Simulation-Selection-Extrapolation: Estimation in High-Dimensional Errors-in-Variables Models

Abstract: This paper considers errors-in-variables models in a high-dimensional setting where the number of covariates can be much larger than the sample size, and there are only a small number of non-zero covariates. The presence of measurement error in the covariates can result in severely biased parameter estimates, and also affects the ability of penalized methods such as the lasso to recover the true sparsity pattern. A new estimation procedure called SIMSELEX (SIMulation-SELection-EXtrapolation) is proposed. This procedure augments the traditional SIMEX approach with a variable selection step based on the group lasso. The SIMSELEX estimator is shown to perform well in variable selection, and has significantly lower estimation error than naive estimators that ignore measurement error. SIMSELEX can be applied in a variety of errors-in-variables settings, including linear models, generalized linear models, and Cox survival models. It is furthermore shown how SIMSELEX can be applied to spline-based regression models. SIMSELEX estimators are compared to the corrected lasso and the conic programming estimator for a linear model, and to the conditional scores lasso for a logistic regression model. Finally, the method is used to analyze a microarray dataset that contains gene expression measurements of favorable histology Wilms tumors.