Why Vary the Block Size?
Permuted blocks control allocation counts within short stretches of enrollment: once a block is complete, its assignments match the target allocation ratio exactly. But if every block has the same size and that size is known, the remaining assignments in a block can become predictable. Randomly varying block sizes addresses this weakness: a participant assigning treatment may not know where the current block ends, even though each completed block remains balanced.
Varying block sizes does not make assignments unpredictable in every circumstance. The current allocation history still provides information, and the final positions within a block may be forced by balance. The design goal is narrower: preserve exact balance at completed-block boundaries while making those boundaries less easy to anticipate.
Compatible Sizes and the Randomization Rule
For a 1:1 allocation ratio, compatible block sizes are even: size 4 contains two assignments per arm, and size 6 contains three per arm. For a 2:1 ratio, compatible sizes are multiples of 3: size 6 contains four assignments to the first arm and two to the second. The size set and the rule for selecting among its members must be fixed before enrollment; they are not changed in response to observed outcomes.
One clear rule is to select independently for each new block from the prespecified set, with stated probabilities, and then choose uniformly among the distinct sequences of assignments having the required counts. For instance, with size 4 and a 1:1 ratio, there are \(\binom{4}{2}=6\) distinct sequences, each with probability \(1/6\) conditional on that size. Randomly choosing a block size and randomly permuting its assignments are separate steps. A design must specify both.
Choose only compatible sizes and state how each new block size is selected. Do not select sizes after examining treatment outcomes or choosing a desired next assignment.
At the start of a new block, draw its size using the prespecified rule. If the schedule is concealed, personnel enrolling participants should not be able to inspect future assignments or infer the chosen size from an exposed schedule.
Assign the required number to each arm in a randomized order, uniformly among distinct valid sequences if that is the specified rule.
After the block is complete, select a size for the next block. Check each completed block against its specified size and allocation counts.
Why Completed Blocks Remain Balanced
The balance property follows directly from the construction; it does not depend on which compatible size is selected. Let block \(j\) have size \(k_j=m_j(a+b)\). By definition, it contains \(m_j a\) assignments to the first arm and \(m_j b\) to the second. Therefore, after \(J\) complete blocks, the totals are \(\sum_{j=1}^{J}m_j a\) and \(\sum_{j=1}^{J}m_j b\), respectively. Their ratio is \(a:b\), and this conclusion holds for every sequence of selected compatible sizes.
This guarantee applies at the end of each complete block. It does not guarantee exact balance at every point within a block, or if enrollment stops partway through a block. Randomly choosing among compatible sizes changes neither of those qualifications.
Worked Example: Building a 1:1 Schedule From Two Sizes
Suppose a hypothetical two-arm trial targets a 1:1 ratio and prespecifies block sizes 4 and 6, selected independently with probability \(1/2\) each. For illustration, imagine the realized sizes are 4, 6, and 4, and the randomized sequences are ABBA, BAABAB, and BAAB. Here A and B denote the two treatment arms.
Check the sizes. Each size is even, so each is compatible with 1:1 allocation. The first and third blocks each contain two assignments to A and two to B. The second contains three assignments to each arm.
Check the schedule. The three blocks together contain \(4+6+4=14\) assignments. Arm A receives \(2+3+2=7\), and arm B receives \(2+3+2=7\). Thus the completed schedule is balanced at 1:1.
This is one possible realization of the random procedure, not a rule to alternate between sizes 4 and 6. Under the stated independent selection rule, another schedule could select size 4 twice in a row or size 6 several times in succession. The randomized permutations also need not resemble the displayed sequences.
How Variation Changes Predictability
If a fixed block size is known, the number of slots remaining in the current block can reveal the next assignment. For example, in a known size-4 block with a 1:1 ratio, a history of A, A, B means that the last slot must be B. Varying the size can reduce this certainty when the size is concealed, because the same observed history may be consistent with more than one block size.
Worked Example: A Concealed Size Makes the Next Assignment Less Certain
Suppose a hypothetical 1:1 design chooses each new block size, 4 or 6, with probability \(1/2\). Within each block, all distinct balanced sequences are equally likely. The first three assignments observed in the current block are A, A, B. Assume the block size has not been revealed.
Find the probability of this history under size 4. A size-4 block has two A and two B assignments. The probability of the ordered prefix AAB is \( (2/4)(1/3)(2/2)=1/6 \). If the size is 4, the remaining assignment must be B.
Find the probability of this history under size 6. A size-6 block has three assignments per arm, so the probability of the same prefix is \( (3/6)(2/5)(3/4)=3/20 \). Given size 6 and prefix AAB, two A assignments and one B assignment remain; the next assignment is A with probability \(2/3\).
Update the probability of the block size. The two possible sizes had equal prior probabilities, so their posterior probabilities are proportional to their prefix probabilities. Thus \(P(\text{size }4\mid\text{AAB})=(1/6)/[(1/6)+(3/20)]=10/19\), and \(P(\text{size }6\mid\text{AAB})=9/19\).
The next assignment is A only if the block size is 6, and then its probability is \(2/3\). Consequently, \(P(\text{next is A}\mid\text{AAB})=(9/19)(2/3)=6/19\), approximately 0.3158. The next assignment is therefore B with probability \(13/19\), approximately 0.6842. With a known size-4 block, B would be certain; with the size concealed and variable, it is not certain. The assignment is still not equally likely to be A or B, so variation reduces predictability here without eliminating it.
This calculation illustrates why the randomization rule and concealment both matter. If enrolling staff can see the selected block size, the uncertainty from not knowing that size disappears. If the size remains hidden, the observed history still changes the probabilities: the history AAB is somewhat more likely under size 4 than under size 6, so it shifts belief toward size 4. A randomly varying design does not justify claiming that each next assignment has probability \(1/2\).
Worked Example: A Compatible Set for Unequal Allocation
Consider a hypothetical trial with target ratio 2:1, with A receiving twice as many assignments as B. Suppose the prespecified size set is \(\{6,9\}\), and each size is selected with probability \(1/2\). Both sizes are multiples of \(2+1=3\), so both are compatible.
A size-6 block contains \(6(2/3)=4\) assignments to A and \(6(1/3)=2\) to B. A size-9 block contains \(9(2/3)=6\) to A and \(9(1/3)=3\) to B. Imagine the selected sizes are 6 and 9, with randomized sequences AABABA and AB AABAAAB, respectively (spaces added only to separate the blocks). The first sequence has four A and two B; the second has six A and three B.
Across the two completed blocks, there are \(6+9=15\) participants: \(4+6=10\) assigned to A and \(2+3=5\) assigned to B. The ratio is \(10:5=2:1\), as required. The particular order within each block is randomized; the displayed sequences simply illustrate one valid realization.
Diagnostics, Alternatives, and Limitations
A schedule audit should check that every block size belongs to the prespecified set, every block has the correct arm counts for the target ratio, and the selected sizes follow the stated randomization rule. For example, a size-5 block is incompatible with a 1:1 ratio, while a size-6 block with four A and two B is incompatible with 1:1 even though the size itself is allowed. Checking only the overall totals can miss these block-level errors.
Keep three design features distinct. Fixed-size permuted blocks also guarantee balance after each complete block, but known fixed sizes can make block completion easier to predict. Randomly varying permuted blocks retain that completed-block guarantee while making the boundary less certain when the size is concealed. Complete randomization assigns participants across the full schedule without requiring each short block to have exact balance; it avoids a known block boundary but can produce greater chance imbalance over a short enrollment period. These approaches address different priorities, so varying block sizes is not automatically preferable in every design.
There are practical limits. A finite set of sizes does not prevent inference from the assignment history, and a participant’s assignment can still become forced near the end of a block. If enrollment ends partway through a block, the final enrolled group need not satisfy the target ratio. Also, choosing very small blocks can improve short-term balance while leaving fewer possible assignment sequences, which may make the within-block order easier to anticipate. These are reasons to specify and review the design carefully, not reasons to claim that varying sizes removes all predictability.
- Do not confuse compatibility with random selection. Compatible sizes make the intended block counts possible; they do not specify how a size or sequence is randomized.
- Do not treat a concealed size as unknowable. Assignment history may provide evidence about the current block size, as the conditional-probability example shows.
- Do not promise exact balance at every enrollment point. The exact guarantee is at completed-block boundaries.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- For a 1:1 target ratio, are sizes 4, 6, and 8 compatible? How many assignments to each arm occur in a size-8 block?
- Is a fixed-total schedule using block sizes 4 and 6 feasible for 18 participants if it uses only complete blocks? Explain.
- For a 2:1 target ratio, what counts belong in a size-12 block? Is size 10 compatible?
- Explain why a varying-size design does not guarantee that the next assignment is equally likely to be either arm.
Solutions. (1) All three sizes are even and hence compatible with 1:1 allocation; a size-8 block contains four assignments to each arm. (2) Yes. Three complete size-6 blocks total \(3(6)=18\) participants, so this schedule is feasible; the available size 4 does not have to be used. (3) Since \(2+1=3\), a size-12 block has \(12(2/3)=8\) assignments to the first arm and \(12(1/3)=4\) to the second. Size 10 is not a multiple of 3 and is incompatible. (4) The history within a block affects how many assignments of each arm remain. Concealing a randomly selected block size can leave uncertainty about which assignments remain, but the history can still make one arm more likely than the other, and an assignment can be forced when only one slot remains.