Gaussian variational Bayesian inference for mixed effect quantile regression model with grouped data

Grouped data arises frequently across various research domains when observations are clustered within higher-level units creating analytical challenges due to within-group correlation. Mixed effects quantile regression model offers a flexible approach for modelling such data especially when responses exhibit non-standard distributional features. Variable selection becomes essential when many covariates are available. The Bayesian spike-and-slab Lasso prior serves as a useful tool for variable selection, yet traditional Markov Chain Monte Carlo algorithms suffer from high computational demands, particularly in high-dimensional settings. This paper develops a Gaussian variational Bayesian inference framework for mixed effects quantile regression model with variable selection. Our approach combines a structured Gaussian variational approximation with a parameter reduction strategy through covariance matrix factorization and low-rank decomposition. An efficient stochastic optimization procedure using reparameterization gradients is implemented for evidence lower bound maximization. Simulation studies and real data analysis demonstrate that the proposed method achieves superior computational efficiency and statistical performance compared to existing methods offering a scalable and accurate alternative for analyzing complex grouped data.