From Site Strata to a Usable Schedule
In a multisite trial, enrollment at one site may proceed faster than at another. If the design calls for balance within each site, each site must have its own randomization sequence: a participant’s assignment is determined by that site’s place in its sequence, not by the pooled enrollment order across all sites. The practical task is to make those sequences long enough, compatible with the allocation ratio, and easy to check.
This tutorial builds site-stratified permuted-block schedules. The focus is the schedule itself: which blocks to use, how treatment labels are ordered within each block, and how to confirm the resulting counts. A schedule can guarantee a specified ratio at the end of each completed block, but it cannot guarantee that ratio at every interim enrollment point.
Let there be \(J\) sites, indexed by \(j=1,\ldots,J\), and let the trial use two treatments, A and B, with allocation ratio \(r:1\). A site’s schedule is used only for participants enrolled at that site. The schedule is prepared with enough assignments for the planned enrollment, including any buffer chosen by the trial team.
Choose Block Sizes Compatible With the Ratio
For an A:B allocation ratio of \(r:1\), a block containing \(r+1\) participants has \(r\) assignments to A and one to B. More generally, every compatible block has size \(m=(r+1)k\) for some positive integer \(k\). Its required counts are \(rk\) assignments to A and \(k\) to B. Thus, the block size must be divisible by the sum of the ratio parts.
For a 2:1 allocation, compatible block sizes include 3, 6, and 9. Their required counts are, respectively, 2:1, 4:2, and 6:3. A block of size 5 cannot give the exact 2:1 ratio, because five is not divisible by three. It should not be used as a block if exact ratio balance at block completion is a design requirement.
Selecting among compatible sizes also involves practical trade-offs. Smaller blocks reach the target ratio more often, but a fixed small block can make upcoming assignments easier to guess if the block structure and previous assignments are known. Larger blocks reduce how often a block boundary restores the target ratio. Randomly varying compatible block sizes can make the boundary less apparent, but each selected size still must support the chosen ratio. The appropriate choice depends on the protocol and operational context; compatibility alone does not settle it.
Worked Example: Selecting Blocks for a 2:1 Allocation
A hypothetical two-treatment trial plans to allocate participants to A and B in a 2:1 ratio. A site expects to enroll 15 participants. The design team is considering block sizes 5, 6, and 9.
For a 2:1 ratio, compatible sizes are multiples of \(2+1=3\). Size 5 is not compatible: it cannot be divided into the required counts in a 2:1 ratio. Size 6 is compatible, with \(k=6/3=2\), so each such block contains \(2(2)=4\) assignments to A and 2 to B. Size 9 is compatible, with \(k=9/3=3\), giving 6 assignments to A and 3 to B.
One way to provide exactly 15 scheduled assignments is to use one block of size 6 and one of size 9. The total counts will be \(4+6=10\) for A and \(2+3=5\) for B, which is 10:5, or 2:1. This determines a compatible plan, not the random order of the assignments; that order must still be randomized within each block.
Construct Separate Sequences for Each Site
Before generating any assignments, specify the design decisions that govern the schedule. At minimum, state the treatment labels and ratio, the sites that define the strata, the permissible block sizes, and the number of assignments to prepare for each site. Also specify that the assignments within each block are randomized and that each site’s sequence is used only at that site.
For a block with \(n_A\) assignments to A and \(n_B\) assignments to B, there are \(\binom{m}{n_A}\) distinct label orders, where \(m=n_A+n_B\). Under uniform random ordering within the block, each of these orders has probability \(1/\binom{m}{n_A}\). For example, a 2:1 block of size 6 has \(\binom{6}{4}=15\) possible orders, each with probability \(1/15\). Blocks are generated separately; if the design specifies independent randomization across blocks and sites, their orders are sampled independently.
The following assumptions make the schedule correspond to the intended design: the site is recorded correctly before assignment; each participant is assigned once and receives the next unused entry in that site’s sequence; treatment labels and the allocation ratio are fixed by the design; and the block orders are generated by a valid random procedure rather than selected to obtain a preferred pattern. The schedule also needs appropriate protection so that future assignments are not available to someone who could influence enrollment or assignment. These are design and implementation conditions, not properties created merely by printing a randomized list.
List each site and the number of assignments to prepare for its expected enrollment. Do not pool sites if the protocol requires site-specific balance.
For each site, choose block sizes that are multiples of the sum of the allocation-ratio parts. Confirm that the planned block sizes provide enough assignments.
Create exactly the required number of each treatment label, then randomize their order using the specified randomization procedure.
Count assignments within each block, sum them within each site, and then sum across sites. Compare every total with the planned ratio and capacity.
Worked Example: Building a Three-Site Schedule
Consider a hypothetical trial with sites North, Central, and South. The allocation ratio is 2:1, and planned capacities are 12, 9, and 6 participants, respectively. Use block sizes 6 and 9 where needed. Suppose the randomized block orders are those shown below; the sequences are hypothetical examples of possible randomized orders.
| Site | Block | Size | Assignment order | A count | B count |
|---|---|---|---|---|---|
| North | 1 | 6 | A, B, A, A, B, A | 4 | 2 |
| North | 2 | 6 | B, A, A, B, A, A | 4 | 2 |
| Central | 1 | 9 | A, A, B, A, B, A, A, B, A | 6 | 3 |
| South | 1 | 6 | B, A, A, B, A, A | 4 | 2 |
Each size-6 block contains four A assignments and two B assignments; the size-9 block contains six A assignments and three B assignments. North therefore has \(4+4=8\) A assignments and \(2+2=4\) B assignments. Central has 6 A and 3 B, and South has 4 A and 2 B.
Across the three sites, the schedule has \(8+6+4=18\) assignments to A and \(4+3+2=9\) to B. The total is \(18+9=27\) assignments, matching the combined capacity \(12+9+6=27\); the overall ratio is \(18:9=2:1\). This overall check is useful, but it does not replace the site-by-site checks: a schedule could have the correct overall totals while one site’s sequence is wrong.
Why Completed Blocks Have the Intended Counts
The count guarantee follows directly from how each block is constructed. If a block uses size \(m=(r+1)k\), it is populated with exactly \(rk\) labels A and \(k\) labels B before those labels are ordered. Randomly permuting the labels changes their positions, not their numbers. Therefore, every possible order of that block has the same specified treatment counts when the block is complete.
If site \(j\) has \(B_j\) completed blocks, with block \(b\) using multiplier \(k_{jb}\), its completed-block totals are \(N_{Aj}=r\sum_{b=1}^{B_j}k_{jb}\) and \(N_{Bj}=\sum_{b=1}^{B_j}k_{jb}\). The site’s completed-block total thus retains the ratio \(r:1\). Summing over sites retains that ratio as well, provided all counts refer to completed blocks and the same ratio is used at every site.
This guarantee is about counts, not outcomes or baseline characteristics. It does not ensure that treatment groups are identical on prognostic variables, nor does it mean that participants are assigned in a predictable order. Randomization determines the order; the block composition determines the completed-block totals.
Worked Example: Interpreting an Incomplete Block
Return to the hypothetical North schedule above. After the first six participants, the first block is complete, with 4 assigned to A and 2 to B. Suppose enrollment then stops after the first five entries of the second block, whose order is B, A, A, B, A.
Among those five entries, there are 3 assignments to A and 2 to B. North’s enrolled total at that point is therefore \(4+3=7\) for A and \(2+2=4\) for B, for 11 participants. The completed first block has the exact 2:1 ratio, but the 11 enrolled participants do not: their observed allocation ratio is \(7:4\), or 1.75:1.
That interim deviation does not show that the block was constructed incorrectly. The second block is incomplete, and its remaining assignment is A. If the block is completed, North’s totals become 8 for A and 4 for B, exactly 2:1. A site that enrolls fewer participants than planned may end with a partial final block; the design should not be described as guaranteeing exact ratio balance among all enrolled participants in that situation.
Checks, Alternatives, and Limitations
A basic schedule check can be done with a count table. For every block, confirm its size and treatment counts; for every site, sum its block counts; then check the combined totals. Also compare each site’s schedule length with its planned capacity. These checks can identify a block with the wrong composition, a sequence assigned to the wrong site, or an insufficient number of entries. They do not establish that the random orders were generated properly or that future assignments were concealed.
Common warning signs include a block size that is not compatible with the allocation ratio, a site sequence shorter than its planned enrollment, or an overall total that appears correct only because errors at different sites cancel. An apparent ratio deviation during enrollment should first be interpreted in light of whether each site has completed its current block. Conversely, an exact total does not by itself establish that assignments were randomized: treatment labels placed in a regular or deliberately chosen order could have correct counts but fail the randomization requirement.
| Approach | What it controls | Key distinction |
|---|---|---|
| Complete randomization | Random assignment counts across the full randomized group, subject to the chosen allocation procedure | Does not organize assignments into site-specific completed blocks. |
| Stratified complete randomization | Randomization within each stratum | Does not necessarily restore the target ratio at regular points during enrollment within a site. |
| Site-stratified permuted blocks | Site-specific assignment sequences and target counts at completed block boundaries | Requires compatible block sizes; an incomplete final block need not have the target ratio. |
The main limitation is that block balance is guaranteed only at block completion. Fixed, known block sizes can also increase predictability near a block’s end, even though the order within the block was randomized. Varying among compatible block sizes can reduce the visibility of block boundaries, but it does not eliminate the need for allocation concealment or make every design equally suitable. Further, treating each site as a stratum creates more separate sequences to manage; sites with little enrollment may finish with partial blocks and limited within-site counts.
Common Mistake
Check Your Understanding
Use the definitions and count rules above. The solutions are included after the questions so that each check can be verified.
- For a 3:1 allocation, what are the treatment counts in a compatible block of size 8? Is size 10 compatible?
- A hypothetical site uses two completed blocks for a 2:1 allocation: one of size 6 and one of size 9. Find the site’s A and B totals.
- Why can the observed ratio among participants enrolled so far differ from the target ratio even when the schedule is correctly constructed?
- Two sites have correct combined totals for a 1:1 schedule, but one site has three more A assignments than B assignments. Does the combined check establish site-level balance?
Solutions. (1) Size 8 is \(2(3+1)\), so \(k=2\); it contains \(3(2)=6\) assignments to A and 2 to B. Size 10 is not compatible because it is not divisible by 4. (2) The size-6 block contains 4 A and 2 B; the size-9 block contains 6 A and 3 B. The totals are 10 A and 5 B. (3) The current count may include only part of the last block. The exact ratio is guaranteed at completed block boundaries, not at every interim point. (4) No. Combined totals can conceal an imbalance at one site, so each site’s counts must be checked separately.