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Shrinkage estimation in linear mixed models for longitudinal data



This paper is concerned with the selection and estimation of fixed effects in linear mixed models while the random effects are treated as nuisance parameters. We propose the non-penalty James–Stein shrinkage and pretest estimation methods based on linear mixed models for longitudinal data when some of the fixed effect parameters are under a linear restriction. We establish the asymptotic distributional biases and risks of the proposed estimators, and investigate their relative performance with respect to the unrestricted maximum likelihood estimator (UE). Furthermore, we investigate the penalty (LASSO and adaptive LASSO) estimation methods and compare their relative performance with the non-penalty pretest and shrinkage estimators. A simulation study for various combinations of the inactive covariates shows that the shrinkage estimators perform better than the penalty estimators in certain parts of the parameter space. This particularly happens when there are many inactive covariates in the model. It also shows that the pretest, shrinkage, and penalty estimators all outperform the UE. We further illustrate the proposed procedures via a real data example.


Asymptotic distributional bias and risk Linear mixed model Likelihood ratio test LASSO Monte Carlo simulation Shrinkage and pretest estimators 



We would like to thank the referees, editor and associate editor for their valuable suggestions in the revision of this paper. The research of Shakhawat Hossain and Ejaz Ahmed was supported by the Natural Sciences and the Engineering Research Council of Canada.


  1. Ahmed SE, Rohatgi VJ (1996) Shrinkage estimation in a randomized response model. Metrika 43:17–30MathSciNetCrossRefMATHGoogle Scholar
  2. Ahmed SE, Hussein A, Sen PK (2006) Risk comparison of some shrinkage M-estimators in linear models. J Nonparametr Stat 18(3):401–415MathSciNetCrossRefMATHGoogle Scholar
  3. Ahmed SE, Hossain S, Doksum KA (2012) LASSO and shrinkage estimation in Weibull censored regression models. J Stat Plan Inference 142(6):1273–1284MathSciNetCrossRefMATHGoogle Scholar
  4. Ahn M, Zhang HH, Lu W (2012) Moment-based method for random effects selection in linear mixed models. Stat Sin 22(4):1539–1562MathSciNetMATHGoogle Scholar
  5. Bondell HD, Krishna A, Ghosh SK (2010) Joint variable selection for fixed and random effects in linear mixed-effects models. Biometrics 66(4):1069–1077MathSciNetCrossRefMATHGoogle Scholar
  6. Datta G, Ghosh M (2012) Small area shrinkage estimation. Stat Sci 27(1):95–114MathSciNetCrossRefMATHGoogle Scholar
  7. Diggle PJ, Heagerty P, Liang KY, Zeger SL (2002) Analysis of longitudinal data. Oxford University Press, OxfordMATHGoogle Scholar
  8. Fan J, Li R (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. J Am Stat Assoc 96(456):1348–1360MathSciNetCrossRefMATHGoogle Scholar
  9. Fan YY, Li RZ (2012) Variable selection in linear mixed effects models. Ann Stat 40(4):2043–2068MathSciNetCrossRefMATHGoogle Scholar
  10. Fitzmaurice GM, Laird NM, Ware JH (2011) Longitudinal data analysis. Wiley, New YorkMATHGoogle Scholar
  11. Gao X, Ahmed SE, Fen Y (2017) Post selection shrinkage estimation for high-dimensional data analysis. Appl Stoch Models Bus Ind 33:97–120MathSciNetGoogle Scholar
  12. Gumedze FN, Dunne TT (2011) Parameter estimation and inference in the linear mixed model. Linear Algebra Appl 435(8):1920–1944MathSciNetCrossRefMATHGoogle Scholar
  13. Gupta A, Saleh AKME, Sen PK (1989) Improved estimation in a contingency table: independence structure. J Am Stat Assoc 84(406):525–532MathSciNetCrossRefMATHGoogle Scholar
  14. Harville D (1977) Maximum likelihood approaches to variance component estimation and to related problems. J Am Stat Assoc 358:320–340MathSciNetCrossRefMATHGoogle Scholar
  15. Hedeker D, Gibbons RD (2006) Longitudinal data analysis. Wiley, New YorkMATHGoogle Scholar
  16. Hossain S, Doksum KA, Ahmed SE (2009) Positive-part shrinkage and absolute penalty estimators in partially linear models. Linear Algebra Appl 430:2749–2761MathSciNetCrossRefMATHGoogle Scholar
  17. Hossain S, Ahmed SE, Doksum KA (2015) Shrinkage, \(l_1\) penalty and pretest estimators for generalized linear models. Stat Methodol 24:52–68MathSciNetCrossRefGoogle Scholar
  18. Hossain S, Ahmed SE, Yi Y, Chen B (2016) Shrinkage and pretest estimators for longitudinal data analysis under partially linear models. J Nonparametr Stat 28(3):531–549MathSciNetCrossRefMATHGoogle Scholar
  19. Ibrahim JG, Zhu HT, Garcia R, Guo RX (2011) Fixed and random effects selection in mixed effects models. Biometrics 67(2):495–503MathSciNetCrossRefMATHGoogle Scholar
  20. Judge GG, Bock ME (1978) The statistical implication of pretest and Stein-rule estimators in econometrics. North-Holland, AmsterdamMATHGoogle Scholar
  21. Judge GG, Mittelhammaer RC (2004) A semiparametric basis for combining estimation problem under quadratic loss. J Am Stat Assoc 99(466):479–487MathSciNetCrossRefMATHGoogle Scholar
  22. Kemper HCG (1995) The Amsterdam Growth Study: a longitudinal analysis of health, fitness and lifestyle. HK Sport Science Monograph Series, vol 6. Human Kinetics Publishers, ChampaignGoogle Scholar
  23. Laird NM, Ware JH (1982) Random-effects models for longitudinal data. Biometrics 38(4):963–974CrossRefMATHGoogle Scholar
  24. Lindstrom MJ, Bates DM (1988) Newton–Raphson and EM algorithms for linear mixed-effects models for repeated measures data. J Am Stat Assoc 83(404):1014–1022MathSciNetMATHGoogle Scholar
  25. Longford NT (1993) Random coefficient models. Oxford University Press, OxfordMATHGoogle Scholar
  26. McCulloch CE, Searle SR, Neuhaus JR (2008) Generalized, linear, and mixed models, 2nd edn. Wiley, Hoboken, New JerseyMATHGoogle Scholar
  27. Müller S, Scealy JL, Welsh AH (2013) Model selection in linear mixed models. Stat Sci 28(2):135–167MathSciNetCrossRefMATHGoogle Scholar
  28. Pan J, Shang J (2017) Adaptive LASSO for linear mixed model selection via profile log-likelihood. Commun Stat Theory Methods. Google Scholar
  29. Peng H, Lu Y (2012) Model selection in linear mixed effect models. J Multivar Anal 109:109–129MathSciNetCrossRefMATHGoogle Scholar
  30. Schelldorfer J, Bühlmann P, van de Geer S (2011) Estimation for high-dimensional linear mixed effects models using \(l_1\)-penalization. Scand J Stat 38(2):197–214MathSciNetCrossRefMATHGoogle Scholar
  31. Searle SR, Casella G, McCulloch CE (2006) Variance components. Wiley, New YorkMATHGoogle Scholar
  32. Thomson T, Hossain S, Ghahramani M (2015) Application of shrinkage estimation in linear regression models with autoregressive errors. J Stat Comput Simul 85(16):1580–1592MathSciNetCrossRefGoogle Scholar
  33. Tibshirani R (1996) Regression shrinkage and selection via the LASSO. J R Stat Soc Ser B 58(1):267–288MathSciNetMATHGoogle Scholar
  34. Twisk JWR (2003) Applied longitudinal data analysis for epidemiology—a practical guide. Cambridge University Press, New YorkGoogle Scholar
  35. Verbeke G, Molenberghs G (2009) Linear mixed models for longitudinal data. Springer Series in Statistics: corrected edition, New YorkGoogle Scholar
  36. Zeng T, Hill RC (2016) Shrinkage estimation in the random parameters logit model. Open J Stat 6:667–674CrossRefGoogle Scholar
  37. Zou H (2006) The adaptive lasso and its oracle properties. J Am Stat Assoc 101(476):1418–1429MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of WinnipegWinnipegCanada
  2. 2.Department of Mathematics and StatisticsBrock UniversitySt. CatharinesCanada

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