🤖 AI Summary
This study addresses the problem of unbiased inference and multiple comparison control for effect modification in factorial experiments under high-dimensional covariates. The proposed method integrates cross-fitting, Lasso penalization, high-dimensional Gaussian approximation, and a Walsh spectrum multiplier bootstrap to residualize Walsh-Hadamard scores. By establishing the global insensitivity of the debiased central limit theorem, it eliminates the need for polynomial convergence rate conditions and heredity assumptions. This work enables simultaneous confidence interval construction and strong family-wise error rate control across all effect units, facilitating the precise identification of significant modifiers. Supporting growing covariate dimensions and encompassing all contrasts, the approach provides a rigorous statistical framework for analyzing high-dimensional causal heterogeneity.
📝 Abstract
In factorial randomized experiments with $K$ binary treatment components, we study which baseline covariates modify component main effects and interactions. We define effect-modifier cells $(j,S)$, $j\in[p]$ and $\emptyset\neq S\subseteq[K]$, through the coefficients $θ^\ast_{jS}$ of population linear projections of conditional Walsh--Hadamard contrasts $τ_{S}$ onto standardized covariates. Using baseline-robust direct scores, we develop frequentist inference for all $p(2^K-1)$ cells. First, we prove a debiased central limit theorem with global insensitivity: the residualization baseline enters the cross-fitted score through an exactly conditionally mean-zero perturbation. The influence-function remainder therefore requires only foldwise weighted $L_2$ consistency of the baseline estimator, with no prescribed polynomial rate or nuisance product-rate condition. A separate, explicit penalty-scale condition controls the baseline's effect on the score Lasso. Second, under explicitly stated uniform nuisance, baseline, studentization, and influence-array conditions derived from bounded sparse-regression primitives, we establish a high-dimensional Gaussian approximation over the full grid, calibrated by a Walsh-spectral multiplier bootstrap. This yields simultaneous confidence bands and Romano--Wolf step-down selection of modified cells with strong familywise error control, without effect-heredity assumptions. The factorial design is fixed in the asymptotic analysis, while the covariate dimension may grow; all $2^K-1$ contrasts are covered jointly.