Local Alternatives and Shifting Allocation Targets

Why a Moving Target Changes the Limit

An adaptive allocation rule may aim to assign a fraction $q(\theta)$ of participants to one treatment, where $\theta$ describes features of the response distributions. If those response features change with sample size, the target itself can move. A change that vanishes as the trial grows can still matter at the same order as the random error in an estimated target: a perturbation of size $1/\sqrt{n}$ produces an order-one shift after multiplication by $\sqrt{n}$.

The essential bookkeeping is to distinguish two reference points. Centering the allocation proportion at the target for the current response model asks how far allocation is from its moving target. Centering at the target under a baseline model also includes the systematic shift from baseline to current target. These are different asymptotic quantities.

Definition: Let $\theta_0$ be a baseline parameter and let $h$ be a fixed vector. The sequence $\theta_n=\theta_0+h/\sqrt{n}$ is a local alternative in direction $h$. It is contiguous to the baseline sequence when the corresponding probability laws are mutually contiguous: any sequence of events whose probability tends to zero under one sequence also has probability tending to zero under the other. Contiguity describes alternatives that cannot be separated from the baseline with probability tending to one by a single sequence of events.

The local form specifies the scale of the parameter change; by itself, it does not establish contiguity for every statistical model. The likelihood behavior and regularity of the response model determine whether contiguity holds. The results below require an appropriate local limit under the sequence $\theta_n$, rather than assuming that a limit established under the baseline automatically remains unchanged. The earlier material on triangular arrays provides the relevant setting for changing response distributions.

Assumptions and the Main Expansion

Suppose the allocation target is a scalar function $q(\theta)$, where $\theta$ may be a vector. Assume that $q$ is differentiable at $\theta_0$, with derivative row vector $J=Dq(\theta_0)$. Assume also that the local parameter is $\theta_n=\theta_0+h/\sqrt{n}$, that $\hat{\theta}_n$ estimates $\theta_n$, and that the trial’s final allocation proportion to the treatment of interest is $A_n$. The following joint limits make the required stochastic behavior explicit:

\sqrt{n}(\hat{\theta}_n-\theta_n)\Rightarrow Z_h, \qquad \sqrt{n}\{A_n-q(\hat{\theta}_n)\}\Rightarrow U_h.

The limits are assumed to hold jointly under the local sequence, so that the pair on the left converges to $(Z_h,U_h)$. In particular, the second limit says that the allocation proportion tracks the estimated target closely enough for its remaining error to be of order $1/\sqrt{n}$. Neither a Gaussian limit nor independence of $Z_h$ and $U_h$ is required for the basic expansion. The subscript $h$ allows the limiting laws to depend on the local alternative.

Theorem: Under these assumptions and differentiability of $q$ at $\theta_0$, $$ \sqrt{n}\{A_n-q(\theta_n)\}\Rightarrow JZ_h+U_h, \qquad \sqrt{n}\{A_n-q(\theta_0)\}\Rightarrow Jh+JZ_h+U_h. $$ The first limit is centered at the moving target; the second is centered at the baseline target.

To justify the result, expand the estimated target around the current parameter $\theta_n$. Since $\hat{\theta}_n-\theta_n=O_p(1/\sqrt{n})$, differentiability and convergence of $\theta_n$ to $\theta_0$ give

q(\hat{\theta}_n)-q(\theta_n) = Dq(\theta_0)(\hat{\theta}_n-\theta_n)+o_p(n^{-1/2}).

Add and subtract $q(\hat{\theta}_n)$, multiply by $\sqrt{n}$, and use the assumed joint convergence:

\sqrt{n}\{A_n-q(\theta_n)\} = \sqrt{n}\{A_n-q(\hat{\theta}_n)\} +J\sqrt{n}(\hat{\theta}_n-\theta_n)+o_p(1) \Rightarrow U_h+JZ_h.

For the fixed baseline, differentiability also gives

\sqrt{n}\{q(\theta_n)-q(\theta_0)\} = \sqrt{n}\left\{\frac{Jh}{\sqrt{n}}+o(n^{-1/2})\right\} \longrightarrow Jh.

Adding this deterministic shift to the moving-target limit proves the second assertion. This derivation shows why the local drift is $Jh$, not $h$ itself: the target function transforms parameter changes according to its derivative.

Worked Examples

Worked Example: A Smooth Target for a Difference in Response Means

Consider two treatments with binary responses and let $\theta=\mu_1-\mu_2$, the difference in response probabilities. Suppose the target fraction assigned to treatment 1 is $q(\theta)=1/(1+\exp(-\theta))$. Under the hypothetical local alternative, the baseline difference is $\theta_0=0$, $h=0.8$, and $n=400$. Thus $\theta_n=0+0.8/\sqrt{400}=0.04$. For example, response probabilities $\mu_1=0.52$ and $\mu_2=0.48$ have this difference.

At baseline, $q(\theta_0)=1/2=0.5$. The derivative is $q'(\theta)=q(\theta)\{1-q(\theta)\}$, so $q'(0)=0.25$. The first-order target drift on the root-$n$ scale is therefore

\sqrt{n}\{q(\theta_n)-q(\theta_0)\} \approx q'(0)h=0.25(0.8)=0.200.

Direct calculation gives $q(0.04)=1/(1+\exp(-0.04))\approx0.509999$. Hence the finite-$n$ target shift is about $0.009999$, and multiplying by $\sqrt{400}=20$ gives about $0.2000$, consistent with the derivative calculation.

To see how estimation adds another component, suppose a hypothetical trial snapshot has 204 assignments to treatment 1 and 196 to treatment 2, with observed response proportions $0.55$ and $0.45$. The estimated difference is $\hat{\theta}_n=0.10$, giving $\hat{q}_n=1/(1+\exp(-0.10))\approx0.52498$. The observed allocation proportion is $A_n=204/400=0.510$. Its plug-in-target discrepancy is $20(0.510-0.52498)\approx-0.300$. This snapshot is only an illustration of the quantities: it does not determine their asymptotic distribution. The limit centered at the moving target contains estimation and allocation noise; the limit centered at 0.5 contains that same noise plus the approximately $0.200$ local drift.

Worked Example: The Joint Covariance Affects the Limit

Suppose $q(\theta)=0.30+0.20\theta$, $\theta_0=0.50$, and the hypothetical local direction is $h=-1.2$ with $n=900$. Then $\theta_n=0.50-1.2/30=0.46$. The baseline target is $q(\theta_0)=0.400$; the moving target is $q(\theta_n)=0.30+0.20(0.46)=0.392$. The root-$n$ drift relative to baseline is $30(0.392-0.400)=-0.240$, also given by $q'(\theta_0)h=0.20(-1.2)=-0.240$.

For illustration, suppose that under this local sequence the joint limit is mean-zero bivariate normal, with $\operatorname{Var}(Z_h)=0.25$, $\operatorname{Var}(U_h)=0.09$, and $\operatorname{Cov}(Z_h,U_h)=0.03$. Here $J=0.20$. The moving-target limit $JZ_h+U_h$ is normal with variance

$$ (0.20)^2(0.25)+0.09+2(0.20)(0.03) =0.010+0.090+0.012 =0.112. $$

Thus $\sqrt{n}\{A_n-q(\theta_n)\}$ has limiting distribution $N(0,0.112)$ under these illustrative assumptions. Relative to the fixed baseline target, the limit is $N(-0.240,0.112)$. Omitting the covariance term would give variance $0.100$, which is incorrect under the stated joint limit. A zero covariance or independence assumption would need justification; it is not implied by the expansion.

Worked Example: A Nonlinear Target and a Finite-Sample Remainder

Let $q(\theta)=\theta/(1+\theta)$ for positive $\theta$, with baseline $\theta_0=1$, local direction $h=2$, and $n=100$. The local parameter is $\theta_n=1+2/10=1.2$. The baseline target is $1/2=0.5$, while the moving target is $1.2/2.2=6/11\approx0.54545$. The exact root-$n$ shift is

10\left(\frac{6}{11}-\frac{1}{2}\right) =10\left(\frac{1}{22}\right) =\frac{5}{11} \approx0.4545.

Since $q'(\theta)=1/(1+\theta)^2$, the first-order local approximation is $q'(1)h=0.25(2)=0.500$. The difference between $0.500$ and $0.4545$ is a finite-$n$ Taylor remainder, not a contradiction: the local approximation describes the limit as $n$ grows. Indeed, $q''(1)=-0.25$; the second-order contribution before root-$n$ scaling is approximately $\tfrac12(-0.25)(0.2)^2=-0.005$, reducing the first-order target change $0.050$ toward the exact change $0.04545$.

Interpretation, Diagnostics, and Alternatives

If the joint limit is Gaussian and mean zero for $(Z_h,U_h)$, then centering at the moving target removes the deterministic first-order target drift. Centering at the baseline target leaves the shift $Jh$. This shift can matter when describing the sampling distribution relative to a baseline reference, even though the response change itself tends to zero. In a vector-parameter model, $Jh$ is the derivative of the target in the local direction; a direction $h$ to which the target is locally insensitive has $Jh=0$.

The expansion relies on specific conditions. Check that the target is differentiable at the baseline, that the local parameter perturbation has the stated $1/\sqrt{n}$ scale, that the estimator has a root-$n$ limit under the local sequence, and that allocation tracks its estimated target at the same scale. Also check the joint limit, including any covariance needed for a variance calculation. A limit under the baseline alone does not establish all these facts under local alternatives.

A kink or boundary. If the target is clipped at a boundary or has a corner at $\theta_0$, an ordinary derivative may not describe the shift. The earlier material on nonsmooth allocation targets is relevant; do not apply the smooth expansion without checking its conditions.
A larger response change. A fixed nonzero change in $\theta$ is not an $h/\sqrt{n}$ perturbation. The local Taylor expansion does not characterize that fixed-alternative regime.
Unverified tracking. If $\sqrt{n}\{A_n-q(\hat{\theta}_n)\}$ does not have a finite limit, allocation error may be too large for the displayed decomposition to yield the claimed root-$n$ limit.
Misleading recentering. A zero-centered limit around $q(\theta_n)$ does not imply a zero-centered limit around $q(\theta_0)$; the latter includes $Jh$.
Ignoring dependence. When computing a variance, the estimation and allocation components may be correlated. Their covariance belongs in the variance unless it is known to be zero.

A fixed-parameter analysis centers at one target and can miss a drift that is of the same order as sampling fluctuations. A local-alternative analysis retains that drift explicitly. Centering at the plug-in target is useful for isolating fluctuation around the current aim, while centering at the baseline target is useful when the baseline is the scientific reference. These are complementary descriptions of the same allocation process, not competing allocation procedures.

The scope of the result is limited. It gives a first-order expansion for a smooth target under stated joint-limit assumptions; it does not establish those limits for a particular adaptive rule, guarantee a Gaussian distribution, or justify a variance estimate. If $Jh=0$, higher-order target changes may matter at a finer scale, but they are not captured by the root-$n$ result here.

Check Your Understanding

Answer each question using the local expansion, then compare with the short solutions.

If $J=0.3$ and $h=-0.5$, what is the first-order root-$n$ drift relative to the baseline target?
Why can the limiting variance of $JZ_h+U_h$ not generally be calculated by adding only the two component variances?
For $q(\theta)=\theta^2$, $\theta_0=2$, and $h=1$, calculate the first-order target drift under $\theta_n=\theta_0+h/\sqrt{n}$.
What additional condition is needed to use the displayed theorem if the allocation proportion is centered at $q(\hat{\theta}_n)$?
Does a local alternative of the form $\theta_0+h/\sqrt{n}$ alone guarantee contiguity? Explain.

Solutions. (1) The drift is $Jh=0.3(-0.5)=-0.15$. (2) The variance also includes $2J\operatorname{Cov}(Z_h,U_h)$, which vanishes only if the covariance is zero. (3) Since $q'(\theta)=2\theta$, $J=q'(2)=4$, so the drift is $Jh=4$. (4) One needs a suitable root-$n$ joint limit for the estimator error and the allocation discrepancy, including the stated tracking behavior. (5) No. The local scale specifies the parameter sequence, but contiguity depends on the response model and its probability laws.

Local Alternatives and Shifting Allocation Targets

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