Detecting confounding in multivariate linear models via spectral analysis
This addresses the issue of confounding bias in causal inference for researchers in statistics and machine learning, representing an incremental advance in spectral methods for detection.
The paper tackles the problem of distinguishing between direct causal influence and hidden confounding in multivariate linear models, and shows that confounding spoils the generic orientation of regression coefficients relative to the covariance eigenspaces in a characteristic way, enabling quantitative estimation of confounding.
We study a model where one target variable Y is correlated with a vector X:=(X_1,...,X_d) of predictor variables being potential causes of Y. We describe a method that infers to what extent the statistical dependences between X and Y are due to the influence of X on Y and to what extent due to a hidden common cause (confounder) of X and Y. The method relies on concentration of measure results for large dimensions d and an independence assumption stating that, in the absence of confounding, the vector of regression coefficients describing the influence of each X on Y typically has `generic orientation' relative to the eigenspaces of the covariance matrix of 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.