Measurement Error in Lasso: Impact and Correction

TL;DR

This paper investigates how measurement error affects lasso regression, proposes a correction method, and demonstrates improved covariate selection accuracy in high-dimensional genomic data through theoretical analysis and simulations.

Contribution

It introduces a simple correction method for measurement error in lasso, showing improved covariate selection consistency and fewer false positives in high-dimensional settings.

Findings

Corrected lasso achieves sign consistency under similar conditions as perfect measurement.

Uncorrected lasso requires stricter conditions for covariate selection.

Corrected lasso reduces false positives in genomic classification tasks.

Abstract

Regression with the lasso penalty is a popular tool for performing dimension reduction when the number of covariates is large. In many applications of the lasso, like in genomics, covariates are subject to measurement error. We study the impact of measurement error on linear regression with the lasso penalty, both analytically and in simulation experiments. A simple method of correction for measurement error in the lasso is then considered. In the large sample limit, the corrected lasso yields sign consistent covariate selection under conditions very similar to the lasso with perfect measurements, whereas the uncorrected lasso requires much more stringent conditions on the covariance structure of the data. Finally, we suggest methods to correct for measurement error in generalized linear models with the lasso penalty, which we study empirically in simulation experiments with logistic regression, and also apply to a classification problem with microarray data. We see that the corrected lasso selects less false positives than the standard lasso, at a similar level of true positives. The corrected lasso can therefore be used to obtain more conservative covariate selection in genomic analysis.