Separating Response Data, Allocation Counts, and Target Shares

Why a DBCD Needs More Than One State Variable

A doubly adaptive biased coin design (DBCD) changes its randomization probabilities as the trial proceeds. To do this coherently, it must keep track of two different kinds of information: which arms participants were assigned to, and what responses have become available. It also needs a target allocation share and a probability distribution for the next assignment. These quantities interact, but they are not interchangeable.

The distinction matters especially when responses are delayed. An assignment occurs at a particular time; its outcome may be recorded later. Allocation counts therefore change at assignment, while response records change only when an outcome is available to the design. A target share is a design output based on information available at a decision time. It is neither an observed count nor necessarily the probability used for the next assignment.

The bookkeeping identities below do not require a particular response distribution or target formula. To interpret response estimates, however, the design must specify what outcome is being measured, when it is considered observed, and how the target is computed from available information. Any assumptions needed for that estimator or target map must be checked separately.

Definition: Suppose there are \(K\) treatment arms and \(t\) participants have been assigned. Let \(A_{j,i}=1\) if participant \(j\) was assigned to arm \(i\), and \(A_{j,i}=0\) otherwise. The allocation count for arm \(i\) is \(N_i(t)=\sum_{j=1}^{t}A_{j,i}\). Let \(R_j(t)=1\) if participant \(j\)'s response is available to the randomization procedure by time \(t\), and \(0\) otherwise. The observed-response count is \(M_i(t)=\sum_{j=1}^{t}A_{j,i}R_j(t)\). The response data for arm \(i\) are the observed outcomes, with their participant or record identifiers, not just their count.

For a binary response, one may also maintain the number of observed successes \(S_i(t)\), summing the observed outcome indicators among participants assigned to arm \(i\). An estimator such as \(\hat p_i(t)=S_i(t)/M_i(t)\) is defined only when \(M_i(t)>0\), unless the design specifies a separate rule for zero observed responses. For other response types, the estimator and the required response records differ, but the distinction between assignments and observed outcomes remains.

Definition: The current allocation share for arm \(i\) is \(q_i(t)=N_i(t)/t\), when \(t>0\). The target share is \(\rho_i(t)\), a design-specified desired share available at time \(t\), with \(\rho_i(t)\geq 0\) and \(\sum_{i=1}^{K}\rho_i(t)=1\). The next assignment probability is \(\pi_i(t+1)=P(A_{t+1,i}=1\mid\mathcal H_t)\), where \(\mathcal H_t\) denotes the information available to the design before participant \(t+1\) is assigned. These are three different quantities.

The target map may use response estimates, other design inputs, or a fixed target specified in advance. The DBCD coin rule then uses the target and the current allocation state to produce the probability vector \((\pi_1(t+1),\ldots,\pi_K(t+1))\). The target share is not itself that probability. Nor does a target share describe the proportion already assigned.

Update Order and Accounting Identities

The timing convention is essential. Immediately before the next assignment, the design uses the state based on assignments and responses available so far. It calculates the next assignment probabilities and makes a random assignment. The allocation count changes as soon as that assignment is made. The response record changes only when the corresponding response becomes available. If response availability is immediate, those updates happen close together; they remain conceptually separate.

\sum_{i=1}^{K}A_{j,i}=1,\qquad N_i(t+1)=N_i(t)+A_{t+1,i},\qquad \sum_{i=1}^{K}N_i(t)=t.

For response records, let \(R_j(t)\) indicate whether participant \(j\)'s outcome is available by the relevant decision time. Then \(M_i(t)\leq N_i(t)\) for every arm, and \(\sum_i M_i(t)\leq t\). The difference \(N_i(t)-M_i(t)\) counts assigned participants in arm \(i\) whose outcomes are not yet available under this definition. It does not mean that they had negative outcomes.

\sum_{i=1}^{K}\pi_i(t+1)=1,\qquad \pi_i(t+1)\geq 0,\qquad \sum_{i=1}^{K}\rho_i(t)=1.

These identities provide a short derivation of the core bookkeeping checks. Each participant contributes exactly one assignment indicator across the arms, so summing \(N_i(t)\) over arms counts exactly the \(t\) assignments. Each next probability vector describes one categorical assignment, so its entries must be nonnegative and sum to one. A target vector is also a composition of shares and must sum to one, but its entries need not equal either the current shares or the next probabilities.

Freeze the pre-assignment state.
Record assignments to date, response records available now, and any outcomes still pending. Calculate estimates only from data permitted by the design.

Compute target and coin probabilities.
Apply the specified target map to the available inputs, then apply the DBCD allocation rule to obtain a probability vector for the next assignment.

Record the assignment.
After the random draw, increment exactly one allocation count. Do not add an outcome merely because the participant was assigned.

Record an outcome when available.
Add the response to the assigned arm’s response data and update its observed-response count and estimator. Recompute target shares at the next decision time if the design requires it.

Worked Updates

Worked Example: Immediate Binary Responses

Consider a hypothetical two-arm trial with binary success outcomes. At \(t=4\), the assignment sequence is arm 1, arm 2, arm 1, arm 1. All four responses are observed: arm 1 has outcomes \(1,0,1\), and arm 2 has outcome \(1\). For illustration, suppose the prespecified target map is \( \rho_i(t)=\sqrt{\hat p_i(t)}/\sum_{k=1}^{2}\sqrt{\hat p_k(t)} \). This is an example target map, not a claim that every DBCD uses this map.

The allocation counts are \(N_1(4)=3\) and \(N_2(4)=1\), so the current shares are \(q_1(4)=3/4=0.750\) and \(q_2(4)=1/4=0.250\). The observed-response counts are \(M_1(4)=3\) and \(M_2(4)=1\). The success counts are \(S_1(4)=2\) and \(S_2(4)=1\), giving \( \hat p_1(4)=2/3=0.667 \) and \( \hat p_2(4)=1/1=1.000 \).

The target map gives \( \rho_1(4)=\sqrt{2/3}/(\sqrt{2/3}+1)=0.449 \) and \( \rho_2(4)=1-0.449=0.551 \), to three significant figures. Suppose the DBCD coin rule, using this target and the current state, gives \((\pi_1(5),\pi_2(5))=(0.600,0.400)\). That probability vector is an input to the next random draw; it is not the target vector.

Suppose the draw assigns participant 5 to arm 2, and the response of 0 is immediately observed. The allocation counts become \(N_1(5)=3\), \(N_2(5)=2\); the response counts become \(M_1(5)=3\), \(M_2(5)=2\); and the success counts become \(S_1(5)=2\), \(S_2(5)=1\). Thus \(\hat p_2(5)=1/2=0.500\). The updated target is \( \rho_1(5)=\sqrt{2/3}/(\sqrt{2/3}+\sqrt{1/2})=0.536 \) and \(\rho_2(5)=0.464\). The observed outcome changes the response data and may change the next target; it does not undo or alter the assignment already recorded.

Worked Example: A Response Is Still Pending

In a hypothetical two-arm trial, three participants have been assigned: two to arm 1 and one to arm 2. The observed binary responses are \(1\) for one arm 1 participant and \(0\) for the arm 2 participant. The second arm 1 participant's outcome is pending. The same illustrative square-root target map is used, and the design estimates success rates from observed responses only.

The allocation counts are \(N_1(3)=2\), \(N_2(3)=1\), so the current shares are \(2/3=0.667\) and \(1/3=0.333\). But the observed-response counts are \(M_1(3)=1\), \(M_2(3)=1\). The observed success rates are \(\hat p_1(3)=1/1=1.000\) and \(\hat p_2(3)=0/1=0\). Under the stated target map, the target is \((1,0)\). If this map or the coin rule needs special handling for a zero target component, that handling must be specified by the design; the bookkeeping alone cannot supply it.

The pending arm 1 participant contributes to \(N_1(3)\), but not to \(M_1(3)\), \(S_1(3)\), or the response estimate. Treating the pending value as a failure would change the estimated rate to \(1/2\), which would silently replace “not yet observed” with “observed zero.” That is a different data state and can produce a different target.

Worked Example: Three Arms and a Delayed Outcome

Suppose a hypothetical three-arm trial has assigned six participants, with counts \(N_1(6)=3\), \(N_2(6)=2\), and \(N_3(6)=1\). Five responses are available: arm 1 has \(1,0\), arm 2 has \(1,1\), and arm 3 has \(0\). The third arm 1 participant is still pending.

The allocation shares are \(3/6=0.500\), \(2/6=0.333\), and \(1/6=0.167\). The response counts are \(M_1(6)=2\), \(M_2(6)=2\), and \(M_3(6)=1\), which sum to five, not six. The success counts are \(S_1(6)=1\), \(S_2(6)=2\), and \(S_3(6)=0\); the observed rates are therefore \(0.500\), \(1.000\), and \(0.000\).

Suppose the pending arm 1 outcome then arrives as 1, without any new assignment. The allocation counts and shares remain unchanged. The response count for arm 1 becomes 3, its success count becomes 2, and its observed rate becomes \(2/3=0.667\). The total observed-response count becomes six. A target map depending on response estimates may now produce a new target vector, but the allocation counts still sum to six and the next assignment probability vector must be computed separately.

Diagnostics, Alternatives, and Limitations

State checks are useful because a single bookkeeping error can affect many later assignments. Before each randomization, verify that allocation counts sum to the number assigned, each participant has exactly one assigned arm, response counts do not exceed allocation counts, and response records are attached to the assigned arm. Confirm that target shares and next assignment probabilities each sum to one, while retaining them as separate stored quantities. For binary outcomes, check that \(0\leq S_i(t)\leq M_i(t)\); if \(M_i(t)=0\), do not calculate \(S_i(t)/M_i(t)\).

A sudden change in a target share should be traceable to a changed input or a changed target rule. A sudden change in a current share should be traceable to an assignment. A change in an observed-response count should be traceable to an outcome becoming available. These are distinct diagnostic signatures. If they do not line up with the recorded event history, investigate the state update before interpreting the allocation trajectory.

A fixed-allocation design can use a prespecified probability vector at every step and may not need response-based target updates. A response-adaptive design adds response records and a target map, while still requiring allocation counts to track actual assignments. These approaches differ in how probabilities are chosen, not in the basic fact that assignment history and response history are separate records.

The state definitions do not resolve every trial-data issue. A design must specify how it treats delayed, missing, censored, or otherwise unavailable outcomes, and whether the response estimator is appropriate for that observation process. If participants can deviate from assigned treatment, assignment records should still describe randomization; treatment received is a separate variable. The target map, the coin rule, and any safeguards are design choices and must be documented rather than inferred from the state variables.

Common Mistake

Common mistake: “Two participants have been assigned to arm 1, so there are two arm 1 responses; a participant with no recorded outcome can be entered as a failure.” This is wrong because allocation count \(N_1(t)\) counts assignments, whereas \(M_1(t)\) counts responses available to the procedure. A pending or missing response is not automatically a zero. Record the assignment immediately, record the outcome only when it is observed under the design’s rule, and use the specified missing-data handling if an outcome will never be observed.

The practical benefit of this separation is reproducibility: given the event history and design specification, another analyst can reconstruct the same pre-assignment state, target, and probability vector without confusing what was assigned, what was observed, and what was desired.

Check Your Understanding

Answer each question using the distinction between assignment state, response state, target shares, and assignment probabilities.

At \(t=5\), arm counts are \(N_1=3\) and \(N_2=2\), but one arm 1 outcome is pending. What are the current allocation shares, and does the pending outcome change them?
For three arms, a proposed target vector is \((0.2,0.5,0.2)\). Identify the problem and state the required check.
An arm has \(N_i=4\), \(M_i=3\), and two observed binary successes. What is its observed success-rate estimate, and what does the difference between \(N_i\) and \(M_i\) mean?
After a new participant is assigned to arm 2, but before that participant’s response is available, which state quantities must change and which need not change?

Solutions. (1) The shares are \(3/5=0.600\) and \(2/5=0.400\). The pending response does not change either share because no assignment changed. (2) The entries sum to \(0.9\), not 1; a valid target-share vector must have nonnegative entries summing to 1. (3) The estimate is \(2/3=0.667\). One assigned arm \(i\) response is not yet available; it is not thereby a failure. (4) Increment \(N_2\) and the total number assigned; update current shares accordingly. Do not increment \(M_2\) or add a response value until the outcome is available. The target and next probability vector may be recomputed at the next decision according to the design.