Cohort-Based Response-Adaptive Randomization

Why Update Between Cohorts?

In response-adaptive randomization, assignment probabilities depend on outcome information accumulated during the trial. With delayed outcomes, a patient may be assigned before the response of an earlier patient is known. A cohort design makes the timing explicit: assign a batch using one set of probabilities, then consider an update before assigning the next batch. This prevents outcomes from patients assigned later in a batch from changing the allocation rule for patients assigned earlier in that same batch.

A cohort design is a protocol for grouping assignments and specifying the information available at each update. It does not, by itself, solve problems caused by delayed or selectively observed outcomes. The design must say exactly which earlier outcomes enter each calculation and when the data used for that calculation are frozen.

Define the Cohorts and Update History

Consider two treatment arms, labelled 1 and 0, and a binary outcome for which 1 denotes a beneficial response. Let cohort \(k\) contain \(b_k\) patients, enrolled and randomized as a batch. The allocation probability for arm 1 in cohort \(k\) is \(q_k\); the probability for arm 0 is \(1-q_k\). The first cohort can use a prespecified starting probability, such as \(q_1=0.5\).

Before cohort \(k\) is randomized, the protocol specifies a data cutoff. For each earlier cohort, an outcome is included only if it has been observed by that cutoff. Let \(m_{jk}\) be the number of patients assigned to arm \(j\) in cohorts earlier than \(k\) whose outcomes are observed by the cutoff, and let \(S_{jk}\) be the number of those observed outcomes that are beneficial. Pending outcomes are not counted as responses or nonresponses in this update.

The protocol needs more than a formula. It should define cohort membership and size, the update cutoffs, the outcome definition, the information included in each update, the starting allocation probability, the update rule, and any bounds on allocation probabilities. These choices should be written down before the response data are used to adapt allocation.

A Specific Bounded Update Rule

The following rule is an illustrative design choice for a beneficial binary endpoint; it is not the only possible response-adaptive rule. For each arm, form a smoothed response estimate from the observed outcomes:

The added one in the numerator and two in the denominator keep the estimate strictly between zero and one, including when no responses or no observed outcomes are available. The rule then sets the raw probability of assigning to arm 1 in the next cohort to the proportion of the two smoothed estimates attributable to arm 1. To retain exploration and prevent extreme allocations, bound that probability between 0.20 and 0.80:

Here the estimates with subscript \(k+1\) use the outcomes available at the cutoff before cohort \(k+1\). The formula assigns a larger probability to arm 1 when its smoothed response estimate is larger, but it does not imply that either estimate is an unbiased estimate of the eventual response rate. In particular, if which outcomes are observed by the cutoff depends on response or differs by treatment arm, the observed outcomes may not represent all assigned patients.

For a concrete implementation, suppose every patient in a cohort is randomized independently with probability \(q_k\) to arm 1 and probability \(1-q_k\) to arm 0. The number assigned to arm 1 is then random, not fixed at exactly \(b_kq_k\). The rule must be clear about whether it uses this independent assignment method or another method, such as randomizing a fixed number of patients to each arm. Those methods have different assignment dependence and variability.

How Outcomes Enter Each Update

The operational sequence is important because a cohort design is defined by both its probability rule and its information timing.

This ordering makes the rule nonanticipating: the assignment probability for a patient is determined without using that patient’s own outcome or outcomes that were not available when the probability was set. It also makes clear that an outcome can be pending at one cutoff and available at a later one. Whether and when to pause enrollment while outcomes accrue is a separate operational decision; a cohort design can be specified without assuming that every outcome from the preceding cohort has matured.

Why the Batch Has Random Allocation Counts

Under independent randomization within a cohort of size \(b_k\), let \(X_k\) denote the number assigned to arm 1. Conditional on the probability \(q_k\) fixed before the cohort, \(X_k\) has a binomial distribution. Its mean and variance describe expected allocation and randomization variability; they do not promise exact balance in the realized cohort.

These results follow by writing \(X_k\) as the sum of \(b_k\) independent indicators, each equal to 1 with probability \(q_k\). Each indicator has mean \(q_k\) and variance \(q_k(1-q_k)\); independence makes the variances add. Because \(q_k\) is frozen before these assignments, the calculation applies conditional on the history used to set it.

Assumptions, Checks, and Failure Modes

The allocation calculation itself requires that the treatment labels, outcome definition, data cutoff, available-outcome counts, and probability rule be unambiguous and applied consistently. The binomial count calculation additionally assumes independent assignments with the same probability for all members of a cohort. If assignments are instead made with fixed arm totals, the binomial variance calculation above does not apply.

To interpret the smoothed rates as informative about response under each arm, more is needed. The response definition should be comparable across arms, assignments should follow the planned randomization, and the outcome data included should not be distorted by treatment-dependent or response-dependent observation in a way that makes the available outcomes unrepresentative. A fixed cutoff alone does not guarantee these conditions. If follow-up delays differ systematically by arm, for example, the observed response proportions at a cutoff may reflect different mixes of early and late outcomes.

Useful implementation checks include verifying that each probability was calculated before its cohort’s assignments, that every assignment used the recorded probability, and that each included outcome was observed by the stated cutoff. Maintain an auditable record of cohort membership, allocation probability, assignment, outcome status, and observation time. Also inspect arm-specific numbers of observed and pending outcomes at each update. A large imbalance in pending proportions can signal that an apparently favorable response estimate is based on a selected subset.

• Updating inside a cohort. If assignments within the same batch use changing probabilities based on newly observed outcomes, the design is no longer the fixed-probability cohort rule described here. Freeze and record the probability before the batch begins.
• Counting pending responses as failures. A missing-at-cutoff outcome is not necessarily a nonresponse. Include it only as specified by an explicit method for pending data; this rule excludes it from the observed response calculation.
• Treating expected counts as quotas. The calculation \(b_kq_k\) is an expectation under independent randomization, not a required arm total.
• Reading the allocation probability as evidence of efficacy. A higher probability reflects the chosen algorithm and available history. It is not, by itself, a hypothesis test or a guarantee that the arm has a better true response rate.
• Allowing a probability rule to become extreme. Sparse outcomes can produce unstable empirical comparisons. Smoothing and bounds reduce extreme assignments in this example rule, but do not eliminate uncertainty or repair biased outcome availability.

Compared with individual response-adaptive randomization, a cohort design reduces the frequency of updates and prevents within-batch feedback. Compared with fixed randomization, it uses accumulated outcome information to change probabilities across cohorts, but adds operational complexity and can increase dependence of later assignments on earlier outcomes. A fixed allocation is simpler and does not respond to early outcome fluctuations; a cohort-adaptive design may improve allocation toward an arm under some conditions, but performance depends on the response model, delay pattern, cohort size, and allocation rule. No single allocation rule is best for every trial objective.