Temporal Blocking to Balance Enrollment Periods

Why Block Enrollment in Calendar Time?

In a response-adaptive trial, assignment probabilities may change as outcomes accumulate. If prognosis also changes over calendar time, an adaptive rule can assign different proportions of participants to each treatment in different enrollment periods. A treatment comparison may then reflect both treatment and the kinds of participants enrolled at different times. Earlier material on calendar-time drift and concurrent controls explains why randomization alone does not guarantee good temporal overlap.

Temporal blocking addresses this design problem by dividing enrollment into prespecified calendar periods and requiring each treatment to receive a minimum number of assignments within each period. The blocks do not remove drift, ensure that participants are identical across treatments, or establish a causal effect by themselves. They provide a design guarantee: each defined period contains representation from every treatment, if the block reaches its planned enrollment and the assignment constraints are followed.

Definition: A calendar-time block is a prespecified interval of enrollment time, such as a week or month. For block \(b\), let \(N_b\) be its planned number of assignments, \(K\) the number of treatments, and \(n_{bk}\) the number assigned to treatment \(k\). A minimum-representation requirement specifies \(m_{bk}\) assignments for treatment \(k\) in block \(b\), with \(\sum_{k=1}^{K}m_{bk}\leq N_b\). The requirement is met when \(n_{bk}\geq m_{bk}\) for every treatment.

Calendar boundaries should be chosen before looking at outcomes. They can be defined by dates or by consecutive enrollment periods, but should not be moved in response to favorable or unfavorable outcomes. The design also needs a planned block size or a clear rule for what happens if enrollment stops before the block is complete. If a block ends early, minimum counts that have not yet been reached cannot be guaranteed.

Constrained Adaptive Assignment Within a Block

One way to combine temporal representation with response adaptation is to calculate an adaptive assignment distribution at each enrollment, then restrict assignment to treatments that leave the minimum-count requirements achievable. The adaptive distribution may use the response information available under the trial’s prespecified rule. It does not change the block boundaries or the representation targets.

Immediately before an assignment in block \(b\), let \(R_b\) be the number of unfilled slots in that block and let \(n_{bk}\) be the current count for treatment \(k\). Define the remaining deficit for treatment \(k\) as

d_{bk}=\max(0,m_{bk}-n_{bk}), \qquad D_b=\sum_{k=1}^{K}d_{bk}.

A treatment is feasible if assigning the next participant to it leaves enough slots to fill every remaining deficit. Let \(F_b\) be the set of feasible treatments. If the response-adaptive rule gives treatment \(k\) a positive weight \(w_{bk}\), one constrained assignment rule is

q_{bk}= \begin{cases} \dfrac{w_{bk}}{\sum_{\ell\in F_b}w_{b\ell}}, & k\in F_b,\\[6pt] 0, & k\notin F_b. \end{cases}

Here \(q_{bk}\) is the conditional probability of assigning the next participant to treatment \(k\), given the allocation history and the information used by the adaptive rule. Every feasible treatment must have positive weight for this rule to retain randomization among feasible choices. As the block fills, a treatment may become infeasible; near the end, the representation requirement can force assignments and reduce adaptation.

Why the Feasibility Rule Works

Before an assignment, the current deficits must be fillable, so \(D_b\leq R_b\). If treatment \(k\) is assigned, its deficit falls by one when \(d_{bk}>0\), and otherwise does not change. Thus the total deficit after that assignment is \(D_b-\mathbf{1}(d_{bk}>0)\), while the number of remaining slots becomes \(R_b-1\). The assignment is feasible precisely when the new total deficit is no greater than the new number of slots.

D_b-\mathbf{1}(d_{bk}>0)\leq R_b-1.

If \(D_b=R_b\), only a treatment with a positive deficit is feasible: every remaining slot is needed to meet a target. If \(D_b<R_b\), any treatment is feasible, because at least one slot remains beyond those needed for the deficits. This also shows why the rule preserves flexibility early in a block but may restrict it as the end approaches.

Guarantee: Suppose a block has \(N_b\) planned assignments, its minimum-count requirements are feasible, and every assignment is made to a feasible treatment. If all \(N_b\) assignments occur, then \(n_{bk}\geq m_{bk}\) for every treatment \(k\) when the block closes. The guarantee follows because the feasibility rule never permits a choice that makes the remaining deficits exceed the remaining slots.

This is a design guarantee, not a claim that every assignment remains genuinely adaptive. When only one treatment is feasible, its conditional assignment probability is one. The probabilities should be calculated and recorded under the actual constrained rule; probabilities from an unconstrained adaptive rule would not describe the assignments that were made.

Worked Example: An Eight-Participant Block

Consider a hypothetical two-treatment trial with \(N_b=8\) assignments in a calendar block. The design requires at least \(m_{bA}=2\) assignments to A and \(m_{bB}=2\) to B. At the start, the adaptive weights are \(w_{bA}=0.70\) and \(w_{bB}=0.30\). Both treatments are feasible, so the assignment probabilities are \(q_{bA}=0.70\) and \(q_{bB}=0.30\).

Suppose the realized first assignment is A. There are now \(R_b=7\) slots and deficits \(d_{bA}=1\), \(d_{bB}=2\), so \(D_b=3<7\). Both treatments remain feasible. If the updated adaptive weights are 0.60 for A and 0.40 for B, the constrained probabilities are \(0.60/(0.60+0.40)=0.60\) for A and \(0.40\) for B.

Suppose the next assignment is B. There are six slots left, with deficits 1 for A and 1 for B. Both remain feasible, and the adaptive rule may continue to favor either treatment. For example, a subsequent weight pair of 0.65 for A and 0.35 for B gives probabilities 0.65 and 0.35 while both treatments are feasible.

A possible full realization is A, B, A, A, B, A, B, B. It gives \(n_{bA}=4\) and \(n_{bB}=4\), satisfying both minimums. The sequence is a hypothetical realization, not a prediction of the allocation probabilities. The probabilities depend on the adaptive weights and the feasible set at each assignment.

Choosing Block Width

Block width determines the time scale on which the design guarantees representation. Narrow blocks give tighter temporal overlap: participants assigned to different treatments are enrolled within relatively similar periods. But each narrow block must be large enough to accommodate its representation targets. If enrollment is slow or unpredictable, targets that are feasible on average may be infeasible in a particular block.

Wider blocks allow more assignments to respond to accumulating outcome information before a representation constraint binds. However, a treatment can be absent from a short subperiod within a wide block while still meeting the block-level minimum. Therefore, a guarantee for a four-week block is not a guarantee for each week in that block.

Worked Example: Comparing Two-Week and Eight-Week Blocks

Suppose a hypothetical trial expects eight participants per week and has two treatments. The design requires at least two assignments to each treatment in every block. With two-week blocks, the planned block size is \(N_b=16\); the minimums require \(2+2=4\) assignments, leaving 12 assignments beyond those minimums. With one eight-week block, \(N_b=64\), leaving \(64-4=60\) assignments beyond the same block-level minimums.

The two-week design guarantees at least two assignments to each treatment every two weeks. The eight-week design guarantees only two to each treatment over the full eight weeks; it does not prevent one treatment from being absent in a particular two-week segment. The wider block also gives more assignments before its block-level minimums can constrain allocation, so it can offer more room for response adaptation across the full period.

If enrollment instead falls to four participants in a two-week period, the same two-week minimums require all four assignments to meet the targets. The block is feasible, but there is no allocation flexibility in that period. A practical block-width decision must therefore consider expected enrollment, the desired time resolution for overlap, and how much adaptive freedom is acceptable—not just the total trial sample size.

What the Design Does and Does Not Guarantee

Temporal blocking is one option among several ways to handle the tension between adaptation and enrollment-time overlap.

Approach	What it provides	Main trade-off
Unrestricted response-adaptive assignment	Can follow adaptive probabilities without within-period minimum counts.	A treatment may have few or no assignments in a calendar period.
Constrained adaptation within calendar blocks	Guarantees prespecified minimum representation in each completed block.	Some assignments may be forced or less adaptive as constraints bind.
Non-adaptive blocked randomization	Can balance treatment counts within blocks using a fixed randomization scheme.	Does not use accumulating response information to change allocation.

Minimum counts establish representation, not equal allocation. A block with minimums of two per treatment could still have very unequal final counts. Nor does representation alone guarantee that allocation probabilities were adequate for every scientific comparison, that measurements were collected consistently, or that outcomes did not drift within the block. The block is a design safeguard at its specified calendar resolution.

Diagnostics and Common Failure Modes

Review implementation by block, rather than relying only on trial-wide totals. A simple arm-by-block enrollment table should show the dates or boundaries, planned and actual block sizes, and treatment counts. Compare each count with its target. Also inspect the recorded conditional probabilities: probabilities should correspond to the feasible set and adaptive weights at the time of each assignment.

Blocks defined after observing outcomes. Moving boundaries to create favorable-looking balance makes the design outcome-dependent. Prespecify calendar boundaries.
Targets too large for the block. If \(\sum_k m_{bk}>N_b\), the requirements cannot all be met. Check feasibility before enrollment.
Unfinished blocks treated as complete. Early trial closure or slow enrollment can leave a block below its minimums. Report this as a shortfall, not as a satisfied guarantee.
Assuming block-level representation means fine-scale overlap. A wide block can contain a subperiod with assignments to only one treatment. Choose a width that matches the time resolution at which overlap is needed.
Ignoring constraints when reporting assignment probabilities. The probabilities must reflect the feasible set, not just the unconstrained adaptive weights.

Common mistake: “Every treatment has several participants in the trial overall, so treatment overlap is adequate throughout enrollment.” Overall counts do not establish within-period representation. The correction is to inspect counts at the prespecified block level and state precisely which period each minimum-count guarantee covers.

Worked Example: Detecting an Infeasible Block Plan

A hypothetical three-treatment trial plans a block of 10 assignments and proposes minimums of four assignments per treatment. The total required is \(4+4+4=12\), which exceeds the block size \(N_b=10\). No assignment algorithm can satisfy all three targets.

One repair is to reduce the minimums—for example, to three per treatment, requiring \(3+3+3=9\) assignments and leaving one assignment unconstrained by the minimums. Another is to increase the planned block size to at least 12. The choice depends on enrollment and the required temporal resolution. It should be made before the block begins, not after observing its outcomes.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A two-treatment block has \(N_b=12\) and minimums of three assignments per treatment. Are the targets feasible? How many assignments remain beyond the minimums?
At a point in a block, \(R_b=3\) slots remain and the deficits are \(d_{bA}=2\), \(d_{bB}=1\). Which treatments are feasible?
Why can an eight-week block fail to guarantee treatment overlap in every week?
What does the minimum-count guarantee fail to establish about outcomes or treatment effects?

Solutions. (1) The targets are feasible because \(3+3=6\leq12\). There are \(12-6=6\) assignments beyond the minimums. (2) Here \(D_b=2+1=3=R_b\), so every remaining slot is needed: A and B are both feasible because each has a positive deficit. Assigning either treatment reduces \(D_b\) to 2 while \(R_b\) becomes 2. (3) The guarantee applies to the eight-week total, not to each week within it; a treatment could receive no assignments during one week and still meet the block minimum. (4) The guarantee establishes only the specified treatment counts in a completed block. It does not prove equal prognosis, eliminate drift within the block, or by itself establish a treatment effect.