Assumption-Lean Post-Integrated Inference with Negative Control Outcomes

Data integration has become increasingly common in aligning multiple heterogeneous datasets. With high-dimensional outcomes, data integration methods aim to extract low-dimensional embeddings of observations to remove unwanted variations, such as batch effects and unmeasured covariates, inherent in data collected from different sources. However, multiple hypothesis testing after data integration can be substantially biased due to the data-dependent integration processes. To address this challenge, we introduce a robust post-integrated inference (PII) method that adjusts for latent heterogeneity using negative control outcomes. By leveraging causal interpretations, we derive nonparametric identification conditions that form the basis of our PII approach. Our assumption-lean semiparametric inference method extends robustness and generality to projected direct effect estimands that account for mediators, confounders, and moderators. These estimands remain statistically meaningful under model misspecifications and with error-prone embeddings. We provide deterministic quantifications of the bias of target estimands induced by estimated embeddings and finite-sample linear expansions of the estimators with uniform concentration bounds on the residuals for all outcomes. The proposed doubly robust estimators are consistent and efficient under minimal assumptions, facilitating data-adaptive estimation with machine learning algorithms. Using random forests, we evaluate empirical statistical errors in simulations and analyze single-cell CRISPR perturbed datasets with potential unmeasured confounders.