Nonlinear models for repeated measurement data: An overview and update

Editor’s Invited Article


Nonlinear mixed effects models for data in the form of continuous, repeated measurements on each of a number of individuals, also known as hierarchical nonlinear models, are a popular platform for analysis when interest focuses on individual-specific characteristics. This framework first enjoyed widespread attention within the statistical research community in the late 1980s, and the 1990s saw vigorous development of new methodological and computational techniques for these models, the emergence of general-purpose software, and broad application of the models in numerous substantive fields. This article presentsan overview of the formulation, interpretation, and implementation of nonlinear mixed effects models and surveys recent advances and applications.

Key Words

Hierarchicalmodel Inter-individual variation Intra-individual variation Nonlinear mixed effects model Random effects Serial correlation Subject-specific 


  1. Beal, S. L., and Sheiner, L. B. (1982), “Estimating Population Pharmacokinetics,” CRC Critical Reviews in Biomedical Engineering, 8, 195–222.Google Scholar
  2. Bennett, J., and Wakefield, J. (2001), “Errors-in-Variables in Joint Population Pharmacokinetic/Pharmacodynamic Modeling,” Biometrics, 57, 803–812.CrossRefMathSciNetGoogle Scholar
  3. Boeckmann, A. J., Sheiner, L. B., and Beal S. L. (1992), NONMEM User’s Guide, Part V, Introductory Guide, San Francisco: University of California.Google Scholar
  4. Breslow, N. E., and Clayton, D. G. (1993), “Approximate Inference in Generalized Linear Mixed Models,” Journal of the American Statistical Association, 88, 9–25.MATHCrossRefGoogle Scholar
  5. Carlin, B. P., and Louis, T. A. (2000) Bayes and Empirical Bayes Methods for Data Analysis (2nd ed.), New York: Chapman and Hall/CRC Press.MATHGoogle Scholar
  6. Chu, K. K., Wang, N. Y., Stanley, S., and Cohen, N. D. (2001), “Statistical Evaluation of the Regulatory Guidelines for Use of Furosemide in Race Horses,” Biometrics, 57, 294–301.CrossRefMathSciNetGoogle Scholar
  7. Clarkson, D. B., and Zhan, Y. H. (2002), “Using Spherical-Radial Quadrature to Fit Generalized Linear Mixed Effects Models,” Journal of Computational and Graphical Statistics, 11, 639–659.CrossRefMathSciNetGoogle Scholar
  8. Clayton, C. A., Starr, T. B., Sielken, R. L. Jr., Williams, R. L., Pontal, P. G., and Tobia, A. J. (2003), “Using a Nonlinear Mixed Effects Model to Characterize Cholinesterase Activity in Rats Exposed to Aldicarb,” Journal of Agricultural, Biological, and Environmental Statistics, 8, 420–437.CrossRefGoogle Scholar
  9. Concordet, D., and Nunez, O. G. (2000), “Calibration for Nonlinear Mixed Effects Models: An Application to the Withdrawal Time Prediction,” Biometrics, 56, 1040–1046.MATHCrossRefMathSciNetGoogle Scholar
  10. Davidian, M., and Gallant, A. R. (1992a), “Nlmix: A Program for Maximum Likelihood Estimation of the Nonlinear Mixed Effects Model With a Smooth Random Effects Density,” Department of Statistics, North Carolina State University.Google Scholar
  11. —, (1992b), “Smooth Nonparametric Maximum Likelihood Estimation for Population Pharmacokinetics, With Application to Quinidine,” Journal of Pharmacokinetics and Biopharmaceutics, 20, 529–556.CrossRefGoogle Scholar
  12. — (1993), “The Nonlinear Mixed Effects Model With a Smooth Random Effects Density,” Biometrika, 80, 475–488.MATHCrossRefMathSciNetGoogle Scholar
  13. Davidian, M., and Giltinan, D. M. (1993), “Some Simple Methods for Estimating Intra-individual Variability in Nonlinear Mixed Effects Models,” Biometrics, 49, 59–73.CrossRefGoogle Scholar
  14. — (1995), Nonlinear Models for Repeated Measurement Data, New York: Chapman and Hall.Google Scholar
  15. Demidenko, E. (1997), “Asymptotic Properties of Nonlinear Mixed Effects Models” in Modeling Longitudinal and Spatially Correlated Data: Methods, Applications, and Future Directions, eds. T. G. Gregoire, D. R. Brillinger, P. J. Diggle, E. Russek-Cohen, W. G. Warren, and R. D. Wolfinger, New York: Springer.Google Scholar
  16. Dey, D. K., Chen, M. H., and Chang, H. (1997), “Bayesian Approach for Nonlinear Random Effects Models,” Biometrics, 53, 1239–1252.MATHCrossRefGoogle Scholar
  17. Diggle, P. J., Heagerty, P., Liang, K.-Y., and Zeger, S. L. (2001), Analysis of Longitudinal Data (2nd ed.), Oxford: Oxford University Press.Google Scholar
  18. Fang, Z., and Bailey, R. L. (2001), “Nonlinear Mixed Effects Modeling for Slash Pine Dominant Height Growth Following Intensive Silvicultural Treatments” Forest Science, 47, 287–300.Google Scholar
  19. Galecki, A. T. (1998), “NLMEM: A New SAS/IML Macro for Hierarchical Nonlinear Models,” Computer Methods and Programs in Biomedicine, 55, 207–216.CrossRefGoogle Scholar
  20. Gelman, A., Bois, F., and Jiang, L. M. (1996), “Physiological Pharmacok inetic Analysis Using Population Modeling and Informative Prior Distributions,” Journal of the American Statistical Association, 91, 1400–1412.MATHCrossRefGoogle Scholar
  21. Gregoire, T. G., and Schabenberger, O. (1996a), “Nonlinear Mixed-Effects Modeling of Cumulative Bole Volume With Spatially-Correlated Within-Tree Data,” Journal of Agricultural, Biological, and Environmental Statistics, 1, 107–119.CrossRefMathSciNetGoogle Scholar
  22. — (1996b), “A Non-Linear Mixed-Effects Model to Predict Cumulative Bole Volume of Standing Trees,” Journal of Applied Statistics, 23, 257–271.CrossRefGoogle Scholar
  23. Hall, D. B., and Bailey, R. L. (2001), “Modeling and Prediction of Forest Growth Variables Based on Multilevel Nonlinear Mixed Models,” Forest Science, 47, 311–321.Google Scholar
  24. Hall, D. B., and Clutter, M. (2003), “Multivariate Multilevel Nonlinear Mixed Effects Models for Timber Yield Predictions”, Biometrics, in press.Google Scholar
  25. Hartford, A., and Davidian, M. (2000), “Consequences of Misspecifying Assumptions in Nonlinear Mixed Effects Models,” Computational Statistics and Data Analysis, 34, 139–164.MATHCrossRefGoogle Scholar
  26. Heagerty, P. (1999), “Marginally Specified Logistic-Normal Models for Longitudinal Binary Data,” Biometrics, 55, 688–698.MATHCrossRefGoogle Scholar
  27. Karlsson, M. O., Beal, S. L., and Sheiner, L. B. (1995), “Three New Residual Error Models for Population PK/PD Analyses,” Journal of Pharmacokinetics and Biopharmaceutics, 23, 651–672.CrossRefGoogle Scholar
  28. Karlsson, M. O., and Sheiner, L. B. (1993), “The Importance of Modeling Inter-Occasion Variability in Population Pharmacokinetic Analyses,” Journal of Pharmacokinetics and Biopharmacentics, 21, 735–750.CrossRefGoogle Scholar
  29. Ke, C., and Wang, Y. (2001), “Semiparametric Nonlinear Mixed Models and Their Applications,” Journal of the American Statistical Association, 96, 1272–1298.MATHCrossRefMathSciNetGoogle Scholar
  30. Ko, H. J., and Davidian, M. (2000), “Correcting for Measurement Error in Individual-Level Covariates in Nonlinear Mixed Effects Models,” Biometrics, 56, 368–375.MATHCrossRefMathSciNetGoogle Scholar
  31. Lai, T. L., and Shih, M.-C. (2003), “Nonparametric Estimation in Nonlinear Mixed Effects Models,” Biometrika, 90, 1–13.MATHCrossRefMathSciNetGoogle Scholar
  32. Law, N. J., Taylor, J. M. G., and Sandler, H. (2002), “The Joint Modeling of a Longitudinal Disease Progression Marker and the Failure Time Process in the Presence of a Cure,” Biostatistics, 3, 547–563.MATHCrossRefGoogle Scholar
  33. Li, L., Brown, M. B., Lee, K. H., and Gupta, S. (2002), “Estimation and Inference for a Spline-Enhanced Population Pharmacokinetic Model,” Biometrics, 58, 601–611.CrossRefMathSciNetGoogle Scholar
  34. Lindstrom, M. J. (1995), “Self-Modeling With Random Shift and Scale Parameters and a Free-Knot Spline Shape Function,” Statistics in Medicine, 14, 2009–2021.CrossRefGoogle Scholar
  35. Lindstrom, M. J., and Bates, D. M. (1990), “Nonlinear Mixed Effects Models for Repeated Measures Data,” Biometrics, 46, 673–687.CrossRefMathSciNetGoogle Scholar
  36. Littell, R. C., Milliken, G. A., Stroup, W. W., and Wolfinger, R. D. (1996), SAS System for Mixed Models, Cary, NC: SAS Institute Inc.Google Scholar
  37. Lopes, H. F., Müller, P., and Rosner, G. L. (2003), “Bayesian Meta-Analysis for Longitudinal Data Models Using Multivariate Mixture Priors,” Biometrics, 59, 66–75.CrossRefMathSciNetGoogle Scholar
  38. Mallet, A. (1986), “A Maximum Likelihood Estimation Method for Random Coefficient Regression Models,” Biometrika, 73, 645–656.MATHCrossRefMathSciNetGoogle Scholar
  39. Mandema, J. W., Verotta, D., and Sheiner L. B. (1992), “Building Population Pharmacokinetic/ Pharmacodynamic Models,” Journal of Pharmacokinetics and Biopharmacentics, 20, 511–529.CrossRefGoogle Scholar
  40. McRoberts, R. E. and Brooks, R. T., and Rogers, L. L. (1998), “Using Nonlinear Mixed Effects Models to Estimate Size-Age Relationships for Black Bears,” Canadian Journal of Zoology, 76, 1098–1106.CrossRefGoogle Scholar
  41. Mentré, F., and Mallet, A. (1994), “Handling Covariates in Population Pharmacokinetics,” International Journal of Biomedical Computing, 36, 25–33.CrossRefGoogle Scholar
  42. Mezzetti, M., Ibrahim, J. G., Bois, F. Y., Ryan, L. M., Ngo, L., and Smith, T. J. (2003), “A Bayesian Compartmental Model for the Evaluation of 1,3-Butadiene Metabolism,” Applied Statistics, 52, 291–305.MATHMathSciNetGoogle Scholar
  43. Mikulich, S. K., Zerbe, G. O., Jones, R. H., and Crowley, T. J. (2003), “Comparing Linear and Nonlinear Mixed Model Approaches to Cosinor Analysis,” Statistics in Medicine, 22, 3195–3211.CrossRefGoogle Scholar
  44. Monahan, J., and Genz, A. (1997), “Spherical-Radial Integration Rules for Bayesian Computation,” Journal of the American Statistical Association, 92, 664–674.MATHCrossRefGoogle Scholar
  45. Morrell, C. H., Pearson, J. D., Carter, H. B., and Brant, L. J. (1995) “Estimating Unknown Transition Times Using a Piecewise Nonlinear Mixed-Effects Model in Men With Prostate Cancer,” Journal of the American Statistical Association, 90, 45–53.CrossRefGoogle Scholar
  46. Müller, P., and Rosner, G. L. (1997), “A Bayesian Population Model With Hierarchical Mixture Priors Applied to Blood Count Data,” Journal of the American Statistical Association, 92, 1279–1292.MATHCrossRefGoogle Scholar
  47. Notermans, D. W., Goudsmit, J., Danner, S. A., de Wolf, F., Perelson, A. S., and Mittler, J. (1998), “Rate of HIV-1 Decline Following Antiretroviral Therapy is Related to Viral Load at Baseline and Drug Regimen,” AIDS, 12, 1483–1490.CrossRefGoogle Scholar
  48. Oberg, A., and Davidian, M. (2000), “Estimating Data Transformations in Nonlinear Mixed Effects Models,” Biometrics, 56, 65–72.MATHCrossRefGoogle Scholar
  49. Pauler, D., and Finkelstein, D. (2002), “Predicting Time to Prostate Cancer Recurrence Based on Joint Models for Non-linear Longitudinal Biomarkers and Event Time,” Statistics in Medicine, 21, 3897–3911.CrossRefGoogle Scholar
  50. Pilling, G. M., Kirkwood, G. P., and Walker, S. G. (2002), “An Improved Method for Estimating Individual Growth Variability in Fish, and the Correlation Between von Bertalanffy Growth Parameters,” Canadian Journal of Fisheries and Aquatic Sciences, 59, 424–432.CrossRefGoogle Scholar
  51. Pinheiro, J. C., and Bates, D. M. (1995), “Approximations to the Log-Likelihood Function in the Nonlinear Mixed Effects Model,” Journal of Computational and Graphical Statistics, 4, 12–35.CrossRefGoogle Scholar
  52. — (2000), Mixed-Effects Models in S and Splus, New York: Springer.Google Scholar
  53. Raudenbush, S. W., Yang, M. L., and Yosef, M. (2000), “Maximum Likelihood for Generalized Linear Models With Nested Random Effects Via High-Order, Multivariate Laplace Approximation,” Journal of Computational and Graphical Statistics, 9, 141–157.CrossRefMathSciNetGoogle Scholar
  54. Rekaya, R., Weigel, K. A., and Gianola, D. (2001), “Hierarchical Nonlinear Model for Persistency of Milk Yield in the First Three Lactations of Holsteins,” Lifestock Production Science, 68, 181–187.CrossRefGoogle Scholar
  55. Rodriguez-Zas, S. L., Gianola, D., and Shook, G. E. (2000), “Evaluation of Models for Somatic Cell Score Lactation Patterns in Holsteins,” Lifestock Production Science, 67, 19–30.CrossRefGoogle Scholar
  56. Rosner, G. L. and Müller, P. (1994), “Pharmacokinetic/Pharmacodynamic Analysis of Hematologic Profiles,” Journal of of Pharmacokinetics and Biopharmaceutics, 22, 499–524.CrossRefGoogle Scholar
  57. SAS Institute (1999) PROC NLMIXED, SAS Online Doc, Version 8, Cary, NC: SAS Institute Inc.Google Scholar
  58. Schabenberger, O., and Pierce, F. J. (2001), Contemporary Statistical Models for the Plant and Soil Sciences, New York: CRC Press.Google Scholar
  59. Schumitzky, A. (1991), “Nonparametric EM Algorithms for Estimating Prior Distributions,” Applied Mathematics and Computation, 45, 143–157.MATHCrossRefMathSciNetGoogle Scholar
  60. Sheiner, L. B., and Ludden, T. M. (1992), “Population Pharmacokinetics/Pharmacodynamics,” Annual Review of Pharmacological Toxicology, 32, 185–209.Google Scholar
  61. Sheiner, L. B., Rosenberg, B., and Marathe, V. V. (1997), “Estimation of Population Characteristics of Population Pharmacokinetic Parameters From Routine Clinical Data,” Journal of Pharmacokinetics and Biopharmaceutics, 8, 635–651.Google Scholar
  62. Steimer, J. L., Mallet, A., Golmard, J. L., and Boisvieux, J. F. (1984), “Alternative Approaches to Estimation of Population Pharmacokinetic Parameters: Comparison with the Nonlinear Mixed Effect Model,” Drug Metabolism Reviews, 15, 265–292.CrossRefGoogle Scholar
  63. Verbeke, G., and Molenberghs, G. (2000), Linear Mixed Models for Longitudinal Data, New York: Springer.MATHGoogle Scholar
  64. Vonesh, E. F. (1992), “Mixed-Effects Nonlinear Regression for Unbalanced Repeated Measures” Biometrics, 48, 1–17.CrossRefMathSciNetGoogle Scholar
  65. — (1996), “A Note on the Use of Laplace’s Approximation for Nonlinear Mixed-Effects Models,” Biometrika, 83, 447–452.MATHCrossRefMathSciNetGoogle Scholar
  66. Vonesh, E. F., and Chinchilli, V. M. (1997), Linear and Nonlinear Models for the Analysis of Repeated Measurements New York: Marcel Dekker.MATHGoogle Scholar
  67. Vonesh, E. F., Chinchilli, V. M., and Pu, K. W. (1996), “Goodness-Of-Fit in Generalized Nonlinear Mixed-Effects Models,” Biometrics, 52, 572–587.MATHCrossRefGoogle Scholar
  68. Vonesh, E. G., Wang, H., Nie, L., and Majumdar, D. (2002), “Conditional Second-Order Generalized Estimating Equations for Generalized Linear and Nonlinear Mixed-Effects Models,” Journal of the American Statistical Association, 97, 271–283.MATHCrossRefMathSciNetGoogle Scholar
  69. Wakefield, J. (1996), “The Bayesian Analysis of Population Pharmacokinetic Models,” Journal of the American Statistical Association, 91, 62–75.MATHCrossRefGoogle Scholar
  70. Wakefield, J., and Rahman, N. (2000), “The Combination of Population Pharmacokinetic Studies,” Biometrics, 56, 263–270.MATHCrossRefGoogle Scholar
  71. Wakefield, J. C., Smith, A. F. M., Racine-Poon, A., and Gelfand, A. E., (1994), “Bayesian Analysis of Linear and Nonlinear Population Models by Using the Gibbs Sampler,” Applied Statistics, 43, 201–221.MATHCrossRefGoogle Scholar
  72. Walker, S. G. (1996), “An EM algorithm for Nonlinear Random Effects Models,” Biometrics, 52, 934–944.MATHCrossRefMathSciNetGoogle Scholar
  73. Wang, N., and Davidian, M. (1996), “A Note on Covariate Measurement Error in Nonlinear Mixed Effects Models,” Biometrika, 83, 801–812.MATHCrossRefMathSciNetGoogle Scholar
  74. Wolfinger, R. (1993), “Laplace’s Approximation for Nonlinear Mixed Models,” Biometrika, 80, 791–795.MATHCrossRefMathSciNetGoogle Scholar
  75. Wolfinger, R. D., and Lin, X. (1997), “Two Taylor-series Approximation Methods for Nonlinear Mixed Models,” Computational Statistics and Data Analysis, 25, 465–490.MATHCrossRefGoogle Scholar
  76. Wu, L. (2002), “A Joint Model for Nonlinear Mixed-Effects Models With Censoring and Covariates Measured With Error, With Application to AIDS Studies,” Journal of the American Statistical Association, 97, 955–964.MATHCrossRefMathSciNetGoogle Scholar
  77. Wu, H. L., and Ding, A. A. (1999), “Population HIV-1 Dynamics in vivo: Applicable Models and Inferential Tools for Virological Data From AIDS Clinical Trials,” Biometrics, 55, 410–418.MATHCrossRefGoogle Scholar
  78. Wu, H. L., and Wu, L. (2002a), “Identification of Significant Host Factors for HIV Dynamics Modelled by Non-Linear Mixed-Effects Models,” Statistics in Medicine, 21, 753–771.CrossRefGoogle Scholar
  79. — (2002b), “Missing Time-Dependent Covariates in Human Immunodeficiency Virus Dynamic Models,” Applied Statistics, 51, 2002.Google Scholar
  80. Yeap, B. Y., Catalano, P. J., Ryan, L. M., and Davidian, M. (2003), “Robust Two-Stage Approach to Repeated Measurements Analysis of Chronic Ozone Exposure in Rats,” Journal of Agricultural, Biological, and Environmental Statistics, 8, 438–454.CrossRefGoogle Scholar
  81. Yeap, B. Y., and Davidian, M. (2001), “Robust Two-Stage Estimation in Hierarchical Nonlinear Models,” Biometrics, 57, 266–272.CrossRefMathSciNetGoogle Scholar
  82. Young, D. A., Zerbe, G. O., and Hay, W. W. (1997), “Fieller’s Theorem, Scheffé Simultaneous Confidence Intervals, and Ratios of Parameters of Linear and Nonlinear Mixed-Effects Models,” Biometrics, 53, 838–347.MATHCrossRefGoogle Scholar
  83. Zeng, Q., and Davidian, M. (1997), “Testing Homgeneity of Intra-run Variance Parameters in Immunoassay,” Statistics in Medicine, 16, 1765–1776.CrossRefGoogle Scholar

Copyright information

© International Biometric Society 2003

Authors and Affiliations

  1. 1.Department of StatisticsNorth Carolina State UniversityRaleigh
  2. 2.Genentech, Inc.South San Francisco

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