Design Specification for a Reproducible DBCD

From a DBCD Idea to an Executable Specification

A description such as “allocate adaptively toward the arms with better estimated outcomes” does not determine a unique trial. It leaves open which outcomes count as responses, how response rates are estimated, how estimates become target shares, when adaptation begins, and how strongly current allocation imbalances are corrected. Different choices can produce different probabilities at the very next assignment.

A reproducible doubly adaptive biased coin design (DBCD) therefore needs a fully specified sequence of calculations. At each step, observed responses update an estimator, the estimator determines a target allocation vector, and a correction rule combines that target with current allocation shares. A random draw from the resulting probabilities assigns the next participant.

Notation and the Sequential Rule

Suppose there are \(K\) treatment arms. Just before participant \(t+1\) is assigned, \(N_k(t)\) is the number assigned to arm \(k\) among the first \(t\) participants, and \(S_k(t)\) is the number of observed responses on that arm. For binary responses, each observed outcome is \(Y_i\in\{0,1\}\), where 1 denotes a response under a prespecified response definition. The allocation share is \(\pi_k(t)=N_k(t)/t\), when \(t>0\). It describes assignments already made; it is not the target share for the next assignment.

Let \(\hat p_k(t)\) be the prespecified estimate of the response probability on arm \(k\), calculated from responses observed by time \(t\). A target map \(R\) turns the vector of estimates and any prespecified design inputs into target shares \(\rho_k(t)\), with \(\rho_k(t)\geq 0\) and \(\sum_{k=1}^K\rho_k(t)=1\). The DBCD assignment probability is

This expression assumes the target shares and current allocation shares are positive wherever the ratio is used, and that the denominator is finite and positive. A design must ensure that these conditions hold before using the formula. The exponent \(\gamma\geq0\) controls the strength of the correction: when an arm’s current share is below its target, its target-to-current ratio is greater than 1. The correction then increases its weight relative to the uncorrected target weight. At \(\gamma=0\), the probabilities equal the target shares, provided the formula is defined.

Specify the Estimator, Target, and Constraints

The target map should follow from the trial’s stated objective, not from a vague preference for an arm that appears better. For example, if the objective is to estimate arm-specific binary response means with equal per-participant costs and minimize a prespecified weighted sum of their sampling variances, a Neyman-type target can be appropriate. The exact objective and target map must still be stated: other objectives can imply different targets.

For a concrete specification in the examples below, use a Beta(1,1) smoothing rule for each binary response rate:

This estimator is defined even if an arm has recorded no responses or no failures. Use the illustrative equal-cost target map \(\tilde\rho_k(t)\propto\sqrt{\hat p_k(t)\{1-\hat p_k(t)\}}\), which assigns shares in proportion to estimated response standard deviations. To impose a prespecified target-share floor \(\varepsilon\), define \(\rho_k(t)=\varepsilon+(1-K\varepsilon)\tilde\rho_k(t)\), where \(0\leq\varepsilon<1/K\). This is an explicit part of the target map: it preserves a positive share while keeping the shares’ sum equal to 1. A different floor, estimator, or scientific objective would define a different design.

For initialization, suppose \(m_0\geq1\) participants are assigned to each arm using a stated randomization procedure, and adaptation begins only after these \(Km_0\) assignments and their required responses have been recorded. During initialization the design must say how assignments are made, how incomplete responses are handled, and whether enrollment pauses until those responses are available. Once adaptation begins, \(\pi_k(t)\) is positive because each arm has at least \(m_0\) assignments.

Worked Calculations

Why the Correction Has This Form

The correction is easiest to audit by comparing two arms. Dividing their probabilities cancels the common normalizing denominator:

If the target shares are equal and arm \(i\) is underrepresented relative to arm \(j\), then \(\pi_i<\pi_j\), so the ratio in braces exceeds 1. For \(\gamma>0\), this factor raises the relative probability of assigning to arm \(i\). The correction is not a guarantee that observed counts will match targets at every time: random assignments can create temporary departures. Nor does the formula alone determine the target; it only combines a target vector with current allocation shares.

Constraints, Comparison, and Diagnostics

A reproducible specification must say whether constraints apply to the target shares, the assignment probabilities, or both. The floor used in the examples constrains \(\rho\), not automatically \(q\). If the design also requires \(q_k\geq\delta\), it must state an exact probability transformation and its conditions—for example, \(q_k^{*}=\delta+(1-K\delta)q_k\) with \(0\leq\delta<1/K\). This is a modified randomization rule and should be reported as such. Maximum sample sizes, temporary arm unavailability, or other restrictions also require explicit rules for recalculating probabilities; there is no universal adjustment that leaves the original design unchanged.

Compared with fixed randomization, a DBCD uses observed responses and allocation history to update future probabilities. Compared with allocation directly at the target (\(\gamma=0\)), positive \(\gamma\) adds correction for current imbalance. Increasing \(\gamma\) strengthens that correction, but it does not alter the response estimator or target map. These are distinct design choices and should not be conflated.

A practical audit should check, at each assignment, that the response data include only outcomes available under the stated rule; all \(N_k(t)\) and \(S_k(t)\) agree with the assignment and outcome log; target shares are nonnegative and sum to 1; the allocation shares are positive; all constraints are applied in the specified order; and the final \(q_k\) values are nonnegative and sum to 1. Record unrounded intermediate values for calculations, the random-number generator and seed or stream, arm order for cumulative probabilities, and the draw used. Such details make the realized sequence auditable without treating a particular random sequence as part of the probability formula.

A common mistake is to say, “arm 1’s target share is 0.3326, so the next participant must have a 0.3326 probability of going to arm 1.” That ignores the allocation-ratio correction. In the second example the target share is about \(0.3326\), but the corrected probability is about \(0.1990\). The target describes the desired allocation proportions; \(q\) is the probability used for the next random assignment. Another mistake is to clip an estimated target after calculation without specifying how the remaining shares are adjusted. A reproducible design defines a transformation, such as the floor mixture used here, and applies it consistently.