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

TL;DR

This paper derives the distribution of maximum spurious correlations in high-dimensional data, providing tools to assess the validity of variable discoveries and test covariate exogeneity.

Contribution

It introduces a Gaussian approximation and bootstrap method to estimate the distribution of maximum spurious correlations, enabling validation of variable selection results.

Findings

Derived distributions for maximum spurious correlations under certain conditions

Proposed a bootstrap procedure for distribution approximation and validation

Applied methods to real data for testing covariate exogeneity

Abstract

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 from these data mining approaches be spurious due to high dimensionality and limited sample size? Can our fundamental assumptions about the exogeneity of the 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 a 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 a finite $s$ and a diverging $s$, using Gaussian approximation and empirical process techniques. However, such a distribution depends on the unknown covariance matrix of $\mathbf{X}$. Hence, we use the multiplier bootstrap procedure to approximate the unknown distributions and establish the consistency of such a simple bootstrap approach. The results are further extended to the situation where the residuals are from regularized fits. Our approach is then used to construct the upper confidence limit for the maximum spurious correlation and to test the exogeneity of the covariates. The former provides a baseline for guarding against false discoveries and the latter tests whether our fundamental assumptions for high-dimensional model selection are statistically valid. Our techniques and results are illustrated with both numerical examples and real data analysis.