Sparse Pairwise Likelihood Estimation for Multivariate Longitudinal Mixed Models

It is becoming increasingly common in longitudinal studies to collect and analyze data on multiple responses. For example, in the social sciences we may be interested in uncovering the factors driving mental health of individuals over time, where mental health is measured using a set of questionnaire items. One approach to analyzing such multi-dimensional data is multivariate mixed models, an extension of the standard univariate mixed model to handle multiple responses. Estimating multivariate mixed models presents a considerable challenge however, let alone performing variable selection to uncover which covariates are important in driving each response. Motivated by composite likelihood ideas, we propose a new approach for estimation and fixed effects selection in multivariate mixed models, called approximate pairwise likelihood estimation and shrinkage (APLES). The method works by constructing a quadratic approximation to each term in the pairwise likelihood function, and then augmenting this approximate pairwise likelihood with a penalty that encourages both individual and group coefficient sparsity. This leads to a relatively fast method of selection, as we can use coordinate ascent type methods to then construct the full regularization path for the model. Our method is the first to extend penalized likelihood estimation to multivariate generalized linear mixed models. We show that the APLES estimator attains a composite likelihood version of the oracle property. We propose a new information criterion for selecting the tuning parameter, which employs a dynamic model complexity penalty to facilitate aggressive shrinkage, and demonstrate that it asymptotically leads to selection consistency, that is, leads to the true model being selected. A simulation study demonstrates that the APLES estimator outperforms several univariate selection methods based on analyzing each outcome separately. Supplementary materials for this article are available online.