State Variables for Single-Arm Predictive Monitoring

Why State Variables Matter

A predictive monitoring rule asks what the data observed so far imply about a decision that will be made at the end of a trial. Before any probability can be calculated, the design must say how many participants are planned, how many have supplied an assessable outcome, how many responses have been observed, and what final result would lead to each terminal action. If these pieces are ambiguous, a predictive calculation may be arithmetically correct but answer the wrong question.

This tutorial specifies those pieces for a single-arm phase 2 trial with a binary response endpoint. It focuses on defining the state that is supplied to a monitoring rule, not on calculating predictive probabilities. Those calculations require additional probability-model and prior-distribution details.

The counts refer to participants, not to visits, doses, or response assessments. “Assessable” must have a protocol-defined meaning: for example, a participant may need to complete a specified assessment window before being counted in \(n\). In the basic formulation here, every planned participant contributes one binary outcome, and the interim data contain only outcomes whose response status has been resolved.

Assumptions and the Monitoring State

The basic state is meaningful under a fixed-horizon design: \(N\) is set in advance, and monitoring does not change the planned total. Each participant has one response indicator, coded 1 for response and 0 for nonresponse, according to a defined endpoint and assessment time. For the simplest model, outcomes are conditionally independent given a common response probability \(\theta\). This common-probability assumption means that participants are treated as having the same response chance within the modeled trial population.

The clinical benchmark response rate is denoted \(p_0\). It is a reference value specified for the trial’s clinical question; it is not the same thing as the unknown response probability \(\theta\). For example, a design may ask whether the treatment’s response performance is promising relative to a benchmark of 20%. Specifying \(p_0\) does not assert that the true response probability equals \(p_0\), nor does it by itself determine a terminal action.

A complete monitoring state also includes the terminal decision rule. It is useful to write this rule abstractly as \(\delta(T)\), where the input is the final response count \(T\) and the output is the prespecified terminal action. For a simple binary-response design, an illustrative rule might declare a result favorable when \(T\) reaches a specified count \(c\). The actual endpoint definition, count threshold or other decision criterion, and actions attached to its possible outcomes must be specified by the design.

The count \(Y\) cannot take arbitrary values. Since it counts responses among \(m\) people, \(Y\) must be an integer from 0 to \(m\). Thus \(T\) can range from \(x\) to \(x+m=N\). If the terminal rule is a count threshold \(c\), the number of additional responses needed to reach it is \(\max(0,c-x)\). This is a statement about arithmetic feasibility, not a probability calculation.

Checking the State Before Using It

The state should pass basic consistency checks before it is used in a monitoring calculation:

• \(N\), \(n\), and \(x\) are whole-number counts with \(0\leq n\leq N\) and \(0\leq x\leq n\).
• The remaining sample size \(m=N-n\) is nonnegative, and the future response count is restricted to \(0\leq Y\leq m\).
• The benchmark \(p_0\) is a response rate on the probability scale, so \(0\leq p_0\leq1\), and its clinical meaning is stated.
• The endpoint, assessment timing, and rules for classifying response are the same for the interim and terminal counts.
• The terminal decision rule is specified before looking at the interim result; it is not improvised to favor the observed data.

If participants can have pending assessments, then “enrolled,” “treated,” and “with assessable outcomes” are not interchangeable counts. The basic state above uses \(n\) as the number with resolved outcomes and does not assign an unresolved participant a response or nonresponse. A design that explicitly models pending outcomes needs additional state information; treating unresolved outcomes as failures or silently excluding them changes the monitoring question.

Why the Final Count Is the Observed Count Plus the Future Count

The relationship \(T=x+Y\) follows by partitioning the planned participants into two disjoint groups: the \(n\) participants whose outcomes are observed at the interim look, and the \(N-n\) participants whose outcomes are still future. Each response counted at the end belongs to exactly one of these groups.

This identity relies on consistent outcome definitions and a fixed planned total. If the design changes the sample size, the rule for what counts as a final outcome must also state which participants contribute to \(T\). If the study records more than one response assessment per person, the count must still be defined at the participant level or the model must explicitly use a different unit.

What the State Captures—and What It Can Miss

Under a common response probability and conditionally independent binary outcomes, the aggregate counts \(n\) and \(x\) summarize the interim response data for the basic single-arm model. In that setting, the order in which the responses occurred does not affect the count-based state. This is one reason to define the endpoint and population clearly: the compact state is useful only insofar as its assumptions are reasonable for the intended model.

The aggregate state may not be adequate when response probabilities differ systematically across groups or over time. For example, if the trial population changes during recruitment or the treatment response probability drifts over calendar time, the order and timing of responses may contain information not represented by \(x\) alone. The earlier material on temporal drift is relevant to recognizing that possibility. A design may then need additional state variables, such as counts by cohort or period, and a model that represents the variation.

Likewise, if the terminal action depends on more than the total response count—such as a safety condition, a subgroup requirement, or a particular follow-up window—then \(\delta(T)\) alone is not a complete representation. The state and terminal rule must include the information those conditions use. A single count should not be treated as sufficient merely because it is easy to record.

Common Mistakes and Alternatives

• Confusing \(p_0\) with the unknown response probability. The benchmark is a design reference; \(\theta\) represents the response probability in the model. They can be compared in a clinical question, but they are not interchangeable quantities.
• Calling all recruited participants “observed.” The interim count \(n\) is defined here by resolved, assessable outcomes. Enrollment alone does not establish a binary response result.
• Using \(N-n\) as if it were the number of future responses. It is the number of future participants with outcomes; their response count \(Y\) is unknown and can range from 0 to \(N-n\).
• Choosing a terminal cutoff after seeing the interim count. Doing so changes the decision rule in response to the data. The terminal criterion should be set as part of the design.
• Assuming aggregate counts always suffice. They are suitable for the basic common-probability model, but a design with relevant time, subgroup, or safety structure may require additional state variables.

A fixed-sample analysis that waits for all \(N\) outcomes has no interim future-count state: at the terminal analysis, \(n=N\) and \(Y=0\). A boundary-based monitoring design may instead express interim actions directly in terms of observed counts. Predictive monitoring uses the same core count definitions but asks about future completion under a specified model. These approaches differ in what they use to guide interim action; none removes the need to define the endpoint, sample counts, benchmark, and terminal rule precisely.

The main limitation of the compact state is therefore structural: it does not repair a poorly defined endpoint, an unsuitable common-probability assumption, or a terminal rule that omits clinically important conditions. It is a carefully organized description of the design inputs, not a guarantee that the model is appropriate.