Confounder Detection in High Dimensional Linear Models using First Moments of Spectral Measures

Abstract

In this paper, we study the confounder detection problem in the linear model, where the target variable $Y$ is predicted using its $n$ potential causes $X_n=(x_1,...,x_n)^T$. Based on an assumption of rotation invariant generating process of the model, recent study shows that the spectral measure induced by the regression coefficient vector with respect to the covariance matrix of $X_n$ is close to a uniform measure in purely causal cases, but it differs from a uniform measure characteristically in the presence of a scalar confounder. Then, analyzing spectral measure pattern could help to detect confounding. In this paper, we propose to use the first moment of the spectral measure for confounder detection. We calculate the first moment of the regression vector induced spectral measure, and compare it with the first moment of a uniform spectral measure, both defined with respect to the covariance matrix of $X_n$. The two moments coincide in non-confounding cases, and differ from each other in the presence of confounding. This statistical causal-confounding asymmetry can be used for confounder detection. Without the need of analyzing the spectral measure pattern, our method does avoid the difficulty of metric choice and multiple parameter optimization. Experiments on synthetic and real data show the performance of this method.