Why the Information Available at the Decision Matters
At an interim review, a trial team may be permitted to change a protocol feature using information available at that point. The statistical question is not merely whether the change was sensible. It is whether the rule choosing the change used only information available to the decision-maker at the interim. If a rule can depend on outcomes that have not yet been observed, it is not an interim adaptation in this precise sense.
The key idea is measurability. It gives a mathematical way to specify the information on which a decision can depend. This matters for error-control arguments: conditioning on interim information is meaningful only when the adaptation is determined by that information. The earlier material on conditional Type I error explains why the conditional error after an interim can be used in an error-control argument; measurability specifies the information that an interim decision is allowed to use.
Information Sets and Measurable Actions
Let \(\Omega\) be the set of possible trial states, and let \(\mathcal F\) be a sigma-field representing the events that can be defined from the full trial data. At an interim time, let \(\mathcal F_I\) be the sigma-field representing all information legitimately available for the decision. Usually \(\mathcal F_I\) is contained in \(\mathcal F\). For example, \(\mathcal F_I\) might be generated by the interim observations, an authorized randomization result already revealed to the team, and other specified information available by that time.
Equivalently, the rule can be written as \(A=g(Z)\), where \(Z\) is a record containing all and only the information the rule is permitted to use, and \(g\) is a function from that record to an action. The record need not consist only of response counts. If authorized information includes a revealed randomization result \(U\), for instance, then \(Z\) must include \(U\) as well. Measurability is always relative to the stated information set, not to one particular summary chosen after the fact.
The definition does not say the action must be fixed before the trial, nor that it must be a deterministic function. A randomized rule can be represented by including an authorized random variable in the information set. That variable must be available to the decision-maker before the action is chosen. A randomization draw made only after the decision cannot be used to justify the decision as measurable at the interim.
A Finite-Record Characterization
Suppose the interim record \(Z\) takes finitely many values. It partitions the possible trial states into groups that share the same record. If a deterministic action \(A=g(Z)\) is measurable with respect to the information generated by \(Z\), then it must be constant within each group. In other words, two states that agree on all of \(Z\) cannot lead to different actions under that rule.
Here is a short justification. Fix an action \(a\). The event \(\{A=a\}\) must be determined by \(Z\). If two states have the same \(Z\), any event determined by \(Z\) either contains both states or contains neither. Thus, if one state leads to \(a\), the other must also lead to \(a\). Applying this to every possible action shows that \(A\) is constant among states with the same record. The converse also holds for a finite record: if the action is constant within every group of states sharing \(Z\), then each action event is a union of those groups and is therefore determined by \(Z\).
This conclusion applies only when \(Z\) contains all information the deterministic rule may use. If the decision-maker also knows a revealed randomization result \(U\), states with the same count table but different \(U\) need not receive the same action. The relevant comparison is between states that agree on the full decision record, including \(U\), not on a partial summary.
Worked Example: A Revealed Randomization Result Is Part of the Record
Consider a hypothetical interim review. The team has a response-count table \(C\) and has been authorized to use a revealed tie-breaking coin \(U\), where \(U\) takes values 0 and 1. The rule chooses protocol option A if the count for A exceeds the count for B, chooses B if B’s count is larger, and in a tie chooses A when \(U=1\) and B when \(U=0\). Check whether the rule is measurable with respect to the decision information.
The full decision record is \(Z=(C,U)\), so the relevant information is \(\mathcal F_I=\sigma(C,U)\). For example, the event that A is chosen is \(\{C_A>C_B\}\cup\{C_A=C_B,\ U=1\}\). Both parts of this event can be determined from \(C\) and \(U\), so the event belongs to \(\mathcal F_I\). Likewise, the event that B is chosen is determined by the same record. The rule is therefore measurable with respect to \(\mathcal F_I\).
Suppose the hypothetical counts are \(C_A=C_B=8\). If \(U=1\), the rule chooses A; if \(U=0\), it chooses B. These are not contradictory decisions from identical interim information: the full records \((C,1)\) and \((C,0)\) differ. If the coin result were not revealed or otherwise available before the decision, however, it could not be included in the decision record.
Why Unrevealed Future Data Cannot Set the Interim Action
Let \(Y\) denote a future outcome that will be collected after the interim, and suppose \(Y\) is not determined by the interim information. A proposed interim action that changes when \(Y\) changes is not measurable with respect to \(\mathcal F_I\). In particular, if the rule is \(A=g(X,Y)\), where \(X\) is interim information, then it is not enough that \(X\) is available: the dependence on \(Y\) must also be absent in the sense that the resulting action is determined by \(\mathcal F_I\).
This is an informational restriction, not a claim that future outcomes are statistically independent of interim data. Future outcomes may be associated with interim outcomes. The restriction is that the realized future value cannot be an input to a decision made before that value is observed. If an outcome is already available at the decision time through a permitted source, it belongs in the information set; if it is not available, it does not.
Worked Example: Detecting a Rule That Uses a Future Outcome
In a hypothetical trial, let \(X\) be the interim response count, and let \(Y\) be a binary response measured only after the interim. Assume \(Y\) is not determined by \(X\). A proposed rule sets the continuation option to A when \(Y=1\) and to B when \(Y=0\). Is this an admissible interim rule with respect to \(\mathcal F_I=\sigma(X)\)?
The event that option A is chosen is \(\{A=\text{A}\}=\{Y=1\}\). Because \(Y\) can vary while \(X\) remains the same, this event cannot in general be determined from \(X\); by assumption it is not in \(\sigma(X)\). Therefore the proposed rule is not measurable with respect to the interim information. For a concrete pair of possible trial paths, suppose both have the same hypothetical interim count \(X=6\), but one has future \(Y=1\) and the other has future \(Y=0\). The proposed rule gives different actions despite identical interim information.
The correction is not to pretend the future value was part of the interim record. Instead, specify a rule based on information actually available at the interim, or make the decision later when \(Y\) is available and define the information set for that later decision. If \(Y\) happened to be completely determined by \(X\), the measurability conclusion could differ; the example explicitly assumes it is not.
Checking a Rule Against the Complete Interim Record
A written rule is easiest to audit when the decision record is explicit. Consider a hypothetical review after 40 participants, where the permitted record is a table \(Z\) with the interim response counts for two protocol options, and no other information is used. The rule selects option A if its count is at least 10 and otherwise selects option B. The rule is \(A=g(Z)\): compare the A count with 10, and choose accordingly. Because each action is determined by the complete permitted record, it is measurable with respect to \(\sigma(Z)\).
Worked Example: Applying a Threshold Rule to a Complete Record
Suppose the hypothetical interim table records 11 responses for A and 9 for B. Under the specified rule, the action is A because \(11\geq10\). The event of choosing A is \(\{Z_A\geq10\}\), which is determined by the record \(Z\); the event of choosing B is \(\{Z_A<10\}\), also determined by \(Z\). Thus both action events belong to \(\sigma(Z)\), establishing measurability.
The threshold calculation is not itself evidence that the rule controls Type I error. It establishes only that the action uses the stated information. If the actual team also used a revealed randomization result, a safety report, or another authorized interim variable, the record would have to include that information. If the rule relied on an unrecorded variable, the stated measurability check would not describe the rule actually used.
Measurability Is Not the Same as Error Control
Measurability answers a focused question: can the action be determined from the information available at the decision time? It does not establish that the adaptation preserves a Type I error bound, has adequate power, or is clinically appropriate. Even a perfectly measurable rule may be paired with a continuation test whose error probability is too large.
The conditional error material establishes the separate role of a conditional bound. In an error-control argument, the chosen continuation procedure must satisfy the relevant conditional restriction as well as being selected using interim information. Measurability lets the action be treated as fixed when conditioning on the full decision record; it does not supply the conditional restriction itself. The subsequent topic on arbitrary adaptations under a conditional bound develops the consequence of that bound and is distinct from the informational definition given here.
This distinction also separates measurability from pre-specification. A protocol may state a rule in advance, but the rule still has to be measurable with respect to the information it claims to use. Conversely, a function of interim data can be measurable even if it was not pre-specified; whether such a change is allowed by the trial’s governance and whether it preserves statistical error control are additional questions.
Diagnostics and Common Failure Modes
A practical audit should identify the decision time, list every input the rule used, and compare that list with the information authorized and available at that time. Preserve the rule, decision record, data-access timing, and any randomization results used. A discrepancy between the rule’s documented inputs and the actual inputs is a warning that the formal information set is incomplete.
- Using a partial summary. A rule may depend on a revealed randomization result as well as the count table. Comparing states with identical counts alone does not test measurability with respect to the full record. Include every authorized input in the record.
- Confusing correlation with access. Interim and future outcomes can be associated. That does not mean the future outcome was available to determine the interim action. Check what was known and used, not whether variables are correlated.
- Calling an unrecorded input “clinical judgment.” If judgment affects the selected action, specify what information informed it. A label does not remove the need to define the decision information.
- Assuming measurability proves validity. A data-dependent action can be measurable and still fail to satisfy an error-control requirement. Verify the inferential condition separately.
The main comparison is between a fixed action and an interim-data-dependent action. A fixed action is measurable with respect to any interim information set because its value does not vary across trial states. A data-dependent action may also be measurable, provided its action events are determined by the full authorized record. An action based on a hidden future outcome fails this test unless that outcome is already determined by the interim information. These categories concern information use; none alone establishes the statistical validity of the final test.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A deterministic rule uses only a complete interim record \(Z\). What must be true of its actions for two trial states with the same \(Z\)?
- A rule uses an interim count table and a revealed, authorized tie-breaking variable \(U\). Is it enough to compare states that agree only on the count table? Why?
- A proposed action changes according to a future response \(Y\), which is not determined by the interim record \(X\). What measurability problem arises for \(\mathcal F_I=\sigma(X)\)?
- Does proving that an adaptation is measurable establish that the final test controls Type I error? Explain.
Solutions. 1. The rule must choose the same action in both states, because its action is determined by the complete record \(Z\). 2. No. The complete record is \((Z,U)\); states with the same count table but different authorized values of \(U\) may receive different actions. 3. The action event can depend on \(Y\) and therefore need not belong to \(\sigma(X)\); if \(Y\) is not determined by \(X\), the proposed rule is not measurable with respect to that interim information. 4. No. Measurability concerns which information determines the action. Type I error control requires a separate valid error-control argument and its conditions.