Predictive Monitoring With Pending Binary Outcomes

Why Pending Outcomes Need Their Own Accounting

At an interim look, some enrolled patients may not yet have a resolved binary response. They are not the same as patients who have not yet enrolled: pending patients have entered the trial and may have different follow-up, exposure, or response-assessment histories. Yet neither group has a known outcome to add to the terminal success count. A predictive calculation must represent both groups without silently turning an unresolved outcome into a failure or omitting a patient from the planned sample.

The calculation below assumes a fixed target of \(N\) evaluable patients and a binary response assessed under a common model. It predicts the number of terminal successes by the planned analysis, combining outcomes pending among enrolled patients with outcomes from patients who have not yet enrolled. This is a model-based probability, not a claim that pending patients have already responded.

Response Model and Posterior Updating

Use the following exchangeable response model: conditional on a common response probability \(p\), each patient's terminal outcome is an independent Bernoulli variable with probability \(p\) of success. Assign \(p\) a beta prior, \(p\sim\operatorname{Beta}(a,b)\), where \(a>0\) and \(b>0\). The resolved outcomes then give the posterior \(p\mid\text{resolved data}\sim\operatorname{Beta}(A,B)\), with \(A=a+r\) and \(B=b+m-r\).

This update uses only resolved outcomes. A pending patient contributes neither a success nor a failure to the likelihood. In this model, pending status is assumed not to provide additional information about that patient's eventual binary outcome after conditioning on \(p\) and the resolved data. That is an assumption about the outcome and follow-up process, not an automatic consequence of calling an outcome “pending.”

Under these assumptions, pending patients and not-yet-enrolled patients have the same conditional outcome model, but different trial statuses. The pending patients are already among the \(N\); the not-yet-enrolled patients still have to enter the trial. If the target is not fixed, enrollment may stop short, or outcomes are only counted if observed by a deadline, the calculation needs additional models for enrollment, follow-up, or missingness.

Predicting Terminal Success

Suppose terminal success is defined as at least \(c\) successes among the \(N\) patients. Let \(K\) be the number of successes among the \(L=N-m\) outcomes still unknown at the interim. The terminal number of successes is \(r+K\), so terminal success occurs when \(K\ge c-r\). Conditional on \(p\), the \(q\) pending outcomes and \(u\) future outcomes are independent binomial counts. Their sum is binomial with size \(L=q+u\) and probability \(p\). Integrating over the posterior uncertainty in \(p\) gives a beta-binomial predictive distribution.

Here \(B(\cdot,\cdot)\) is the beta function. A sum with lower limit greater than \(L\) is zero, and if \(c-r\le 0\), terminal success is already secured under the stated criterion. The beta-binomial probability for one value \(k\) follows by averaging the conditional binomial probability over the posterior:

The pending and not-yet-enrolled parts can also be displayed separately. If \(J\) is the number of pending successes and \(F\) is the number of successes among not-yet-enrolled patients, then conditional on \(p\), \(J\sim\operatorname{Binomial}(q,p)\) and \(F\sim\operatorname{Binomial}(u,p)\), independently. Marginally, however, \(J\) and \(F\) are generally dependent because they share the uncertain \(p\). Thus one should not multiply two beta-binomial probabilities as though the groups had independent response probabilities. The total \(K=J+F\) has the beta-binomial distribution above.

Worked Calculations and Checks

Diagnostics, Alternatives, and Limitations

Before using a result, check the bookkeeping and the model. The counts should satisfy \(0\le r\le m\) and \(m+q+u=N\). The threshold must match the stated terminal definition, and the predictive tail must include all values from the required number of additional successes through \(L\). A quick check is that the answer lies between zero and one; if \(c-r>L\), the probability must be zero. Also verify that the analysis target really is all \(N\) patients, rather than only patients with outcomes observed by a particular interim or final cutoff.

The common-response assumption can be questionable if pending patients differ from future patients in prognosis, treatment exposure, assessment time, or eligibility. Pending status may itself be informative—for example, if early progression makes a response assessment impossible, or if patients with early improvement are assessed sooner. In those settings, the unresolved status is not safely ignored. A model may need to condition on baseline covariates, time to response, dropout, or a competing event; alternatively, a prespecified sensitivity analysis can show how predictions change under plausible response probabilities for pending and future patients. Do not adjust the beta posterior merely by pretending pending patients have a particular outcome.

A rule based only on resolved outcomes is a different approach: it can summarize current evidence, but it does not predict what the remaining patients may contribute. A pessimistic analysis that treats pending outcomes as failures is a possible sensitivity scenario, not the neutral calculation under the exchangeable Bernoulli model. The earlier material on predictive monitoring for a continuous normal endpoint uses a different endpoint model; here the discrete beta-binomial tail is appropriate for the stated binary-response model. The calculation also does not account for uncertain enrollment, variable follow-up, or informative missingness unless those processes are added explicitly.