Detecting Confounding in Multivariate Linear Models via Spectral Analysis

Abstract

Abstract We study a model where one target variable Y $Y$ is correlated with a vector X := ( X 1 , … , X d ) $\textbf{X}:=(X_1,\dots,X_d)$ of predictor variables being potential causes of Y $Y$ . We describe a method that infers to what extent the statistical dependences between X $\textbf{X}$ and Y $Y$ are due to the influence of X $\textbf{X}$ on Y $Y$ and to what extent due to a hidden common cause (confounder) of X $\textbf{X}$ and Y $Y$ . The method relies on concentration of measure results for large dimensions d $d$ and an independence assumption stating that, in the absence of confounding, the vector of regression coefficients describing the influence of each X $\textbf{X}$ on Y $Y$ typically has ‘generic orientation’ relative to the eigenspaces of the covariance matrix of X $\textbf{X}$ . For the special case of a scalar confounder we show that confounding typically spoils this generic orientation in a characteristic way that can be used to quantitatively estimate the amount of confounding (subject to our idealized model assumptions).