The NLMIXED Procedure

References

  • Abramowitz, M. and Stegun, I. A. (1972), Handbook of Mathematical Functions, New York: Dover Publications.

  • Anderson, D. A. and Aitkin, M. (1985), “Variance Component Models with Binary Response: Interviewer Variability,” Journal of the Royal Statistical Society, Series B, 47, 203–210.

  • Beal, S. L. and Sheiner, L. B. (1982), “Estimating Population Kinetics,” CRC Critical Reviews in Biomedical Engineering, 8, 195–222.

  • Beal, S. L. and Sheiner, L. B. (1988), “Heteroscedastic Nonlinear Regression,” Technometrics, 30, 327–338.

  • Beal, S. L. and Sheiner, L. B. (1992), NONMEM User’s Guide, San Francisco: NONMEM Project Group, University of California, San Francisco.

  • Beale, E. M. L. (1972), “A Derivation of Conjugate Gradients,” in F. A. Lootsma, ed., Numerical Methods for Nonlinear Optimization, London: Academic Press.

  • Beitler, P. J. and Landis, J. R. (1985), “A Mixed-Effects Model for Categorical Data,” Biometrics, 41, 991–1000.

  • Billingsley, P. (1986), Probability and Measure, Second Edition, New York: John Wiley & Sons.

  • Booth, J. G. and Hobert, J. P. (1998), “Standard Errors of Prediction in Generalized Linear Mixed Models,” Journal of the American Statistical Association, 93, 262–272.

  • Breslow, N. E. and Clayton, D. G. (1993), “Approximate Inference in Generalized Linear Mixed Models,” Journal of the American Statistical Association, 88, 9–25.

  • Burnham, K. P. and Anderson, D. R. (1998), Model Selection and Inference: A Practical Information-Theoretic Approach, New York: Springer-Verlag.

  • Cox, C. (1998), “Delta Method,” Encyclopedia of Biostatistics, 1125–1127.

  • Cox, D. R. and Oakes, D. (1984), Analysis of Survival Data, London: Chapman & Hall.

  • Cramer, J. S. (1986), Econometric Applications of Maximum Likelihood Methods, Cambridge: Cambridge University Press.

  • Crouch, E. A. C. and Spiegelman, D. (1990), “The Evaluation of Integrals of the Form $\int _{-\infty }^{\infty } f(t) \exp (-t^2) dt$: Application to Logistic-Normal Models,” Journal of the American Statistical Association, 85, 464–469.

  • Davidian, M. and Gallant, R. A. (1993), “The Nonlinear Mixed Effects Model with a Smooth Random Effects Density,” Biometrika, 80, 475–488.

  • Davidian, M. and Giltinan, D. M. (1995), Nonlinear Models for Repeated Measurement Data, New York: Chapman & Hall.

  • Dennis, J. E., Gay, D. M., and Welsch, R. E. (1981), “An Adaptive Nonlinear Least-Squares Algorithm,” ACM Transactions on Mathematical Software, 7, 348–368.

  • Dennis, J. E. and Mei, H. H. W. (1979), “Two New Unconstrained Optimization Algorithms Which Use Function and Gradient Values,” Journal of Optimization Theory Applications, 28, 453–482.

  • Dennis, J. E. and Schnabel, R. B. (1983), Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Englewood, NJ: Prentice-Hall.

  • Diggle, P. J., Liang, K.-Y., and Zeger, S. L. (1994), Analysis of Longitudinal Data, Oxford: Clarendon Press.

  • Draper, D. (1996), “Discussion of the Paper by Lee and Nelder,” Journal of the Royal Statistical Society, Series B, 58, 662–663.

  • Draper, N. R. and Smith, H. (1981), Applied Regression Analysis, Second Edition, New York: John Wiley & Sons.

  • Engel, B. and Keen, A. (1992), A Simple Approach for the Analysis of Generalized Linear Mixed Models, volume LWA-92-6, Wageningen, Netherlands: Agricultural Mathematics Group (GLW-DLO).

  • Eskow, E. and Schnabel, R. B. (1991), “Algorithm 695: Software for a New Modified Cholesky Factorization,” ACM Transactions on Mathematical Software, 17, 306–312.

  • Ezzet, F. and Whitehead, J. (1991), “A Random Effects Model for Ordinal Responses from a Crossover Trial,” Statistics in Medicine, 10, 901–907.

  • Fletcher, R. (1987), Practical Methods of Optimization, Second Edition, Chichester, UK: John Wiley & Sons.

  • Galecki, A. T. (1998), “NLMEM: New SAS/IML Macro for Hierarchical Nonlinear Models,” Computer Methods and Programs in Biomedicine, 55, 207–216.

  • Gallant, A. R. (1987), Nonlinear Statistical Models, New York: John Wiley & Sons.

  • Gaver, D. P. and O’Muircheartaigh, I. G. (1987), “Robust Empirical Bayes Analysis of Event Rates,” Technometrics, 29, 1–15.

  • Gay, D. M. (1983), “Subroutines for Unconstrained Minimization,” ACM Transactions on Mathematical Software, 9, 503–524.

  • Gilmour, A. R., Anderson, R. D., and Rae, A. L. (1985), “The Analysis of Binomial Data by Generalized Linear Mixed Model,” Biometrika, 72, 593–599.

  • Goldstein, H. (1991), “Nonlinear Multilevel Models, with an Application to Discrete Response Data,” Biometrika, 78, 45–51.

  • Golub, G. H. and Welsch, J. H. (1969), “Calculation of Gaussian Quadrature Rules,” Mathematical Computing, 23, 221–230.

  • Harville, D. A. and Mee, R. W. (1984), “A Mixed-Model Procedure for Analyzing Ordered Categorical Data,” Biometrics, 40, 393–408.

  • Hedeker, D. and Gibbons, R. D. (1994), “A Random Effects Ordinal Regression Model for Multilevel Analysis,” Biometrics, 50, 933–944.

  • Hurvich, C. M. and Tsai, C.-L. (1989), “Regression and Time Series Model Selection in Small Samples,” Biometrika, 76, 297–307.

  • Lin, X. and Breslow, N. E. (1996), “Bias Correction in Generalized Linear Mixed Models with Multiple Components of Dispersion,” Journal of the American Statistical Association, 91, 1007–1016.

  • Lindstrom, M. J. and Bates, D. M. (1990), “Nonlinear Mixed Effects Models for Repeated Measures Data,” Biometrics, 46, 673–687.

  • Littell, R. C., Milliken, G. A., Stroup, W. W., Wolfinger, R. D., and Schabenberger, O. (2006), SAS for Mixed Models, Second Edition, Cary, NC: SAS Press.

  • Liu, Q. and Pierce, D. A. (1994), “A Note on Gauss-Hermite Quadrature,” Biometrika, 81, 624–629.

  • Longford, N. T. (1994), “Logistic Regression with Random Coefficients,” Computational Statistics and Data Analysis, 17, 1–15.

  • McCulloch, C. E. (1994), “Maximum Likelihood Variance Components Estimation for Binary Data,” Journal of the American Statistical Association, 89, 330–335.

  • McGilchrist, C. E. (1994), “Estimation in Generalized Mixed Models,” Journal of the Royal Statistical Society, Series B, 56, 61–69.

  • Moré, J. J. (1978), “The Levenberg-Marquardt Algorithm: Implementation and Theory,” in G. A. Watson, ed., Lecture Notes in Mathematics, volume 30, 105–116, Berlin: Springer-Verlag.

  • Moré, J. J. and Sorensen, D. C. (1983), “Computing a Trust-Region Step,” SIAM Journal on Scientific and Statistical Computing, 4, 553–572.

  • Ochi, Y. and Prentice, R. L. (1984), “Likelihood Inference in a Correlated Probit Regression Model,” Biometrika, 71, 531–543.

  • Pierce, D. A. and Sands, B. R. (1975), Extra-Bernoulli Variation in Binary Data, Technical Report 46, Department of Statistics, Oregon State University.

  • 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.

  • Polak, E. (1971), Computational Methods in Optimization, New York: Academic Press.

  • Powell, M. J. D. (1977), “Restart Procedures for the Conjugate Gradient Method,” Mathematical Programming, 12, 241–254.

  • Pringle, R. M. and Rayner, A. A. (1971), Generalized Inverse Matrices with Applications to Statistics, New York: Hafner Publishing.

  • Rodriguez, G. and Goldman, N. (1995), “An Assessment of Estimation Procedures for Multilevel Models with Binary Response,” Journal of the Royal Statistical Society, Series A, 158, 73–89.

  • Roe, D. J. (1997), “Comparison of Population Pharmacokinetic Modeling Methods Using Simulated Data: Results from the Population Modeling Workgroup,” Statistics in Medicine, 16, 1241–1262.

  • Schall, R. (1991), “Estimation in Generalized Linear Models with Random Effects,” Biometrika, 78, 719–727.

  • Schittkowski, K. and Stoer, J. (1979), “A Factorization Method for the Solution of Constrained Linear Least Squares Problems Allowing Subsequent Data Changes,” Numerische Mathematik, 31, 431–463.

  • Self, S. G. and Liang, K.-Y. (1987), “Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests under Nonstandard Conditions,” Journal of the American Statistical Association, 82, 605–610.

  • Serfling, R. J. (1980), Approximation Theorems of Mathematical Statistics, New York: John Wiley & Sons.

  • Sheiner, L. B. and Beal, S. L. (1980), “Evaluation of Methods for Estimating Population Pharmacokinetic Parameters. I. Michaelis-Menten Model: Routine Clinical Pharmacokinetic Data,” Journal of Pharmacokinetics and Biopharmaceutics, 8, 553–571.

  • Sheiner, L. B. and Beal, S. L. (1985), “Pharmacokinetic Parameter Estimates from Several Least Squares Procedures: Superiority of Extended Least Squares,” Journal of Pharmacokinetics and Biopharmaceutics, 13, 185–201.

  • Smith, S. P. (1995), “Differentiation of the Cholesky Algorithm,” Journal of Computational and Graphical Statistics, 4, 134–147.

  • Stiratelli, R., Laird, N. M., and Ware, J. H. (1984), “Random Effects Models for Serial Observations with Binary Response,” Biometrics, 40, 961–971.

  • Vonesh, E. F. (1992), “Nonlinear Models for the Analysis of Longitudinal Data,” Statistics in Medicine, 11, 1929–1954.

  • Vonesh, E. F. (1996), “A Note on Laplace’s Approximation for Nonlinear Mixed-Effects Models,” Biometrika, 83, 447–452.

  • Vonesh, E. F. and Chinchilli, V. M. (1997), Linear and Nonlinear Models for the Analysis of Repeated Measurements, New York: Marcel Dekker.

  • Weil, C. S. (1970), “Selection of the Valid Number of Sampling Units and Consideration of Their Combination in Toxicological Studies Involving Reproduction, Teratogenesis, or Carcinogenesis,” Food and Cosmetic Toxicology, 8, 177–182.

  • White, H. (1982), “Maximum Likelihood Estimation of Misspecified Models,” Econometrica, 50, 1–25.

  • Williams, D. A. (1975), “The Analysis of Binary Responses from Toxicological Experiments Involving Reproduction and Teratogenicity,” Biometrics, 31, 949–952.

  • Wolfinger, R. D. (1993), “Laplace’s Approximation for Nonlinear Mixed Models,” Biometrika, 80, 791–795.

  • Wolfinger, R. D. (1997), “Comment: Experiences with the SAS Macro NLINMIX,” Statistics in Medicine, 16, 1258–1259.

  • Wolfinger, R. D. and Lin, X. (1997), “Two Taylor-Series Approximation Methods for Nonlinear Mixed Models,” Computational Statistics and Data Analysis, 25, 465–490.

  • Wolfinger, R. D. and O’Connell, M. (1993), “Generalized Linear Mixed Models: A Pseudo-likelihood Approach,” Journal of Statistical Computation and Simulation, 48, 233–243.

  • Yuh, L., Beal, S., Davidian, M., Harrison, F., Hester, A., Kowalski, K., Vonesh, E., and Wolfinger, R. D. (1994), “Population Pharmacokinetic/Pharmacodynamic Methodology and Applications: A Bibliography,” Biometrics, 50, 566–575.