Inferring causal relationships or related associations from observational data can be invalidated by the existence of hidden confounding. We focus on a high-dimensional linear regression setting, where the measured covariates are affected by hidden confounding and propose the doubly debiased lasso estimator for individual components of the regression coefficient vector. Our advocated method simultaneously corrects both the bias due to estimation of high-dimensional parameters as well as the bias caused by the hidden confounding. We establish its asymptotic normality and also prove that it is efficient in the Gauss–Markov sense. The validity of our methodology relies on a dense confounding assumption, that is, that every confounding variable affects many covariates. The finite sample performance is illustrated with an extensive simulation study and a genomic application. The method is implemented by the package available from CRAN.
The research of Z. Guo was supported in part by the NSF Grants DMS-1811857, 2015373 and NIH-1R01GM140463-01; Z. Guo also acknowledges financial support for visiting the Institute of Mathematical Research (FIM) at ETH Zurich.
D. Ćevid and P. Bühlmann received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 786461).
We thank Yuansi Chen for providing the code to preprocess the raw data from the GTEx project. We also thank Matthias Löffler for his help and useful discussions about random matrix theory.
Z. Guo and D. Ćevid contributed equally to this work.
"Doubly debiased lasso: High-dimensional inference under hidden confounding." Ann. Statist. 50 (3) 1320 - 1347, June 2022. https://doi.org/10.1214/21-AOS2152