Journal of Statistical Theory and Practice

, Volume 12, Issue 1, pp 12–22 | Cite as

A semiparametric mixture regression model for longitudinal data

  • Tapio NummiEmail author
  • Janne Salonen
  • Lasse Koskinen
  • Jianxin Pan


A normal semiparametric mixture regression model is proposed for longitudinal data. The proposed model contains one smooth term and a set of possible linear predictors. Model terms are estimated using the penalized likelihood method with the EM algorithm. A computationally feasible alternative method that provides an approximate solution is also introduced. Simulation experiments and a real data example are used to illustrate the methods.


Curve clustering EM algorithm finite mixtures growth curves 

AMS Subject Classification

62G05 62B99 62J07 


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  1. Basford, Κ. Ε., and G. J. McLachlan. 1985. Likelihood estimation with normal mixture models. Applied Statistics 34 (3):282–89.MathSciNetCrossRefGoogle Scholar
  2. Dempster, Α., Ν. Laird, and D. Rubin. 1977. Maximum likelihood estimation for incomplete data via the EM algorithm. Journal of the Royal Statistical Society Β 39:1–38.zbMATHGoogle Scholar
  3. Diggle, P., P. Heagerty, K.-Y. Liang, and S. Zeger. 2013. Analysis of longitudinal data, 2nd ed. Oxford, UK: Oxford University Press.zbMATHGoogle Scholar
  4. Fariaa, S., and G. Soromenho. 2010. Fitting mixtures of linear regressions. Journal of Statistical Computation and Simulation 80 (2):201–25.MathSciNetCrossRefGoogle Scholar
  5. Fitzmaurize, G. M., N. M. Laird, and J. H. Ware. 2011. Applied longitudinal analysis, 2nd ed. Hoboken, NJ: Wiley.Google Scholar
  6. Gasser, T., H. G. Muller, W. Kohler, L. Molinari, and A. Prader. 1984. Nonparametric Regression analysis of growth curves. Annals of Statistics 12:210–29.MathSciNetCrossRefGoogle Scholar
  7. Green, P., and B. Silverman. 1994. Nonparametric regression and generalized linear models. A roughness penalty approach. Monographs on Statistics and Applied Probability, 58. Boca Raton, FL: Chapman Hall/CRC.CrossRefGoogle Scholar
  8. Huang, M., and W. Yao. 2012. Mixture regression models with varying mixing proportions: A semiparametric approach. Journal of the American Statistical Association 107:711–24.MathSciNetCrossRefGoogle Scholar
  9. Huang, M., R. Li, and S. Wang. 2013. Nonparametric mixture regression models. Journal of the American Statistical Association 108:929–41.MathSciNetCrossRefGoogle Scholar
  10. Johnson, W. 2015. Human biology toolkit: Analytical strategies in human growth research. American Journal of Human Biology. 27:69–83.CrossRefGoogle Scholar
  11. Jones, B., D. Nagin, and K. Roeder. 2001. A SAS procedure based on mixture models for estimating developmental trajectories. Sociological Methods & Research 29:374–393.MathSciNetCrossRefGoogle Scholar
  12. Karlberg, J. 1987. On the modeling of human growth. Statistics in Medicine 6:185–92.CrossRefGoogle Scholar
  13. Leisch, F. 2004. FlexMix: A general framework for finite mixture models and latent class regression in R. Journal of Statistical Software 11 (8):1–18.CrossRefGoogle Scholar
  14. Leng, C., W. Zhang, and J. Pan. 2010. Semiparametric mean-covariance regression analysis for longitudinal data. Journal of the American Statistical Association 105 (489):181–93. doi:10.1198/ jasa.2009.tm08485MathSciNetCrossRefGoogle Scholar
  15. McLachlan, G., and D. Peel. 2000. Finite mixture models. New York, NY: John Wiley and Sons.CrossRefGoogle Scholar
  16. McLachlan, G., and S. Rathnayake. 2014. On the Number of Components in a Gaussion mixture. WIREs Data Mining and Knowledge Discovery 4 (5):341–55. Hoboken, NJ: John Wiley & Sons. doi:10.1002/widm.1135CrossRefGoogle Scholar
  17. Muthen, B., and S. T. Khoo. 1998. Longitudinal studies of achievement growth using latent variable modeling. Learning and Individual Differences 10:73–101.CrossRefGoogle Scholar
  18. Muthen, L., and B. Muthen. 2007. Mplus user’s guide, 6th ed. Los Angeles, CA: Muthen & Muthen.Google Scholar
  19. Nagin, D. 1999. Analyzing developmental trajectories: Semiparametric, group-based approach. Psychological Methods 4:39–177.CrossRefGoogle Scholar
  20. Nagin, D. 2005. Group-based modeling of development. Cambridge, MA: Harvard University Press.CrossRefGoogle Scholar
  21. Nummi, T., J. Pan, T. Siren, and K. Liu. 2011. Testing for cubic smoothing splines under dependent data. Biometrics 67 (3):871–75.MathSciNetCrossRefGoogle Scholar
  22. Nummi, T., J. Pan, and N. Mesue. 2013. Testing linearity in semiparametric regression models. Statistics and Its Interface 6:3–8.MathSciNetCrossRefGoogle Scholar
  23. Nummi, T., T. Hakanen, L. Lipiäinen, U. Harjunmaa, M. Salo, M.-T. Saha, and N. Vuorela. 2014. A trajectory analysis of body mass index for Finnish children. Journal of Applied Statistics 41 (7):1422–35.MathSciNetCrossRefGoogle Scholar
  24. Poortema, K. 1984. On the statistical analysis of growth. PhD thesis, Groningen University, Groningen, The Netherlands.Google Scholar
  25. Ruppert, D., M. P. Wand, and R. J. Carrol. 2005. Semiparametric regression. New York, NY: Cambridge University Press.Google Scholar
  26. Titterington, D. M., A. F. M. Smith, and U. E. Makov. 1985. Statistical analysis of finite mixture distribution. Wiley, New York.Google Scholar
  27. Verbeke, G., and E. Lesaffre. 1996. A linear mixed-effects model with heterogeneity in the random-effects population. Journal of the American Statistical Association 91 (433):217–21.CrossRefGoogle Scholar
  28. Vuorela, N. 2011. Body mass index, overweight and obesity among children in Finland — A retrospective epidemilogical study in Pirkanmaa District spanning over four decades. Acta Universitatis Tamperensis 1611, Tampere University Press, Tampere.Google Scholar
  29. Ye, H., and J. Pan. 2006. Modelling covariance structures in generalized estimating equations for longitudinal data. Biometrika 93:927–41.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing, 20 Middlefield Ct, Greensboro, NC 27455 2018

Authors and Affiliations

  • Tapio Nummi
    • 1
    Email author
  • Janne Salonen
    • 2
  • Lasse Koskinen
    • 3
  • Jianxin Pan
    • 4
  1. 1.Faculty of Natural SciencesUniversity of TampereTampereFinland
  2. 2.Research DepartmentThe Finnish Centre for PensionsHelsinkiFinland
  3. 3.Faculty of ManagementUniversity of TampereTampereFinland
  4. 4.School of MathematicsThe University of ManchesterManchesterUnited Kingdom

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