Doubly debiased lasso: High-dimensional inference under hidden confounding

Abstract

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 DDL package available from CRAN.

A Decorrelating and Debiasing Approach to Simultaneous Inference for High-Dimensional Confounded Models

“…Then F T Y is regressed on F T X through Lasso procedure to estimate the sparse coefficient β. Next, to reduce bias and perform inference on a single component β j , Guo et al (2022) applies the Trim transform as well as the debiased Lasso method in Zhang and Zhang (2014) to construct an asymptotically unbiased estimator of β j . Nevertheless, the dependence structure among the covariates after such transformation is unknown and the predictors may still be highly correlated.…”

“…It will be shown in Section 3.4 that any initial estimator β that satisfies Assumption 6 can reach the desired theoretical properties and hence can be employed in the construction of β j . As an example, we apply the Trim transform in (7) together with the Lasso procedure to obtain a biased initial estimator β ( Ćevid et al, 2020;Guo et al, 2022).…”

“…It is worthwhile to note that, the asymptotic distribution of β j is not conditioning on any observed data, and the dependency structure of null T j 's can be approximated through the correlations among η T j ξ's. In comparison, Guo et al (2022) derives some conditional asymptotic distribution of their debiased estimator based on additional Gaussian noise assumption, and it is unclear whether the multiple testing procedure based on their statistics can provably control the error rates.…”

“…For example, to identify differentially expressed genes with respect to different disease states, confounding may occur in the form of batch effects and the confounders may include processing dates and other unrecorded laboratory conditions (Leek et al, 2010). On the other hand, the confounded model may appear in the forms of perturbed model, measurement error model and linear structural equation model as reported and discussed in Guo et al (2022). As an example, to quantify the effects of particulate matters and gaseous pollutants on the human mortality rate, measurement errors of some pollutants may serve as additional confounding effects (Schwartz and Coull, 2003).…”

“…Based on Model (1), we further adopt the factor model structure to characterize the dependency between the predictors and the confounders ( Ćevid et al, 2020;Guo et al, 2022), i.e.,…”