GLMMLasso: An Algorithm for High-Dimensional Generalized Linear Mixed Models Using ℓ₁-Penalization

Abstract

We propose an ℓ₁-penalized algorithm for fitting high-dimensional generalized linear mixed models (GLMMs). GLMMs can be viewed as an extension of generalized linear models for clustered observations. Our Lasso-type approach for GLMMs should be mainly used as variable screening method to reduce the number of variables below the sample size. We then suggest a refitting by maximum likelihood based on the selected variables only. This is an effective correction to overcome problems stemming from the variable screening procedure that are more severe with GLMMs than for generalized linear models. We illustrate the performance of our algorithm on simulated as well as on real data examples. Supplementary materials are available online and the algorithm is implemented in the R package glmmixedlasso.

Key Words:

• Coordinate gradient descent
• Laplace approximation
• Random-effects model
• Variable selection

SUPPLEMENTARY MATERIALS

Appendices: Details of the PIRLS algorithm, the comparison of the exact and approximate GLMMLasso algorithms, and additional simulation studies. (glmmlasso_sm.pdf)

Dataset: The extended epilepsy dataset used in Section 6. (epilepsy.txt)

R-package for GLMMLasso: R-package glmmixedlasso containing code to perform the GLMMLasso algorithm. (glmmixedlasso-0.1-2.tar.gz)

ACKNOWLEDGMENTS

The research is supported in part by the Swiss National Science Foundation (grant no. 20PA21-120043/1, “Forschergruppe FOR 916”). The authors thank the members of the DFG-SNF Forschergruppe 916 for many stimulating discussions. In particular, we thank Stephan Dlugosz from the ZEW Mannheim for insisting on studying this particular kind of problem. Moreover, the authors thank the associate editor and two referees for their helpful comments.