Component-Based Regularization of Multivariate Generalized Linear Mixed Models

Abstract

We address the component-based regularization of a multivariate Generalized Linear Mixed Model (GLMM) in the framework of grouped data. A set $\mathit{Y}$ of random responses is modelled with a multivariate GLMM, based on a set $\mathit{X}$ of explanatory variables, a set $\mathit{A}$ of additional explanatory variables, and random effects to introduce the within-group dependence of observations. Variables in $\mathit{X}$ are assumed many and redundant so that regression demands regularization. This is not the case for $\mathit{A}$, which contains few and selected variables. Regularization is performed building an appropriate number of orthogonal components that both contribute to model $\mathit{Y}$ and capture relevant structural information in $\mathit{X}$. To estimate the model, we propose to maximize a criterion specific to the supervised component-based generalized linear regression (SCGLR) within an adaptation of Schall’s algorithm. This extension of SCGLR is tested on both simulated and real grouped data, and compared to ridge and LASSO regularizations. Supplementary material for this article is available online.

Keywords:

• Generalized linear regression
• Supervised components
• Random effects
• Structural relevance

Acknowledgments

The extended data Genus required the arrangement and the inventory of 140,000 developed plots across four countries: Central African Republic, Gabon, Cameroon, and Democratic Republic of Congo. The authors thank the members of the CoForTips project for allowing the use of these data. We are also grateful to the editor, the associate editor, and to the referees for their thorough and constructive review of this work.