Are Discoveries Spurious? Distributions of Maximum Spurious Correlations and Their Applications

Over the last two decades, many exciting variable selection methods have been developed for finding a small group of covariates that are associated with the response from a large pool. Can the discoveries by such data mining approaches be spurious due to high-dimensionality and limited sample size? Can our fundamental assumptions on exogeneity of covariates needed for such variable selection be validated with the data? To answer these questions, we need to derive the distributions of the maximum spurious correlations given certain number of predictors, namely, the distribution of the correlation of a response variable $Y$ with the best $s$ linear combinations of $p$ covariates $\mathbf{X}$, even when $\mathbf{X}$ and $Y$ are independent. When the covariance matrix of $\mathbf{X}$ possesses the restricted eigenvalue property, we derive such distributions for both finite $s$ and diverging $s$, using Gaussian approximation and empirical process techniques. However, such a distribution depends on the unknown covariance matrix of $\mathbf{X}$. Hence, we propose a multiplier bootstrap method to approximate the unknown distributions and establish the consistency of such a simple bootstrap approach. The results are further extended to the situation where residuals are from regularized fits. Our approach is then applied to construct the upper confidence limit for the maximum spurious correlation and testing exogeneity of covariates. The former provides a baseline for guiding false discoveries due to data mining and the latter tests whether our fundamental assumptions for high-dimensional model selection are statistically valid. Our techniques and results are illustrated by both numerical examples.