Proving the Restoring Direction of the Biased Coin

What “Restoring Direction” Means

A doubly adaptive biased coin (DBCD) uses both a target allocation and the allocation already realized to determine the probability of the next assignment. This creates a feedback effect: if an arm currently has a smaller share than its target, the rule should raise that arm’s next-assignment probability above the target. The claim is about the conditional probability of the next assignment, not a guarantee that the next patient will be assigned to that arm or that future allocation proportions will move monotonically.

Here we prove that direction algebraically for two arms while holding the estimated target fixed. Freezing the target matters: if the target changes at the same time as the allocation proportions, a comparison with the current target alone does not establish whether the assignment rule is correcting imbalance or following a moving target.

The positivity conditions ensure that both ratios in the rule are defined and that both weights are positive. In this tutorial, the target estimate is treated as fixed during the comparison; write \(r=\hat\rho_1\) and \(q=\hat\pi_{n,1}\). Thus \(1-r=\hat\rho_2\) and \(1-q=\hat\pi_{n,2}\). Let \(p_1\) denote the conditional probability of assigning the next patient to arm 1, given the current trial history and these fixed values.

The denominator normalizes the two positive weights so that the assignment probabilities sum to one. The central question is whether \(p_1\) lies above or below \(r\), the fixed target share for arm 1. Comparing probabilities directly involves a fraction; comparing their odds gives a simpler algebraic route.

Proof by Comparing Odds

Because both weights are positive, \(0<p_1<1\), so the odds \(p_1/(1-p_1)\) are well defined. The odds of assignment to arm 1 under the DBCD rule are the ratio of its weight to arm 2’s weight:

The target odds for arm 1 are \(r/(1-r)\). For any two probabilities strictly between zero and one, comparing the probabilities is equivalent to comparing their odds: the function \(x/(1-x)\) is strictly increasing on that interval. Consequently, \(p_1>r\) precisely when the DBCD odds exceed the target odds. Dividing both sides of that odds comparison by the positive target odds gives the following condition:

Since \(\gamma>0\), raising a positive number to the power \(\gamma\) preserves whether it is greater than, equal to, or less than one. The expression inside braces is greater than one exactly when \(r(1-q)>q(1-r)\). Expanding both sides and cancelling the common term \(-rq\) reduces this to \(r>q\). The same steps give equality when \(r=q\) and the reverse inequality when \(r<q\).

This conclusion is a statement about the direction of the probability adjustment. It does not say how large the adjustment must be, nor does it guarantee that the realized next assignment will reduce the imbalance. The next patient is still assigned randomly according to the calculated probabilities.

Worked Examples

Why the Result Holds and What It Does Not Establish

The proof depends on a simple relationship between target odds and current allocation odds. In the arm 1 expression, the multiplier on the target odds is

When \(q<r\), the numerator-to-denominator comparison inside braces is greater than one, so the multiplier is greater than one. The DBCD therefore raises arm 1’s odds above its target odds, and hence raises its probability above its target probability. When \(q>r\), the multiplier is less than one, reversing the direction. The result follows from normalization and the ratio correction; it does not require a claim about outcomes or about the quality of the target estimate.

The fixed-target condition is important in applications. An adaptive design may update its target estimate as response data accrue. The theorem applies at a given allocation state when that target is held fixed for the comparison. If the target is updated between assignments, the next probability may change both because the realized allocation has changed and because the target itself has moved. The fixed-target proof alone does not separate those effects or guarantee correction relative to a later target.

Likewise, “restoring” describes the direction of the conditional assignment probability relative to target. Random variation can still move the realized allocation share farther from target on a particular assignment. The probability is not a deterministic correction to the observed count.

Diagnostics, Alternatives, and Failure Modes

A practical algebra check is to calculate the current share \(q\), compare it with the fixed target \(r\), and then verify that the computed \(p_1-r\) has the corresponding sign. The odds form provides a second check without requiring a full normalization:

For \(r,q\in(0,1)\) and \(\gamma>0\), the factor in braces must be above one if and only if \(q<r\). If a calculation instead gives a probability below target in that case, check that the target and allocation-share values have not been swapped, that the two weights were normalized correctly, and that the same target was used for both arms.

The comparison with simpler approaches helps clarify what is specific to the DBCD feedback. Under simple randomization to a fixed target, the next-assignment probability for arm 1 is always \(r\), regardless of \(q\); it does not adjust for current allocation imbalance. The DBCD rule instead uses the current shares in its correction weights. At exact balance it agrees with the target probability, while away from balance it adjusts in the restoring direction under the theorem’s conditions. This comparison concerns the direction, not a claim that one design is uniformly preferable.

Several situations fall outside the proof’s assumptions:

• A boundary allocation share. If \(q=0\) or \(q=1\), a target-to-share ratio is undefined. The theorem does not prescribe how to initialize or modify the design at such a state.
• A boundary target. If \(r=0\) or \(r=1\), target odds are not finite and the interior-target argument does not apply.
• No positive correction exponent. The direction proof uses \(\gamma>0\). Without that condition, the odds multiplier need not be greater than one when the arm is underrepresented.
• A moving target. If the target changes during the comparison, the result is not a proof about correction relative to one fixed target.
• Confusing probability with outcome. A probability above target raises the chance of the next assignment to that arm; it does not determine the realized assignment.

These are limits of this particular statement, not evidence that the DBCD formula is algebraically inconsistent. They identify which conditions must be checked before using the restoring-direction conclusion.