Allocation Using Only Matured Outcomes

Why Update Only From Matured Outcomes?

In a trial with delayed outcomes, an allocation decision may be due before every enrolled patient’s response is known. One transparent policy is to base each update only on outcomes that have fully resolved by that decision. This avoids treating an unobserved response as though it were known. It does not, by itself, make the resolved patients representative of all patients: that depends on how outcomes become available.

The central question is therefore twofold. First, can a rule be defined that uses only information genuinely available at each randomization? Second, under what conditions does the resulting set of resolved outcomes provide a defensible basis for comparing arms? The earlier material on complete-case adaptation explains why selecting only observed outcomes can mislead; here the focus is how to specify the rule and assess its assumptions.

Notation, Model, and Allocation Rule

Consider \(K\) treatment arms and patients enrolled in sequence. Let \(A_i\) denote the arm assigned to patient \(i\), and let \(Y_i\) be that patient’s binary outcome, coded \(1\) for a beneficial response and \(0\) otherwise. Let \(D_i\) be the time at which the outcome is fully resolved. At the decision for patient \(n+1\), define \(M_i(n+1)=1\) if patient \(i\)’s outcome is resolved and available before that decision, and \(M_i(n+1)=0\) otherwise. An outcome with \(M_i(n+1)=0\) is pending and is not used in the update.

For arm \(a\), let \(m_a(n+1)\) be its number of matured outcomes and \(s_a(n+1)\) its number of beneficial matured outcomes available at that decision. When \(m_a(n+1)>0\), the matured-case response estimate is \( \hat p_a^{\,M}(n+1)=s_a(n+1)/m_a(n+1) \). The superscript \(M\) emphasizes that this is calculated from resolved outcomes only. It is not automatically an estimate of the response rate for all patients; the conditions needed for that interpretation are considered below.

Here is one explicit rule for binary beneficial outcomes. It is a design choice, not a uniquely required formula. It uses a positive score offset \(\delta\) to keep an arm’s score above zero and an exploration fraction \(\varepsilon\) to ensure that every arm retains a minimum chance of assignment. Activate the rule after each arm has at least one matured outcome; before then, use a prespecified initial randomization scheme.

Here \(q_a(n+1)\) is the response-based share before exploration is added, and \(\pi_{n+1,a}\) is the probability of assigning patient \(n+1\) to arm \(a\). With \(0<\varepsilon\leq 1\) and \(\delta>0\), the probabilities sum to one and satisfy \(\pi_{n+1,a}\geq\varepsilon/K\). The random assignment must actually be drawn using these probabilities; assigning deterministically to the arm with the largest estimate is a different procedure.

When Does Excluding Pending Patients Give a Defensible Update?

The key requirement is that, within an arm and after accounting for relevant observed factors, whether an outcome has matured by the decision must not select patients according to their response. For the unadjusted arm-specific matured mean to represent the arm’s response rate, a simple sufficient condition is \(Y_i \perp M_i(n+1)\mid A_i=a\): among patients assigned to arm \(a\), maturation by that decision is independent of the binary outcome. If the independence holds only after conditioning on covariates or enrollment time, an unadjusted matured-case rate need not be representative; the analysis or allocation rule must account for that dependence.

Other design assumptions also matter. The response definition and follow-up requirement should be prespecified and applied consistently. Outcomes should be defined for the trial population of interest, and any differences in patient mix or calendar time between the matured cases and the patients currently being randomized should not make the historical response rates irrelevant. Finally, each arm must continue to have a positive chance of assignment; the exploration component in the displayed rule provides that property for this allocation policy.

The basic justification can be seen from conditional expectations. Write \(R=M_i(n+1)\) for the maturation indicator at a particular decision and fix an arm \(a\). If \(P(R=1\mid A_i=a)>0\) and \(Y_i\) is independent of \(R\) within that arm, then

For a binary outcome, the conditional expectation is the response probability. Thus, under this independence condition, restricting attention to matured patients does not change the arm-specific response probability in expectation. The sample proportion still varies from sample to sample, especially when few outcomes have matured, but it is not shifted simply because of selection by maturation.

This is a justification for using resolved outcomes as an input to adaptation under the stated conditions; it is not a general proof that a trial’s final treatment comparison is unbiased. Adaptive assignment, changing patient populations, covariate imbalance by chance, and the analysis method can all matter for final inference. Allocation probabilities should be recorded, and the analysis should reflect the actual design.

Implementation Checks, Alternatives, and Limitations

A matured-only rule is simple to audit: at each decision, the input data should be exactly the outcomes that were resolved before that decision. A practical record should include each patient’s assignment, outcome-resolution status and time, and the randomization probabilities in effect. Useful checks include:

• Check the maturity rule. Confirm that “resolved” has a prespecified operational meaning and that unresolved records are not silently coded as response failures.
• Compare resolution patterns. Examine resolution proportions and times by treatment arm and by relevant measured patient characteristics. Differences can flag a need to question the assumption that resolved cases represent the arm.
• Check calendar-time relevance. Compare the characteristics and outcome experience of earlier matured patients with those of current enrollees. Shifts in care, eligibility, or patient mix can make earlier outcomes a poor guide.
• Check the update inputs and probabilities. Verify the matured counts and response counts, confirm the probabilities sum to one, and ensure the planned exploration minimum is maintained.

These diagnostics can reveal problems, but they cannot establish independence of maturation and outcome using only resolved outcomes. In particular, if the response is unknown for pending patients, a comparison of observed cases cannot directly show whether those pending responses differ. The rule also updates more slowly when few outcomes mature, and its estimates can be unstable with small matured counts. The offset and exploration fraction reduce certain allocation extremes; they do not repair biased response estimates.

The main alternatives answer different practical needs. Fixed allocation does not use outcome updates, so it avoids feedback from a selected set of matured responses but does not adapt assignment to observed outcomes. A rule that uses only matured outcomes is more direct than one that incorporates values for pending responses, but it can be less timely. Waiting for additional outcomes, using a cohort schedule, or incorporating information about pending responses are distinct approaches with their own assumptions and design consequences; they should not be treated as equivalent to the matured-only rule defined here.

The rule is also specific to a beneficial binary endpoint and to its chosen score transformation. For a different outcome scale, the score and its interpretation would need to be specified anew. Finally, probabilities that adapt to outcomes do not guarantee that more patients receive the better treatment, nor do they replace prespecified methods for final treatment comparisons. Their operating behavior depends on the outcome process, delay process, and randomization history.