Ethical Allocation for Binary Response Outcomes

Binary Successes and the Allocation Decision

When an outcome is binary, each patient either has the defined success or does not. If success is the patient benefit of interest, the expected number of successes depends on both the response probabilities and the numbers assigned to each arm. A treatment-favored allocation may increase expected successes, but assigning too few patients to one arm can make the treatment-effect estimate too imprecise. The allocation decision can therefore be framed as maximizing expected successes subject to a precision requirement.

This tutorial considers two arms, a treatment arm and a control arm. The objective assigns equal value to each success and no additional value or penalty to either arm. That is a deliberately specific ethical objective: it is not a general claim that success is the only relevant patient outcome. The allocation is a planning target, not a rule for deciding which treatment an individual patient should receive.

The variance expression is the planning variance of the difference between two independent sample proportions, using the specified response probabilities. Its reciprocal square root is not used here: the standard error is the square root of the variance. A bound \(V_{\max}\) is therefore a variance bound; equivalently, a standard-error bound \(S_{\max}\) corresponds to \(V_{\max}=S_{\max}^2\).

Assumptions Behind the Planning Formula

The calculation assumes a fixed total sample size, two treatment groups, and binary outcomes defined consistently across arms. For planning, each patient's outcome is treated as a Bernoulli response with the relevant arm-specific probability: \(p_T\) under treatment and \(p_C\) under control. Outcomes are assumed independent across patients, and the response probabilities are treated as known planning inputs. In practice, they are estimates or assumptions and may be uncertain.

The variance formula also presumes that the treatment and control sample proportions are based on their respective assigned sample sizes and that all outcomes are observed. It does not account for missing outcomes, clustering, covariate adjustment, or dependence among patients. Those features can alter the variance and must be handled in the model used to set a target. In particular, if the probabilities are not credible planning values, a target that satisfies the calculated constraint may fail to satisfy the actual precision requirement.

Write \(a=p_T(1-p_T)\) and \(b=p_C(1-p_C)\). For a proposed allocation \(x\), the planning variance is \(V(x)=a/x+b/(n-x)\). The objective is linear:

Thus, if \(p_T>p_C\), increasing \(x\) increases expected successes; if \(p_T<p_C\), increasing \(x\) decreases expected successes; and if the probabilities are equal, the expected-success objective is constant across allocations. The precision constraint determines which values of \(x\) are allowed. This simple direction-of-change argument is the key to selecting the ethical target once the feasible allocations are known.

Deriving the Feasible Allocation Range

Because \(x\) and \(n-x\) are positive for an allocation with both arms represented, multiplying the precision inequality by \(x(n-x)\) does not reverse its direction. The constraint becomes a quadratic inequality:

When \(V_{\max}>0\), the left side is an upward-opening quadratic. If it has real roots, the continuous values satisfying the inequality lie between the roots, inclusive. The actual target must be an integer in that interval and between 1 and \(n-1\). A real-valued interval alone is not enough: if it contains no integer, no allocation meets the stated constraint.

A useful feasibility check comes from minimizing \(V(x)\) over continuous \(x\). Differentiating gives \(V'(x)=-a/x^2+b/(n-x)^2\). Setting this to zero yields \(x/(n-x)=\sqrt{a/b}\) when \(a,b>0\). This is the precision-minimizing allocation, not generally the patient-benefit-maximizing allocation. If even the minimum achievable variance exceeds \(V_{\max}\), the constraint is infeasible. If the constraint is feasible, the benefit objective chooses an endpoint of the feasible range, not necessarily the precision-minimizing point.

Worked Allocations

Diagnostics, Alternatives, and Limitations

Several checks help detect a faulty target calculation. Confirm that \(x\) and \(n-x\) are both positive; verify that the variance terms use the correct arm's probability and sample size; and substitute the chosen integer back into the original variance formula. Check the adjacent integer in the direction of greater patient benefit as well. If that adjacent value violates the constraint, the boundary choice is supported. If it does not, the proposed allocation may not be the best feasible integer.

The continuous quadratic interval can be empty, or it can contain no allowable integer after rounding and enforcing arm-size restrictions. In either case, there is no target satisfying the stated requirement. The correct response is to revise a design input transparently—for example, increase the total sample size or reconsider the precision requirement—not to report an infeasible allocation as if it met the constraint.

The principal alternative targets answer different questions. Equal allocation is simple and may be preferred for reasons outside this objective, but it need not maximize expected successes or minimize variance. A precision-minimizing allocation selects the point that minimizes the variance of the treatment-effect estimator; it may sacrifice expected successes relative to the feasible welfare-maximizing endpoint. A purely welfare-maximizing allocation would assign everyone to the arm with the higher assumed response probability, but it cannot estimate a between-arm difference when an arm has no participants. The precision constraint rules out such unsupported extremes when they fail the bound.

These calculations depend on the response probabilities. If they are misspecified, both the expected-success comparison and the variance constraint can be misleading. Further, equal value for each success omits possible differences in the importance of outcomes, treatment-related harm, costs, and patient preferences. Those considerations require a different utility specification rather than an informal adjustment to the response-probability calculation. Finally, a target allocation is not itself a complete randomization procedure: it specifies a desired overall allocation, while the mechanism for assigning each arriving participant is a separate design choice.

A common mistake is to select the Neyman, or precision-minimizing, ratio and call it the ethical target because it gives the smallest variance. That reasoning optimizes precision alone. Under a precision constraint, first determine the feasible allocations, then use the sign of \(p_T-p_C\) to select the feasible endpoint that maximizes expected successes. Another mistake is to round a continuous endpoint without checking it. Since the constraint can fail at the next integer, always evaluate the original variance at the final allocation.