# Component-based regularisation of multivariate generalised linear mixed models

## TL;DR

This paper introduces a component-based regularisation method for multivariate GLMMs that effectively handles high-dimensional, redundant explanatory variables by constructing orthogonal components, improving model estimation for grouped data.

## Contribution

It proposes an extension of SCGLR for regularising multivariate GLMMs, combining component construction with an adapted estimation algorithm, tested on simulated and real data.

## Key findings

- The method outperforms ridge and LASSO regularisations in grouped data scenarios.
- Orthogonal components effectively capture relevant structural information.
- The approach is validated on both simulated and real datasets.

## Abstract

We address the component-based regularisation of a multivariate Generalised Linear Mixed Model (GLMM) in the framework of grouped data. A set Y of random responses is modelled with a multivariate GLMM, based on a set X of explanatory variables, a set A of additional explanatory variables, and random effects to introduce the within-group dependence of observations. Variables in X are assumed many and redundant so that regression demands regularisation. This is not the case for A, which contains few and selected variables. Regularisation is performed building an appropriate number of orthogonal components that both contribute to model Y and capture relevant structural information in X. To estimate the model, we propose to maximise a criterion specific to the Supervised Component-based Generalised 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 regularisations. Supplementary material for this article is available online.