Covariance From Estimator-Allocation Feedback

Why Feedback Creates Covariance

In response-adaptive randomization, an estimated response parameter can influence the probability of assigning the next participant to each arm. The resulting allocation is therefore connected to the response estimates that helped determine it. Even when the randomization draw and the current response residual are conditionally independent, an estimate’s error can persist into later allocation decisions. This is a source of covariance between allocation and estimation fluctuations.

The purpose here is to isolate that covariance using a stable, linearized two-arm recursion. The recursion is a local approximation: its coefficients summarize how allocation deviations persist, how strongly an estimated response parameter changes later allocation, and how the estimation error evolves. It is not a universal formula for every response-adaptive design. In a particular design, the coefficients and innovation covariances must come from that design’s allocation rule and estimator.

A Linearized Two-Arm Recursion

Consider the local model

Here \(r\) describes persistence of allocation deviation in the absence of new inputs. The coefficient \(\beta\) is the local sensitivity of future allocation deviation to the current response-estimation error. For example, if a differentiable allocation rule assigns arm 1 with probability \(g(\hat\theta_m)\), then its local sensitivity at the target parameter \(\theta\) is \(g'(\theta)\); the coefficient in a chosen recursion can include additional scaling. The coefficient \(\kappa\) describes persistence of the estimation error on this local scale. The random terms \(\xi_{m+1}\) and \(\eta_{m+1}\) are new allocation and estimation innovations.

Assume the innovations have mean zero, are uncorrelated with the state \((U_m,V_m)\) before step \(m+1\), and have constant second moments

The innovation covariance matrix must be positive semidefinite; in particular, \(q_U\geq 0\), \(q_V\geq 0\), and \(q_{UV}^2\leq q_Uq_V\). Assume also \(|r|<1\) and \(|\kappa|<1\). These conditions make the triangular recursion stable and allow its second moments to settle to finite stationary limits. They are assumptions about this linearized model, not automatic facts about every adaptive design.

Deriving the Covariance Contribution

Write \(S_m=\operatorname{Var}(V_m)\), \(C_m=\operatorname{Cov}(U_m,V_m)\), and \(R_m=\operatorname{Var}(U_m)\). Expand the second recursion’s variance and use the fact that \(\eta_{m+1}\) is uncorrelated with the state at step \(m\):

For the cross-covariance, multiply out the two updates. The terms involving a current innovation and a previous state vanish by the innovation assumption, while the two same-step innovations contribute \(q_{UV}\):

At stationarity, set \(S_{m+1}=S_m=S\) and \(C_{m+1}=C_m=C\). Solving the first equation gives \(S=q_V/(1-\kappa^2)\). Substitution into the cross-covariance equation gives the central result:

This formula separates two contributions. The term \(\beta\kappa S/(1-r\kappa)\) is the covariance created by feedback and persistence: estimation error persists through \(\kappa\), influences later allocation through \(\beta\), and the allocation-estimation relationship persists through \(r\kappa\). The term \(q_{UV}/(1-r\kappa)\) propagates same-step innovation covariance. Either contribution can be positive or negative.

For completeness, the allocation variance obeys

The cross-term \(2r\beta C\) matters: adding the separate variances of the persistence term, feedback term, and innovation while omitting their covariance generally gives the wrong allocation variance.

Using the Calculation Responsibly

A practical derivation starts by writing down the state and its scale. A centered allocation count, an allocation proportion, and a square-root-scaled proportion are different quantities and generally have different recursion coefficients and innovation variances. Likewise, \(V_m\) must be the specific response-estimation error whose effect enters the allocation rule, not an unspecified “response noise.”

Diagnostics should focus on the assumptions as well as the arithmetic. A fitted or derived recursion with \(|r|\) or \(|\kappa|\) near 1 can produce large stationary moments, so small coefficient errors can matter substantially. A negative calculated variance, or a covariance matrix violating \(C^2\leq RS\), signals an algebra error, an infeasible set of inputs, or a mismatch between the proposed recursion and the process being approximated. Persistent residual dependence on past states also contradicts the assumed innovation structure.

The recursion calculation is useful because it identifies which mechanism creates covariance and how that contribution propagates. It is not a substitute for deriving the design-specific joint limit law. Directly setting \(C=0\) discards feedback and any shared innovations; treating all fluctuations as independent can therefore give incorrect variance calculations. Simulation can check a derived approximation, but simulation alone does not establish the recursion’s assumptions or explain the source of a covariance.

The main limitation is scope: this is a stable, time-homogeneous, linearized two-state calculation. If the allocation target is at a boundary, the rule is nonsmooth, the coefficients change with time, or the relevant state is not stable, these stationary equations may not describe the asymptotic covariance. In such cases the recursion itself and its limiting regime must be reconsidered rather than forcing the same formula onto the problem.