Causal inference for group-contaminated structured outcomes: observable quotients, lossless reduction and exact randomization inference

2026-08-12

AI Summary

This study addresses the challenge that structured latent outcomes—such as microscopy images—are often confounded by unknown unit-specific transformations that may depend on treatment assignment, covariates, or intrinsic outcomes, thereby entangling biological effects with acquisition geometry in conventional analyses. To resolve this, the authors model observed data through the lens of group actions and introduce the “Quotient Faithful Reconstruction Theorem,” which rigorously distinguishes observability from statistical sufficiency. They construct maximal invariants tailored to finite-support, multi-channel lattice images, circumventing the loss of cross-location relative information caused by component-wise normalization. Integrating lossless quotient reduction with exact randomization inference, the method achieves a type I error rate of 0.052 and power of 0.992 in simulations, and yields a primary contrast p-value of 0.0078 in the RxRx1 HUVEC study.

Abstract

Structured potential outcomes such as microscopy images may be recorded after an unknown, unit-specific transformation. If that transformation can depend on treatment, covariates or the intrinsic outcome, raw-coordinate analyses may mix biological effects with acquisition geometry. We study the unrestricted observation model X = Γ . Y(A) and characterize its observable information: a target is uniformly recoverable exactly when it is constant on group orbits, while a Borel maximal invariant retains every measurable invariant target. We then distinguish observability from statistical losslessness. A quotient-faithful reconstruction theorem shows that quotient reduction is sufficient for the full transformed experiment exactly when the conditional law of the raw observation given treatment, covariates and the quotient has a parameter-free version. Conditional Haar contamination on a compact group yields Blackwell equivalence as a special case; it is not imposed in the main model. We also separate independent site-specific product actions from shared diagonal actions and show why componentwise canonicalization can discard relative cross-site information. Under explicit metric and kernel regularity, an approximate-contamination theorem bounds quotient-law Wasserstein error and the induced perturbation of population maximum mean discrepancy. For finite-support multichannel lattice images, we construct a maximal invariant under integer translations and quarter turns, combine its characteristic Gaussian kernel with a complete paired-swap test, and retain the original simulations and RxRx1 HUVEC study. Under the sharp null, the quotient test rejected in 0.052 of simulation replicates; at unit effect strength its power was 0.992. The primary RxRx1 contrast had an enumerated paired-swap p-value of 0.0078.

Problem

• causal inference
• structured outcomes
• group contamination
• observability
• invariance

Innovation

• causal inference
• group invariance
• quotient reduction
• maximal invariant
• randomization test