When Neyman Allocation Reduces to Equal Allocation

Why Equal Allocation Can Be Optimal

Equal allocation is often chosen because it is simple and gives the two arms the same number of participants. That simplicity does not make it universally optimal. For a two-arm comparison of means, the variance-minimizing allocation depends on the arms’ outcome standard deviations. When those standard deviations are equal, the Neyman target reduces to equal allocation. When they differ, the target assigns more participants to the arm with greater outcome variability.

This is a statement about a particular objective: minimizing the sampling variance of the difference between two arm means when the total sample size is fixed. It is not a general claim that equal allocation is best for every trial objective, patient-benefit criterion, or allocation constraint.

The model assumptions for this result are that the outcome is measured on a common scale in both arms; observations are independent within and across arms; each arm has a finite, well-defined outcome variance; the total sample size is fixed; and participants have equal per-participant costs. The formula for \(V\) is the variance of the difference in sample means under these assumptions. The continuous allocation calculation also assumes \(n_1>0\) and \(n_2>0\). For the standard proportional Neyman result, both \(\sigma_1\) and \(\sigma_2\) must be strictly positive.

The allocation target concerns arm sizes, not a particular randomization procedure. In a response-adaptive design, a target allocation proportion can guide assignment probabilities, but the realized arm counts need not equal that target exactly. The variance calculation here describes the fixed-arm-size allocation objective; it does not by itself establish the operating characteristics of every adaptive estimator or randomization rule.

Deriving the Equal-Variance Case

Suppose the two arms have the same strictly positive outcome variance, so \(\sigma_1^2=\sigma_2^2=\sigma^2\), with \(\sigma^2>0\). Substituting \(n_2=N-n_1\) into the variance objective gives a one-variable expression. Its minimum can be seen directly by writing the product \(n_1(N-n_1)\) as a square-completion identity.

The numerator \(\sigma^2N\) is fixed and positive. Therefore \(V(n_1)\) is smallest when its positive denominator is largest. The displayed inequality shows that the denominator is largest exactly when \(n_1=N/2\), which also gives \(n_2=N/2\). Substitution yields the minimum variance and its corresponding standard error:

This establishes the special case without relying on a comparison of approximate sample-size rules: under the stated model and objective, equal arm sizes attain the minimum. The conclusion depends on equality of the variances, not equality of treatment effects. The treatment-effect magnitude does not appear in this variance objective.

How Unequal Variances Shift the Target

When both arm standard deviations are positive but unequal, the target changes. The allocation ratio follows by differentiating the variance objective with respect to \(n_1\), while keeping \(n_2=N-n_1\). Setting the derivative to zero gives the interior candidate:

Because both standard deviations are positive, the derivative equation has an interior solution. The second derivative is \(2\sigma_1^2/n_1^3+2\sigma_2^2/n_2^3\), which is positive throughout the feasible interior. Thus the objective is strictly convex there, and the stationary allocation is its unique minimum. Solving the ratio together with \(n_1+n_2=N\) gives the continuous target shares \(n_1/N=\sigma_1/(\sigma_1+\sigma_2)\) and \(n_2/N=\sigma_2/(\sigma_1+\sigma_2)\).

The ratio uses standard deviations, not variances. A larger standard deviation means that, per participant, the corresponding sample mean contributes more variance. Assigning more participants to that arm reduces its contribution. If the two standard deviations are equal, the ratio is one and the target shares are equal; if one is twice the other, its target sample size is twice as large.

Boundary Cases, Diagnostics, and Limitations

Strict positivity matters. If exactly one arm has zero variance and both sample sizes must be positive, the usual proportional-allocation result does not give an attained minimum. For example, let \(\sigma_1=0\), \(\sigma_2=3\), and \(N=100\). Then \(V=9/n_2=9/(100-n_1)\). For any \(0<n_1<100\), \(n_2<100\), so \(V>9/100=0.0900\) outcome-units squared. As \(n_1\) approaches zero from above, \(V\) approaches \(0.0900\), but no allocation with both arm sizes positive attains that value. The standard-deviation ratio would require \(n_1=0\), outside the positive-allocation domain. If both variances are zero, \(V=0\) for every positive allocation, and the standard-deviation ratio is undefined. These are separate cases, not instances of the ordinary positive-variance formula.

In practice, the variances used to set a target are often not known exactly. A target based on uncertain variance estimates can therefore differ from the target implied by the true variances. Before interpreting an apparent imbalance as a design improvement, check that the variances refer to the same outcome and scale, that the observations and variance estimates are not being distorted by data or coding errors, and that the allocation being assessed is the intended target rather than a random fluctuation in realized arm counts. Large differences in estimated variability deserve particular scrutiny; a few unusual observations can materially affect an estimated variance.

Neyman allocation and equal allocation answer different practical needs. Equal allocation is transparent and can be a sensible default when variances are similar, are poorly known, or other design considerations matter. Neyman allocation is justified when the stated assumptions hold and the primary design objective is to minimize the variance of the two-arm mean difference for a fixed total sample size. Neither conclusion alone establishes which design is ethically preferable or best for every scientific goal. The formula also treats allocation as continuous; constraints and integer sample sizes can prevent exact attainment of its target.