Worked Example: Pending Binary Responses

Why Pending Responses Matter

In a trial with a binary endpoint, an enrolled participant may not yet have completed the fixed follow-up period at the time the next allocation decision is made. The participant has been assigned to treatment, but the outcome is not yet known. If the randomization rule uses only outcomes currently observed, the allocation probabilities depend on a mixture of resolved and pending records. Keeping those records distinct is essential: a response that is not yet observed is neither a success nor a failure.

This tutorial works through one explicit allocation rule. The purpose is to calculate its successive probabilities correctly, not to claim that this rule is optimal for every trial objective. The earlier material on delayed binary endpoints and calculating allocation probabilities with delayed data provides the setting; here the emphasis is on a concrete sequence of decisions.

Model and Allocation Rule

Assume outcomes are Bernoulli within each arm, with a constant response probability over the period considered. Use independent uniform priors, \(p_A\sim\mathrm{Beta}(1,1)\) and \(p_B\sim\mathrm{Beta}(1,1)\). Conditional on the resolved data, the posterior distribution for arm \(j\) is \(\mathrm{Beta}(1+s_j,1+n_j-s_j)\). Its posterior mean, also the posterior predictive probability of success for the next participant assigned to that arm, is

For this worked example, set the target probability of assigning the next participant to A proportional to the two posterior mean response probabilities, with a lower and upper bound. Specifically, let \(\ell=0.20\) and \(u=0.80\), and define

Here \(\pi_A\) and \(\pi_B\) are the probabilities used for the next random assignment. The bounds prevent the rule from assigning less than 20% or more than 80% of the next allocation probability to either arm. The rule uses resolved outcomes only: \(r_A\) and \(r_B\) do not appear in the posterior means. That is appropriate when pending status is determined by time since enrollment under a fixed follow-up schedule and response probabilities are stable; these assumptions are examined below.

Why the Posterior Mean Enters the Rule

With a Beta prior and Bernoulli outcomes, each resolved success adds one to the prior success parameter and each resolved failure adds one to the prior failure parameter. The posterior mean follows directly from the mean of a Beta distribution. Under the uniform prior, arm \(j\) therefore has posterior parameters \(1+s_j\) and \(1+n_j-s_j\), giving

The allocation ratio then compares these two predictive success probabilities. If both are positive, the unbounded target proportion for A satisfies

This algebra justifies the stated ratio rule. It does not establish that the rule maximizes patient benefit, statistical power, or decision quality. Those objectives can differ, and the choice of allocation rule and bounds must be prespecified for the trial’s aims.

Successive Updates as Responses Resolve

An allocation probability is calculated from the information available at that decision time. When follow-up is completed and outcomes are recorded, the resolved counts change; the next probability is recalculated from those updated counts. A pending response contributes only when its outcome becomes available. If allocations occur in cohorts, every participant in a cohort can be assigned using the probability calculated before that cohort, with the next update made after the cohort’s outcomes have matured.

Assumptions, Diagnostics, and Failure Modes

The calculations above are valid as applications of the specified rule under the following assumptions:

• Binary endpoint at a defined time: success or failure refers to the same prespecified endpoint and fixed follow-up time for every participant.
• Stable arm-specific response probabilities: within the period analyzed, each arm’s outcomes follow a Bernoulli model with a constant \(p_j\). Changes in case mix or calendar-time outcome patterns can undermine this assumption.
• Pending status is not itself an unrecorded outcome: unresolved participants have not yet reached the assessment time or have not yet had the result recorded. Pending does not mean failure, and it is not dropout.
• Resolution is not selectively related to the unobserved response after accounting for the fixed schedule: if outcomes likely to be successes resolve faster than failures, the resolved subset can be unrepresentative of all enrolled participants.
• Specified prior and rule: the independent uniform priors, posterior-mean ratio, and allocation bounds are fixed as part of the design rather than chosen after seeing favorable results.

Useful operational checks include comparing pending counts and follow-up ages across arms, confirming that each resolved outcome has reached the same endpoint definition, and independently recalculating the posterior means and probabilities from a dated data snapshot. A rapidly growing pending backlog can make the allocation decisions rely heavily on older resolved observations. If resolution is outcome-related or the underlying response probabilities change over time, the posterior based on resolved records may not represent current response prospects.

A common mistake is to say, “The pending responses have not succeeded yet, so count them as failures.” That substitutes an unobserved outcome with a known failure and changes both the posterior means and allocation probabilities. The correction is to keep a separate pending count and update the success and failure totals only when each binary response is observed. A second mistake is to use the total number enrolled as \(n_j\) in the posterior formula: here \(n_j\) is the number with resolved outcomes, not the number assigned to that arm.

How This Rule Differs From Alternatives

Equal randomization is simple and does not react to early outcome differences, but it does not direct more allocations toward the arm with the larger current posterior mean. The plug-in alternative is also a possible design choice, but with few observations an observed rate of 0 or 1 can produce extreme targets unless additional safeguards are specified. The posterior-mean rule moderates that behavior, while its bounds further limit extreme allocations. Neither smoothing nor bounding removes bias caused by selective resolution or changing response probabilities.

The rule is deliberately a calculation based on resolved data, not a model for predicting unresolved responses. Its limitations include sensitivity to the chosen prior, allocation bounds, and target formula, as well as potential loss of allocation efficiency when feedback is stale. Its calculated probabilities describe the randomization procedure at a decision time; they do not establish that one treatment is clinically superior.