Gradient Boosted Mixed Models: Flexible Estimation of Mean and Variance Components for Clustered Data

Abstract

We introduce Gradient Boosted Mixed Models (GBMixed), a framework which extends boosting to clustered data by jointly modeling the mean and variance components in a linear mixed model via likelihood-based gradients. GBMixed estimates a nonparametric fixed effects function characterizing the overall mean of the response, while also allowing the random effects covariance matrix along with the residual variance to depend on covariates in a flexible manner. We demonstrate how GBMixed facilitates covariate-dependent random effect predictions, and subsequently point predictions and prediction intervals for individual treatment effects, that can adapt between population-level and cluster-level information. Simulations and applications to two real-world datasets demonstrate that GBMixed can accurately recover complex nonlinear fixed effect functions and covariate-dependent covariances in a linear mixed model, while also improving point and probabilistic predictive performance compared with several existing approaches such as parametric linear mixed models, Natural Gradient Boosting, and Gaussian Process Boosting.