Identifiability of covariance parameters in linear mixed effects models

Building a linear mixed model often involves selection of the parametrized covariance matrix structures for the random components of the model. Parameters in the covariance matrix of the response then consist of those from the random effects and from the random residual error. However, some specifications of the structures can result in the parameters not identifiable, even if the model is not over-parametrized. Software output can look normal with no indication of error when fitting non-identifiable models. In our simulation studies, we found no implication of model non-identifiable about half of the times. We derive model identifiability conditions which only rely on properties of the known design matrix associated with the random effects and the specific structures being used. The results can be applied to study identifiability for commonly used covariance matrix structures.