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

B. Arendacká () and S. Puntanen ()

Statistical Papers, 2015, vol. 56, issue 4, 1235-1247

Abstract: 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

Keywords: Best linear unbiased estimator; Best linear unbiased predictor; Linear mixed model; Linear fixed effects model; Henderson’s mixed model equation; Frisch–Waugh–Lovell theorem; 62J05; 62J10 (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (5)

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DOI: 10.1007/s00362-014-0634-2

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