Why the Allocation Ratio Constrains Block Size
A block for two treatments has a finite number of assignment positions. If its treatment counts must match a specified allocation ratio exactly, those counts must be integers in the same proportion as the ratio. Consequently, not every block size can support every target ratio. The block size and target ratio need to be chosen together.
This is a simple arithmetic requirement, but overlooking it can produce a schedule that cannot have the intended composition. The key is to reduce the ratio to its smallest whole-number terms, then check whether the block size is a multiple of the sum of those terms. Once that check passes, the treatment counts are determined.
The Divisibility Rule
If treatment 1 is to receive \(a\) assignments for every \(b\) assignments to treatment 2, one complete set of ratio units contains \(a+b\) assignments. A block can contain one such set, two such sets, or another whole-number multiple. It cannot contain a fractional set of ratio units and still have integer treatment counts in the specified proportion.
These equations give both a test and a construction. A proposed block size \(B\) is valid for an exact-ratio block if and only if \(a+b\) divides \(B\). When it does, \(k=B/(a+b)\), and the required counts follow by multiplying each ratio term by \(k\).
Why the Rule Is Necessary and Sufficient
The result follows directly from the requirement that the block counts be integers. It is useful to see why reducing the ratio matters and why the divisibility condition works in both directions.
For positive integer counts \(n_1,n_2\), the equality \(n_1:n_2=a:b\) means \(bn_1=an_2\), while \(B=n_1+n_2\).
Because \(a\) and \(b\) are coprime, \(bn_1=an_2\) implies that \(a\) divides \(n_1\) and \(b\) divides \(n_2\). Thus \(n_1=ka\) and \(n_2=kb\) for a positive integer \(k\).
Adding the counts gives \(B=ka+kb=k(a+b)\), so \(a+b\) must divide \(B\).
If \(B=k(a+b)\) for a positive integer \(k\), choosing \(n_1=ka\) and \(n_2=kb\) gives integer counts that sum to \(B\) and have ratio \(a:b\). Therefore every such block size is valid.
The proof is a counting argument, not a claim about which assignment order is used. Once the counts are valid, a block can be represented by any assignment sequence containing exactly \(n_1\) copies of \(T_1\) and \(n_2\) copies of \(T_2\). The earlier material on constructing permuted blocks by enumeration addresses how to list such sequences; the present question is which counts, and therefore which sizes, are possible.
Worked Example: A Two-to-One Allocation Ratio
A hypothetical trial plans to allocate participants to \(T_1\) and \(T_2\) in the ratio \(2:1\). Determine whether a block of size 12 supports the exact ratio, and construct one valid assignment sequence.
The ratio is already reduced, and \(a+b=2+1=3\). Since \(12/3=4\) is a positive integer, the block size is valid, with \(k=4\). The treatment counts are
The counts sum to \(8+4=12\), and \(8:4=2:1\). One possible sequence is \(T_1,T_1,T_2,T_1,T_1,T_2,T_1,T_1,T_2,T_1,T_1,T_2\). It contains eight assignments to \(T_1\) and four to \(T_2\), so it is a valid composition. Other orderings with the same counts are also valid compositions.
Finding a Feasible Size and Constructing Counts
A reliable workflow is to reduce the requested ratio first, check divisibility second, and calculate counts only after the size passes the check. If the proposed size fails, choose a permitted multiple that fits the design constraints, such as a required minimum block size or a planned total number of assignments. A block-size choice should not be made by rounding the treatment counts independently: that can change the ratio.
Worked Example: A Three-to-Two Ratio With a Minimum Size
A hypothetical study specifies a \(3:2\) allocation ratio and requires blocks of at least 12 assignments. What is the smallest valid block size at or above 12, and what treatment counts does it require?
The ratio is reduced, and the sum of its terms is \(3+2=5\). Valid sizes are positive multiples of 5: \(5,10,15,20,\ldots\). The first one that is at least 12 is \(B=15\), not 12. Here \(k=15/5=3\), so
The counts total \(9+6=15\), and \(9:6=3:2\). A size of 12 cannot be repaired by choosing nearby integer counts such as 7 and 5: those counts sum to 12, but \(7:5\ne3:2\). Choosing 15 meets the minimum-size constraint while preserving the exact target ratio.
Worked Example: Reduce the Ratio Before Checking the Size
A hypothetical protocol describes its target allocation as \(4:2\). Determine whether a block of size 18 can match the target exactly, and give its required counts.
First reduce \(4:2\) to \(2:1\). The sum of the reduced terms is \(2+1=3\), and \(18/3=6\), so the size is valid. The counts are
They sum to 18 and have ratio \(12:6=2:1=4:2\). Checking the unreduced sum, \(4+2=6\), would also give a divisor of 18, but reducing first provides the clearest general rule and avoids mistaking a redundant description of the ratio for a different target.
What to Check in a Proposed Block
For a proposed size and ratio, check both the arithmetic and the interpretation. The divisibility test determines whether the desired exact composition exists. The count calculation verifies that the resulting block has the stated size and ratio. These checks are especially useful when reviewing a schedule specification or translating a protocol’s allocation proportion into whole-number counts.
| Check | Question | What a failure indicates |
|---|---|---|
| Ratio reduction | Are the target terms written in lowest whole-number terms? | The divisibility calculation may be harder to interpret or may use a redundant unit. |
| Divisibility | Is \(B\) a multiple of \(a+b\)? | No integer counts in the exact target ratio can sum to \(B\). |
| Count total | Do \(ka\) and \(kb\) add to \(B\)? | The block size or count calculation is wrong. |
| Ratio check | Does \(n_1:n_2\) reduce to \(a:b\)? | The constructed counts do not match the target. |
If a target is stated as allocation probabilities \(p_1\) and \(p_2\), with \(p_1+p_2=1\), an exact finite block requires those probabilities to correspond to a rational ratio. For example, \(p_1=0.6\) and \(p_2=0.4\) give \(3:2\), so valid exact-ratio block sizes are multiples of 5. In contrast, an irrational target proportion cannot be represented exactly by any finite pair of integer treatment counts. In practice, a protocol needs a specified whole-number ratio or another explicit rule for handling integer counts.
Exact Block Composition Is Not the Only Allocation Rule
Complete randomization and exact-composition block randomization address allocation in different ways. Under complete randomization, assignments are drawn for the planned set of participants according to the design’s allocation rule; the realized counts need not match a target ratio exactly in every subset of participants. With exact-composition blocks, each completed block is restricted to counts that realize the target ratio within that block. The arithmetic here determines which block sizes can implement that restriction.
A block’s exact composition says nothing by itself about how its valid sequences are selected or about properties of the allocation process beyond those counts. It also does not mean that every interim enrollment total has the target ratio: a sequence may be observed partway through a block. The choice of size therefore solves a count-feasibility problem, not every design question. In particular, an exact ratio is not a reason to select a block size without considering the study’s broader randomization requirements.
Common Mistakes and Limitations
- Testing the unreduced sum without understanding the ratio. For \(4:2\), the clearest rule uses the reduced ratio \(2:1\), whose unit sum is 3. Reduce first, then apply the test.
- Rounding the counts to fit a desired size. If 12 positions cannot be divided in a \(3:2\) ratio, counts such as 7 and 5 do not solve the problem. Select a valid size or specify a different allocation rule.
- Confusing a target proportion with an exact count. A target of 60% for \(T_1\) can be written \(3:2\), but a block of size 12 cannot contain exactly 60% of assignments to \(T_1\). The fact that 60% is a valid proportion does not make every block size valid.
- Assuming a valid size determines assignment order. The rule determines counts, not a unique sequence. The order must still be generated according to the randomization procedure.
- Assuming each partial block already has the target ratio. Exact composition describes the full block’s counts. At an enrollment point before the block is complete, the counts may differ from the target.
The rule also has a narrow scope: it concerns two treatments and a specified exact ratio within each block. It does not select the target ratio, establish that a particular ratio is ethically or statistically desirable, or decide how to handle a study that stops before a block is complete.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A design targets \(1:1\). Is a block of size 7 valid for an exact ratio? If not, give the smallest valid size greater than 7 and its counts.
- A design targets \(5:3\). Is a block of size 24 valid? If so, calculate the counts.
- A protocol states a target of \(6:4\) and proposes a block of size 15. Explain why reducing the ratio is important and determine whether the block is valid.
- For a \(2:1\) target, a proposed block of size 10 has counts 7 and 3. Does it meet the exact target? Show a check.
Solutions. (1) No. The reduced ratio is \(1:1\), so the size must be a multiple of \(1+1=2\). The smallest valid size greater than 7 is 8, with \(n_1=4\) and \(n_2=4\). (2) Yes. The ratio is reduced and \(5+3=8\), which divides 24. Here \(k=24/8=3\), giving \(n_1=3(5)=15\) and \(n_2=3(3)=9\); these sum to 24 and reduce to \(5:3\). (3) Reduce \(6:4\) to \(3:2\), whose terms sum to 5. Since \(15/5=3\), the block is valid, with counts \(3(3)=9\) and \(3(2)=6\). (4) No. Although the counts sum to 10, \(7:3\ne2:1\), since \(7\ne2(3)\). Also, a \(2:1\) block size must be a multiple of 3, and 10 is not.