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Multilevel Generalized Linear Models

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References

  • O. O. Aalen. Heterogeneity in survival analysis. Statistics in Medicine, 7:1121–1137, 1988.

    Article  Google Scholar 

  • M. Abramowitz and I. A. Stegun, editors. Handbook of Mathematical Functions. Number 55 in National Bureau of Standards Applied Mathematics Series. U.S. Government Printing Office, Washington, DC, 1964.

    Google Scholar 

  • M. Aitkin, D. Anderson, B. Francis, and J. Hinde. Statistical Modelling in GLIM. Clarendon Press, Oxford, 1989.

    MATH  Google Scholar 

  • M. Aitkin and D. G. Clayton. The fitting of exponential, Weibull and extreme value distributions to complex censored survival data using GLIM. Applied Statistics, 29:156–163, 1980.

    Article  Google Scholar 

  • P. D. Allison. Discrete-time methods for the analysis of event histories. Sociological Methodology, 13:61–98, 1982.

    Article  Google Scholar 

  • D. A. Anderson and M. Aitkin. Variance component models with binary response: Interviewer variability. Journal of the Royal Statistical Society, Series B, 47:203–210, 1985.

    MathSciNet  Google Scholar 

  • J. S. Barber, S. A. Murphy, W. G. Axinn, and J. Maples. Discrete-time multilevel hazard analysis. Sociological Methodology, 30:201–235, 2000.

    Article  Google Scholar 

  • S. Bennet and J. Whitehead. Fitting logistic and log-logistic regression models to censored data using GLIM. GLIM Newsletter, 4:12–19, 1981.

    Google Scholar 

  • N. G. Best, M. K. Cowles, and S. K. Vines. CODA: Convergence Diagnosis and Output Analysis Software for Gibbs Sampling Output, version 0.40. Medical Research Council Biostatistics Unit, Cambridge, UK, 1997.

    Google Scholar 

  • J. G. Booth, J. P. Hobert, and W. Jank. A survey of Monte Carlo algorithms for maximizing the likelihood of a two-stage hierarchical model. Statistical Modelling, 1:333–349, 2001.

    Article  MATH  Google Scholar 

  • N. E. Breslow and D. G. Clayton. Approximate inference in generalized linear mixed models. Journal of the American Statistical Association, 88:9–25, 1993.

    Article  MATH  Google Scholar 

  • N. E. Breslow and X. Lin. Bias correction in generalised linear mixed models with a single component of dispersion. Biometrika, 82:81–91, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  • W. J. Browne and D. Draper. A comparison of Bayesian and likelihood-based methods for fitting multilevel models. Bayesian Analysis, 1:473–549, 2006. (with discussion)

    Article  MathSciNet  Google Scholar 

  • D. G. Clayton. A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 65:141–151, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  • D. G. Clayton. The analysis of event history data: Review of progress and outstanding problems. Statistics in Medicine, 7:819–841, 1988.

    Article  Google Scholar 

  • D. G. Clayton. Generalized linear mixed models. In W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, editors, Markov Chain Monte Carlo in Practice, pages 275–301. Chapman & Hall, London, 1996.

    Google Scholar 

  • D. G. Clayton and J. Cuzick. The EM algorithm for Cox’s regression model using GLIM. Applied Statistics, 34:148–156, 1985.

    Article  Google Scholar 

  • D. G. Clayton and J. Cuzick. Multivariate generalizations of the proportional hazards model. Journal of the Royal Statistical Society, Series B, 148:82–117, 1985. (with discussion).

    Article  MATH  MathSciNet  Google Scholar 

  • C. Corcoran, B. Coull, and A. Patel. Egret for Windows User Manual. Cytel Software Corporation, Cambridge, MA, 1999.

    Google Scholar 

  • D. R. Cox. Regression models and life tables. Journal of the Royal Statistical Society, Series B, 34:187–220, 1972. (with discussion).

    MATH  Google Scholar 

  • D. R. Cox. Partial likelihood. Biometrika, 62:269–276, 1975.

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  • M. J. Crowder. Beta-binomial Anova for proportions. Applied Statistics, 27:34–37, 1978.

    Article  Google Scholar 

  • M. Davidian and D. M. Giltinan. Nonlinear Models for Repeated Measurement Data. Chapman & Hall, London, 1995.

    Google Scholar 

  • P. J. Diggle, K.-Y. Liang, and S. L. Zeger. Analysis of Longitudinal Data. Oxford University Press, Oxford, UK, 1994.

    Google Scholar 

  • P. Feigl and M. Zelen. Estimation of exponential survival probabilities with concomitant information. Biometrics, 21:826–838, 1967.

    Article  Google Scholar 

  • A. Gelman and D. B. Rubin. Inference from iterative simulation using multiple sequences. Statistical Science, 7:457–511, 1992. (with discussion).

    Article  Google Scholar 

  • J. Geweke. Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, editors, Bayesian Statistics 4, pages 169–194. Oxford University Press, Oxford, UK, 1992.

    Google Scholar 

  • W. R. Gilks and P. Wild. Adaptive rejection sampling for Gibbs sampling. Applied Statistics, 41:337–348, 1992.

    Article  MATH  Google Scholar 

  • H. Goldstein. Nonlinear multilevel models, with an application to discrete response data. Biometrika, 78:45–51, 1991.

    Article  MathSciNet  Google Scholar 

  • H. Goldstein. Multilevel models and generalised estimating equations. Multilevel Modelling Newsletter, 5(2):2, 1993.

    Google Scholar 

  • H. Goldstein. Multilevel unit specific and population average generalised linear models. Multilevel Modelling Newsletter, 7(3):4–5, 1995.

    Google Scholar 

  • H. Goldstein. Consistent estimators for multilevel generalised linear models using an iterated bootstrap. Multilevel Modelling Newsletter, 8(1):3–6, 1996.

    Google Scholar 

  • H. Goldstein and J. Rasbash. Improved approximations for multilevel models with binary responses. Journal of the Royal Statistical Society, Series A, 159:505–513, 1996.

    MATH  MathSciNet  Google Scholar 

  • C. Gouriéroux and A. Monfort. Simulation-Based Econometric Methods. Oxford University Press, Oxford, UK, 1996.

    Google Scholar 

  • P. J. Green. Penalized likelihood for general semi-parametric regression models. International Statistical Review, 55:245–259, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  • G. Guo and G. Rodríguez. Estimating a multivariate proportional hazards model for clustered data using the EM algorithm, with an application to child survival in Guatemala. Journal of the American Statistical Association, 87:969–976, 1992.

    Article  Google Scholar 

  • D. Hedeker and R. D. Gibbons. A random-effects ordinal regression model for multilevel analysis. Biometrics, 50:933–944, 1994.

    Article  MATH  Google Scholar 

  • D. Hedeker and R. D. Gibbons. MIXOR: A computer program for mixed-effects ordinal regression analysis. Computer Methods and Programs in Biomedicine, 49:157–176, 1996.

    Article  Google Scholar 

  • T. R. Holford. The analysis of rates and survivorship using log-linear models. Biometrics, 36:299–306, 1980.

    Article  MATH  Google Scholar 

  • P. Hougaard. Life table methods for heterogeneous populations: Distributions describing the heterogeneity. Biometrika, 71:75–83, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  • P. Hougaard. Survival models for heterogeneous populations derived from stable distributions. Biometrika, 73:387–396, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  • J. D. Kalbfleisch and R. L. Prentice. The Statistical Analysis of Failure Time Data, 2nd edition. Wiley, New York, 2002.

    MATH  Google Scholar 

  • M. P. Keane. Simulation estimation for panel data models with limited dependent variables. In G. S. Maddala, C. R. Rao, and H. D. Vinod, editors, Handbook of Statistics, volume 11, pages 545–571. North-Holland, Amsterdam, 1993.

    Google Scholar 

  • A. Y. C. Kuk. Asymptotically unbiased estimation in generalized linear models with random effects. Journal of the Royal Statistical Society, Series B, 57:395–407, 1995.

    MATH  MathSciNet  Google Scholar 

  • N. M. Laird. Empirical Bayes methods for two-way contingency tables. Biometrika, 65:581–590, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  • N. M. Laird and D. Olivier. Covariance analysis of censored survival data using log-linear analysis techniques. Journal of the American Statistical Association, 76:231–240, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  • J. F. Lawless. Regression methods for Poisson process data. Journal of the American Statistical Association, 82:808–815, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  • Y. Lee and J. A. Nelder. Hierarchical generalized linear models. Journal of the Royal Statistical Society, Series B, 58:619–678, 1996.

    MATH  MathSciNet  Google Scholar 

  • S. R. Lerman and C. F. Manski. On the use of simulated frequencies to approximate choice probabilities. In C. F. Manski and D. McFadden, editors, Structural Analysis of Discrete Data with Econometric Applications, pages 305–319. MIT Press, Cambridge, MA, 1981.

    Google Scholar 

  • L. A. Lillard and C. W. A. Panis. aML: Multilevel Multiprocess Statistical Software, Version 2.0. EconWare, Los Angeles, CA, 2003.

    Google Scholar 

  • X. Lin and N. E. Breslow. Bias correction in generalized linear mixed models with multiple components of dispersion. Journal of the American Statistical Association, 91:1007–1016, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  • Q. Liu and D. A. Pierce. A note on Gauss-Hermite quadrature. Biometrika, 81:624–629, 1994.

    MATH  MathSciNet  Google Scholar 

  • N. T. Longford. A quasi-likelihood adaptation for variance component analysis. In American Statistical Association Proceedings of the Statistical Computing Section, pages 137–142. 1988.

    Google Scholar 

  • N. T. Longford. VARCL: Software for Variance Component Analysis of Data with Nested Random Effects (Maximum Likelihood). Educational Testing Service, Princeton, NJ, 1988.

    Google Scholar 

  • N. T. Longford. Logistic regression with random coefficients. Computational Statistics & Data Analysis, 17:1–15, 1994.

    Article  MATH  Google Scholar 

  • T. A. Louis. Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society, Series B, 44:226–233, 1982.

    MATH  MathSciNet  Google Scholar 

  • K. G. Manton, E. Stallard, and J. W. Vaupel. Alternative models for the heterogeneity of mortality risks among the aged. Journal of the American Statistical Association, 81:635–644, 1986.

    Article  Google Scholar 

  • P. McCullagh and J. A. Nelder. Generalized Linear Models, 2nd edition. Chapman & Hall, London, 1989.

    MATH  Google Scholar 

  • C. E. McCulloch. Maximum likelihood algorithms for generalized linear mixed models. Journal of the American Statistical Association, 92:162–170, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  • D. McFadden. A method of simulated moments for estimation of discrete response models without numerical integration. Econometrica, 57:995–1026, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  • W. H. Mosley and L. C. Chen. An analytical framework for the study of child survival in developing countries. Population and Development Review, 10:25–45, 1984.

    Article  Google Scholar 

  • J. Myles and D. G. Clayton. GLMMGibbs: An R Package for Estimating Bayesian Generalised Linear Mixed Models by Gibbs Sampling. Comprehensive R Archive Network (devel section), 2001. URL http://cran.r-project.org

    Google Scholar 

  • J. C. Naylor and A. F. M. Smith. Applications of a method for the efficient computation of posterior distributions. Applied Statistics, 31:214–225, 1980.

    Article  MathSciNet  Google Scholar 

  • J. A. Nelder and R. Wedderburn. Generalized linear models. Journal of the Royal Statistical Society, Series B, 135:370–384, 1972.

    Article  Google Scholar 

  • J. M. Neuhaus, J. D. Kalbfleisch, and W. W. Hauck. A comparison of cluster-specific and population-averaged approaches to analyzing correlated binary data. International Statistical Review, 59:25–35, 1991.

    Google Scholar 

  • E. S. W. Ng, J. R. Carpenter, H. Goldstein, and J.Rasbash. Estimation in generalised linear mixed models with binary outcomes by simulated maximum likelihood. Statistical Modelling, 6:23–42, 2006.

    Article  MathSciNet  Google Scholar 

  • D. Oakes. A model for association in bivariate survival data. Journal of the Royal Statistical Society, Series B, 44:414–422, 1982.

    MATH  MathSciNet  Google Scholar 

  • A. R. Pebley, N. Goldman, and G. Rodríguez. Prenatal and delivery care and childhood immunization in Guatemala: Do family and community matter? Demography, 33:231–247, 1996.

    Article  Google Scholar 

  • J. C. Pinheiro and D. M. Bates. Approximations to the log-likelihood function in the nonlinear mixed-effects model. Journal of Computational and Graphical Statistics, 4:12–35, 1995.

    Article  Google Scholar 

  • J. C. Pinheiro and D. M. Bates. Mixed-Effects Models in S and S-PLUS. Springer, New York, 2000.

    MATH  Google Scholar 

  • R. L. Prentice and L. A. Gloeckler. Regression analysis of grouped survival data with application to breast cancer data. Biometrics, 34:57–67, 1978.

    Article  MATH  Google Scholar 

  • W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in C, 2nd edition. Cambridge University Press, Cambridge, MA, 1992.

    MATH  Google Scholar 

  • S. Rabe-Hesketh, A. Skrondal, and A. Pickles. Reliable estimation of generalized linear mixed models using adaptive quadrature. The Stata Journal, 2:1–21, 2002.

    Google Scholar 

  • A. E. Raftery and S. M. Lewis. How many iterations in the Gibbs sampler? In J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, editors, Bayesian Statistics 4, pages 763–773. Oxford University Press, Oxford, UK, 1992.

    Google Scholar 

  • A. E. Raftery and S. M. Lewis. Implementing MCMC. In W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, editors, Markov Chain Monte Carlo in Practice, pages 115–130. Chapman & Hall, London, 1996.

    Google Scholar 

  • S. W. Raudenbush, M.-L. Yang, and M. Yosef. 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, 2000.

    Article  MathSciNet  Google Scholar 

  • G. O. Roberts. Markov chain concepts related to sampling algorithms. In W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, editors, Markov Chain Monte Carlo in Practice, pages 45–57. Chapman & Hall, London, 1996.

    Google Scholar 

  • G. Rodríguez. Event history analysis. In S. Kotz, C. B. Read, and D. L. Banks, editors, Encyclopedia of Statistical Sciences, Update Volume, pages 222–230. Wiley, New York, 1997.

    Google Scholar 

  • G. Rodríguez and I. Elo. Intra-class correlation in random-effects models for binary data. The Stata Journal, 3:32–46, 2003.

    Google Scholar 

  • G. Rodríguez and N. Goldman. An assessment of estimation procedures for multilevel models with binary responses. Journal of the Royal Statistical Society, Series A, 158:73–89, 1995.

    Google Scholar 

  • G. Rodríguez and N. Goldman. Improved estimation procedures for multilevel models with binary response: A case-study. Journal of the Royal Statistical Society, Series A, 164:339–355, 2001.

    MATH  Google Scholar 

  • N. Sastry. A nested frailty model for survival data, with an application to the study of child survival in northeast Brazil. Journal of the American Statistical Association, 92:426–435, 1997.

    Article  MATH  Google Scholar 

  • R. Schall. Estimation in generalized linear models with random effects. Biometrika, 78:719–727, 1991.

    Article  MATH  Google Scholar 

  • S. G. Self and K.-Y. Liang. Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under non-standard conditions. Journal of the American Statistical Association, 82:605–610, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  • D. J. Spiegelhalter, A. Thomas, N. G. Best, and W. R. Gilks. UGS: Bayesian Inference Using Gibbs Sampling. Medical Research Council Biostatistics Unit, Cambridge, UK, 1996.

    Google Scholar 

  • R. Stiratelli, N. M. Laird, and J. H. Ware. Random-effects models for serial observations with binary response. Biometrics, 40:961–971, 1984.

    Article  Google Scholar 

  • D. O. Stram and J. W. Lee. Variance components testing in the longitudinal mixed-effects model. Biometrics, 50:1171–1177, 1994.

    Article  MATH  Google Scholar 

  • T. M. Therneau and P. M. Grambsch. Modeling Survival Data: Extending the Cox Model. Springer, New York, 2000.

    MATH  Google Scholar 

  • R. A. Thisted. Elements of Statistical Computing: Numerical Computation. Chapman & Hall, London, 1988.

    MATH  Google Scholar 

  • J. Vaupel, K. G. Manton, and E. Stallard. The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography, 16:439–454, 1979.

    Article  Google Scholar 

  • J. Vaupel and A. Yashin. Heterogeneity’s ruses: Some surprising effects of selection on population dynamics. American Statistician, 39:176–185, 1985.

    Article  MathSciNet  Google Scholar 

  • J. Whitehead. Fitting Cox's regression model to survival data using GLIM. Applied Statistics, 29:268–275, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  • G. Y. Wong and W. M. Mason. The hierarchical logistic regression model for multilevel analysis. Journal of the American Statistical Association, 80:513–524, 1985.

    Article  Google Scholar 

  • S. L. Zeger and M. R. Karim. Generalized linear models with random effects: A Gibbs sampling approach. Journal of the American Statistical Association, 86: 79–86, 1991.

    Article  MathSciNet  Google Scholar 

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Rodríguez, G. (2008). Multilevel Generalized Linear Models. In: Leeuw, J.d., Meijer, E. (eds) Handbook of Multilevel Analysis. Springer, New York, NY. https://doi.org/10.1007/978-0-387-73186-5_9

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