Further remarks on the connection between fixed linear model and mixed linear model

The linear mixed model $$\mathbf {y}= \mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}+ \varvec{\varepsilon }$$ y = X β + Z u + ε , where $$\varvec{\beta }$$ β is the vector of fixed effects and $$\mathbf {u}$$ u the vector of random effects, has strong links with a particular augmented linear model including only fixed effects. This goes back to Henderson’s mixed model equations and was recently exploited by Haslett and Puntanen (Stat Papers 51:465–475, 2010 ), who restated Henderson’s result in a model with possibly singular covariance matrices. In this paper we point out that the connection between the two models is actually very straightforward: a mixed linear model can be obtained from the augmented model by a simple linear transformation. This, for example, immediately opens up a new viewpoint for studying the relationship between the BLUEs and BLUPs in the two models. In doing so we discuss a modification of the Frisch–Waugh–Lovell theorem as well as some not so well established theorems about the BLUPs of random errors. Copyright Springer-Verlag Berlin Heidelberg 2015