Maximum Likelihood for Generalized Linear Models with Nested Random Effects via High-Order, Multivariate Laplace Approximation

Journal of Computational and Graphical Statistics

Volumn 9, Issue 1, 2000, Pages 141-157

  Raudenbush, Stephen W ^a   Yang, Meng Li ^b   Yosef, Matheos ^c  

^a UNIVERSITY OF MICHIGAN   (United States)

^b SOUTHERN TAIWAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Taiwan)

^c MICHIGAN STATE UNIVERSITY   (United States)

Author keywords

Hierarchical models; Mixed models; Numerical integration

Indexed keywords

EID: 77950035264     PISSN: 10618600     EISSN: 15372715     Source Type: Journal    
DOI: 10.1080/10618600.2000.10474870     Document Type: Article

References (58)

1. Aitken, M., and Longford, N. (1986), “Statistical Modeling Issues in School Effectiveness Studies,” Journal of Royal Statistical Society, Ser. A, 149, 1-43.
2. Anderson, D. A., and Aitkin, M. (1985), “Variance Component Models with Binary Response: Interviewer Variability,” Journal of the Royal Statistical Society, Ser. B, 47, 204-210.
3. Belin, T., Diffendal, J., Mack, S., Rubin, D., Schafer, J., and Zazlavsky, A. (1993), “Hierarchical Logistic Regression Models for Imputation of Unresolved Enumeration Status in Undercount Estimation,” Journal of the American Statistical Association, 88, 1149-1159.
4. Berkey, D. C., Hoaglin, F., Mosteller, F., and Colditz, G. A. (1995), “A Random Effects Regression Model for Meta-Analysis,” Statistics in Medicine, 14, 395-411.
5. Bock, R. D. (1989), Multilevel Analysis of Educational Data, New York: Academic Press.
6. Breslow, N. E., and Clayton, D. G. (1993), “Approximate Inference in Generalized Linear Mixed Models,” Journal of the American Statistical Association, 88, 9-25.
7. Breslow, N. E., and Lin, X. (1995), “Bias Correction in Generalised Linear Mixed Models With a Single Component of Dispersion," Biometrika, 82, 81-91.
8. Bryk, A., and Raudenbush, S. (1992), Hierarchical Linear Models for Social and Behavioral Research: Applications and Data Analysis Methods, Newbury Park, CA: Sage.
9. Bryk, A. S., Raudenbush, S. W., and Congdon, R. (1996), HLM: Hierarchical Linear and Nonlinear Modeling with the HLM/2L and HLM/3L Programs, Chicago: Scientific Software International, Inc.
10. Chan, W. C. (1994), “Toward a Multilevel Model: The Case for Poisson Distributed Data,” unpublished Ph.D. dissertation, Michigan State University, Department of Counseling, Educational Psychology, and Special Education.
11. Cox, D., and Hinkley, D. (1974), Theoretical Statistics, Boca Raton, FL: CRC Press.
12. DeLeeuw, J., and Kreft, I. (1986), “Random Coefficient Models for Multilevel Analysis,” Journal of Educational Statistics, 11, 57-85.
13. Dempster, A., Laird, N., and Rubin, D. (1977), “Maximum Likelihood From Incomplete Data via the EM Algorithm,” Journal of the Royal Statistical Society, Ser. B, 39, 1-8.
14. Dempster, A. P., Rubin, D. B., and Tsutakawa, R. K. (1981), “Estimation in Covariance Components Models," Journal of American Statistical Association, 76, 341-353.
15. DerSimonian, R., and Laird, N. (1986), “Meta-Analysis in Clinical Trials," Controlled Clinical Trials, 7, 177-188.
16. Diggle, P. J., Liang, K. Y., and Zeger, S. L. (1994), Analysis of Longitudinal Data, Oxford: Clarendon Press.
17. Gilks, W. R. (1987), “Some Applications of Hierarchical Models in Kidney Transplantation,” The Statistician, 36, 127-136.
18. Goldstein, H. (1986), “Multilevel Mixed Linear Model Analysis Using Iterative Generalized Least Squares,” Biometrika, 73, 43-56.
19. Goldstein, H. (1991), “Nonlinear Multilevel Models, With an Application to Discrete Response Data,” Biometrika, 78, 45-51.
20. Goldstein, H. (1995), Multilevel Statistical Models (2nd ed.), New York: Wiley.
21. Goldstein, H., and Rasbash, J. (1996), “Improved Approximations for Multilevel Models with Binary Responses," Journal of the Royal Statistical Society, Ser. B, 159, 505-513.
22. Green, P. J. (1984), “Iteratively Reweighted Least Squares for Maximum Likelihood Estimation, and Some Robust and Resistant Alternatives,” Journal of the Royal Society, Set. B, 46, 149-192.
23. Hedeker, D., and Gibbons, R. D. (1994), “A Random Effects Ordinal Regression Model for Multilevel Analysis,” Biometrics, 50, 933-944.
24. Hedeker, D., and Gibbons, R. D. (1996), “MIXOR: A Computer Program for Mixed-Effects Ordinal Probit and Logistic Regression Analysis,” Computer Methods and Programs in Biomedicine, 49, 157-176.
25. Karim, M. R. (1991), “Generalized Linear Models with Random Effects,” unpublished Ph.D. dissertation, Department of Biostatistics, Johns Hopkins University.
26. Kass, R. E., Tierney, L., and Kadane, J. B. (1990), “The Validity of Posterior Expansions Based on Laplace’s Method,” in Bayesian and Likelihood Methods in Statistics and Econometrics, eds. S. Geisser, J. S. Hodges, S. J. Press, and A. Zellner, North-Holland: Elsevier.
27. Laird, N., and Ware, J. (1982), “Random Effects Models for Longitudinal Data,” Biometrika, 65, 581-590.
28. Lee, Y., and Nelder, J. A. (1996), “Hierarchical Generalized Linear Models,” Journal of the Royal Statistical Society, Series B, 58, 619-678.
29. 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.
30. Lindley, D. (1980), “Approximate Bayesian Methods,” in Bayesian Statistics, eds. J. Bernardo, M. H. DeGroot, D. V. Lindley, and A. F. M. Smith, Valencia: Valencia University Press, 223-245.
31. Lindley, D., and Smith, A. (1972), “Bayes Estimates for the Linear Model,” Journal of the Royal Statistical Society, Ser. B, 34, 1-41.
32. Lindstrom, M. J., and Bates, D. M. (1988), “Newton-Raphson and EM Algorithms for Linear Mixed-Effects Models for Repeated-Measures Data,” Journal of the American Statistical Association, 83, 1014-1022.
33. Liu, Q., and Pierce, D. A. (1993), “Heterogeneity in Mantel-Haenszel-type Models,” Biometrika, 80, 543-556.
34. Longford, N. (1987), “A Fast Scoring Algorithm for Maximum Likelihood Estimation in Unbalanced Mixed Models With Nested Random Effects,” Biometrika, 74, 817-827.
35. Longford, N. (1993), Random Coefficient Models, Oxford: Clarendon Press.
36. Magnus, J. R., and Neudecker, H. (1988), Matrix Differential Calculus with Applications in Statistics and Econometrics, New York: Wiley.
37. McCullagh, P., and Nelder, J. A. (1989), Generalized Linear Models (2nd ed.), London: Chapman and Hall.
38. McGilchrist, C. E. (1994), “Estimation in Generalized Mixed Models,” Journal of the Royal Statistical Society, Ser. B, 56, 61-69.
39. Pinheiro, J. C., and Bates, D. M. (1995), “Approximations to the Log-Likelihood Function in the Non-Linear Mixed-Effects Model,” Journal of Computational and Graphical Statistics, 4, 12-35.
40. Raudenbush, S. W., and Bryk, A.S. (1985), “Empirical Bayes Meta-Analysis,” Journal of Educational Statistics, 10, 75-98.
41. Raudenbush, S. W., and Bryk, A.S. (1986), “A Hierarchical Model for Studying School Effects,” Sociology of Education, 59, 1-17.
42. Raudenbush, S. W., and Bhumirat, C. (1992), “The Distribution of Resources for Primary Education and its Consequences for Educational Achievement in Thailand,” International Journal of Educational Research, 143-164.
43. Rodriguez, G., and Goldman, N. (1995), “An Assessment of Estimation Procedures for Multilevel Models with Binary Responses,” Journal of the Royal Statistical Society, Ser. A, 158, 73-89.
44. Rubin, D. (1981), “Estimation in Parallel Randomized Experiments,” Journal of Educational Statistics, 6, 337-401.
45. Schall, R. (1991), “Estimation in Generalized Linear Models With Random Effects," Biometrika, 40, 719-727.
46. Seltzer, M. H. (1993), “Sensitivity Analysis for Fixed Effects in Hierarchical Model: A Gibbs Sampling Approach,” Journal of Educational Statistics, 18, 207-235.
47. Shun, Z. (1997), “Another Look at the Salamander Mating Data: A Modified Laplace Approximation Approach,” Journal of American Statistical Association, 92, 341-349.
48. Shun, Z., and McCullagh, P. (1995), “Laplace Approximation of High Dimensional Integrals,” Journal of the Royal Statistical Society, Ser. B, 57, 749-760.
49. Solomon, P. J., and Cox, D. R. (1992), “Nonlinear Components of Variance Models,” Biometrika, 79, 1-11.
50. Strenio, J., Weisberg, H., and Bryk, A. (1983), “Empirical Bayes Estimation of Individual Growth Curve Parameters and Their Relationship to Covariates,” Biometrics, 39, 71-86.
51. Stiratelli, R., Laird, N., and Ware, J. (1984), “Random Effects Models for Serial Observations With Binary Responses," Biometrics, 40, 961-971.
52. Thum, Y. (1997), “Hierarchical Linear Models for Multivariate Outcomes,” Journal of Educational and Behavioral Statistics, 22, 77-108.
53. Tsutakawa, R. K. (1985), “Estimation of Cancer Mortality Rates: A Bayesian Analysis of Small Frequencies,” Biometrics, 41, 69-79.
54. Wei, G. C. G., and Tanner, M. A. (1990), “A Monte Carlo Implementation of the EM Algorithm and the Poor Man’s Augmentation Algorithms,” Journal of the American Statistical Association, 86, 669-704.
55. Wolfinger, R. (1993), “Laplace’s Approximation for Nonlinear Mixed Models,” Biometrika, 80, 791-795.
56. Wolfinger, R. (1999), “Nonlinear Mixed Models: A Future Direction,” presented at the Annual Interface Conference, Shaumberg, IL, June 10, 1999.
57. Wong, G. Y., and Mason, W. M. (1985), “The Hierarchical Logistic Regression Model for Multilevel Analysis,” Journal of the American Statistical Association, 80, 513-524.
58. Yang, M. (1998), “Increasing the Efficiency in Estimating Multilevel Bernoulli Models,” unpublished Ph.D. dissertation, Michigan State University, Department of Counseling, Educational Psychology, and Special Education.