Setting Decision Thresholds for Stochastic Curtailment

From a Probability Threshold to an Interim Rule

At an interim analysis, a trial may have little chance of meeting its final efficacy criterion even if it continues. Stochastic curtailment uses this idea to decide whether to stop for futility: calculate the conditional probability of eventual success, then stop if that probability is below a prespecified threshold. The threshold controls how willing the design is to stop early; it does not, by itself, change what counts as efficacy.

This tutorial uses a one-sided normal test and one interim look to make the translation explicit. Assume a fixed final information level and a prespecified final efficacy boundary \(c\): success occurs if the final statistic \(Z_T\) is at least \(c\). For example, a fixed one-sided 2.5% test uses \(c=1.96\). In a group sequential design, \(c\) can instead denote the already chosen final-look boundary. The futility threshold considered here does not revise that boundary.

Here, \(\theta\) is the effect expressed in units of the standard deviation of the final test statistic. It is an assumed effect for this calculation, not an estimate forced by the interim result. The rule therefore depends on three choices: the efficacy boundary \(c\), the assumed effect \(\theta\), and the conditional-success threshold \(q\). A different assumed effect can produce a different rule even when \(q\) is unchanged.

Model and Cutoff Derivation

Use the canonical normal model for information accumulating over the trial. Under the assumed effect \(\theta\), the interim statistic has distribution \(Z_t\sim N(\theta\sqrt{t},1)\). Independent future information contributes a normal increment. Conditional on the observed interim value \(Z_t=z\), the final statistic has distribution

The variance \(1-t\) is the remaining information fraction on the canonical scale. These statements assume a correctly specified normal approximation, independent increments of information, a known or appropriately handled variance structure, a fixed future information target, and no unmodelled change in the analysis plan. The effect \(\theta\) is held fixed when computing this conditional power.

The conditional probability of crossing \(c\) is the upper-tail probability under that conditional normal distribution. Writing \(\Phi\) for the standard normal cumulative distribution function gives

Because \(\Phi\) is increasing, the rule \(\operatorname{CP}(z;\theta)\le q\) can be rearranged into a rule on the observed interim statistic. Let \(z_q=\Phi^{-1}(q)\). First compare the arguments of \(\Phi\); then isolate \(z\):

Thus the operational futility cutoff is \(z_{\mathrm{fut}}(q)=\{c+\sqrt{1-t}\Phi^{-1}(q)-\theta(1-t)\}/\sqrt{t}\). Stop when the observed \(Z_t\) is at or below this cutoff; otherwise continue. This equivalence holds because conditional power is strictly increasing in \(z\) for \(0<t<1\). The equality convention does not matter under a continuous model, but specifying “at or below” makes the rule operationally unambiguous.

What Changing the Threshold Does

Holding \(t\), \(c\), and \(\theta\) fixed, increasing \(q\) increases \(\Phi^{-1}(q)\), and therefore raises \(z_{\mathrm{fut}}(q)\). A higher threshold labels more interim outcomes as insufficiently promising and makes early stopping more likely. A lower threshold is more permissive: it requires conditional power to be especially small before stopping. This is a direct mathematical consequence of the cutoff formula, not a change in the efficacy criterion.

The following comparison keeps the same hypothetical \(t=0.50\), \(c=1.96\), and \(\theta=2.80\). The interim statistic has mean \(\theta\sqrt{t}=1.9799\) under the assumed effect and mean zero under the null. The stopping probabilities are the probabilities that \(Z_t\) falls at or below the listed cutoff.

For instance, with \(q=0.10\), the probability of stopping under the assumed effect is \(P(Z_t\le-0.4896)\), where \(Z_t\sim N(1.9799,1)\). Standardizing gives \(\Phi(-0.4896-1.9799)=\Phi(-2.4695)\approx0.0068\). Under the null, the corresponding probability is \(\Phi(-0.4896)\approx0.312\). These probabilities describe interim stopping, not final rejection. A high null stopping probability can be entirely compatible with a low false-positive rate, because stopping for futility does not declare efficacy.

Applying the Rule to an Observed Interim Result

Error Rates, Alternatives, and Practical Checks

Suppose the original test rejects only when the final statistic crosses its fixed efficacy boundary. Adding a futility rule that can only stop the trial and never declare efficacy cannot increase the probability of rejection under the null: the rejection event with curtailment is a subset of the rejection event without curtailment. In this one-boundary example,

Consequently, if the uncurtailed final test has null rejection probability \(\alpha\), the curtailment rule has null rejection probability no greater than \(\alpha\), provided the efficacy analysis and boundary are not altered. This does not mean every operating characteristic is unchanged: power, expected sample size, and the probability of stopping can change. The threshold should be selected with those consequences in mind, rather than treated as a universal conventional constant.

A fixed futility boundary is an alternative way to specify an interim stopping rule directly on \(Z_t\). Under the stated model, a conditional-power threshold is equivalent to such a boundary once \(t\), \(c\), and \(\theta\) are fixed; the conditional-power version gives the boundary an interpretation in terms of a chosen assumed effect. By contrast, predictive probability integrates over uncertainty about effects or other parameters. It is a different probability calculation and should not be substituted for conditional power without explicitly changing the rule’s definition.

Before adopting the rule, check that the effect scale and information fraction match the analysis plan, that the future information target is realistic, and that the assumed effect is scientifically defensible. Recompute the cutoff if the planned information fraction, final boundary, assumed effect, or threshold changes. If the design has multiple future looks, changing information accumulation, or a conditional distribution that is not adequately approximated by the canonical model, the one-look formula may not describe the actual future success probability; calculate that probability under the actual design instead.

A useful diagnostic is monotonicity: at fixed design inputs, larger interim \(Z_t\) must imply larger conditional power, and a larger \(q\) must imply a higher futility cutoff. If calculations violate either ordering, check the sign of the quantile, the direction of the efficacy tail, and whether remaining information \(1-t\) has been confused with interim information \(t\). Also check that the assumed effect is on the final-statistic scale; inserting an unstandardized treatment difference where \(\theta\) is required gives a meaningless cutoff.

• Wrong reasoning: “A 20% conditional-power threshold means the trial has a 20% chance of a false-positive result, so raising it changes the efficacy level.”
• Correction: The threshold is a conditional probability under the specified effect model and observed interim data. It governs futility stopping, not the null rejection probability or the efficacy boundary. With a futility-only rule and unchanged efficacy analysis, stopping cannot add null rejections.