Worked Example of a Precision-Constrained Ethical Target

Why a Precision Constraint Changes the Welfare Target

An allocation that sends nearly every participant to the arm with the higher success probability can maximize expected successes, but it may leave too few participants in the other arm to estimate a treatment-control contrast precisely. A precision-constrained ethical target makes this trade-off explicit: maximize expected successes subject to an upper bound on the contrast variance. This tutorial works through that calculation, including feasibility checks and cases where a variance coefficient is zero.

The target is a planned allocation share, not by itself a rule for assigning each arriving participant. The calculation treats the response probabilities as known planning values. In practice they are estimates, so the resulting target depends on the assumptions used to construct them.

Model and Allocation Objective

Consider a fixed total sample size \(N\) and two arms: treatment and control. Let \(q\) be the fraction assigned to treatment, so the planned arm sizes are \(Nq\) and \(N(1-q)\). Assume \(0<q<1\), so both arms are represented. Each participant has a binary success outcome; outcomes are independent across participants, with success probabilities \(p_T\) on treatment and \(p_C\) on control. Assume these probabilities are fixed across participants within each arm, and that there is no interference between participants.

For the planning calculation, treat \(Nq\) and \(N(1-q)\) as continuous target sizes. Integer sample sizes may require rounding, which can change the achieved variance slightly. The estimator considered is the difference between the treatment and control sample proportions. Under the stated model, its variance is the sum of the two independent sample-proportion variances.

The objective is linear in \(q\):

Thus, if \(p_T>p_C\), expected successes increase with \(q\); if \(p_T<p_C\), they decrease with \(q\); and if \(p_T=p_C\), they are constant in \(q\). Without a precision constraint, the welfare objective therefore pushes toward an endpoint. The contrast variance, however, requires observations from both arms, which is why the optimization is restricted to feasible shares that retain adequate representation.

Feasibility and the Shape of the Constraint

First suppose \(A>0\) and \(B>0\), equivalently \(0<p_T<1\) and \(0<p_C<1\). Then \(V(q)\) diverges as \(q\) approaches either endpoint. Its minimum over \(0<q<1\) occurs at the Neyman share \(q_N=\sqrt{A}/(\sqrt{A}+\sqrt{B})\). Consequently, a feasible allocation exists if and only if the cap is at least this minimum variance:

The cap is not automatically feasible: if \(\bar V<V_{\min}\), no allocation can meet it. If the cap equals the minimum, only \(q_N\) is feasible. If it is larger, the feasible shares lie between two roots, as follows by multiplying the variance inequality by the positive quantity \(Nq(1-q)\).

When \(A>0\), \(B>0\), and the cap exceeds the minimum, the two roots are

The feasible interval is \([q_-,q_+]\). Because \(W(q)\) is linear, maximizing expected successes over this interval is direct: choose \(q_+\) when \(p_T>p_C\), choose \(q_-\) when \(p_T<p_C\), and any feasible share when \(p_T=p_C\). This is the complete justification for the constrained optimum in the nondegenerate case: the variance inequality determines the feasible interval, and a linear function reaches its maximum at the endpoint in the direction of its slope.

Zero-Variance Coefficients Need Separate Treatment

The two-root description applies only when \(A>0\) and \(B>0\). If a success probability is zero or one, its arm’s binary outcome variance is zero, so the corresponding term does not diverge as that arm’s share approaches zero. In these cases, check the inequality directly rather than assuming two roots inside \((0,1)\).

For example, if \(A=0\) and \(B>0\), the constraint reduces to \(B/(1-q)\leq C\). There is a feasible share strictly between zero and one only if \(C>B\), in which case \(q\leq 1-B/C\). If \(C=B\), equality is reached only at \(q=0\), which is excluded because it leaves no treatment participants; if \(C<B\), no share is feasible. Symmetrically, if \(A>0\) and \(B=0\), an interior feasible share requires \(C>A\), and then \(q\geq A/C\). If \(A=B=0\), the variance is zero for every interior share, so every \(q\in(0,1)\) satisfies any nonnegative cap. In such cases the welfare objective may have only a supremum, not an attained maximum, if it favors an excluded endpoint.

How This Target Differs From Alternatives

Unconstrained welfare maximization chooses the arm with the higher success probability exclusively (or any allocation if the probabilities are equal). It maximizes the stated success objective but does not preserve information about the between-arm contrast. The precision-constrained target instead chooses the welfare-best share among those meeting a specific variance requirement.

Neyman allocation minimizes the variance of the contrast without regard to expected successes. When both \(A\) and \(B\) are positive, it uses \(q_N=\sqrt{A}/(\sqrt{A}+\sqrt{B})\). It is the constrained target only when the variance cap is exactly the minimum, or when the welfare objective is flat and that share is selected from the feasible set. With a looser cap and unequal success probabilities, the welfare-constrained target generally moves toward the better arm until the variance cap binds.

A minimum-arm-size rule is another possible design constraint, but it is not the same as a variance cap: a fixed lower bound on each arm’s share does not automatically guarantee a specified contrast variance. Conversely, a variance cap does not necessarily enforce a chosen minimum share in zero-variance boundary cases. The appropriate constraint depends on the design requirement being protected.

Diagnostics, Limitations, and a Common Mistake

Before reporting a target, check that \(0<q<1\), verify that the cap is feasible, and substitute the selected share back into \(V(q)\). For \(A,B>0\), the variance should be at or below the cap; when the welfare slope is nonzero and the cap is loose enough to create an interval, the selected endpoint usually makes the constraint bind. If the computed roots are nonreal, outside the interval, or inconsistent with the direct variance calculation, recheck the algebra and the units. Here \(\bar V\) is a variance of a proportion difference, so it is measured in squared proportion units.

The calculation assumes known, constant response probabilities and a fixed total sample size. Misspecified probabilities can produce a target whose actual variance exceeds the planned cap. Rounding \(Nq\) to whole participants can also alter the variance, so the rounded allocation should be checked directly. In an adaptive trial, realized allocation counts and response estimates may differ from their planning values; this calculation alone does not specify how assignments should change over time.

• Mistake: “The variance diverges at both endpoints, so every feasible set must be bounded away from zero and one.” This conclusion requires \(A>0\) and \(B>0\). If one coefficient is zero, its variance contribution is zero, and the constraint may have a boundary root or a different feasible range.
• Mistake: “A quadratic inequality always gives two roots inside \((0,1)\).” That is true for a cap strictly above the minimum when both coefficients are positive, not for every allowed probability. In degenerate cases, use the original variance inequality and enforce \(0<q<1\).
• Mistake: “The unconstrained allocation is acceptable because it maximizes expected successes.” It does so only for the welfare objective; it may leave no estimable treatment-control contrast. State the precision requirement and check it separately.