Worked Design Example With Midtrial Calendar-Time Blocks

Why a Known Midtrial Change Needs a Design Plan

Suppose a trial compares an experimental treatment with standard care, and standard care is scheduled to change on a known date. Patients enrolled before that date receive one version of standard care; those enrolled after it receive another. The change may alter outcomes even if the experimental treatment itself is unchanged. A design that adapts allocation across the whole trial without accounting for the date can therefore base later assignments on responses that are no longer relevant to the current comparison.

The design goal is to maintain concurrent comparisons: within each period of standard care, enroll enough patients to both the experimental treatment and the standard-care control. Calendar-time blocks make the allocation plan operational. An adaptive target can still influence the treatment ratio, but block-level rules prevent that target from eliminating one arm during either period.

The plan below assumes the date of the standard-care change is known in advance, the treatment assignment is randomized, and the outcome used for adaptation is defined consistently across patients. For the binary-outcome illustration, responses within an enrollment epoch are treated as comparable for the allocation update. If substantial drift is expected even within an epoch, blocks should be shortened or the adaptation rule should account for that drift; a known care change does not make all other time trends disappear.

Epochs, Blocks and an Adaptive Allocation Rule

For clarity, call the experimental treatment \(A\) and standard care \(C\). Let a favorable binary outcome be coded as 1 and an unfavorable outcome as 0. One possible response-adaptive rule is to estimate each arm’s favorable-outcome probability within the current epoch and form the target

Here, \(\tilde p_A\) and \(\tilde p_C\) are the current epoch’s estimated probabilities of a favorable outcome. This square-root rule is an illustrative ethical tilting rule: a higher estimated favorable-outcome probability tends to give that arm a larger share, without assigning everyone to it. It is not a universal optimum. The rule, outcome, update schedule and safeguards must be selected before enrollment.

For example, a simple estimation scheme begins with independent Beta(1,1) priors for the two favorable-outcome probabilities. If an arm has \(s\) favorable outcomes and \(f\) unfavorable outcomes observed in the current epoch, its posterior distribution is Beta(1 + s, 1 + f), with posterior mean \((1+s)/(2+s+f)\). Only outcomes available before a block’s allocation is set are used. This defines an update rule; it does not assert that a particular prior or target is required for every trial.

Let \(B\) be the block size, and require at least \(m\) assignments to each arm in every complete block. First clip \(q_A\) to a prespecified interval, such as \([0.35,0.65]\), to limit extreme target shifts. Then set the number assigned to \(A\) to the nearest integer to \(Bq_A\), restricted to the feasible range \(m\) through \(B-m\). Assign the remaining block slots to \(C\). Randomly permute the resulting assignments over the block’s enrollment positions. All block rules, including how rounding ties are handled, should be fixed in advance.

The resulting realized block ratio need not equal \(q_A\) exactly: the target is translated into an integer count and bounded by the minimum arm counts. Random permutation determines which eligible patient receives each assignment. The target is set before the block; it is not recalculated after each response within that block.

Worked Design Example: Six Blocks Around a Care Change

What the Block Randomization Guarantees

A block of \(B\) positions with a prespecified count \(n_A\) for treatment \(A\) contains exactly \(n_A\) assignments to \(A\) and \(B-n_A\) to \(C\). Randomly permuting that assignment list protects the order from being chosen deterministically by the investigator. At the start of the block, each position has the same marginal chance of receiving \(A\), namely \(n_A/B\).

To see why, let \(I_j\) equal 1 if position \(j\) receives \(A\) and 0 otherwise. Every permutation position is symmetric, so each has the same expectation, say \(r\). But the block has exactly \(n_A\) assignments to \(A\), so \(\sum_{j=1}^{B} I_j=n_A\). Taking expectations gives \(Br=n_A\), hence \(r=n_A/B\). After some assignments have been revealed, the conditional probability for a remaining position is the number of remaining \(A\) assignments divided by the number of remaining positions. Thus probabilities can approach 0 or 1 near the end of a fixed-count block even though the block-start marginal probability is within the planned range.

This distinction matters when describing the design. The clipped target \(q_A\) is not necessarily the probability of assignment for every patient: integer rounding, block constraints and the revealed assignments all matter. The actual allocation mechanism should be recorded, including the target used, block counts and randomized order. If a later analysis uses assignment probabilities, it must use probabilities appropriate to that mechanism rather than simply substituting the unclipped target.

Worked Example: Updating the Target Within an Epoch

Worked Example: Closing a Block at the Change Date

Diagnostics, Alternatives and Limits

Before enrollment, check that every planned block size can satisfy its minimums: \(B\geq 2m\). Check that the clipped target and rounding rule always produce a feasible count. Confirm that the care-change date, block boundaries, outcome definition, update times and rule for incomplete blocks are documented. A simulated run of the allocation algorithm can reveal coding errors, unexpectedly extreme realized ratios, or too few planned assignments in one epoch.

During the trial, monitor whether blocks cross the change date, whether assignments match the recorded block plan, and whether outcomes used in an update were available at the time the target was set. Also check that each epoch has the intended arm representation. If outcomes are delayed, a target may reflect only a small and potentially unrepresentative set of observed responses; the update schedule should account for that delay rather than treating unavailable outcomes as observed.

The main alternatives answer different design priorities. Fixed 1:1 randomization within every block gives predictable concurrent representation and does not adapt to observed outcomes. Unrestricted response-adaptive allocation can change the arm ratio more quickly, but may leave too few concurrent controls in a period, especially if probabilities become extreme. The bounded, block-level plan trades some responsiveness for prespecified minimum representation and clear epoch boundaries.

This plan does not remove every source of bias or guarantee that observed outcomes are comparable. Randomization protects the allocation mechanism, while the concurrent assignments protect against comparing patients enrolled only under different care versions. A change in patient mix, other calendar-time drift, or inconsistent outcome measurement can still complicate interpretation. The target rule is also a design choice, not evidence that the treatment with the larger current estimate is superior. In particular, the plan should not be used to treat outcomes from different epochs as though they necessarily estimate one unchanged treatment contrast.

Common Mistake