The NLMIXED Procedure

Abramowitz, M., and Stegun, I. A., eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 10th printing. New York: Dover.

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 Numerical Methods for Nonlinear Optimization, edited by F. A. Lootsma, 39–43. 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. 2nd ed. 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.” In Encyclopedia of Biostatistics, edited by P. Armitage, and T. Colton. New York: John Wiley & Sons.

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 : Application to Logistic-Normal Models.” Journal of the American Statistical Association 85:464–469.

Davidian, M., and Gallant, A. R. (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 and Applications 28:453–482.

Dennis, J. E., and Schnabel, R. B. (1983). Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Englewood Cliffs, 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. 2nd ed. New York: John Wiley & Sons.

Engel, B., and Keen, A. (1992). A Simple Approach for the Analysis of Generalized Linear Mixed Models, vol. 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. 2nd ed. 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.

Ghebretinsae, A. H., Faes, C., Molenberghs, G., De Boeck, M., and Geys, H. (2013). “A Bayesian, Generalized Frailty Model for Comet Assays.” Journal of Biopharmaceutical Statistics 23:618–636.

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. 2nd ed. Cary, NC: SAS Institute Inc.

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

Molenberghs, G., Verbeke, G., Demetrio, C. G. B., and Vieira, A. M. C. (2010). “A Family of Generalized Linear Models for Repeated Measures with Normal and Conjugate Random Effects.” Statistical Science 25:325–347.

Moré, J. J. (1978). “The Levenberg-Marquardt Algorithm: Implementation and Theory.” In Lecture Notes in Mathematics, vol. 30, edited by G. A. Watson, 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, Part 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. A. (1993). “Generalized Linear Mixed Models: A Pseudo-likelihood Approach.” Journal of Statistical Computation and Simulation 48:233–243.

Yuh, L., Beal, S. L., 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.