The Remaining Beta Budget After An Interim Look

What Remains After a Futility Look?

A futility plan assigns part of a design’s tolerated probability of stopping for futility to each scheduled analysis. After an interim look, investigators may ask how much of that planned allowance has not yet been assigned. This is an accounting question about the design’s cumulative spending schedule. It is not, by itself, a probability calculated from the interim statistic that has just been observed.

The distinction matters when planning later boundaries. A later allocation cannot use more than the unspent amount if the design is to stay within its specified cumulative futility target. But the remaining amount does not directly determine a boundary: boundary calibration also depends on the joint distribution of the test statistics across looks, the stopping-rule convention, and the design alternative.

The earlier material on cumulative and incremental beta spending established the basic schedule. Here the focus is on what its remainder means, how to calculate it, and what it can and cannot tell us after observing interim data.

Define the Budget Under a Specified Design

Let \(\beta\) be the design’s target cumulative probability of stopping for futility under a specified design alternative. Consider analyses at looks \(1,\ldots,K\). Write \(\mathcal{F}_k\) for the event that the design has stopped for futility by look \(k\), using the prespecified stopping-rule convention. For example, that convention must specify how a futility rule is treated if an efficacy rule could also stop the trial.

Definition: The cumulative futility spending through look \(k\) is \(B_k=P_A(\mathcal{F}_k)\), where \(P_A\) denotes probability under the specified design alternative and the specified stopping-rule convention. Set \(B_0=0\). The incremental spending at look \(k\) is \(\Delta B_k=B_k-B_{k-1}\). If the total target is \(\beta\), the remaining planned beta budget after look \(j\) is \(R_j=\beta-B_j\).

Because “stopped for futility by look \(k\)” includes stopping at earlier looks, the events are nested: \(\mathcal{F}_{k-1}\subseteq\mathcal{F}_k\). Therefore \(B_k\) cannot decrease, and each incremental allocation is nonnegative:

0=B_0\le B_1\le \cdots \le B_K\le\beta,\qquad \Delta B_k\ge 0,\qquad R_j=\beta-B_j.

A schedule that is designed to use the entire target has \(B_K=\beta\). A design may instead specify \(B_K<\beta\), leaving some of the target unallocated. In either case, \(R_j\) is calculated from the prespecified target and the cumulative spending through look \(j\); it is not recalculated by subtracting the observed \(Z\)-statistic or an observed conditional probability.

The subscript \(A\) is important: these probabilities are evaluated under the chosen design alternative. They describe a design operating characteristic. The futility-crossing probability is not generally the same as the overall type II error probability, which concerns failure to reject the null hypothesis. A trial may fail to reject without stopping for futility, and the relationship between the two quantities depends on the full design and its stopping rules.

Why the Remaining Amount Constrains Later Spending

Each future incremental allocation uses part of the same total target. If the design has cumulative spending \(B_j\) through look \(j\), then the future increments must satisfy:

\sum_{k=j+1}^{K}\Delta B_k =B_K-B_j \le \beta-B_j =R_j.

This is a direct consequence of adding the increments: the sum telescopes to \(B_K-B_j\), and the final cumulative target must not exceed \(\beta\). If the design is intended to exhaust the target, then the future increments sum to \(R_j\). If it is intended to leave some of the target unused, their sum is less than \(R_j\).

The increment \(\Delta B_k\) is a design-level probability allocation, not automatically the probability of crossing a boundary conditional on having reached look \(k\). The probability of a futility stop at a later look is evaluated under the design’s joint sampling distribution and stopping rules. Since test statistics across looks are dependent, a boundary calibrated for an increment generally cannot be found by treating that increment as a fresh, unconditional normal-tail probability.

Key result: After look \(j\), later cumulative futility spending must remain at or below \(\beta\). Equivalently, the total of all later incremental allocations must be no greater than \(R_j=\beta-B_j\). This is a constraint on the spending schedule; it does not specify a unique set of later boundaries.

Worked Examples

Worked Example: Calculating the Remaining Budget

A hypothetical design sets a total futility target of \(\beta=0.20\). Its planned cumulative spending is \(B_1=0.04\) after the first look and \(B_2=0.11\) after the second. The trial reaches the second look without stopping for futility. Calculate the remaining planned budget.

By definition, the remainder after look 2 is the target minus cumulative spending through that look:

R_2=\beta-B_2=0.20-0.11=0.09.

Thus, \(0.09\), or 9 percentage points of probability under the specified design alternative, remains available for later planned futility spending. The incremental amounts already assigned are \(\Delta B_1=0.04\) and \(\Delta B_2=0.11-0.04=0.07\); together they equal \(B_2=0.11\). Reaching the second look does not change this arithmetic. The observed interim statistic determines whether the trial continues under its rule, but it does not replace \(B_2\) in the prespecified spending schedule.

Worked Example: Checking a Proposed Later Allocation

A second hypothetical design has \(\beta=0.18\) and has spent \(B_2=0.10\) cumulatively by look 2. A proposed schedule assigns future increments of \(0.05\) at look 3 and \(0.04\) at look 4. Is this schedule compatible with the target?

The remainder after look 2 is:

$$ R_2=0.18-0.10=0.08. $$

The proposed future allocations total \(0.05+0.04=0.09\), which exceeds \(0.08\). Equivalently, the proposed final cumulative spending would be \(0.10+0.05+0.04=0.19\), greater than the target \(0.18\). The schedule is therefore incompatible with the stated target.

For instance, if the look-3 increment remains \(0.05\), the look-4 increment can be at most \(0.08-0.05=0.03\). That revised allocation would use the remaining budget exactly. This arithmetic establishes feasibility of the spending totals; a separate calibration is still needed to find boundaries that achieve the chosen increments under the joint model.

Worked Example: A Conditional Probability Is Not the Budget Remainder

Consider a hypothetical two-look canonical normal model under the design alternative. Let the final information be \(I_2=100\), the first-look information be \(I_1=50\), and the standardized effect per square root of information be \(\theta_A=2.5\). Suppose the observed first-look statistic is \(Z_1=0.8\), and the lower futility boundary at look 2 is \(l_2=-0.5\). Calculate the conditional chance of crossing that boundary at look 2, given \(Z_1=0.8\).

Under the canonical model, \(Z_k\) has mean \(\theta_A\sqrt{I_k}\), variance 1, and correlation \(\sqrt{I_1/I_2}\) between these two looks. The conditional distribution of \(Z_2\) given \(Z_1=z\) is normal with mean \(\theta_A\sqrt{I_2}+\sqrt{I_1/I_2}(z-\theta_A\sqrt{I_1})\) and variance \(1-I_1/I_2\). Substituting the values gives:

\begin{aligned} E_A(Z_2\mid Z_1=0.8) &=2.5\sqrt{100}+\sqrt{50/100}\left(0.8-2.5\sqrt{50}\right)\\ &=25+0.7071(0.8-17.68)\\ &\approx 13.07,\\ \operatorname{Var}_A(Z_2\mid Z_1=0.8)&=1-50/100=0.5. \end{aligned}

The conditional standard deviation is \(\sqrt{0.5}\approx0.7071\). Hence:

P_A(Z_2\le -0.5\mid Z_1=0.8) =\Phi\left(\frac{-0.5-13.07}{0.7071}\right) \approx \Phi(-19.19) \approx 0.

This conditional probability is effectively zero for the stated model and observed value. It is not the remaining beta budget. If the schedule had, for example, \(R_1=0.08\), that \(0.08\) would remain the design’s unspent cumulative allowance even though this particular conditional crossing probability is nearly zero. The example is deliberately illustrative: the boundary was specified to demonstrate the distinction, not calibrated to spend a particular increment.

Why an Observed Interim Value Does Not Reset the Schedule

A futility spending plan is ordinarily specified before the trial, with cumulative targets at the scheduled information times under a design alternative. Observing the trial’s interim data does not erase earlier planned spending or grant a new allowance of \(\beta\). If \(B_j\) has already been assigned, the remaining planned amount is still \(\beta-B_j\), whether the observed statistic is high, low, or close to the futility boundary.

There is a separate conditional question: given the interim data, what is the probability of crossing a future boundary under a model? Under the canonical normal model, for \(I_k>I_j\),

Z_k\mid Z_j=z \sim N\left( \theta_A\sqrt{I_k} +\sqrt{\frac{I_j}{I_k}}\left(z-\theta_A\sqrt{I_j}\right), \;1-\frac{I_j}{I_k} \right).

This conditional distribution can help describe the outlook from a particular observed statistic. It does not make the conditional probability equal to the planned remainder. The latter is defined by cumulative probabilities under the design distribution, while the former conditions on a specific observed history. Replacing the original schedule with a data-dependent recalibration is a design change; its operating characteristics cannot be inferred merely by subtracting the observed conditional crossing probability from \(\beta\).

Calibration, Checks, and Alternatives

The remaining budget constrains boundary calibration without dictating it. For any proposed continuation, first verify that its cumulative targets are nondecreasing and do not exceed \(\beta\). Then calibrate the future boundaries under the specified design alternative and the same joint model and stopping-rule convention used to define \(B_k\). A boundary is chosen to achieve a target crossing probability in that joint setting; it is not generally obtained by applying a one-look normal quantile to \(\Delta B_k\).

This approach differs from using one fixed lower boundary at every look. A fixed boundary may be simple to state, but its cumulative crossing probability is an outcome of the design, not necessarily the prespecified spending schedule. A spending schedule instead states the desired cumulative probability over time and uses model-based calibration to choose boundaries. A conditional prediction based on the observed interim value answers a different question: it describes future probability conditional on that data value, rather than how much of the unconditional design target remains.

Several checks can reveal an incoherent schedule:

Decreasing cumulative targets. If \(B_k<B_{k-1}\), the implied incremental allocation is negative, which is not a probability allocation.
Exceeding the target. If \(B_k>\beta\), or proposed later increments sum to more than \(R_j\), the schedule overspends.
Mixing probability definitions. Cumulative probabilities must use the same design alternative and stopping-rule convention at every look. Changing the convention partway through makes the subtraction meaningless.
Calling the remainder conditional risk. \(R_j\) is not the probability of future futility given the observed interim data, and it is not automatically the overall type II error.
Assuming a unique boundary. A feasible cumulative target does not itself give a boundary. The joint distribution and the design’s crossing rules are needed for calibration.

The schedule also has limitations. It depends on the selected design alternative: changing that alternative can change the crossing probabilities and hence the calibrated boundaries. It depends on the chosen definition of futility crossing when other stopping rules are present. Finally, a mathematically feasible remainder is only an accounting condition; it does not guarantee that a proposed boundary is operationally desirable or that the design achieves its intended power.

A Short Justification of the Remainder Formula

Let \(T_F\) denote the look at which the design first stops for futility, with no such stop treated as occurring after look \(K\). The events \(T_F=1,\ldots,T_F=k\) are mutually exclusive and together form the event of stopping for futility by look \(k\). Therefore:

B_k=P_A(T_F\le k) =\sum_{m=1}^{k}P_A(T_F=m) =\sum_{m=1}^{k}\Delta B_m.

If the total planned target is \(\beta\), the amount left after look \(j\) is the difference between that target and the amount already allocated. Thus \(R_j=\beta-B_j\). Since every future increment is nonnegative, a continuation satisfying \(B_K\le\beta\) must have:

\sum_{m=j+1}^{K}\Delta B_m=B_K-B_j\le\beta-B_j=R_j.

This proves the budget constraint. It does not prove that any particular set of boundaries achieves those increments; that requires calibration under the design’s sampling model.

Key takeaway: The remaining planned beta budget after look \(j\) is \(\beta-B_j\). It limits the sum of all later planned futility increments, but it is neither a conditional probability given the observed interim statistic nor a boundary-calibration formula.

Check Your Understanding

Answer each question using the stated definitions. Short solutions follow.

A design has \(\beta=0.15\) and \(B_3=0.09\). What is the remaining planned budget after look 3?
After look 2, a design has \(R_2=0.06\). A proposed pair of later increments is \(0.02\) and \(0.05\). Is the proposal feasible under the target?
True or false: if the observed interim statistic is unusually favorable, the remaining planned budget automatically increases.
Explain in one sentence why a conditional future crossing probability is not generally equal to \(R_j\).

Solutions. (1) \(R_3=0.15-0.09=0.06\). (2) No: the proposed increments sum to \(0.07\), which exceeds \(R_2=0.06\) by \(0.01\). (3) False: \(R_j=\beta-B_j\) is determined by the prespecified target and cumulative schedule, not by the observed statistic. (4) A conditional future crossing probability conditions on the observed interim data, whereas \(R_j\) is the unspent part of the design-level cumulative target under the specified alternative and stopping convention.