Debiased and Simultaneous Inference for Heterogeneous Factorial Effect Modifiers via Residualized Walsh-Hadamard Scores
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∈[p] and ∅≠ S⊆[K], through the coefficients θ∗jS of population linear projections of conditional Walsh--Hadamard contrasts τS onto standardized covariates. Using baseline-robust direct scores, we develop frequentist inference for all p(2K−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 L2 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 2K−1 contrasts are covered jointly.
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BibTeX
@misc{nakamura2026debiased,
title = {{Debiased and Simultaneous Inference for Heterogeneous Factorial Effect Modifiers via Residualized Walsh-Hadamard Scores}},
author = {Tomoshige Nakamura and Ryo Emoto},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37382},
eprint = {2609.37382},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37382},
}
RIS
TY - UNPB
TI - Debiased and Simultaneous Inference for Heterogeneous Factorial Effect Modifiers via Residualized Walsh-Hadamard Scores
AU - Tomoshige Nakamura
AU - Ryo Emoto
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37382
UR - https://arxiv.org/abs/2609.37382
AB - 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∈[p] and ∅≠ S⊆[K], through the coefficients θ^∗_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.
ER -