Comparing Ethical and Neyman Allocation Ratios

Two Targets, Two Objectives

An allocation target describes the intended share of trial participants assigned to each arm. In a two-arm trial, the treatment-arm share and the treatment-to-control allocation ratio are two ways to state the same target. Ethical and Neyman targets can use the same outcomes and still differ because they answer different questions: an ethical target concerns expected patient utility, while a Neyman target concerns the precision of an estimated contrast.

The distinction matters when interpreting a proposed allocation. A target that assigns more participants to the arm with higher expected utility may increase expected welfare among trial participants, subject to the design constraints. It does not automatically give the most precise estimate of the treatment-control contrast. Conversely, a target that minimizes the contrast variance is not, by that fact alone, the target that maximizes patient utility.

Definition: Let \(q_1\) be the target proportion assigned to treatment and \(q_0=1-q_1\) the target proportion assigned to control. For \(0<q_1<1\), the treatment-to-control allocation ratio is \(r=q_1/q_0=q_1/(1-q_1)\). Write \(\mu_i\) and \(\sigma_i^2\) for the mean and variance of the measured outcome in arm \(i\), where \(i=0\) denotes control and \(i=1\) treatment. Let \(U_i\) be the expected utility for a patient assigned to arm \(i\).

What the Two Targets Optimize

For an ethical allocation objective, the expected utility per randomized participant at treatment share \(q_1\) is

$$ W(q_1)=q_1U_1+(1-q_1)U_0. $$

This expression is linear in \(q_1\). If \(U_1>U_0\), increasing the treatment share increases expected utility; without a constraint, the maximum is at \(q_1=1\). If \(U_1<U_0\), the maximum is at \(q_1=0\). If the utilities are equal, all shares give the same expected utility. Thus, pure expected-utility maximization does not generally specify an interior ethical ratio. An interior ethical target must reflect a constraint or an additional objective, such as a required minimum allocation to each arm or a precision requirement.

For comparison, a common constrained ethical target maximizes \(W(q_1)\) subject to a minimum control share \(q_0\ge c\), where \(0<c<1\). If treatment has higher expected utility, the maximizing target is \(q_1=1-c\), giving \(r_E=(1-c)/c\). The value of this ethical ratio depends on the constraint \(c\), not just on the observed outcome variability.

The Neyman target instead minimizes the variance of the difference in sample means, \(\bar{x}_1-\bar{x}_0\), for a fixed total sample size \(N=n_1+n_0\). Under independent sampling within arms, with both arm variances strictly positive and an interior allocation, that variance is

\operatorname{Var}(\bar{x}_1-\bar{x}_0) =\frac{\sigma_1^2}{Nq_1}+\frac{\sigma_0^2}{N(1-q_1)}.

This objective uses the standard deviations of the outcome being analyzed. If patient utility is a different quantity from that outcome, its expected value determines the ethical objective, but it does not replace the outcome standard deviations in this variance formula.

Deriving the Neyman Ratio

Assume a fixed \(N\), independent observations within and across arms, finite arm variances, and \(\sigma_0>0\) and \(\sigma_1>0\). Both arms must receive participants, so \(0<q_1<1\). Differentiating the variance with respect to \(q_1\) gives

\frac{d}{dq_1}\operatorname{Var}(\bar{x}_1-\bar{x}_0) =-\frac{\sigma_1^2}{Nq_1^2} +\frac{\sigma_0^2}{N(1-q_1)^2}.

Setting this derivative to zero gives \(\sigma_1/q_1=\sigma_0/(1-q_1)\), since the standard deviations and shares are positive. Therefore

q_{1,N}=\frac{\sigma_1}{\sigma_0+\sigma_1}, \qquad q_{0,N}=\frac{\sigma_0}{\sigma_0+\sigma_1}, \qquad r_N=\frac{q_{1,N}}{q_{0,N}}=\frac{\sigma_1}{\sigma_0}.

The second derivative is \(2\sigma_1^2/(Nq_1^3)+2\sigma_0^2/[N(1-q_1)^3]\), which is positive throughout the interior. Hence this stationary point is the unique interior minimum. The result allocates a larger share to the arm with the larger outcome standard deviation, because additional observations in that arm reduce its contribution to the contrast variance.

Boundary cases: The interior Neyman formula requires both standard deviations to be positive. If \(\sigma_1=0\) and \(\sigma_0>0\), the contrast variance decreases toward its infimum as \(q_1\) approaches 0; if \(\sigma_0=0\) and \(\sigma_1>0\), it decreases toward its infimum as \(q_1\) approaches 1. In either case the infimum is not attained among strictly interior shares. If both variances are zero, every interior share gives zero variance. These cases do not have a unique interior Neyman ratio.

Worked Comparisons

Worked Example: Higher Treatment Utility, Unequal Outcome Variability

Suppose a hypothetical trial has a binary response, coded 1 for response and 0 otherwise. The response probabilities are \(p_0=0.30\) under control and \(p_1=0.60\) under treatment. Let patient utility equal the response, so \(U_0=0.30\) and \(U_1=0.60\). The design requires at least 25% of participants in control. Compare the constrained ethical target with the Neyman target for estimating the difference in response probabilities.

Because \(U_1>U_0\), expected utility increases with the treatment share. The ethical target uses the smallest permitted control share: \(q_{0,E}=0.25\) and \(q_{1,E}=0.75\). Thus

r_E=\frac{q_{1,E}}{q_{0,E}}=\frac{0.75}{0.25}=3.

For a binary outcome, the variance is \(p_i(1-p_i)\). Therefore \(\sigma_0=\sqrt{0.30(0.70)}=\sqrt{0.21}\approx0.4583\), and \(\sigma_1=\sqrt{0.60(0.40)}=\sqrt{0.24}\approx0.4899\). Both standard deviations are positive, so the interior Neyman formula applies:

r_N=\frac{\sigma_1}{\sigma_0} =\sqrt{\frac{0.24}{0.21}} =\sqrt{\frac{8}{7}} \approx1.069.

The corresponding treatment share is \(q_{1,N}=r_N/(1+r_N)\approx1.069/(2.069)=0.5167\). The ethical target assigns 75% to treatment; the Neyman target assigns about 51.7%. Their difference is not a calculation conflict: the ethical target uses higher expected utility plus a control minimum, while the Neyman target minimizes variance for the response-probability contrast.

For illustration, with \(N=100\), the variance at the ethical target is \(0.24/75+0.21/25=0.01160\). At the Neyman target it is approximately \(0.24/51.67+0.21/48.33=0.00899\), in squared response-probability units. The latter is smaller, as the Neyman target is designed to minimize this variance.

Worked Example: Utility Changes Without Changing the Neyman Ratio

Keep the same hypothetical response probabilities, \(p_0=0.30\) and \(p_1=0.60\), but suppose treatment has an additional fixed utility cost of 0.40 units per assigned patient. Define net utility as response minus this cost in the treatment arm; control has no such cost. A design requires at least 20% allocation to treatment. Find the constrained ethical target and compare it with the Neyman ratio for the response contrast.

The expected net utilities are \(U_0=0.30\) and \(U_1=0.60-0.40=0.20\). Since \(U_1<U_0\), expected utility is maximized by minimizing treatment allocation while satisfying the constraint. Hence \(q_{1,E}=0.20\), \(q_{0,E}=0.80\), and

r_E=\frac{0.20}{0.80}=0.2500.

The cost changes the utility comparison, but it does not change the response-outcome variances: \(\sigma_0=\sqrt{0.21}\) and \(\sigma_1=\sqrt{0.24}\). For estimating the response-probability contrast, the Neyman ratio remains \(r_N=\sqrt{0.24/0.21}\approx1.069\). The ratios point in different directions because one objective incorporates the treatment cost and the other is based on response variability alone. If the scientific contrast itself were defined on net utility rather than response, its variance would need to be calculated for that net-utility outcome.

Worked Example: When the Targets Coincide

Return to the response probabilities \(p_0=0.30\) and \(p_1=0.60\), with response as utility. Suppose the design requires at least \(c=\sigma_0/(\sigma_0+\sigma_1)\) of participants in control. Calculate the ethical ratio and show why it matches the Neyman ratio.

Using \(\sigma_0=\sqrt{0.21}\approx0.4583\) and \(\sigma_1=\sqrt{0.24}\approx0.4899\),

c=\frac{0.4583}{0.4583+0.4899}\approx0.4833, \qquad q_{1,E}=1-c\approx0.5167.

The treatment-to-control ratio is

r_E=\frac{q_{1,E}}{c} =\frac{\sigma_1/(\sigma_0+\sigma_1)}{\sigma_0/(\sigma_0+\sigma_1)} =\frac{\sigma_1}{\sigma_0} \approx1.069.

This equals the Neyman ratio because the control constraint was set to the Neyman control share. The agreement is created by that design choice; it is not a general consequence of treatment having higher expected utility.

How to Compare Targets and Diagnose Problems

A useful comparison starts by naming the estimand and the ethical objective separately. For a response contrast, the Neyman target depends on the response standard deviations. An ethical target depends on the specified utility assigned to outcomes, including any stated harm or cost, and on constraints such as minimum arm shares. If the utility definition changes while the response distributions stay fixed, the ethical target may change while the Neyman target for the response contrast does not.

The following distinctions help detect an incorrect comparison:

Confusing shares and ratios. A treatment share \(q_1=0.75\) corresponds to a treatment-to-control ratio of \(0.75/0.25=3\), not 0.75.
Treating a pure welfare maximum as automatically interior. A linear expected-utility objective selects a boundary unless constraints or additional objectives restrict the allocation.
Using utilities in place of outcome standard deviations. Neyman allocation uses the variability of the outcome defining the contrast. Utility values enter only if utility itself is the analysis outcome.
Ignoring the design conditions. The usual Neyman ratio presumes a fixed total sample size and positive variances in both arms for a unique interior minimum. Zero-variance cases require separate treatment.
Rounding a share before recovering its ratio. Compute the ratio from the unrounded expressions when possible; rounding intermediate shares can slightly change the reported ratio.

Neither target is universally preferable. Neyman allocation is optimal for the stated variance objective under its assumptions, but it does not account for participant utility unless that is built into a different objective. An ethical target makes its utility and constraints explicit, but it need not minimize contrast variance. Practical designs may impose minimum arm sizes, safety requirements, or precision requirements; those constraints can prevent either unconstrained optimum from being used directly.

The key question is therefore not which ratio is “the” correct one, but what objective and constraints justify the target. Report the outcome defining the contrast, the utility definition, the allocation constraints, and whether the ratio is an unconstrained optimum or a constrained target. These details make it possible to understand why two principled targets differ.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

For a treatment share \(q_1=0.60\), what is the treatment-to-control ratio?
If \(U_1>U_0\) and the only ethical constraint is \(q_0\ge0.30\), what is the constrained ethical treatment share and ratio?
If \(\sigma_1=0.30\) and \(\sigma_0=0.45\), calculate the interior Neyman ratio and treatment share.
Why can adding a fixed treatment cost change the ethical ratio without changing the Neyman ratio for a response contrast?

Solutions. (1) \(q_0=1-0.60=0.40\), so \(r=0.60/0.40=1.500\). (2) Higher treatment utility favors the largest permitted treatment share: \(q_0=0.30\), \(q_1=0.70\), and \(r_E=0.70/0.30\approx2.333\). (3) Both standard deviations are positive, so \(r_N=0.30/0.45=0.6667\). The treatment share is \(q_{1,N}=0.30/(0.30+0.45)=0.4000\). (4) A fixed cost changes expected net utility and can change which arm the ethical objective favors. It does not change the response variances, so the Neyman ratio for estimating the response contrast remains based on the same response standard deviations.