Plug-In Target Error and Allocation Fluctuations

Why the Target Estimate and the Allocation Both Matter

In a response-adaptive trial, the allocation rule may aim for a proportion determined by unknown response parameters. The rule must estimate those parameters from the accumulating outcomes, so the target itself is random. Meanwhile, the realized allocation proportion can fluctuate around the target currently being used. A limit calculation that includes only one source of variation can therefore give an incorrect asymptotic variance.

This tutorial separates these effects for a two-arm design. Arm 1 receives a proportion of assignments determined by a smooth target function \(\rho(\theta)\), where \(\theta\) is a vector of unknown response parameters and \(0<\rho(\theta)<1\). Arm 2 receives the remaining assignments. The same reasoning applies to a scalar target coordinate in a larger allocation problem.

The distinction is important: the second term is not necessarily just the noise of independent coin tosses. In an adaptive design it can include the effects of the allocation rule, sequential feedback, and the way the current estimate is used. Its joint dependence with \(\hat{\theta}_n\) must be retained.

Assumptions and the Basic Decomposition

Write \(g=\nabla\rho(\theta)\), the column vector of first derivatives of the target function at the true parameter. For the standard smooth-target calculation, assume the following:

• The true parameter \(\theta\) is an interior point of its parameter space, and \(\rho\) is differentiable in a neighbourhood of \(\theta\), with \(0<\rho(\theta)<1\).
• The estimator has a root-\(n\) limit: \(\sqrt n(\hat{\theta}_n-\theta)\) converges in distribution to a mean-zero Gaussian vector \(T\) with covariance matrix \(\Sigma\).
• The allocation residual has a root-\(n\) limit, and the two terms have a joint limit: \(\big(\sqrt n\{N_{1,n}/n-\rho(\hat{\theta}_n)\},\sqrt n(\hat{\theta}_n-\theta)\big)\) converges to \((A,T)\).
• The joint limit \((A,T)\) is mean-zero Gaussian, with \(\operatorname{Var}(A)=\tau^2\), \(\operatorname{Cov}(T)=\Sigma\), and \(\operatorname{Cov}(A,T)=c\), where \(c\) is a column vector.

These assumptions summarize the limit information needed here. In applications, they must be justified for the particular response model and allocation rule. For example, the arm-specific estimators must have enough observations for their stated limits, and the sequential randomization and response process must satisfy the conditions used to establish the joint limit. Consistency of target proportions alone does not establish a root-\(n\) limit or its covariance.

The equality is exact for every \(n\). Differentiability and the root-\(n\) estimator limit give the first-order expansion \(\sqrt n\{\rho(\hat{\theta}_n)-\rho(\theta)\}=g^\mathsf{T}\sqrt n(\hat{\theta}_n-\theta)+o_p(1)\). Thus the allocation proportion has the following limit.

Why the Covariance Term Appears

The derivation uses two facts. First, the exact decomposition expresses the total error as the sum of the residual and target error. Second, the differentiability expansion replaces the target error, at root-\(n\) scale, by a linear function of the estimator error. The joint limit then gives \(A+g^\mathsf{T}T\). For any jointly distributed variables with finite second moments, the variance of a sum includes twice their covariance:

Here \(\operatorname{Cov}(A,g^\mathsf{T}T)=g^\mathsf{T}\operatorname{Cov}(T,A)=g^\mathsf{T}c\). The covariance can be positive, negative, or zero. A negative value can reduce the total variance; it does not make the total variance negative, since the expression is the variance of the combined limit. The joint covariance matrix of \(A\) and \(T\) must be positive semidefinite.

Worked Example: A Smooth Neyman-Allocation Target

For two arms with response standard deviations \(\sigma_1\) and \(\sigma_2\), consider the target \(\rho(\theta)=\sigma_1/(\sigma_1+\sigma_2)\), where \(\theta=(\sigma_1,\sigma_2)^\mathsf{T}\). This target is smooth when both standard deviations are positive. Suppose, hypothetically, that the true values are \(\sigma_1=2\) and \(\sigma_2=3\), and that \[ \Sigma= \begin{pmatrix} 0.25&0.06\\ 0.06&0.36 \end{pmatrix}, \qquad \tau^2=0.010,\qquad c= \begin{pmatrix}0.005\\-0.002\end{pmatrix}. \] The target is \(2/(2+3)=0.400\). Differentiating the target gives \[ g= \begin{pmatrix} \sigma_2/(\sigma_1+\sigma_2)^2\\ -\sigma_1/(\sigma_1+\sigma_2)^2 \end{pmatrix} = \begin{pmatrix}0.12\\-0.08\end{pmatrix}. \]

The plug-in target variance is \[ g^\mathsf{T}\Sigma g =(0.12)^2(0.25)+(-0.08)^2(0.36) +2(0.12)(-0.08)(0.06) =0.003600+0.002304-0.001152=0.004752. \] The covariance contribution is \[ 2g^\mathsf{T}c =2\{(0.12)(0.005)+(-0.08)(-0.002)\} =2(0.000760)=0.001520. \] Thus the total limiting variance is \(0.010+0.004752+0.001520=0.016272\). Under the stated joint Gaussian assumptions, \(\sqrt n\{N_{1,n}/n-0.400\}\) converges to \(N(0,0.016272)\). The corresponding limiting standard deviation on the root-\(n\) scale is approximately \(0.1276\).

This calculation uses the target gradient and the full covariance information. It does not assert that these hypothetical covariance values apply to every design using this target; they must be obtained or justified for the particular response-adaptive procedure.

When the Target Estimator Is the Only First-Order Source

Some allocation rules track the current plug-in target so closely that the residual is negligible at the root-\(n\) scale. In that case \(A=0\) in the limit, so \(\tau^2=0\) and \(c=0\). The allocation proportion still fluctuates because the estimated target fluctuates: \[ \sqrt n\{N_{1,n}/n-\rho(\theta)\} \ \Longrightarrow\ g^\mathsf{T}T, \qquad \operatorname{Var}(g^\mathsf{T}T)=g^\mathsf{T}\Sigma g. \] This conclusion requires the stronger tracking condition \(\sqrt n\{N_{1,n}/n-\rho(\hat{\theta}_n)\}\to_p0\). Merely showing that the allocation proportion and plug-in target differ by a quantity that converges to zero is not enough to discard their difference at root-\(n\) scale.

Comparisons, Diagnostics, and Limits

The decomposition clarifies how this result differs from several tempting benchmarks. An oracle design that knows \(\theta\) has no plug-in target error, though its realized allocation can still fluctuate around the known target. A fixed allocation chosen in advance does not incur this particular plug-in error, but it may target a different proportion. A plug-in calculation that uses only \(g^\mathsf{T}\Sigma g\) omits allocation residual variation, while one that adds the two variances but assumes independence also omits their covariance.

In practice, check that the target is interior and differentiable, that the estimator has a root-\(n\) limit, and that the allocation residual has a limit at the same scale. Check the joint covariance, not just the marginal variances. A large estimated covariance contribution, or a calculated total variance inconsistent with a valid joint covariance matrix, is a reason to revisit the assumptions or calculations. Also examine whether the design maintains enough assignments to each arm for its response-parameter estimators to behave as assumed.

The result is a first-order approximation. It does not by itself establish finite-sample normality, give a valid standard error estimator, or account for bias that remains at order \(n^{-1/2}\). It also relies on a smooth target. When differentiability fails, the linear expansion used here may not hold, and a different limit argument is needed. The formula should not be applied by substituting a gradient where no valid gradient-based approximation exists.

Common Mistake

A common mistake is to say: “The estimator is asymptotically normal, and allocation noise is random, so their variances simply add.” That reasoning silently assumes the two limiting terms are uncorrelated. In an adaptive trial, the same responses affect both the parameter estimate and subsequent assignments, so that assumption is not automatic. The correction is to derive or estimate the joint limit and include \(2g^\mathsf{T}c\); omit it only when independence or zero covariance is justified.