone.
Worked Example: Evaluating a Smooth Spending Schedule
Suppose a hypothetical trial has design alternative \(\delta^*\), an endpoint target of 0.20 for futility stopping under that alternative, and planned information fractions \(t_1=0.25\), \(t_2=0.60\), and \(t_3=1\). Consider the candidate schedule \(B(t)=0.20t^{1.5}\).
At the first look, \(B(0.25)=0.20(0.25)^{1.5}=0.20(0.125)=0.0250\). The target probability of having stopped for futility by that look is 0.0250, or 2.50%, under the specified design alternative.
At the second look, \(B(0.60)=0.20(0.60)^{1.5}\). Since \((0.60)^{1.5}\approx0.4648\), this gives \(B(0.60)\approx0.0930\), or 9.30%.
At the endpoint, \(B(1)=0.20(1)^{1.5}=0.20\), matching the stated futility-stopping target. The schedule is nondecreasing on \(0\leq t\leq1\), because \(t^{1.5}\) increases on that interval. These calculations assess the proposed targets. They do not show that any particular lower boundaries achieve them.
What the Schedule Represents
A spending function separates a design preference from the procedure used to realize it. The schedule answers: by each planned information time, what probability of a futility stop is targeted under the design alternative? The futility boundaries answer a different question: what interim evidence will trigger a stop? To link the two, a design must evaluate the probability of the stopping events under the joint distribution of the statistics and the complete stopping policy. Boundary calibration is a separate step.
The value of specifying a schedule is that timing becomes a deliberate part of the design. Two schedules can share the same endpoint target yet allocate different target probabilities to earlier looks. That can lead to different early-stopping behavior and different expected sample sizes. Whether those differences are acceptable depends on the trial’s scientific and operational goals; the target curve alone does not determine the realized power, expected sample size, or boundary locations.
Worked Example: Checking a Proposed Set of Targets
Consider a hypothetical three-look design with information fractions \(0.20\), \(0.55\), and \(1.00\). A team proposes target probabilities of 0.03, 0.08, and 0.15 for having stopped for futility by those looks under its design alternative.
The targets satisfy the necessary ordering: \(0.03\leq0.08\leq0.15\). They also end at the stated futility-stopping target of 0.15. Therefore, they form a logically ordered set of targets for the planned looks. In percentage terms, the targets are 3%, 8%, and 15%.
Now suppose the team instead proposes 0.03, 0.02, and 0.15. This fails the ordering check because \(0.03>0.02\). The probability of having stopped by the second look cannot be lower than the probability of having stopped by the first look, since the first event is contained in the second. The proposal must be corrected or clarified before it can serve as a spending schedule.
Passing the ordering check is necessary, not sufficient. It says nothing about whether boundaries exist that attain all the targets under the joint model, whether those boundaries are operationally sensible, or whether the resulting design meets its power requirements.
Assumptions, Diagnostics, and Alternatives
A beta-spending schedule is meaningful only with its context attached. At a minimum, report the design alternative \(\delta^*\), the planned information fractions, the endpoint futility-stopping target, the efficacy and futility stopping conventions, and the model used to calculate probabilities. A probability under one alternative is not automatically the probability under another. If the true effect differs from \(\delta^*\), the observed rate of futility stopping may also differ.
Several checks help identify a schedule or its implementation going wrong:
- Check ordering and range. Targets should be nondecreasing and lie between 0 and 1. A target that declines across looks contradicts the meaning of “stopped by this look.”
- Check the endpoint definition. Confirm whether the final target refers to futility stopping or to overall type II error. They should not be treated as interchangeable without justification.
- Check the conditioning model. A schedule evaluated under the null, or under a different effect from the stated design alternative, does not represent the stated design target.
- Check the full stopping policy. Changing how efficacy stops or binding futility rules are handled can change the probability of stopping for futility, even if the planned looks and target curve are unchanged.
- Check the joint model and timing. Using incorrect information fractions or an unsuitable dependence model can make calculated probabilities inconsistent with the intended schedule.
The main alternatives answer different design questions. A fixed futility boundary applies a prespecified evidence cutoff at each look; it need not deliver a chosen probability schedule under \(\delta^*\). A design with no interim futility stopping avoids allocating any such early stopping probability, though it may forgo reductions in sample size when results are unpromising. A beta-spending approach instead states probability targets over information time, after which a separate procedure must determine whether and how boundaries implement them. These approaches should be compared using their operating characteristics, not by assuming that matching one endpoint makes their interim behavior equivalent.
Worked Example: Interpreting a Model-Based Check
In a hypothetical design, a probability calculation under the stated design alternative gives probabilities of having stopped for futility by the three planned looks of 0.025, 0.071, and 0.150. The proposed schedule targets 0.020, 0.070, and 0.150 at those looks.
Compare each calculated probability with its target: the differences are \(0.025-0.020=0.005\), \(0.071-0.070=0.001\), and \(0.150-0.150=0\). The calculated endpoint matches the target, while the first-look probability is 0.005, or 0.50 percentage points, above its target. The second-look probability is 0.001, or 0.10 percentage points, above its target.
This comparison shows that the calculated stopping probabilities do not exactly match every target, even though both sequences are nondecreasing and the endpoint agrees. A design team would need to assess whether the differences are acceptable for its intended use. The check does not, by itself, identify why the mismatch occurred or prescribe a boundary adjustment.
Common Mistake
A second practical mistake is to treat a valid-looking curve as a complete design. A nondecreasing schedule only defines targets. It does not provide boundaries, establish that the targets are attainable, or verify the design’s power and error rates.
Check Your Understanding
Answer each question using the definition and assumptions above.
- What does \(B(t)\) target, and under which hypothesis or parameter value is that probability evaluated?
- A schedule has targets 0.04, 0.03, and 0.12 at successive looks. What property fails, and why?
- For \(B(t)=0.16t^2\), find the target at information fraction \(t=0.50\).
- Why does a valid beta-spending schedule not specify the futility boundaries?
- Explain why a 0.20 endpoint futility-stopping target need not equal the overall type II error.
Solutions: (1) It targets the probability of having stopped for futility by information time \(t\), evaluated under the specified design alternative. (2) The targets decrease from 0.04 to 0.03, but events of stopping by successive looks are nested, so their probabilities cannot decrease. (3) \(B(0.50)=0.16(0.50)^2=0.16(0.25)=0.040\), or 4.00%. (4) The schedule gives probability targets; boundaries are evidence cutoffs that require a separate procedure to relate them to those probabilities. (5) Overall type II error concerns failure to reject the null by the end, whereas the endpoint target concerns stopping for futility under the design alternative; the events and stopping policies need not coincide.