Variance Penalties Away From the Neyman Ratio

Why quantify departure from the Neyman share?

The Neyman allocation identifies the two-arm sample-size share that minimizes the variance of a difference in sample means when the total sample size is fixed. In practice, a design may use a different share: equal allocation is common, a target may be chosen for operational reasons, or estimated variances may lead to a share that differs from the variance-minimizing one. Knowing only that an allocation is “not optimal” is not enough to judge its consequences. The exact variance penalty shows how much precision is lost under the stated model.

This tutorial studies a fixed total of \(n\) participants assigned between treatment 1 and treatment 0. It holds the outcome variances fixed and compares allocations for estimating the contrast \(\mu_1-\mu_0\). It does not assign a value to other design considerations: the calculation is about contrast variance alone.

Model and assumptions

For the calculation, assume that the outcome in arm \(j\) has mean \(\mu_j\) and finite variance \(\sigma_j^2\), for \(j=0,1\). Outcomes are independent within and between arms, and the arm sample sizes are fixed in advance. Each arm uses the sample mean to estimate its mean. The total sample size \(n\) is fixed, and every participant has the same cost, so fixing \(n\) is also fixing the total cost. Both arms must have positive sample sizes: \(0<q<1\).

Under these assumptions, the two sample means are independent, with variances \(\sigma_1^2/n_1\) and \(\sigma_0^2/n_0\). Therefore the variance of their difference is the sum of those variances. For an interior Neyman share and the relative-penalty formula below, assume in addition that \(\sigma_1>0\) and \(\sigma_0>0\). This positivity condition matters: zero-variance cases need separate treatment.

The formula is an exact sampling variance under the stated model; it does not require normally distributed outcomes. Normality may matter for some inferential procedures, but it is not needed to add the variances of independent sample means. In a response-adaptive design, arm sizes may depend on previous outcomes. The expression here describes the fixed-allocation comparison; it is not, without further justification, a complete variance calculation for every outcome-dependent allocation process.

Deriving the minimizing share and the penalty identity

When both standard deviations are strictly positive, differentiate \(V(q)\) over \(0<q<1\). The second derivative is positive throughout the interval, so any stationary point is the unique minimum. The first derivative vanishes when the two positive terms in the resulting equality are equal.

Because \(V(q)\) tends to infinity as \(q\) approaches either endpoint when both variances are positive, the unique stationary point is the global minimum. Its allocation ratio is \(n_1/n_0=q^*/(1-q^*)=\sigma_1/\sigma_0\), and its variance is \((\sigma_1+\sigma_0)^2/n\).

A useful result is an exact expression for the variance away from this share. Divide \(V(q)\) by \(V(q^*)\), substitute \(q^*=\sigma_1/(\sigma_1+\sigma_0)\), and combine terms over the common denominator \(q(1-q)\). Subtracting 1 leaves a squared difference, so the excess is nonnegative.

Thus the relative variance penalty is exactly \((q-q^*)^2/[q(1-q)]\), and the variance itself is the minimum variance multiplied by one plus that penalty. The penalty is zero at the Neyman share and positive at every other interior share. Its denominator also explains why a fixed absolute departure can be especially costly near an allocation boundary.

Worked numerical comparisons

Reading the penalty and diagnosing a poor comparison

The penalty identity separates the effect of allocation mismatch from the scale of the outcome variances. The minimum variance \(V(q^*)\) sets the scale; the dimensionless factor \(1+(q-q^*)^2/[q(1-q)]\) tells how much the chosen share inflates it. Because the mismatch is squared, shares equally far above and below \(q^*\) have the same penalty only when their denominators \(q(1-q)\) are also equal. In general, departures on opposite sides need not cost the same amount.

A sound numerical comparison should report the target and alternative shares, the arm counts or their continuous equivalents, the variance in squared outcome units, and the standard-error ratio if discussing precision on the original outcome scale. Check that shares sum to one, that the sample sizes add to \(n\), and that the direct variance calculation agrees with the penalty identity. A disagreement usually signals a switched arm label, an incorrect denominator, or confusion between a variance ratio and a standard-error ratio.

The comparison is only as informative as its inputs. If the variances used to calculate \(q^*\) are uncertain or misspecified, the share called “Neyman” may not minimize the true variance. Likewise, if arm sizes are rounded to integers, the realized share may differ slightly from the theoretical continuous target; use the actual counts in the variance formula to assess that allocation. The displayed exact identity applies to the continuous share formulation. It does not replace evaluating feasible integer allocations or a design with additional constraints.

Boundary cases and limits

The assumption \(\sigma_1,\sigma_0>0\) is essential to the interior minimization argument. If \(\sigma_1=0\) and \(\sigma_0>0\), then \(V(q)=\sigma_0^2/[n(1-q)]\), which increases with \(q\). Its infimum over \(0<q<1\) is approached as \(q\) tends to zero, but is not attained at any share in that interval. At \(q=0\), the treatment-1 mean cannot be estimated from a sample, so that endpoint is not a valid two-arm contrast estimator. If \(\sigma_0=0\) and \(\sigma_1>0\), the corresponding infimum is approached as \(q\) tends to one and is likewise not attained in the interior. If both variances are zero, \(V(q)=0\) for every interior share, and the positive-variance formula for \(q^*\) does not determine a unique target.

In applications, minimum arm sizes, integer counts, unequal costs, or other design objectives can alter the relevant comparison. Those cases require their own constraints or objective functions; the unconstrained, equal-cost result here should not be applied as if those features were absent. In particular, a variance-optimal share is not automatically the preferred allocation for every scientific or operational purpose.

Common mistake

A second useful check is not to substitute a zero standard deviation into the interior formula and claim an interior optimum at an endpoint. The derivation requires both standard deviations to be positive; the zero-variance cases have the boundary behavior described above.