Identifiability of linear mixed effects models

Abstract

In linear mixed efiects models, the covariance matrix of the response is modeled as the sum of two matrices: the product of the covariance matrix of the random efiects with the associated design matrix, and the covariance matrix of the residual error. Building a linear mixed model usually involves selection of the parametrized covariance matrix structures for the random efiects and the residual error. However, even if the covariance matrix of the response is not over-parametrized, some specifications of covariance structures can result in the non-identifiability of parameters. When fitting such models, software may or may not indicate a problem with model identifiability. Consequently, it is useful to have a way to check if a model is identifiable which does not rely on the software output. We derive conditions for identifiability of the covariance parameters of the response and study commonly used covariance structures. The derived conditions only rely on the covariance structures being used and properties of the design matrix associated with the random efiects and are easy to check.