What do we mean by identifiability in mixed effects models?

Abstract

We discuss the question of model identifiability within the context of nonlinear mixed effects models. Although there has been extensive research in the area of fixed effects models, much less attention has been paid to random effects models. In this context we distinguish between theoretical identifiability, in which different parameter values lead to non-identical probability distributions, structural identifiability which concerns the algebraic properties of the structural model, and practical identifiability, whereby the model may be theoretically identifiable but the design of the experiment may make parameter estimation difficult and imprecise. We explore a number of pharmacokinetic models which are known to be non-identifiable at an individual level but can become identifiable at the population level if a number of specific assumptions on the probabilistic model hold. Essentially if the probabilistic models are different, even though the structural models are non-identifiable, then they will lead to different likelihoods. The findings are supported through simulations.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

References

  1. 1.

    Bellman R, Åström KJ (1970) On structural identifiability. Math Biosci 7(3):329–339

    Article  Google Scholar 

  2. 2.

    Bonate PL (2011) Pharmacokinetic–pharmacodynamic modeling and simulation. Springer, New York

  3. 3.

    Brun R, Reichert P, Künsch HR (2001) Practical identifiability analysis of large environmental simulation models. Water Resour Res 37(4):1015–1030

    Article  Google Scholar 

  4. 4.

    Chappell MJ, Godfrey KR, Vajda S (1990) Global identifiability of the parameters of nonlinear systems with specified inputs: a comparison of methods. Math Biosci 102(1):41–73

    PubMed  CAS  Article  Google Scholar 

  5. 5.

    Cobelli C, Distefano JJ 3rd (1980) Parameter and structural identifiability concepts and ambiguities: a critical review and analysis. Am J Physiol-Regul Integr Comp Physiol 239(1):R7–R24

    CAS  Google Scholar 

  6. 6.

    Evans ND, Godfrey KR, Chapman MJ, Chappell MJ, Aarons L, Duffull SB (2001) An identifiability analysis of a parent-metabolite pharmacokinetic model for ivabradine. J Pharmacokinet Pharmacodynam 28(1):93–105

    CAS  Article  Google Scholar 

  7. 7.

    Fröhlich, F., Theis, F.J., Hasenauer, J.: Uncertainty analysis for non-identifiable dynamical systems: Profile likelihoods, bootstrapping and more. In: Computational Methods in Systems Biology, pp. 61–72. Springer (2014)

  8. 8.

    Garcia RI, Ibrahim JG, Wambaugh JF, Kenyon EM, Setzer RW (2015) Identifiability of PBPK models with applications to dimethylarsinic acid exposure. J Pharmacokinet Pharmacodynam 42(6):591–609

    CAS  Article  Google Scholar 

  9. 9.

    Gargash B, Mital D (1980) A necessary and sufficient condition of global structural identifiability of compartmental models. Comput Biol Med 10(4):237–242

    PubMed  CAS  Article  Google Scholar 

  10. 10.

    Godfrey KR, Chapman MJ, Vajda S (1994) Identifiability and indistinguishability of nonlinear pharmacokinetic models. J Pharmacokinet Biopharm 22(3):229–251

    PubMed  CAS  Article  Google Scholar 

  11. 11.

    Guedj J, Thiébaut R, Commenges D (2007) Practical identifiability of HIV dynamics models. Bull Math Biol 69(8):2493–2513

    PubMed  CAS  Article  Google Scholar 

  12. 12.

    Lavielle M (2014) Mixed Effects Models for the Population Approach: Models, Tasks, Methods and Tools. Chapman and Hall/CRC

  13. 13.

    Petersen B, Gernaey K, Vanrolleghem PA (2001) Practical identifiability of model parameters by combined respirometric-titrimetric measurements. Water Sci Technol 43(7):347–356

    PubMed  CAS  Google Scholar 

  14. 14.

    Raue A, Kreutz C, Maiwald T, Bachmann J, Schilling M, Klingmüller U, Timmer J (2009) Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25(15):1923–1929

    PubMed  CAS  Article  Google Scholar 

  15. 15.

    Shivva V, Korell J, Tucker I, Duffull S (2013) An approach for identifiability of population pharmacokinetic-pharmacodynamic models. CPT: Pharmacometrics Syst Pharmacol 2(6):e49

    CAS  Google Scholar 

  16. 16.

    Shivva V, Korell J, Tucker IG, Duffull SB (2014) Parameterisation affects identifiability of population models. J Pharmacokinet Pharmacodynam 41(1):81–86

    CAS  Article  Google Scholar 

  17. 17.

    Tikhonov AN, Goncharsky A, Stepanov VV, Yagola AG (1995) Numerical methods for the solution of ill-posed problems. Springer Science & Business Media, Dordrecht

  18. 18.

    Walter E, Pronzato L (1996) On the identifiability and distinguishability of nonlinear parametric models. Math Comput Simul 42(2):125–134

    Article  Google Scholar 

  19. 19.

    Wang W et al (2013) Identifiability of linear mixed effects models. Electron J Stat 7:244–263

    Article  Google Scholar 

  20. 20.

    Wu L (2010) Mixed effects models for complex data. CRC Press, Boca Raton

  21. 21.

    Xia X, Moog CH (2003) Identifiability of nonlinear systems with application to HIV/AIDS models. IEEE Trans Autom Control 48(2):330–336

    Article  Google Scholar 

  22. 22.

    Yates J, Jones R, Walker M, Cheung S (2009) Structural identifiability and indistinguishability of compartmental models. Expert Opinion Drug Metab Toxicol 5(3):295–302

    Article  Google Scholar 

Download references

Acknowledgments

The research leading to these results received support from the Innovative Medicines Initiative Joint Undertaking under Grant agreement 115156, resources of which are composed of financial contributions from the European Union’s Seventh Framework Programme (FP7/2007-2013) and EFPIA companies’ in kind contribution. The DDMoRe project is also supported by financial contribution from Academic and SME partners.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Marc Lavielle.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lavielle, M., Aarons, L. What do we mean by identifiability in mixed effects models?. J Pharmacokinet Pharmacodyn 43, 111–122 (2016). https://doi.org/10.1007/s10928-015-9459-4

Download citation

Keywords

  • Model identifiability
  • Practical identifiability
  • Structural identifiability
  • Parameter estimation
  • Mixed effects model
  • Pharmacokinetics