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Journal of Statistical Theory and Practice

, Volume 4, Issue 4, pp 727–742 | Cite as

Statistical Inference for Non-linear Models involving Ordinary Differential Equations

  • Sujit K. GhoshEmail author
  • Lovely Goyal
Article
  • 1 Downloads

Abstract

In the context of nonlinear fixed effect modeling, it is common to describe the relationship between a response variable and a set of explanatory variables by a system of nonlinear ordinary differential equations (ODEs). More often such a system of ODEs does not have any analytical closed form solution, making parameter estimation for these models quite challenging and computationally very demanding. Two new methods based on Euler’s approximation are proposed to obtain an approximate likelihood that is analytically tractable and thus making parameter estimation computationally less demanding than other competing methods. These methods are illustrated using a data on growth colonies of paramecium aurelium and simulation studies are presented to compare the performances of these new methods to other established methods in the literature.

AMS Subject Classification

62F03 62F15; and 62P10 

Keywords

Bayesian inference Ordinary differential equations MCMC Non-linear models Splines 

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References

  1. Atkinson, K., 1978. An Introduction to Numerical Analysis. Wiley, New York.zbMATHGoogle Scholar
  2. Butcher, J.C., 2003. Numerical Methods for Ordinary Differential Equations. John Wiley & Sons.CrossRefGoogle Scholar
  3. Davidian, M., Giltinan, D.M., 1995. Nonlinear Models for Repeated Measurement Data. Chapman & Hall, London.Google Scholar
  4. Diggle, P.J., 1990. Time Series A Biostatistical Introduction. Oxford Science Publications.zbMATHGoogle Scholar
  5. Ding, A.A., Wu, H., 1999. Relationships between antiviral treatment effects and biphasic viral decay rates in modelling HIV dynamics. Mathematical Biosciences, 160, 63–82.CrossRefGoogle Scholar
  6. Gelman, A., Bois, F., Jing, J., 1996. Physiological pharmacokinetic analysis using population modeling and informative prior distributions. Journal of the American Statistical Association, 85, 398–409.zbMATHGoogle Scholar
  7. Geman, S., Geman, D., 1984. Stochastic relaxation, Gibbs distributions and the Bayesian resolution of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.CrossRefGoogle Scholar
  8. Han, C., Chaloner, K., Perelson, A.S., 2002. Bayesian analysis of a population HIV dynamic model. Case Sudies in Bayesian Statistics, 6, 223–237.MathSciNetCrossRefGoogle Scholar
  9. Hastings, W.K., 1970. Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97–109.MathSciNetCrossRefGoogle Scholar
  10. Ho, D.D., Neumann, A.U., Perelson, A.S., Chen, W., Leonard, J.M. Markowitz, M., 1995. Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection. Nature, 373, 123–126.CrossRefGoogle Scholar
  11. Holte, S.E., Cornelisse, P., Heagerty, P., Self, S., 2003. An alternative to nonlinear least-squares regression for estimating parameters in ordinary differential equations models (personal communication).Google Scholar
  12. Huang, Y., Liu, D., Wu, H. (2004). Hierarchical bayesian methods for estimation of parameters in a longitudinal HIV dynamic system. Biometrics, 62(2), 413–423.MathSciNetCrossRefGoogle Scholar
  13. Lambert, J.D., 1987. Numerical Methods for Ordinary Differential Equations. John Wiley, Chichester.Google Scholar
  14. Lunn, D.J., Best, N., Thomas, A., Wakefield, J., Spiegelhalter, D., 2002. Bayesian analysis of population PK/PD models: General concepts and software. Journal of Pharmacokinetics and Pharmacodynamics, 29, 271–307.CrossRefGoogle Scholar
  15. Natarajan, R., Kass, R., 2000. Reference Bayesian methods for generalized linear mixed models. Journal of the American Statistical Association, 95, 227–237.MathSciNetCrossRefGoogle Scholar
  16. Perelson, A.S., Neumann, A.V., Markowitz, M., Leonard, J.M., Ho, D.H., 1996. HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science, 271, 1582–1587.CrossRefGoogle Scholar
  17. Petzold, L.R., 1987. Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations. Journal of Scientific and Statistical Computing, 4, 136–148.CrossRefGoogle Scholar
  18. Putter, H., Heisterkamp, S.H., Lange, J.M.A., de Wolf, F., 2002. A Bayesian approach to parameter estimation in HIV dynamical models. Statistics in Medicine, 21, 2199–2214.CrossRefGoogle Scholar
  19. Racine-Poon, A., Wakefield, J., 1998. Statistical methods for population pharmacokinetic modelling. Statistical Methods in Medical Research, 7, 63–84.CrossRefGoogle Scholar
  20. Robert, C.P., Casella, G., 2005. Monte Carlo Statistical Methods, Springer.zbMATHGoogle Scholar
  21. Schafer, J.L., 1997. Analysis of Incomplete Multivariate Data. Chapman & Hall/CRC.CrossRefGoogle Scholar
  22. Shampine, L.F., 1994. Numerical Solutions of Ordinary Differential Equations. Chapman & Hall, New York.zbMATHGoogle Scholar
  23. Wahba, G., 1990. Splines for Observational Data. Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
  24. Wakefield, J.C., 1996. The Bayesian analysis to population pharmacokinetic models. Journal of the Amre-ican Statistical Association, 91, 62–75.CrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2010

Authors and Affiliations

  1. 1.Department of StatisticsNC State UniversityRaleighUSA
  2. 2.Medical Sciences Biostatistics, Amgen Inc.Thousand OaksUSA

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