Linear mixed models (LMMs) are widely used to analyze repeated, longitudinal, or clustered data in many disciplines, such as biology, medicine, psychology, sociology, economics, etc. One of the essential components of a linear mixed model is its covariance structure, i.e., the covariance matrices of the random part and the error part, respectively, and the relationship between them. Specifying the covariance structure can be very challenging in a given situation. An important issue concerns the best linear unbiased estimators (BLUE) and the best linear unbiased predictors (BLUPs) under a given LMM and if two LMMs with different covariance structures generate equal BLUEs and/or BLUPs.

This thesis proposes a new approach to determine the equality of the BLUEs and the BLUPs of the fixed and random effects under two linear mixed models with different covariance structures, and we present straightforward criteria to evaluate conditions that lead to equal BLUEs and BLUPs under these two models.  

Papers I and II focus on the equality of the BLUEs, while Papers II and IV study equal BLUPs under two LMMs with different covariance structures. More specifically, Paper I considers two different LMMs concerning their covariance matrices and whether the models generate equal BLUEs. In Paper II, we study the conditions for invariant BLUEs under a partitioned linear fixed effects model and a corresponding linear mixed model. Paper III develops the results to obtain the common BLUPs under two LMMs considering uncorrelated random effects and random errors. Finally, Paper IV extends the results of Paper III to obtain equal BLUPs in two linear mixed models, allowing for a correlation between the random effects and the random errors.