Resetting Adaptive Targets Across Enrollment Epochs

Why Reset Targets at Prescheduled Boundaries?

In response-adaptive randomization, allocation probabilities can change as trial responses accumulate. Updating after every outcome may make the allocation rule react to noise; keeping one probability throughout a long trial may leave it poorly matched to later information. An epoch-based policy divides enrollment into prespecified calendar periods and sets a target allocation at the start of each period. The target remains in force until the next boundary.

This approach separates when the rule is allowed to update from what the update calculates. The calendar boundaries are fixed before outcomes are examined. At each boundary, the design uses the outcomes available under a prespecified data rule to calculate a new target. Within an epoch, each participant is randomized using that target. An epoch target is not a promise that the realized treatment counts will equal their target counts.

A complete rule needs more detail than “update periodically.” It should state the boundary schedule, whether the update uses cumulative outcomes or a recent window, how pending outcomes are handled, how response estimates become allocation probabilities, and what minimum and maximum probabilities are allowed. The probability for each arriving participant must be determined before that participant’s outcome is observed.

A Concrete Target-Update Rule

Consider two arms, A and B, and a binary outcome for which a larger response rate is preferred. Write \(\hat{p}_{A,k}\) and \(\hat{p}_{B,k}\) for the response-rate estimates available at the start of epoch \(k\). A simple illustrative rule is

Here \(q_k\) is the probability of assigning a participant in epoch \(k\) to A; the probability of assignment to B is \(1-q_k\). The constant \(\gamma>0\) controls how strongly the target responds to the estimated difference, and \(0<\varepsilon<1/2\) imposes allocation floors and ceilings. The rule is an example, not a universally optimal choice. Its parameters and response-estimation method must be selected in advance for the trial’s aims.

For a simple interpretation of the calculations, suppose responses are independent Bernoulli outcomes within each arm, the response probabilities are stable over the period used to estimate them, and current assignment is randomized with the specified probability conditional on the trial history. In that setting, an observed response rate is a natural estimate of the corresponding response probability. If outcomes drift over calendar time, a cumulative rate may combine different periods; the design must then prespecify an appropriate time-aware estimation method rather than assume that this simple calculation resolves drift.

What an Epoch Target Guarantees—and What It Does Not

Assume an epoch contains \(m\) participants, its target \(q\) is fixed at the boundary, and assignments are generated independently with probability \(q\) for A. Conditional on the information available at that boundary, the number \(N_A\) assigned to A has a binomial distribution. Consequently,

These results follow by writing \(N_A\) as the sum of \(m\) assignment indicators, each with mean \(q\) and variance \(q(1-q)\). Under independent randomization, the indicators have zero pairwise covariance, so their means and variances add. The realized count can differ from \(mq\), even when the randomization procedure is implemented correctly.

For example, if \(m=20\) and \(q=0.60\), then \(E(N_A)=12\) and \(SD(N_A)=\sqrt{20(0.60)(0.40)}=\sqrt{4.8}=2.191\) participants. This is variation around the target, not a failure of the probability rule. More importantly, if \(q\ne 0.5\), the intended allocation is not equal allocation. “Within-period balance” should mean consistency with the prespecified target, not necessarily equal arm counts.

An epoch-based target is also not a quota. A quota or a blocking procedure can constrain counts within a block; independent random assignment at a fixed probability does not. The distinction matters if the trial specifically needs close count balance in every short time period. Temporal blocking is an alternative when that is the design objective, but it changes the assignment mechanism.

Updating at Boundaries: Information, Timing, and Justification

An update rule is only as meaningful as its data rule. Cumulative estimates use more observations and can be less noisy when response probabilities are stable, but may react slowly if those probabilities change. A recent-window estimate can respond more quickly to a changing environment, but usually uses fewer observations and may be more variable. The design should specify which data are eligible at each boundary; it should not choose a convenient window after inspecting the results.

Outcome availability also matters. If response is measured weeks after enrollment, many participants enrolled before a boundary may not yet have an outcome at the boundary. One transparent policy is to calculate each update from outcomes recorded by a prespecified cutoff, using only outcomes then available. Another is to delay a boundary until a prespecified minimum information rule is met. Either policy must be stated in advance and applied consistently. Treating pending outcomes as nonresponses would change the response-rate calculation and is not a neutral substitute.

The timing rule helps avoid a subtle form of outcome-dependent assignment: the next target is based on data available at a boundary fixed in advance, not on a boundary chosen because an interim pattern looks persuasive. It does not remove bias from a poorly specified estimator or ensure that response estimates are stable. When prognosis or treatment response changes with calendar time, concurrent treatment comparisons and appropriate time-aware estimation remain important. Allocation changes alone do not remove temporal imbalance or make outcomes comparable across different enrollment periods.

Choosing Epoch Length and Comparing Alternatives

Shorter epochs permit more frequent target updates. They can respond sooner to newly observed information or a changing allocation objective, but each update may be based on little data. That can make targets jump in response to random variation. More frequent updates can also make the allocation probability history more complex to monitor and communicate.

Longer epochs hold the target steady for more participants, so the realized allocation proportion is generally less variable around that target under independent randomization. The trade-off is that a target may become stale before the next boundary. If outcomes are delayed, a nominally short epoch may still update using old information. Epoch length therefore needs to be considered together with outcome latency, expected enrollment, and the amount of information needed for a stable update.

A practical design should examine its rule before enrollment: calculate how targets respond to plausible response differences, check that floors and ceilings preserve adequate assignment to each arm, and check what occurs when few outcomes are available. After enrollment begins, useful monitoring includes the planned versus realized allocation proportion in each epoch, the number of outcomes available at each update, and whether the scheduled boundaries and update inputs were followed. Large deviations from the target can occur by chance, especially in small epochs; they should prompt a check of implementation and randomization records, not an automatic, unplanned alteration of the rule.

Common Mistake and Limitations

• Mistake: “The target is 60% for A, so exactly 60% of this epoch will be assigned to A.” Correction: A target probability determines the expected count under the specified randomization, not a guaranteed count. A design needing constrained counts must specify a count-balancing mechanism.
• Mistake: “Shorter epochs always improve balance.” Correction: Shorter epochs allow more frequent updates but have more variable realized proportions around each epoch’s target. They do not automatically improve balance or the quality of response estimates.
• Mistake: “A higher observed response rate proves the arm is better, so its target should increase.” Correction: Response estimates are uncertain, and the target rule is an allocation decision, not an efficacy test. The estimation method, update schedule, and probability bounds should be set in advance.
• Mistake: “Resetting the target removes calendar-time drift.” Correction: A new target does not itself adjust outcome comparisons for time. If outcomes or prognosis vary over enrollment, the design and analysis must account for that variation.

Epoch-based updating also has limits when there are very few observed outcomes at a boundary, substantial outcome delay, or abrupt changes between boundaries. A prespecified rule can make these conditions explicit, but cannot create information that has not yet been observed. Nor does one particular epoch length suit every trial: the choice depends on the intended adaptation, enrollment speed, outcome timing, and the need to keep treatment options available throughout calendar time.