Hierarchical Models: A Current Computational Perspective

References

• Besag , J. , Green , P. , Higdon , D. and Mengersen , K. 1995 . “Bayesian Computation and Stochastic Simulation” . Statistical Science , 10 : 3 – 66 . (with discussion)
   Web of Science ®Google Scholar
• Booth , J. G. and Hobert , J. P. 1999 . “Maximizing Generalized Linear Mixed Model Likelihoods With an Automated Monte Carlo EM Algorithm” . Journal of the Royal Statistical Society , 61 : 265 – 285 . Ser. B
   Google Scholar
• Booth , J. G. , Casella , G. and Hobert , J. P. 2000 . “Negative Binomial Log-Linear Mixed Models,” , University of Florida, Dept. of Statistics . technical report
   Google Scholar
• Breslow , N. E. and Clayton , D. G. 1993 . “Approximate Inference in Generalized Linear Mixed Models” . Journal of the American Statistical Association , 88 : 9 – 25 .
   Web of Science ®Google Scholar
• Chan , J. S. K. and Kuk , A. Y. C. 1997 . “Maximum Likelihood Estimation for Probit-Linear Mixed Models With Correlated Random Effects” . Biometrics , 53 : 86 – 97 .
   Web of Science ®Google Scholar
• Chan , K. S. 1993 . “Asymptotic Behavior of the Gibbs Sampler” . Journal of the American Statistical Association , 88 : 320 – 326 .
   Web of Science ®Google Scholar
• Chan , K. S. and Geyer , C. J. 1994 . Comment on “Markov Chains for Exploring Posterior Distributions” . The Annals of Statistics , 22 : 1747 – 1757 . by L. Tierney
   Web of Science ®Google Scholar
• Cox , D. R. 1983 . “Some Remarks on Overdispersion” . Biometrika , 70 : 269 – 274 .
   Web of Science ®Google Scholar
• Dempster , A. P. , Laird , N. M. and Rubin , D. B. 1977 . “Maximum Likelihood From Incomplete Data via the EM Algorithm” . Journal of the Royal Statistical Society , 39 : 1 – 38 . (with discussion), Ser. B
   Google Scholar
• Gelfand , A. E. , Hills , S. E. , Racine-Poon , A. and Smith , A. F. M. 1990 . “Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling” . Journal of the American Statistical Association , 85 : 972 – 985 .
   Web of Science ®Google Scholar
• Gelfand , A. E. and Smith , A. F. M. 1990 . “Sampling-Based Approaches to Calculating Marginal Densities” . Journal of the American Statistical Association , 85 : 398 – 409 .
   Web of Science ®Google Scholar
• Geyer , C. J. 1992 . “Practical Markov Chain Monte Carlo” . Statistical Science , 7 : 473 – 511 . (with discussion)
   Google Scholar
• Hobert , J. P. and Casella , G. 1996 . “The Effect of Improper Priors on Gibbs Sampling in Hierarchical Linear Mixed Models” . Journal of the American Statistical Association , 91 : 1461 – 1473 .
   Web of Science ®Google Scholar
• Hobert , J. P. and Geyer , C. J. 1998 . “Geometric Ergodicity of Gibbs and Block Gibbs Samplers for a Hierarchical Random-Effects Model” . Journal of Multivariate Analysis , 67 : 414 – 430 .
   Web of Science ®Google Scholar
• Karim , M. R. and Zeger , S. L. 1992 . “Generalized Linear Models With Random Effects; Salamander Mating Revisited” . Biometrics , 48 : 631 – 644 .
   PubMed Web of Science ®Google Scholar
• Kass , R. E. and Steffey , D. 1989 . “Approximate Bayesian Inference in Conditionally Independent Hierarchical Models (Parametric Empirical Bayes Models)” . Journal of the American Statistical Association , 84 : 717 – 726 .
   Web of Science ®Google Scholar
• Lee , Y. and Nelder , J. A. 1996 . “Hierarchical Generalized Linear Models” . Journal of the Royal Statistical Society , 58 : 619 – 678 . (with discussion), Ser. B
   Google Scholar
• Levine , R. A. and Casella , G. 1998 . “Implementations of the Monte Carlo EM Algorithm,” , University of California, Davis, Intercollege Division of Statistics . technical report
   Google Scholar
• McCullagh , P. and Nelder , J. A. 1989 . Generalized Linear Models , London : Chapman and Hall, 2nd ed .
   Google Scholar
• McCulloch , C. E. 1994 . “Maximum Likelihood Variance Components Estimation for Binary Data” . Journal of the American Statistical Association , 89 : 330 – 335 .
   Web of Science ®Google Scholar
• McCulloch , C. E. 1997 . “Maximum Likelihood Algorithms for Generalized Linear Mixed Models” . Journal of the American Statistical Association , 92 : 162 – 170 .
   Web of Science ®Google Scholar
• McLachlan , G. J. and Krishnan , T. 1997 . The EM Algorithm and Extensions , New York : Wiley .
   Google Scholar
• Mengersen , K. L. and Tweedie , R. L. 1996 . “Rates of Convergence of the Hastings and Metropolis Algorithms” . The Annals of Statistics , 24 : 101 – 121 .
   Web of Science ®Google Scholar
• Meyn , S. P. and Tweedie , R. L. 1993 . Markov Chains and Stochastic Stability , New York : Springer-Verlag .
   Google Scholar
• Mykland , P. , Tierney , L. and Yu , B. 1995 . “Regeneration in Markov Chain Samplers” . Journal of the American Statistical Association , 90 : 233 – 241 .
   Web of Science ®Google Scholar
• Natarajan , R. and Kass , R. E. 2000 . “Reference Bayesian Methods for Generalized Linear Mixed Models” . Journal of the American Statistical Association , 95 : 227 – 237 .
   Web of Science ®Google Scholar
• Natarajan , R. and McCulloch , C. E. 1995 . “A Note on the Existence of the Posterior Distribution for a Class of Mixed Models for Binomial Responses” . Biometrika , 82 : 639 – 643 .
   Web of Science ®Google Scholar
• Natarajan , R. and McCulloch , C. E. 1998 . “Gibbs Sampling with Diffuse Proper Priors: A Valid Approach to Data-Driven Inference?” . Journal of Computational and Graphical Statistics , 7 : 267 – 277 .
   Web of Science ®Google Scholar
• Roberts , G. O. and Rosenthal , J. S. 1998 . “Markov Chain Monte Carlo: Some Practical Implications of Theoretical Results” . Canadian Journal of Statistics , 26 : 5 – 31 . (with discussion)
   Web of Science ®Google Scholar
• Roberts , G. O. and Rosenthal , J. S. 1999 . “Convergence of Slice Sampler Markov Chains” . Journal of the Royal Statistical Society , 61 : 643 – 660 . Ser. B
   Google Scholar
• Roberts , G. O. and Tweedie , R. L. 1996 . “Geometric Convergence and Central Limit Theorems for Multidimensional Hastings and Metropolis Algorithms” . Biometrika , 83 : 95 – 110 .
   Web of Science ®Google Scholar
• Rosenthal , J. S. 1995 . “Minorization Conditions and Convergence Rates for Markov Chain Monte Carlo” . Journal of the American Statistical Association , 90 : 558 – 566 .
   Web of Science ®Google Scholar
• Strawderman , W. E. 1971 . “Proper Bayes Minimax Estimators of the Multivariate Normal Mean” . The Annals of Mathematical Statistics , 42 : 385 – 388 .
   Google Scholar
• Thall , P. F. and Vail , S. C. 1990 . “Some Covariance Models for Longitudinal Count Data With Overdispersion” . Biometrics , 46 : 657 – 671 .
   PubMed Web of Science ®Google Scholar
• Tierney , L. 1995 . “Markov Chains for Exploring Posterior Distributions” . The Annals of Statistics , 22 : 1701 – 1762 . (with discussion)
   Web of Science ®Google Scholar
• Tierney , L. , Kass , R. E. and Kadane , J. B. 1989 . “Fully Exponential Laplace Approximations to Expectations and Variances of Nonpositive Functions” . Journal of the American Statistical Association , 84 : 710 – 716 .
   Web of Science ®Google Scholar
• Vaida , F. and Meng , X-L. “A Flexible Gibbs–EM Algorithm for Generalized Linear Mixed Models With Binary Response,” . Proceedings of the 13th International Workshop on Statistical Modeling . Edited by: Marx , B. and Friedl , H. pp. 374 – 379 .
   Google Scholar
• Wei , G. C. G. and Tanner , M. A. 1990 . “A Monte Carlo Implementation of the EM Algorithm and the Poor Man's Data Augmentation Algorithms” . Journal of the American Statistical Association , 85 : 699 – 704 .
   Web of Science ®Google Scholar