Uniformly Valid Inference Based on the Lasso in Linear Mixed Models

TL;DR

This paper develops a method for constructing valid confidence sets for fixed effects in Gaussian linear mixed models using Lasso estimators, ensuring uniform validity over parameters and accounting for variable selection uncertainty.

Contribution

It introduces a novel approach for uniformly valid inference on fixed effects in LMMs with Lasso, including confidence sets and a proof of uniform consistency for REML estimators.

Findings

Confidence sets outperform naive post-selection methods.

Method validated through simulations and real lake data study.

Provides uniform validity over parameter spaces.

Abstract

Linear mixed models (LMMs) are suitable for clustered data and are common in biometrics, medicine, survey statistics and many other fields. In those applications, it is essential to carry out valid inference after selecting a subset of the available variables. We construct confidence sets for the fixed effects in Gaussian LMMs that are based on Lasso-type estimators. Aside from providing confidence regions, this also allows to quantify the joint uncertainty of both variable selection and parameter estimation in the procedure. To show that the resulting confidence sets for the fixed effects are uniformly valid over the parameter spaces of both the regression coefficients and the covariance parameters, we also prove the novel result on uniform Cramer consistency of the restricted maximum likelihood (REML) estimators of the covariance parameters. The superiority of the constructed confidence sets to naive post-selection procedures is validated in simulations and illustrated with a study of the acid neutralization capacity of lakes in the United States.