Why Arm Labels Should Not Affect the Design
In a multi-arm response-adaptive design, the labels “Arm 1,” “Arm 2,” and “Arm 3” are identifiers, not scientific properties. If the same trial state is recorded in a different coordinate order, the probability assigned to each treatment should travel with that treatment. For example, swapping the labels of two arms should swap their allocation probabilities, not change the underlying randomization rule.
This is a useful mathematical property and a practical implementation check. It detects mistakes such as using one coordinate’s target share with another coordinate’s allocation count, or applying a special adjustment to whichever treatment happens to be listed first. The result below concerns relabeling consistently: target shares, current allocation counts, response summaries, and any arm-indexed inputs must all move together.
The Multi-Arm DBCD Rule
Suppose there are \(K\) treatment arms. At a given allocation step, let \(N_i\) be the number already assigned to arm \(i\), and let \(n=\sum_{i=1}^K N_i\) be the total number assigned. The current allocation share for arm \(i\) is \(\pi_i=N_i/n\), when \(n>0\). Let \(\rho_i\) be the target share for arm \(i\), with \(\sum_{i=1}^K\rho_i=1\), and let \(\gamma\geq 0\) be the common biasing parameter.
For positive target and current shares, the ordinary multi-arm DBCD rule assigns probabilities by first forming weights and then normalizing them:
Here \(q_i\) is the probability that the next participant is assigned to arm \(i\). The rule combines the target share with a correction based on the ratio of target share to current share. The present tutorial uses this formula only to establish its behavior under relabeling; it does not require the targets to be fixed across allocation steps.
For the formula above, assume that all \(\rho_i\) and \(\pi_i\) are positive and that the same \(\gamma\) is used for every arm. Positivity makes every weight well-defined. If a design uses a zero-count convention, a positive assignment floor, or another safeguard, the corresponding convention must itself treat arms symmetrically for the same invariance conclusion to hold. A common floor, for example, can be added identically to every coordinate without privileging a label.
Permutation Result and Justification
A permutation \(\sigma\) is a one-to-one rearrangement of the arm labels. For any vector \(x=(x_1,\ldots,x_K)\), define its relabeled vector \(x^\sigma\) by placing the old arm \(i\)'s value in the new coordinate \(\sigma(i)\). Equivalently, \(x^\sigma_j=x_{\sigma^{-1}(j)}\). This convention applies to counts, current shares, targets, response estimates, and probabilities.
The justification follows directly from the coordinatewise weight formula and its common normalizing denominator.
For the new coordinate \(\sigma(i)\), the target and current share are \(\rho^\sigma_{\sigma(i)}=\rho_i\) and \(\pi^\sigma_{\sigma(i)}=\pi_i\).
Applying the same formula gives \(w^\sigma_{\sigma(i)}=\rho_i(\rho_i/\pi_i)^\gamma=w_i\). Hence the new weight vector is exactly the old weight vector with its coordinates permuted.
A permutation does not change a sum, so \(\sum_{j=1}^K w^\sigma_j=\sum_{i=1}^K w_i\). Dividing each relabeled weight by this common sum gives \(q^\sigma_{\sigma(i)}=q_i\) for every arm \(i\), proving the result.
This proof also explains why normalization matters. Even though the individual weights depend on the target and current shares, their total cannot acquire a different value merely because the terms are listed in a different order.
Worked Examples
Worked Example: Swapping Two Arms
Consider three arms with target shares \((0.2,0.5,0.3)\), current shares \((0.4,0.4,0.2)\), and \(\gamma=1\). Calculate the allocation probabilities, then swap Arms 1 and 3 in every input.
For the original ordering, the weights are \(w_1=0.2(0.2/0.4)=0.1\), \(w_2=0.5(0.5/0.4)=0.625\), and \(w_3=0.3(0.3/0.2)=0.45\). Their sum is \(0.1+0.625+0.45=1.175\). Therefore
After swapping Arms 1 and 3, the target shares are \((0.3,0.5,0.2)\) and the current shares are \((0.2,0.4,0.4)\). The new weights are \(0.3(0.3/0.2)=0.45\), \(0.5(0.5/0.4)=0.625\), and \(0.2(0.2/0.4)=0.1\). Their sum remains \(1.175\), giving probabilities \((18/47,25/47,4/47)\). This is exactly the original vector with its first and third coordinates swapped.
Worked Example: A Nontrivial Biasing Parameter
Let the target shares be \((0.25,0.5,0.25)\), the current shares be \((0.5,0.25,0.25)\), and \(\gamma=2\). Find the probabilities and then interchange Arms 1 and 2.
The weights are \(w_1=0.25(0.25/0.5)^2=1/16\), \(w_2=0.5(0.5/0.25)^2=2\), and \(w_3=0.25(0.25/0.25)^2=1/4\). The total is \(1/16+2+1/4=37/16\), so
After interchanging the first two arms, the target vector becomes \((0.5,0.25,0.25)\) and the current vector becomes \((0.25,0.5,0.25)\). The weights are now \(2,1/16,1/4\), with the same total \(37/16\). Thus the probabilities are \((32/37,1/37,4/37)\), precisely the required interchange. The argument does not rely on \(\gamma=1\); it holds for this common value \(\gamma=2\) as well.
Worked Example: Relabeling a Response-Based Target
Suppose three arms have hypothetical binary-response summaries of 8 successes in 10 participants, 6 in 10, and 4 in 10. For illustration, define the target shares as each arm’s observed response proportion divided by the sum of the three proportions. Let current allocation counts be \((12,9,9)\) and \(\gamma=1\). Calculate the probabilities, then reorder the arms as C, A, B.
The response proportions are \(0.8,0.6,0.4\), whose sum is \(1.8\). The corresponding target shares are \(4/9,1/3,2/9\). The current shares, from 30 allocations, are \(0.4,0.3,0.3\). The weights are \(w_1=(4/9)((4/9)/0.4)=40/81\), \(w_2=(1/3)((1/3)/0.3)=10/27=90/243\), and \(w_3=(2/9)((2/9)/0.3)=40/243\). Since \(40/81=120/243\), their total is \(250/243\). Normalizing gives
In the order C, A, B, the response summaries, target shares, and counts must all be reordered together. The new probability vector is \((0.16,0.48,0.36)\). This example illustrates an important condition: it is not enough to permute the final target vector while leaving the response data or counts in their old positions. Every arm-indexed input must continue to describe the same treatment.
Targets, Safeguards, and the Scope of the Result
Targets may be computed from response estimates rather than specified in advance. In that setting, invariance requires that the target-construction rule be label-equivariant: relabeling the arm-specific response data must relabel the resulting target shares in the same way. Computing a separate estimate by the same method for each arm has this property. A procedure that gives a special adjustment to “Arm 1,” or breaks ties according to a fixed label, may not.
The same principle applies to implementation safeguards. A shared rule that treats coordinates identically can preserve invariance. For instance, if a design mixes the DBCD probabilities with a uniform assignment component, the resulting vector remains permuted when its inputs are permuted. In contrast, a different minimum probability for one named arm introduces label-specific behavior unless that arm-specific distinction is itself part of the intended design.
The theorem concerns allocation probabilities at a given state. It does not say that two realized trial histories must be numerically identical after relabeling. Random assignments differ by chance. Rather, conditional on corresponding relabeled states, each treatment has the same conditional assignment probability as before. Consequently, the distribution of the next assigned treatment is carried along by the relabeling. A particular software sampling method may use a coordinate ordering internally, but its output probabilities must still agree with the permuted vector.
Diagnostics and Common Failure Modes
A practical invariance check is to calculate probabilities, permute all arm-indexed inputs, calculate them again, and compare the second result with the correspondingly permuted first result. Check both that each probability vector sums to one and that each treatment’s probability matches its relabeled counterpart. This test can expose indexing errors even when each individual calculation appears plausible.
- Permuting only the targets. This pairs a treatment’s target with another treatment’s current share. Relabel target shares, counts, estimates, and any other arm-specific inputs together.
- Using different parameters by coordinate. If \(\gamma\), a floor, or a correction differs by label rather than by a treatment characteristic that is relabeled too, the common-permutation proof no longer applies.
- Using a label-based tie-break. A response-based target rule that resolves tied arms by always favoring the smallest numeric label can fail to relabel with the data.
- Comparing vectors in different orders. The correct requirement is not that the displayed vectors remain identical. They should be identical only after their coordinates are placed in corresponding arm order.
When This Property Is Useful
Arm-label invariance is a structural check, not a claim that all arms receive equal probabilities. The examples show that target and current shares can differ substantially across arms while the rule remains invariant. It is useful when writing or reviewing multi-arm calculations, comparing two implementations that store arms in different orders, and checking that treatment names do not silently influence an allocation rule.
The property also has limits. It does not establish that the chosen targets are scientifically appropriate, that response estimates are reliable, or that a design meets other trial objectives. It only guarantees consistent behavior under a specified relabeling, given a symmetric target procedure and symmetric allocation rule. If the arms have intentional, treatment-specific constraints, those constraints must be represented and relabeled as part of the design; otherwise, the invariance claim is not applicable.
Check Your Understanding
Use the permutation result to answer each question. Solutions follow the questions.
- A three-arm rule has probabilities \((0.2,0.5,0.3)\). What probability vector should result when Arms 1 and 2 are swapped consistently?
- Why does the denominator in the DBCD formula stay unchanged under a permutation?
- A target-estimation procedure always adds \(0.02\) to the estimate for the arm stored in the first coordinate before normalizing. Does the theorem apply? Explain.
- For weights \((0.2,0.3,0.5)\), normalize the vector and then swap the first and third arms. What should the relabeled normalized vector be?
Solutions. (1) The required vector is \((0.5,0.2,0.3)\), because probabilities move with their arms. (2) The denominator is a sum of the weights, and a permutation changes only their order, not their sum. (3) No. The adjustment singles out a coordinate; after relabeling, the same treatment may no longer receive the added \(0.02\), so the target procedure need not be label-equivariant. (4) The original weights sum to \(1\), so the normalized vector is \((0.2,0.3,0.5)\); swapping its first and third coordinates gives \((0.5,0.3,0.2)\).