Asymptotics With Bounded Outcome Delays

Why a Fixed Delay Bound Can Preserve Long-Run Behavior

In a response-adaptive trial, allocation probabilities can be updated as outcomes become available. At any enrollment time, however, some recently assigned participants may not yet have resolved outcomes. The key distinction is between a fixed delay bound and a delay that grows with the trial: if every outcome is observed within at most a fixed number of later enrollments, the number of pending outcomes is bounded even while the trial grows without bound. This makes their contribution potentially important early on but asymptotically negligible.

That conclusion is not automatic for every adaptive design. Response estimates need enough assigned participants in each arm, and the rule translating estimates into allocation probabilities needs to settle as the estimates settle. The results below make those conditions explicit.

Setup and Assumptions

Index participants by enrollment time \(t=1,2,\ldots\). There are \(K\) treatment arms. Let \(A_t\) denote the arm assigned to participant \(t\), and let \(Y_t(k)\) be that participant’s potential response under arm \(k\). For each arm \(k\), assume the potential responses \(Y_t(k)\) are identically distributed across participants, with mean \(\theta_k\) and finite variance. Participants’ potential-response vectors are independent across \(t\). Assignment is randomized using information available before the current participant’s outcome is generated, so the current potential responses are independent of the assignment randomization conditional on the past.

Write \(\pi_{t,k}=P(A_t=k\mid\text{information available just before assignment }t)\). The delay \(D_t\) is the number of later enrollments by which participant \(t\)’s outcome is resolved. For example, \(D_t=2\) means the outcome is available by the time participant \(t+2\) has enrolled. Assume that, for a fixed finite integer \(L\),

The bound need not make delay independent of response: delays may depend on outcomes or treatment, provided every outcome is eventually observed within this same fixed bound. This is a statement about a stationary response target \(\theta_k\); it does not cover a changing outcome distribution over calendar time.

The bounded-delay condition implies that every participant enrolled by time \(n-L\) has resolved by time \(n\). Thus only participants among the most recent \(L\) enrollments can be pending. In particular,

For the main convergence result, also assume a positivity condition: there is a fixed \(\varepsilon>0\) such that \(\pi_{t,k}\ge\varepsilon\) for every arm \(k\) and every enrollment time \(t\). This prevents the design from starving an arm of assignments. An allocation rule can satisfy this condition by clipping its probabilities to a range bounded away from zero.

Consistency of the Observed Response Estimates

First consider the response average among everyone assigned to arm \(k\), whether or not their outcomes have resolved by time \(n\):

Under the assumptions above, the assigned outcomes in arm \(k\) have long-run average \(\theta_k\). Informally, assignment may depend on previous outcomes, but it cannot anticipate the current participant’s potential response. The assignment-weighted deviations \(I(A_t=k)\{Y_t(k)-\theta_k\}\) therefore have conditional mean zero. A martingale law of large numbers, together with the positive lower bound on assignment probabilities, gives \(N_k(n)\) of order \(n\) and \(\bar{Y}^{\,\mathrm{all}}_k(n)\to\theta_k\) almost surely.

The observed average differs from this full assigned average only through the pending outcomes and the corresponding denominator. Since at most \(L\) outcomes are missing, and the outcomes have finite first moments, the total contribution of those last \(L\) outcomes divided by \(n\) tends to zero almost surely. Also, \(M_k(n)\) and \(N_k(n)\) differ by at most \(L\), while both grow in proportion to \(n\). These facts yield the following result.

The central comparison can be seen directly by writing the observed numerator as the full assigned numerator minus the pending outcomes. There are at most \(L\) terms in that subtraction. With finite first moments, the magnitude of the pending sum divided by \(n\) vanishes; the denominator difference is at most \(L\), whereas \(N_k(n)\) grows at least linearly in \(n\). Consequently, neither the pending sum nor the denominator change affects the limiting average.

When Allocation Proportions Converge

Consistency of response estimates does not by itself specify the limiting allocation. That also depends on the rule used to convert estimates into assignment probabilities. Suppose, for example, that the rule is a fixed continuous function \(g\) of the vector of observed estimates, so that \(\pi_{t,k}=g_k(\hat{\theta}^{\,\mathrm{obs}}(t-1))\), and suppose the rule keeps every probability at least \(\varepsilon\). If the response estimates converge to \(\theta=(\theta_1,\ldots,\theta_K)\), continuity gives \(\pi_{t,k}\to g_k(\theta)\).

The realized allocation proportion also converges to that limit. To see why, subtract the conditional assignment probability from each assignment indicator. The differences \(I(A_t=k)-\pi_{t,k}\) are bounded martingale differences, so their sum divided by \(n\) tends to zero almost surely. Therefore,

The first term converges to \(g_k(\theta)\) because the average of a convergent sequence has the same limit; the second term vanishes by the martingale law of large numbers. Thus bounded delays preserve the limiting allocation prescribed by a stable, continuous response-adaptive rule under the assumptions above. Bounded delays alone do not guarantee allocation convergence if the rule itself oscillates or lacks a well-defined limit.

Derivation of the Pending-Outcome Bound

The finite-pending argument is short but does the essential work. At update \(n\), any participant with \(t\le n-L\) satisfies \(t+D_t\le t+L\le n\), so that participant’s outcome is observed. Hence pending outcomes can only come from the indices \(n-L+1,\ldots,n\), a set with at most \(L\) members. For arm \(k\), the number pending is therefore no greater than \(L\).

For a direct comparison of averages, let \(Q_k(n)\) be the sum of the pending responses in arm \(k\). Then the observed numerator equals the full assigned numerator minus \(Q_k(n)\), and \(M_k(n)=N_k(n)-R_k(n)\), where \(0\le R_k(n)\le L\). Under finite first moments, the sum of at most \(L\) responses from the most recent enrollment indices divided by \(n\) tends to zero almost surely. Since positivity implies \(N_k(n)/n\) is bounded away from zero asymptotically, both \(Q_k(n)/N_k(n)\) and the denominator perturbation vanish. This justifies replacing the full assigned average by the observed average in the limit.

Diagnostics, Comparisons, and Limitations

In practice, the assumptions can be checked or monitored in several ways. Record the age of each unresolved outcome and verify that the maximum observed resolution lag is consistent with the protocol’s fixed bound. Track assigned and resolved counts separately by arm; their difference should not exceed \(L\). Also inspect the smallest assignment probabilities over time and confirm that no arm’s probability is drifting toward zero. Finally, compare successive response estimates and allocation probabilities: persistent oscillation may reflect an unstable rule, unusual data, or a violation of the stationary-response assumptions rather than a delay that is asymptotically harmless.

The result is a long-run statement, not a promise of small finite-sample bias or good finite-sample precision. It also relies on a stationary response distribution, a fixed bound on delays, sufficient randomization to every arm, and a stable allocation rule when allocation convergence is claimed. If the delay bound grows with enrollment, the number of pending outcomes need not remain bounded; that setting requires separate conditions. If outcomes are not eventually resolved, if assignment probabilities vanish, or if the response distribution changes over time, the conclusions here need not hold.