Assumption-Lean Post-Integrated Inference with Negative Control Outcomes

Data integration methods aim to extract low-dimensional embeddings from high-dimensional outcomes to remove unwanted variations, such as batch effects and unmeasured covariates, across heterogeneous datasets. However, multiple hypothesis testing after integration can be biased due to data-dependent processes. We introduce a robust post-integrated inference (PII) method that adjusts for latent heterogeneity using negative control outcomes. Leveraging causal interpretations, we derive nonparametric identifiability of the direct effects, which motivates our semiparametric inference method. Our method extends to projected direct effect estimands, accounting for hidden mediators, confounders, and moderators. These estimands remain statistically meaningful under model misspecifications and with error-prone embeddings. We provide bias quantifications and finite-sample linear expansions with uniform concentration bounds. The proposed doubly robust estimators are consistent and efficient under minimal assumptions and potential misspecification, facilitating data-adaptive estimation with machine learning algorithms. Our proposal is evaluated with random forests through simulations and analysis of single-cell CRISPR perturbed datasets with potential unmeasured confounders.