Representing Outcomes From a Stopped Trial

What Is the Outcome of a Stopped Trial?

A group sequential trial can end before its planned maximum because a statistic crosses an efficacy or futility boundary. The realized result is therefore not simply the statistic at the maximum planned look: that statistic may never have been calculated from collected data. To describe the trial’s statistical outcome correctly, record when it stopped and the statistic observed at that look, along with the decision or stopping reason when relevant.

This distinction is more than a matter of data formatting. A sequential analysis describes a path of statistics across planned looks, but a stopped trial reveals only the part of that path observed before stopping. The unobserved tail of a hypothetical path can be useful for probability calculations under a model; it is not additional data from the realized trial.

Model and Stopping Rule

Suppose a trial has \(K\) planned analyses at fixed, increasing information levels \(I_1<\cdots<I_K\). Let \(Z_k\) be the standardized test statistic calculated at look \(k\). Under the canonical group sequential model under \(H_0\), the statistics are jointly normal with mean zero and covariance

This model assumes that the analysis looks and information levels are specified, the statistic is standardized according to the stated analysis plan, and the process has independent increments on the information scale. The covariance formula follows from those assumptions. The representation of the stopped outcome below depends principally on a different condition: stopping is determined by the observed history according to a specified rule. It does not require that every observed statistic follow the canonical model.

For a common two-sided stopping rule, let \(l_k\) and \(u_k\) be the lower and upper boundaries at interim look \(k<K\), with \(l_k<u_k\). Stop at the first interim look with \(Z_k\leq l_k\) or \(Z_k\geq u_k\). If no interim boundary is crossed, stop at look \(K\) by design. One-sided rules are covered by omitting one boundary.

Formally, if \(\mathcal{F}_k\) denotes the information available through look \(k\), then a stopping rule is a stopping time when the event \(\{T\leq k\}\) can be determined from \(\mathcal{F}_k\). In the boundary rule, the decision at look \(k\) uses only statistics observed up to that look. Thus a later statistic cannot be used to redefine which look was the first stop.

Representing the Stopped Outcomes

The rule partitions possible sequential paths into disjoint stopping regions. At an interim look \(k\), the path must have remained between the boundaries at every previous interim look and cross a boundary at look \(k\). At the final look, it must have remained between the boundaries at all earlier looks; the trial then stops whether or not \(Z_K\) crosses a boundary.

These events cannot both occur for the same path: once an interim boundary is crossed, the trial stops, so it cannot also first stop at a later look. The stopped-outcome space can consequently be written as a disjoint union of look-specific pieces. For an interim upper-boundary stop, for example, the observed record has the form \((k,z)\) with \(z\geq u_k\), and the preceding observed statistics must satisfy their continuation conditions. The pair is a compact representation of the result, not a claim that the complete history consists of only one number.

The distinction between the pair and the observed prefix matters. If a report needs the full sequence of statistics collected before stopping, it should retain that sequence. The pair \((T,Z_T)\) identifies the stopping look and the statistic there, but does not reproduce every earlier value. Likewise, if the operational rule can stop for an external reason or uses an auxiliary variable, that trigger and the reason for stopping must be recorded; the pair alone may not explain the decision.

Worked Examples

Why the Representation Is Useful

The stopped-outcome representation keeps the data aligned with the actual decision process. A table of outcomes can record the look, observed statistic, boundary status, and stopping reason. It also prevents a common analytical error: treating an imagined full vector of statistics as if every component had been observed. The distinction is important when documenting trial results and when defining events for probability calculations under a sequential model.

A concise stopped record and a full analysis dataset serve different purposes. The record \((T,Z_T)\) captures the stopping look and its statistic. A full dataset may include the entire observed prefix, treatment assignments, estimates, standard errors, and operational details. Which fields are needed depends on the analysis and reporting question; the pair should not be presented as a substitute for all trial data.

Checks, Alternatives, and Limitations

• Verify the first stop. Check that every earlier observed look was in its continuation region. A statistic beyond a boundary at look 1 cannot be reported as a look-2 stop.
• Separate the stopping reason from the look. A final-look stop is not automatically an efficacy or futility stop. Record whether a boundary was crossed or the maximum was reached without an interim crossing.
• Do not append unobserved values. Later statistics may be generated in simulations or discussed as hypothetical continuations, but label them clearly and keep them out of the realized stopped record.
• Check what triggered stopping. If the rule also depends on safety, data quality, or another variable, recording only \(T\) and \(Z_T\) may omit information needed to explain the stop.

A fixed-sample analysis typically emphasizes the statistic at one prespecified final analysis. That is simpler when no interim stopping changes which look is observed, but it does not describe the realized result of a trial that actually stopped early. At the other extreme, a full-path representation retains \((Z_1,\ldots,Z_K)\) as a model-based random vector. It is useful for expressing the planned joint distribution, but after early stopping its later components are hypothetical rather than observed. The stopped-outcome representation records the realized look and statistic without confusing these two roles.

The representation is limited by the stopping rule it describes. It assumes a specified set of looks and a rule that can be applied to observed information. If the design changes its boundaries, analysis timing, or stopping criteria during conduct, the record should preserve the rule actually used and relevant design changes. A pair \((T,Z_T)\) alone cannot summarize such changes or all features of the collected data.

A Short Derivation: Why the Look-Specific Events Are Disjoint

For \(k<K\), stopping at \(k\) requires continuation at every earlier look and a boundary crossing now. If a path were in both \(\{T=k\}\) and \(\{T=m\}\) for \(k<m\), then \(\{T=k\}\) would require a boundary crossing at \(k\), while \(\{T=m\}\) would require continuation at \(k\). Those conditions cannot both hold. The final event similarly requires continuation through every interim look, so it cannot overlap an earlier stopping event.

where \(\Omega\) is the set of paths obeying the planned-look rule. The union covers every such path because either a first interim crossing occurs or the trial reaches the planned maximum. This establishes that the stopping look is well defined and that the look-specific outcome records partition the possible realized outcomes.

Common Mistake

Wrong reasoning: “The design planned four looks, so the result should include four statistics. If the trial stopped at look 2, I can fill in looks 3 and 4 from the joint normal model.”

Correction: The joint model describes the distribution of a planned sequence, not additional measurements collected after the trial stopped. For an early stop at look 2, report the observed stopping look and statistic, and retain any other actually observed data through look 2. Statistics at looks 3 and 4 are unobserved for that realized trial; model-based hypothetical values must not be presented as its data.