Expected Imbalance Under a Biased- Coin Rule

Expected Improvement Is Not a Promise of Balance

Minimization compares the imbalance that would result from assigning the next participant to each treatment arm. A deterministic rule always selects the arm with the better score; a biased-coin rule favors that arm but retains a positive probability of selecting the other. This preserves some allocation uncertainty, but it also means an assignment can move away from the lower-scoring option. The useful mathematical question is therefore not whether every assignment improves balance, but how randomization changes the expected score of the next allocation.

The derivation below concerns two treatment arms, labelled A and B. It assumes that the minimization score is a prespecified scalar measure of imbalance, with lower values indicating better balance. It does not require a particular choice of covariates or a particular formula for combining their imbalances.

This definition presumes that the score is specified before allocation, that the participant’s relevant covariates are recorded and used consistently, and that the random draw follows the stated probabilities. The random draw is made conditional on the allocation history and current covariates; assignments are therefore generally dependent across participants. The rule is not the same as assigning independently with a fixed probability at every enrollment.

When the candidate scores are equal, neither arm has a lower score. The implementation must have a prespecified convention for that situation. The calculations that follow either assume unequal candidate scores or explicitly state a tie convention; a tie does not create a lower-scoring arm.

Expected Candidate Score Under the Rule

Write \(S_L\) for the lower candidate score and \(S_H\) for the higher one, so that \(S_L < S_H\). The biased coin selects the lower-scoring option with probability \(q\). Conditional on the current allocation history and the next participant’s covariates, the expected score after assignment is therefore

Because \(0 < 1-q < 1/2\), this expected score lies between the two candidate scores and is closer to the lower one. For comparison, a fair coin between the same options has expected score \((S_L+S_H)/2\). The biased coin lowers that expectation by

This is the precise sense in which the rule favors balance: for the current participant and current history, it has a lower expected candidate score than fair random choice between those same options. The score is still random after assignment. The rule does not promise that the realized score will be \(S_L\), nor does the inequality establish that the score must be lower than its value before this participant is assigned. If both candidate assignments increase the score, the rule chooses the less unfavorable one more often, but the next realized score may nevertheless be worse than the current score.

A Direct Result for Arm-Count Imbalance

A simple special case makes the direction of the expected change especially clear. Let \(X=n_A-n_B\) be the current difference in the numbers assigned to arms A and B, and measure imbalance by the absolute difference \(|X|\). Suppose the current difference is \(d>0\), so A has \(d\) more participants than B. Assigning the next participant to B changes the difference to \(d-1\), while assigning to A changes it to \(d+1\). The rule selects B with probability \(q\) and A with probability \(1-q\).

To verify the result, take \(d>0\). If the lower-imbalance assignment is made, the new absolute difference is \(d-1\); otherwise it is \(d+1\). Since \(d-1\geq 0\), including when \(d=1\), the conditional expectation is

If \(X<0\), the labels A and B exchange roles, and the same calculation applies with \(d=|X|\). This completes the justification for every nonzero count difference. At a tie, \(X=0\), either assignment produces \(|X_{\text{next}}|=1\); a fair coin between the arms does not change that fact. Thus the expected absolute imbalance rises from 0 to 1 at that step. A preference for the lower score cannot prevent a temporary increase when no assignment can reduce the current count difference.

Why Randomization Does Not Guarantee Exact Balance

Randomization governs probabilities, not the outcome of an individual random draw. Even when the rule favors the better-balancing assignment at every non-tied step, it gives the worse-balancing assignment positive probability whenever \(q<1\). A sequence of such draws can therefore produce a final allocation that is not exactly balanced. Moreover, with covariate minimization, balance is usually represented by a score combining several discrepancies; a low total score does not necessarily mean that every covariate level or every marginal count is exactly equal across arms.

Assumptions, Checks, and Limitations

The expectation calculations are conditional statements: they hold for the candidate scores and probabilities actually calculated at a given enrollment step. They rely on the following operational conditions:

• The imbalance score and the value of \(q\) are specified before the assignment, and lower scores consistently represent the preferred allocation.
• The participant’s relevant covariates are available before randomization and are entered accurately into the score calculation.
• The allocation system applies probability \(q\) to the lower-scoring option and \(1-q\) to the higher-scoring option when scores differ.
• The random draw is implemented as specified. Conditional probabilities do not imply that treatment assignments are independent across participants.
• The same score definition and tie convention are used as prespecified. When candidate scores tie, the lower-versus-higher calculation does not apply.

Practical checks should focus on whether the algorithm is doing what the mathematics assumes. Recompute candidate scores for a sample of enrollment steps and confirm that the selected arm had the stated probability of selection. Audit stored covariates, score weights, and the randomization record. Check that software does not silently convert missing covariates into a level or use a different score for one arm. If observed allocations seem repeatedly to favor the higher score, first check the coding of the arm labels and the direction of the score before interpreting the pattern as chance.

A common mistake is to reason, “The rule always favors the less imbalanced arm, so the final arms must be balanced.” This confuses a preference with a guarantee. The correction is to state the conditional probability: with unequal scores, the lower-scoring assignment occurs with probability \(q\), while the higher-scoring one still occurs with probability \(1-q\). Another mistake is to interpret a fractional expected score or expected count difference as a possible realized allocation. An expectation averages over repeated randomizations; one trial produces one assignment at each step.

The main alternatives differ in how strongly they trade allocation uncertainty for balance. Deterministic minimization always selects the lower-scoring option when scores differ, while a fair choice between the options ignores which score is lower. The biased coin lies between them: it favors the lower score without making the assignment fully deterministic. As \(q\) approaches 1, the expected score approaches the lower candidate score, but the protection against predictability also diminishes. As \(q\) approaches 0.5, the expected-score advantage over a fair coin approaches zero.

These comparisons are conditional on the candidate scores for the current participant. They do not establish that any method will balance every covariate, eliminate all chance imbalances, or improve treatment-effect estimation in every trial. The result also depends on what the score measures: a rule can favor a lower prespecified score while leaving an unweighted or unrecorded feature less balanced. Randomization should therefore be viewed as a way to manage allocation probabilities and expected imbalance, not as a substitute for sound score design, data quality, or later analysis.