Covariate Selection in High-Dimensional Generalized Linear Models With Measurement Error

TL;DR

This paper introduces novel correction methods for covariate selection in high-dimensional generalized linear models with measurement error, improving accuracy and reducing false positives without needing error covariance estimates.

Contribution

It proposes a generalized matrix uncertainty selector and an alternative lasso-based method that effectively handle measurement error in high-dimensional GLMs, with efficient algorithms and superior performance.

Findings

Proposed methods outperform standard lasso and Dantzig selector in simulations.

Significant reduction in false positives in covariate selection.

Effective in logistic and Poisson regression with noisy data.

Abstract

In many problems involving generalized linear models, the covariates are subject to measurement error. When the number of covariates p exceeds the sample size n, regularized methods like the lasso or Dantzig selector are required. Several recent papers have studied methods which correct for measurement error in the lasso or Dantzig selector for linear models in the p>n setting. We study a correction for generalized linear models based on Rosenbaum and Tsybakov's matrix uncertainty selector. By not requiring an estimate of the measurement error covariance matrix, this generalized matrix uncertainty selector has a great practical advantage in problems involving high-dimensional data. We further derive an alternative method based on the lasso, and develop efficient algorithms for both methods. In our simulation studies of logistic and Poisson regression with measurement error, the proposed methods outperform the standard lasso and Dantzig selector with respect to covariate selection, by reducing the number of false positives considerably. We also consider classification of patients on the basis of gene expression data with noisy measurements.