Sequential randomization for mediation with intermediate covariates

CI-35 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

First randomize binary $W$ conditional on baseline $X$, observe a post-treatment covariate $L$, and then randomize a feasible mediator setting $M$ conditional on $(X,W,L)$. Let $L(w)$ and $Y(w,l,m)$ be well-defined potential variables with consistency. A separate natural-mediator arm estimates a reference distribution $g_{w'}(m\mid X)$ of the mediator under treatment $w'$. Use a finite mediator state space, bounded outcomes, and known randomization probabilities $q(w\mid X)$ and $r(m\mid X,W,L)$ bounded away from zero. Let $X\in[0,1]^d$, $L\in[0,1]^q$. Require outcome means, $L$ densities, and reference mediator probabilities to have Hölder indices $\beta,\alpha,\gamma>0$, respectively, with radius $B$, and bounded intervention costs.

Formal question

Define a stochastic mediator effect using an independent reference draw $G_{w'}\mid X\sim g_{w'}(\cdot\mid X)$: \[ \psi(w,w')=\E[Y(w,L(w),G_{w'})]. \] Find cost-constrained sequential designs $(q,r)$ minimizing worst-case risk for contrasts among these means. Establish identification and efficient estimation when $L$ both responds to $W$ and predicts the outcome, and characterize adaptive allocation rules retaining uniform inferential validity. Determine what extra structural restrictions, if any, would make these stochastic contrasts coincide with specified natural-effect contrasts.

Scope and status

This precise design question is editorial. Randomization of the mediator setting is an intervention on a stated mechanism; an enhancement that merely shifts the mediator requires additional exclusion and compliance analysis. Sequential randomization identifies the displayed interventions under consistency and positivity. It does not impose independence between $Y(w,l,m)$ and a mediator counterfactual under another treatment, nor determine their cross-world coupling. Accordingly, natural effects such as $\E[Y(w,L(w),M(w'))]$ are not claimed identified merely from two randomized assignments. Post-treatment $L$ is propagated under its treatment-specific law rather than held fixed without explanation.

References for the source of the problem

Carlos Cinelli, Avi Feller, Guido Imbens, Edward Kennedy, Sara Magliacane, and Jose Zubizarreta, Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference, arXiv:2508.17099v1 [stat.ME], 23 August 2025, §3.4.3, "Designing experiments for mediation analysis"; §3.4.2, "Identification & estimation," for cross-world restrictions. https://arxiv.org/html/2508.17099v1#S3.SS4.SSS3.

Common conventions: Source mathematical conventions

Well-defined objects. All probability spaces and measurable variables are specified or inherited from a local statistical model. Expectations used as finite targets require the stated integrability. A decision rule or estimator is measurable with respect to its available information; randomized rules may use an auxiliary independent random variable. Norms without a subscript are Euclidean in finite-dimensional spaces. An optimizer is requested only when attainment is assured; otherwise the question concerns the optimal value and $\varepsilon$-optimal admissible rules for every $\varepsilon>0$.

Identification. A structural model $m\in\mathcal M$ implies an observable law $P_m^{\rm obs}$ and target $\theta(m)$. The target is point identified on $\mathcal M$ if \[ P_{m_1}^{\rm obs}=P_{m_2}^{\rm obs} \quad\Longrightarrow\quad\theta(m_1)=\theta(m_2) \qquad(m_1,m_2\in\mathcal M). \] When this fails, the sharp identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M, P_m^{\rm obs}=P\}. \] Specifying an intervention or estimand does not establish identification. Positivity, exclusion, causal sufficiency, measurement restrictions, and transport assumptions are substantive local inputs, never automatic consequences of notation.

Minimax risk and nontrivial inference. For a statistical experiment with data law $P^{(n)}$, target $\theta(P)$, and nonnegative loss $\ell$, define \[ \mathcal R_n(\mathcal P,\ell) =\inf_{\widehat\theta_n}\sup_{P\in\mathcal P} \E_{P^{(n)}}\ell(\widehat\theta_n,\theta(P)). \] The infimum is over measurable estimators. A sharp rate is a sequence $r_n$ for which $0<c\leq C<\infty$, independent of $n$, satisfies $c r_n\leq\mathcal R_n\leq C r_n$ for all sufficiently large $n$. Squared-error parametric risk is of order $n^{-1}$; the corresponding root-mean-squared error is of order $n^{-1/2}$. These different conventions are distinguished locally. A uniform asymptotic confidence region $C_n$ must satisfy \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P} \Pr_{P^{(n)}}\{\theta(P)\in C_n\}\geq 1-\alpha, \qquad 0<\alpha<1, \] together with an informative diameter or expected-width requirement. Coverage alone permits the vacuous whole parameter space. Testing questions similarly impose both size control and power against a declared separated alternative class.

Binary causal baseline. When an entry explicitly invokes this baseline, one observes independent copies of $O=(X,W,Y)$, with $W\in\{0,1\}$ and potential outcomes $Y(0),Y(1)$. Assume consistency $Y=Y(W)$, no interference, conditional exchangeability $(Y(0),Y(1))\perp W\mid X$, and overlap $\epsilon\leq e(X)\leq1-\epsilon$ almost surely for a fixed $0<\epsilon<1/2$, where \[ e(x)=\Pr(W=1\mid X=x),\qquad m_w(x)=\E[Y\mid W=w,X=x],\qquad \tau(x)=m_1(x)-m_0(x). \] These assumptions identify conditional mean effects almost everywhere under the covariate law. Pointwise or uniform effects require appropriate regularity and a specified support. Continuous treatment, longitudinal data, adaptive experiments, and network interference require their own assumptions in place of this baseline. Bounded outcomes, densities, and smoothness are additional assumptions only where stated.

Function classes. On $[0,1]^d$, $\mathcal H_d^s(L)$ is the isotropic Hölder ball of radius $L>0$: put $k=\lceil s\rceil-1$ and $\gamma=s-k\in(0,1]$, bound derivatives through order $k$, and require the order-$k$ derivatives to be $\gamma$-Hölder with constant at most $L$. Thus integer orders use a Lipschitz final derivative convention. The class $\operatorname{BV}_d(L)$ consists of functions $f\in L^1((0,1)^d)$ whose distributional derivative is a finite vector measure and for which $\|f\|_{L^1}+|Df|((0,1)^d)\leq L$. An $L^\infty$ bound or a particular pointwise version is imposed separately when needed. Sparse and neural-network classes require a declared dictionary or architecture and coefficient bounds.

An anisotropic Hölder class with declared block smoothness indices means coordinatewise Hölder regularity in each block, uniformly in the other blocks, with all corresponding derivative and Hölder bounds at most the stated radius. Derivative orders and the integer-order convention in each block are those just given. Mixed-derivative bounds are additional restrictions only when explicitly stated.

Decision and computational questions. Policy regret compares admissible feasible policies under the same law and constraints. In computational entries, finite data and thresholds have rational encodings; running time is measured in their total bit length. Real-valued equilibrium search uses the explicitly declared algebraic or FIXP encoding. An approximation ratio for maximization is written $\operatorname{OPT}/\operatorname{ALG}$ and for minimization as $\operatorname{ALG}/\operatorname{OPT}$, with zero optima and infeasible instances handled locally. Historical approximation bounds are not automatically current bounds.