Statistics and Computing

, Volume 24, Issue 3, pp 443–460 | Cite as

LASSO-type estimators for semiparametric nonlinear mixed-effects models estimation

  • Ana Arribas-GilEmail author
  • Karine Bertin
  • Cristian Meza
  • Vincent Rivoirard


Parametric nonlinear mixed effects models (NLMEs) are now widely used in biometrical studies, especially in pharmacokinetics research and HIV dynamics models, due to, among other aspects, the computational advances achieved during the last years. However, this kind of models may not be flexible enough for complex longitudinal data analysis. Semiparametric NLMEs (SNMMs) have been proposed as an extension of NLMEs. These models are a good compromise and retain nice features of both parametric and nonparametric models resulting in more flexible models than standard parametric NLMEs. However, SNMMs are complex models for which estimation still remains a challenge. Previous estimation procedures are based on a combination of log-likelihood approximation methods for parametric estimation and smoothing splines techniques for nonparametric estimation. In this work, we propose new estimation strategies in SNMMs. On the one hand, we use the Stochastic Approximation version of EM algorithm (SAEM) to obtain exact ML and REML estimates of the fixed effects and variance components. On the other hand, we propose a LASSO-type method to estimate the unknown nonlinear function. We derive oracle inequalities for this nonparametric estimator. We combine the two approaches in a general estimation procedure that we illustrate with simulations and through the analysis of a real data set of price evolution in on-line auctions.


LASSO Nonlinear mixed-effects model On-line auction SAEM algorithm Semiparametric estimation 



The authors would like to thank the anonymous Associate Editor and two referees for valuable comments and suggestions.

The research of Ana Arribas-Gil is supported by projects MTM2010-17323 and ECO2011-25706, Spain.

The research of Karine Bertin is supported by projects FONDECYT 1090285 and ECOS/CONICYT C10E03 2010, Chile.

The research of Cristian Meza is supported by project FONDECYT 11090024, Chile.

The research of Vincent Rivoirard is partly supported by the french Agence Nationale de la Recherche (ANR 2011 BS01 010 01 projet Calibration).

Supplementary material

11222_2013_9380_MOESM1_ESM.pdf (130 kb)
(PDF 130 kB)


  1. Bertin, K., Le Pennec, E., Rivoirard, V.: Adaptive Dantzig density estimation. Ann. Inst. Henri Poincaré 47, 43–74 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  2. Bickel, P.J., Ritov, Y., Tsybakov, A.B.: Simultaneous analysis of lasso and Dantzig selector. Ann. Stat. 37(4), 1705–1732 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  3. Bühlmann, P., van de Geer, S.: Statistics for High-Dimensional Data. Springer Series in Statistics. Springer, Heidelberg (2011) CrossRefzbMATHGoogle Scholar
  4. Bunea, F.: Consistent selection via the Lasso for high dimensional approximating regression models. In: Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh. Inst. Math. Stat. Collect., vol. 3, pp. 122–137. Inst. Math. Statist., Beachwood (2008) CrossRefGoogle Scholar
  5. Bunea, F., Tsybakov, A.B., Wegkamp, M.H.: Aggregation and sparsity via l 1 penalized least squares. In: Learning Theory. Lecture Notes in Comput. Sci., vol. 4005, pp. 379–391. Springer, Berlin (2006) CrossRefGoogle Scholar
  6. Bunea, F., Tsybakov, A., Wegkamp, M.: Sparsity oracle inequalities for the Lasso. Electron. J. Stat. 1, 169–194 (2007a) CrossRefzbMATHMathSciNetGoogle Scholar
  7. Bunea, F., Tsybakov, A.B., Wegkamp, M.H.: Aggregation for Gaussian regression. Ann. Stat. 35(4), 1674–1697 (2007b) CrossRefzbMATHMathSciNetGoogle Scholar
  8. Comte, F., Samson, A.: Nonparametric estimation of random effects densities in linear mixed-effects model. J. Nonparametr. Stat. 24, 951–975 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  9. Delyon, B., Lavielle, M., Moulines, E.: Convergence of a stochastic approximation version of the EM algorithm. Ann. Stat. 27, 94–128 (1999) CrossRefzbMATHMathSciNetGoogle Scholar
  10. Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum-likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. B 39, 1–38 (1977) zbMATHMathSciNetGoogle Scholar
  11. Ding, A.A., Wu, H.: Assessing antiviral potency of anti-HIV therapies in vivo by comparing viral decay rates in viral dynamic models. Biostatistics 2, 13–29 (2001) CrossRefzbMATHGoogle Scholar
  12. Foulley, J.L., Quaas, R.: Heterogeneous variances in Gaussian linear mixed models. Genet. Sel. Evol. 27, 211–228 (1995) CrossRefGoogle Scholar
  13. Ge, Z., Bickel, P., Rice, J.: An approximate likelihood approach to nonlinear mixed effects models via spline approximation. Comput. Stat. Data Anal. 46, 747–776 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  14. van de Geer, S.: 1-regularization in high-dimensional statistical models. In: Proceedings of the International Congress of Mathematicians, vol. IV, pp. 2351–2369. Hindustan Book Agency, New Delhi (2010) Google Scholar
  15. Hartford, A., Davidian, M.: Consequences of misspecifying assumptions in nonlinear mixed effects models. Comput. Stat. Data Anal. 34, 139–164 (2000) CrossRefzbMATHGoogle Scholar
  16. Harville, D.: Bayesian inference for variance components using only error contrasts. Biometrika 61, 383–385 (1974) CrossRefzbMATHMathSciNetGoogle Scholar
  17. Jank, W.: Implementing and diagnosing the stochastic approximation EM algorithm. J. Comput. Graph. Stat. 15(4), 803–829 (2006) CrossRefMathSciNetGoogle Scholar
  18. Jank, W., Shmueli, G.: Functional data analysis in electronic commerce research. Stat. Sci. 21, 155–166 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  19. Ke, C., Wang, Y.: Semiparametric nonlinear mixed-effects models and their applications (with discussion). J. Am. Stat. Assoc. 96(456), 1272–1298 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  20. Kuhn, E., Lavielle, M.: Coupling a stochastic approximation version of EM with an MCMC procedure. ESAIM Probab. Stat. 8, 115–131 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  21. Kuhn, E., Lavielle, M.: Maximum likelihood estimation in nonlinear mixed effects models. Comput. Stat. Data Anal. 49(4), 1020–1038 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  22. Liu, B., Müller, H.G.: Functional data analysis for sparse auction data. In: Jank, W., Shmueli, G. (eds.) Statistical Methods in E-commerce Research, pp. 269–290. Wiley, New York (2008) CrossRefGoogle Scholar
  23. Liu, W., Wu, L.: Simultaneous inference for semiparametric nonlinear mixed-effects models with covariate measurement errors and missing responses. Biometrics 63, 342–350 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  24. Liu, W., Wu, L.: A semiparametric nonlinear mixed-effects model with non-ignorable missing data and measurement errors for HIV viral data. Comput. Stat. Data Anal. 53, 112–122 (2008) CrossRefzbMATHGoogle Scholar
  25. Liu, W., Wu, L.: Some asymptotic results for semiparametric nonlinear mixed-effects models with incomplete data. J. Stat. Plan. Inference (2009). doi: 10.1016j.jspi.2009.06.006 Google Scholar
  26. Luan, Y., Li, H.: Model-based methods for identifying periodically expressed genes based on time course microarray gene expression data. Bioinformatics 20(3), 332–339 (2004) CrossRefGoogle Scholar
  27. Meza, C., Jaffrézic, F., Foulley, J.L.: Estimation in the probit normal model for binary outcomes using the SAEM algorithm. Biom. J. 49(6), 876–888 (2007) CrossRefMathSciNetGoogle Scholar
  28. Meza, C., Jaffrézic, F., Foulley, J.L.: Reml estimation of variance parameters in nonlinear mixed effects models using the SAEM algorithm. Comput. Stat. Data Anal. 53(4), 1350–1360 (2009) CrossRefzbMATHGoogle Scholar
  29. Patterson, H.D., Thompson, R.: Recovery of inter-block information when block sizes are unequal. Biometrika 58, 545–554 (1971) CrossRefzbMATHMathSciNetGoogle Scholar
  30. Pinheiro, J., Bates, D.: Mixed-Effects Models in S and S-PLUS. Springer, New York (2000) CrossRefzbMATHGoogle Scholar
  31. Ramos, R., Pantula, S.: Estimation of nonlinear random coefficient models. Stat. Probab. Lett. 24, 49–56 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  32. Reithinger, F., Jank, W., Tutz, G., Shmueli, G.: Modelling price paths in on-line auctions: smoothing sparse and unevenly sampled curves by using semiparametric mixed models. Appl. Stat. 57, 127–148 (2008) zbMATHMathSciNetGoogle Scholar
  33. Schelldorfer, J., Bühlmann, P., van de Geer, S.: Estimation for high-dimensional linear mixed-effects models using l1-penalization. Scand. J. Stat. 38, 197–214 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  34. Shmueli, G., Jank, W.: Visualizing online auctions. J. Comput. Graph. Stat. 14, 299–319 (2005) CrossRefMathSciNetGoogle Scholar
  35. Shmueli, G., Russo, R.P., Jank, W.: The BARISTA: a model for bid arrivals in online auctions. Ann. Appl. Stat. 1, 412–441 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  36. Sklar, J.C., Wu, J., Meiring, W., Wang, Y.: Non-parametric regression with basis selection from multiple libraries. Technometrics (2012, accepted) Google Scholar
  37. Tibshirani, R.: Regression shrinkage and selection via the Lasso. J. R. Stat. Soc. B 58, 267–288 (1996) zbMATHMathSciNetGoogle Scholar
  38. Vonesh, E.F.: A note on the use of Laplace’s approximation for nonlinear mixed-effects models. Biometrika 83, 447–452 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  39. Wang, Y., Brown, M.B.: A flexible model for human circadian rhythms. Biometrics 52, 588–596 (1996) CrossRefzbMATHGoogle Scholar
  40. Wang, Y., Ke, C.: Assist: A suite of s functions implementing spline smoothing techniques (2004). http://wwwpstatucsbedu/faculty/yuedong/assistpdf
  41. Wang, Y., Ke, C., Brown, M.B.: Shape-invariant modeling of circadian rhythms with random effects and smoothing spline ANOVA decompositions. Biometrics 59, 804–812 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  42. Wang, Y., Eskridge, K., Zhang, S.: Semiparametric mixed-effects analysis of PKPD models using differential equations. J. Pharmacokinet. Pharmacodyn. 35, 443–463 (2008) CrossRefGoogle Scholar
  43. Wei, G.C., Tanner, M.A.: A Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation algorithm. J. Am. Stat. Assoc. 85, 699–704 (1990) CrossRefGoogle Scholar
  44. Wu, H., Zhang, J.: The study of longterm HIV dynamics using semi-parametric non-linear mixed-effects models. Stat. Med. 21, 3655–3675 (2002) CrossRefGoogle Scholar
  45. Yuan, M., Lin, Y.: Model selection and estimation in regression with grouped variables. J. R. Stat. Soc. B 68(1), 49–67 (2006) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Ana Arribas-Gil
    • 1
    Email author
  • Karine Bertin
    • 2
  • Cristian Meza
    • 2
  • Vincent Rivoirard
  1. 1.Departamento de EstadísticaUniversidad Carlos III de MadridGetafeSpain
  2. 2.CIMFAV-Facultad de IngenieríaUniversidad de ValparaísoValparaísoChile

Personalised recommendations