Abstract

Linear mixed models (LMMs) extend fixed effects linear models by allowing both fixed and random effects. LMMs are an important class of statistical models that can be used to analyze, e.g., clustered, panel, and longitudinal data with complex structures and various sources of variations.

The covariance matrix structure is of utmost importance for the inference about LMM. However, determining the correct covariance matrix can be challenging, even in some relatively simple situations. Hence, extensive research has focused on finding covariance matrices that lead to equivalent best linear unbiased estimators (BLUEs) and best linear unbiased predictors (BLUPs) for the fixed and random effects, respectively.

The thesis derives necessary and sufficient conditions for obtaining equal BLUEs and BLUPs under two LMMs with different covariance structures. These conditions are derived via a system of matrix equations. It has been shown that the consistency conditions for these systems of matrix equations are connected to the existing necessary and sufficient conditions for obtaining equal BLUEs and BLUPs under two LMMs with different covariance structures derived by using different approaches.