Skip to main content
Log in

The link between the mixed and fixed linear models revisited

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript


Haslett and Puntanen (Stat Pap 51:465–475, 2010) studied the links between the linear mixed model and a particular extended linear model including only fixed effects. This paper is a follow-up article to that paper clarifying some central concepts appearing therein and allowing the random effect and error term to be correlated. We also show an interesting connection between the big extended model with fixed effects and a particular mixed model obtained by transforming the extended model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  • Arendacká B, Puntanen S (2014) Further remarks on the connection between fixed linear model and mixed linear model (in press)

  • Baksalary JK, Rao CR, Markiewicz A (1992) A study of the influence of the “natural restrictions” on estimation problems in the singular Gauss-Markov model. J Stat Plan Inf 31:335–351

    Article  MathSciNet  MATH  Google Scholar 

  • Christensen R (2011) Plane answers to complex questions: the theory of linear models, 4th edn. Springer, New York

  • Groß J (2004) The general Gauss–Markov model with possibly singular dispersion matrix. Stat Pap 45:311–336

    Article  MATH  Google Scholar 

  • Haslett SJ, Puntanen S (2010) Equality of BLUEs or BLUPs under two linear models using stochastic restrictions. Stat Pap 51:465–475

    Article  MathSciNet  MATH  Google Scholar 

  • Henderson CR (1950) Estimation of genetic parameters. Ann Math Stat 21:309–310

    Google Scholar 

  • Henderson CR (1963) Selection index and expected genetic advance. Statistical genetics and plant breeding. National Academy of Sciences - National Research Council Publication No. 982, pp 141–163

  • Henderson CR, Kempthorne O, Searle SR, von Krosigh CN (1959) The estimation of environmental and genetic trends from records subject to culling. Biometrics 15:192–218

    Article  Google Scholar 

  • Herzberg A, Vere-Jones D (1990) Some notes on design sensitivity to changes in covariance structure. Util Math 3:13–27

    MathSciNet  Google Scholar 

  • Isotalo J, Puntanen S (2006) Linear prediction sufficiency for new observations in the general Gauss-Markov model. Commun Stat Theory Methods 35:1011–1023

    Article  MathSciNet  MATH  Google Scholar 

  • Liu X, Wang Q-W (2013) Equality of the BLUPs under the mixed linear model when random components and errors are correlated. J Multivar Anal 116:297–309

    Article  Google Scholar 

  • McCulloch CE, Searle SR, Neuhaus JM (2008) Generalized, linear, and mixed models, 2nd edn. Wiley, New York

  • Puntanen S, Styan GPH, Isotalo J (2011) Matrix tricks for linear statistical models: our personal top twenty. Springer, Heidelberg

    Book  Google Scholar 

  • Rao CR (1967) Least squares theory using an estimated dispersion matrix and its application to measurement of signals. In: Le Cam LM, Neyman J (eds) Proceedings of the fifth Berkeley symposium on mathematical statistics and probability: Berkeley, California, 1965/1966, vol 1. University of California Press, Berkeley, pp 355–372

    Google Scholar 

  • Robinson GK (1991) That BLUP is a good thing: the estimation of random effects (discussion: pp. 32–51). Stat Sci 6:15–51

    Article  MATH  Google Scholar 

  • Zyskind G (1967) On canonical forms, non-negative covariance matrices and best and simple least squares linear estimators in linear models. Ann Math Stat 38:1092–1109

    Article  MathSciNet  MATH  Google Scholar 

Download references


Thanks go the anonymous referee for helpful remarks.

Author information

Authors and Affiliations


Corresponding author

Correspondence to B. Arendacká.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Haslett, S.J., Puntanen, S. & Arendacká, B. The link between the mixed and fixed linear models revisited. Stat Papers 56, 849–861 (2015).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification