Worked Single-Arm Monitoring Design Case

From a Terminal Rule to Interim Decisions

A single-arm phase 2 trial may have a clear final criterion for success but still need a plan for what to do before all participants have outcomes. Predictive monitoring connects those decisions: at each scheduled look, it asks how likely the trial is to satisfy its prespecified terminal success rule if enrollment and follow-up continue as planned. A very low probability can support stopping for futility; a sufficiently high probability can support stopping early for efficacy.

This worked case uses a deliberately small hypothetical trial so that every probability can be checked by hand. Six participants have fully observed binary response outcomes, with looks after participants 2 and 4. The numbers illustrate the mechanics, not a recommended clinical design. In practice, sample size, timing, prior distribution, terminal criterion, and stopping cutoffs require design justification appropriate to the trial.

Design Specification and Assumptions

Suppose each participant has a binary outcome, response or no response, and outcomes are modeled as conditionally independent Bernoulli observations with a common response probability \(p\). Assign \(p\) the prespecified prior distribution \(\operatorname{Beta}(1,1)\). After observing \(x\) responses among \(n\) participants, the posterior distribution is \(\operatorname{Beta}(x+1,n-x+1)\). This simple prior is used to make the arithmetic transparent; choosing it does not establish that it is appropriate for a real trial.

The maximum sample size is \(N=6\), and the trial's terminal success rule is \(X_6\geq 3\). Interim analyses occur only after complete cohorts of two participants: at \(n=2\) and \(n=4\). At each interim look, stop for early efficacy if the predictive probability is at least 0.90, stop for futility if it is at most 0.15, and otherwise continue. These cutoffs are prespecified design inputs for this example. They are not universal thresholds.

The calculation assumes outcomes are available and correctly classified at each look; there are no pending outcomes, delayed assessments, or changes to the sample size or analysis plan. It also assumes the response model and prior remain the specified ones at each look. If these conditions do not hold, the calculation may not describe the actual decision problem. Pending outcomes, for example, require an explicit plan for how they enter the analysis rather than silently treating them as failures or successes.

For a look at \(n\), let \(m=N-n\) be the number of participants remaining and \(Y\) their future response count. Given the observed data, \(Y\) has a beta-binomial posterior predictive distribution. The predictive probability for this design is \(P(x+Y\geq 3\mid x,n)\). In particular, the probability integrates uncertainty about \(p\); it is not simply the posterior probability that \(p\) exceeds a fixed value.

Here \(B(\cdot,\cdot)\) is the beta function. The predictive probability is found by summing this distribution over future counts that meet the terminal rule:

If \(3-x>m\), the sum has no eligible future counts and the predictive probability is zero: even responses from every remaining participant could not satisfy the terminal rule. If \(x\geq 3\), the terminal count criterion is already met, so this probability is one.

Worked Example: The First Interim Look

Worked Example: Applying Both Boundaries at the Second Look

Worked Example: Tracing a Trial Through to Its Final Outcome

Why the Calculation Works, and What to Check

The beta-binomial calculation follows from averaging the future binomial response distribution over the posterior uncertainty in \(p\). Conditional on a particular \(p\), \(Y\) has a binomial distribution with \(m\) trials and response probability \(p\). After observing \(x\) responses and \(n-x\) nonresponses, the posterior density for \(p\) is beta with parameters \(a=x+1\) and \(b=n-x+1\). Integrating the conditional probability over that density gives the predictive probability:

The integral is the beta-function identity applied to the product of the binomial likelihood and posterior density. Summing over the future counts compatible with the terminal rule yields the probability used at the look. The posterior distribution reflects uncertainty about \(p\); the predictive distribution additionally accounts for the randomness of the remaining participants' outcomes.

Several checks help catch implementation or interpretation errors:

• Check the remaining sample size. At each look, \(m=N-n\). The future count must range from zero through \(m\), not through the original maximum sample size.
• Check the terminal event. For the rule \(X_6\geq3\), the required future count is \(Y\geq3-x\). A coding error in this threshold can reverse a decision.
• Check impossible and guaranteed outcomes. If \(x+m<3\), terminal success is impossible and the probability is zero. If \(x\geq3\), it is guaranteed by the count rule and the probability is one.
• Check the probability distribution. The predictive probabilities over all \(y=0,\ldots,m\) should sum to one, up to numerical rounding.
• Check the schedule and data status. Use only outcomes available under the prespecified look plan. If outcomes are pending or assessments are delayed, an analysis that treats them as observed can misstate the information available.

As the observed response count increases, the terminal threshold can become easier to reach because fewer additional responses are required and the posterior for \(p\) changes. Separately, fewer participants remaining reduce the available opportunities for future responses and can eventually make success impossible. These are distinct effects; with the current response count and terminal threshold fixed, simply reducing the number of remaining participants cannot make the threshold easier to reach.

Interpreting Boundaries and Comparing Approaches

The efficacy and futility cutoffs form a paired decision rule: a predictive probability at or above the efficacy cutoff triggers early efficacy, one at or below the futility cutoff triggers futility, and a value between them leads to continuation. These are boundaries on the probability of a future terminal event under the specified model. They are not, by themselves, frequentist type I error or power guarantees, and they are not posterior probabilities that a null hypothesis is true or false.

A fixed-sample analysis waits until the maximum sample size and applies the terminal rule once. It is simpler to operate but cannot stop early under this monitoring plan. Predictive monitoring can reduce exposure or resource use when evidence is already compelling in either direction, but its decisions depend on the prior, response model, terminal rule, look schedule, and cutoffs. Those design choices should be evaluated together, including the probability of each stopping decision under plausible response rates.

A different Bayesian decision rule might assess a posterior probability about \(p\) directly, for example whether \(p\) exceeds a clinically relevant benchmark. That is a different target from the probability of satisfying a finite-sample terminal count rule. The two probabilities answer different questions and need not lead to the same decision. This case uses the predictive target because it was specified as the design's stopping quantity.

The simple model also has limitations. A common response probability may be inadequate when outcomes vary by site, dose, subgroup, or calendar time. A prior that is poorly aligned with the clinical context can affect interim predictive probabilities, especially with little data. Changes in eligibility, follow-up, response assessment, or accrual can undermine the assumptions used to calculate the future distribution. Sensitivity analyses and design simulations can expose some of these concerns, but no calculation removes the need to define the target and assumptions carefully.