Randomization Lists for Replicate Sequences

Why the Allocation List Matters

In a replicate crossover study, each participant is assigned to a treatment sequence: an ordered schedule of formulations across periods. The randomization list turns the planned sequence proportions into an operational rule for assigning participants as they enroll. A useful list is not merely a random ordering. It should also support the intended allocation balance over the course of enrollment, be reproducible, and be auditable.

The key design choice is the unit being randomized. Here, it is the participant, and each participant receives one sequence assignment. For example, in a three-period partial-replicate design, the available sequences might be \(T\!R\!R\), \(R\!T\!R\), and \(R\!R\!T\). The randomization list assigns participants to these sequence labels; the period-by-period treatment schedule is determined by the sequence. The list does not change the design or its intended sequence proportions.

Blocking addresses a practical limitation of simple randomization: by chance, enrollment can temporarily favor one sequence. It does not guarantee that every prefix of the list has exactly the target proportions. That is generally impossible when the desired counts are not integers, and even when they are, exact balance at every enrollment position would constrain or eliminate randomness.

From Target Proportions to Blocks

Let \(K\) be the number of sequences, and let \(p_j\) be the intended allocation proportion for sequence \(j\), where \(j=1,\ldots,K\), \(p_j>0\), and \(\sum_{j=1}^{K}p_j=1\). Choose a block size \(b\) for which every \(b p_j\) is an integer. The block contains \(m_j=b p_j\) assignments to sequence \(j\). Randomly permute those assignments to obtain the block's order.

For equal allocation to three sequences, a block size of 6 gives two assignments to each sequence. For a target ratio of \(2:1:1\), a block size of 8 gives counts of 4, 2, and 2. The block size must be compatible with the intended proportions; choosing a size that cannot produce integer counts cannot deliver the exact target at the block boundary.

Within a block, each possible ordering of the prescribed multiset should have the same probability if the method is uniform random permutation. Assignments within that block are dependent: once a sequence's allotted positions have been used, it cannot appear again in that block. This dependence is part of the balancing design, not evidence that the list is incorrectly randomized.

Why Block Balance Holds—and What It Does Not Promise

At the end of each complete block, the construction itself guarantees the target. If there are \(q\) completed blocks, sequence \(j\) has count \(q m_j=qbp_j\), and the total enrollment is \(qb\). Its proportion is therefore \(qbp_j/(qb)=p_j\).

Within an unfinished block, let \(r\) be the number of positions already assigned in that block, and let \(x_j\) be the number of those positions assigned to sequence \(j\). The target count for those \(r\) positions is \(rp_j\). Because the completed blocks are exactly balanced, the total count difference from the target at that enrollment point is \(x_j-rp_j\). This difference need not be zero.

A simple bound is \(\lvert x_j-rp_j\rvert\leq r\): both \(x_j\) and \(rp_j\) lie between 0 and \(r\). The bound is deliberately broad; the exact possible differences also depend on the remaining composition of the block. The important operational conclusion is that the discrepancy comes only from the current incomplete block, while earlier completed blocks contribute no discrepancy. Relative imbalance can still matter early in enrollment, when \(r\) may be a substantial fraction of the total enrolled.

No blocked method can promise exact target proportions at every prefix of a genuinely randomized list. For instance, with three equally allocated sequences and one enrolled participant, the realized counts must be \(1,0,0\) in some order, not \(1/3,1/3,1/3\). Blocking controls where balance is recovered; it does not remove all interim variation.

Building a Reproducible List

A reproducible list requires more than a statement such as “randomize in blocks.” The allocation plan should specify the sequence labels and their treatment schedules, target proportions, allowed block sizes and their composition, any strata, the random-number method and software details, and how the generated list will be checked. Record the list version and generation date, as well as the seed and any information needed to reproduce the output. A seed alone may not reproduce the same list if the software, random-number algorithm, or implementation changes.

For unequal ratios, each block size must generate integer counts in the stated proportions. For equal allocation among three sequences, block sizes such as 6 or 9 meet that condition; block sizes such as 4 do not. If more than one compatible block size is allowed, the procedure must state how a block size is selected. Choosing among compatible sizes can reduce the ease of predicting the next assignment, but the possible sizes and their use should be prespecified, not selected in response to observed enrollment.

A planned total sample size should also be considered. If the total is not divisible by the denominator implied by the intended proportions, the final allocation cannot have exactly those proportions using whole participants. One option is to plan a final block specifically for the remaining enrollment, if its integer counts meet the target ratio. Otherwise, the protocol should state how a truncated final block is handled and what deviation is acceptable. Quietly adding or removing assignments to make observed totals look balanced undermines the prespecified procedure.

Stratification, Diagnostics, and Alternatives

If the design calls for balancing sequence allocation within prespecified enrollment strata, such as study center, generate a separate list for each stratum. Each stratum then has its own block progression and interim imbalance. This can support balance within centers, but it does not guarantee that the pooled proportions are exact at every point: centers may enroll at different rates, and some may be partway through blocks. Stratification should be limited to factors justified in advance; many small strata can create numerous incomplete blocks and complicate implementation.

Several checks can reveal a problem before or during use. Confirm that every sequence label maps to the intended replicate schedule, that the number of assignments in each block agrees with the target ratio, and that block lengths follow the prespecified rule. At enrollment, compare cumulative counts with the list position and the current block boundary. A discrepancy may indicate a skipped position, duplicate assignment, incorrect stratum list, or a report that counts screened rather than randomized participants. The audit should reconcile the discrepancy against the source list rather than generate replacement assignments informally.

Blocked randomization is one of several approaches. Simple randomization assigns each participant independently according to the target probabilities; it is straightforward and less structured, but can produce larger chance imbalances, particularly in small studies. Fixed-size permuted blocks recover exact balance at regular boundaries, but if the block size and prior assignments are known, later assignments may become predictable. Variable compatible block sizes make the boundary less obvious, though they do not guarantee concealment if the list is accessible. Other covariate-balancing approaches may be considered when additional balance goals matter, but they require their own prespecified rules and are not interchangeable with a simple block list.

The allocation list has limitations. It cannot ensure that all enrolled participants complete the study, that each sequence has equal evaluable data, or that sequence groups are similar on every baseline characteristic. It controls the assignment counts under the planned mechanism, not all sources of imbalance or missingness. Nor does blocked allocation justify changing the sequence design after enrollment begins.

Common Mistakes

• Demanding exact balance after every assignment. That is usually incompatible with random ordering and may be arithmetically impossible. Check the intended balance at prespecified block boundaries and assess interim deviations as partial-block variation.
• Using an incompatible block size. A block of 4 cannot give equal counts to three sequences. Choose a size for which every target count \(bp_j\) is an integer.
• Assuming a seed is the whole reproducibility record. Record the algorithm, software details, settings, and list version as well as the seed.
• Changing the list to repair an imbalance. An incomplete block can be imbalanced by design. Do not add, swap, or skip assignments outside the prespecified method.
• Ignoring predictability. A fixed block can make future assignments easier to infer near its end. Protect access to the list and consider prespecified variable compatible block sizes where appropriate.