Bayes Factors and Approximations for Variance Component Models

Journal of the American Statistical Association

Volumn 94, Issue 448, 1999, Pages 1242-1253

  Pauler, Donna K ^a   Wakefield, Jonathan C ^b   Kass, Robert E ^c  

^a HARVARD SCHOOL OF PUBLIC HEALTH   (United States)

^b ST MARY S HOSPITAL   (United Kingdom)

^c CARNEGIE MELLON UNIVERSITY   (United States)

Author keywords

Boundary problem; Importance sampling; Laplace's method; Reference prior; Rejection method; Schwarz criterion

Indexed keywords

EID: 0442293879     PISSN: 01621459     EISSN: 1537274X     Source Type: Journal    
DOI: 10.1080/01621459.1999.10473877     Document Type: Article

References (46)

1. Berger, J. O., and Pericchi, L. R. (1996), "The Intrinsic Bayes Factor for Model Selection and Prediction," Journal of the American Statistical Association, 91, 109-122.
2. Box, G. E. P., and Tiao, G. C. (1973), Bayesian Inference in Statistical Analysis, New York: Wiley.
3. Chen, C. (1985), "On Asymptotic Normality of Limiting Density Functions With Bayesian Implications," Journal of the Royal Statistical Society, Ser B, 47, 540-546.
4. Chen, M. H., and Shao, Q. M. (1997), "On Monte Carlo Methods for Estimating Ratios of Normalizing Constants," The Annals of Statistics, 25, 1563-1594.
5. Chernoff, H. (1954), "On the Distribution of the Likelihood Ratio," Annals of Mathematical Statistics, 25, 573-578.
6. Christensen, R. (1996), "Exact Tests for Variance Components," Biometrics, 52, 309-314.
7. Chib, S. (1995), "Marginal Likelihood from Gibbs Output " Journal of the American Statistical Association, 90, 1313-1321.
8. Diciccio, T. J., Kass, R. E., Raftery, A. E., and Wasserman, L. (1997), "Computing Bayes Factors by Combining Simulation and Asymptotic Approximations," Journal of the American Statistical Association, 92, 903-915.
9. Erkanli, A. (1994), "Laplace Approximations for Posterior Expectations When the Mode Occurs at the Boundary of the Parameter Space," Journal of the American Statistical Association, 89, 250-258.
10. Evans, M., and Swartz, T. (1995), Rejoinder to "Methods for Approximating Integrals in Statistics With Special Emphasis on Bayesian Integration Problems," Statistical Science, 11, 54-64.
11. Feng, Z., and McCulIoch, C. E. (1992), "Statistical Inference Using Maximum Likelihood Estimatin and the Generalized Likelihood Ratio When the True Parameter is on the Boundary of the Parameter Space," Statistics & Probability Letters, 13, 325-332.
12. Gilks, W. R., and Roberts, G. O. (1996), "Strategies for Improving MCMC," in Markov Chain Monte Carlo in Practice, eds. W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, New York: Chapman and Hall.
13. Harville, D. A. (1977), "Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems," Journal of the American Statistical Association, 72, 320-340.
14. Hill, B. M. (1965), "Inference About Variance Components in the One-Way Model," Journal of the American Statistical Association, 60, 806-825.
15. Hsiao, C. K. (1997), "Approximate Bayes Factors when a Mode Occurs on the Boundary," Journal of the American Statistical Association, 92, 656-663.
16. Jeffreys, H. (1961), Theory of Probability. (3rd ed). U.K.: Oxford University Press.
17. Kass, R. E., and Raftery, A. E. (1995), "Bayes Factors and Model Uncer-tainty," Journal of the American Statistical Association, 90, 773-795.
18. Kass, R. E., Tierney, L., and Kadane, J. B. (1990), "The Validity of Posterior Expansions Based on Laplace's Method," in Essays in Honor of George Barnard, eds. S. Geisser, J. Hodges, S. Press, and A. Zellner, Amsterdam, North-Holland, pp. 473-488.
19. Kass, R. E., and Vaidyanathan, S. K. (1992), "Approximate Bayes Factors and Orthogonal Parameters, With Application to Testing Equality of Two Binomial Proportions," Journal of the Royal Statistical Society. Ser. B, 54, 129-144.
20. Kass, R. E.f and Wasserman, L. (1995), "A Reference Bayesian Test for Nested Hypotheses and Its Relationship to the Schwarz Criterion," Journal of the American Statistical Association, 90, 928-934.
21. Khuri, A. I., and Littell, R. C. (1987), "Exact Tests for the Main Effects Variance Components in an Unbalanced Random Two-Way Model," Bio-metrics, 43, 545-560.
22. Lecam, L., and Yang, G. L. (1990), Asymptotics in Statistics, New York: Springer-Verlag.
23. Lin, X. (1997), "Variance Component Testing in Generalised Linear Models with Random Effects," Biometrika, 84, 309-326.
24. Meng, X. L., and Wong, W. H. (1996), "Simulating Ratios of Normalizing Constants via a Simple Identity: A Theoretical Exploration," Statistica Sinica, 6, 831-860.
25. Miller, J. J. (1977), "Asymptotic Properties of Maximum Likelihood Estimates in the Mixed Model of the Analysis of Variance," The Annals of Statistics, 5, 746-762.
26. Nclder, J. A. and Mead, R (1965), "A Simplex Method for Function Min-imization," Computer J., 7, 308-313.
27. O'Hagan, A. (1995), "Fractional Bayes Factors for Model Comparison," Journal of the Royal Statistical Society, Ser. B, 57, 99-138.
28. Ofversten, J. (1993), "Exact Tests for Variance Components in Unbalanced Mixed Linear Models," Biometrics, 49, 45-57.
29. Pauler, D. K. (1996), "The Schwarz Criterion for Mixed Effects Models," Ph.D. thesis, Carnegie Mellon University.
30. Pauler, D. K. (1998), 'The Schwarz Criteria and Related Methods for Normal Linear Models," Biometrika, 85, 13-27.
31. Ripley, B. (1987), Shastic Simulation, New York: Wiley.
32. SAS Institute, Inc. (1992), SAS/STAT Software: Changes and Enhancements, Release 6.12.
33. Schervish, M. J. (1995), Theory of Statistics, New York: Springer-Verlag.
34. Schwarz, G. (1978), "Estimating the Dimension of a Model," The Annals of Statistics, 6, 461-464.
35. Searle, S. R., Casella, G., and McCulloch, C. E. (1992), Variance Components, New York: Wiley.
36. Self, S. G., and Liang, K. Y. (1987), "Asymptotic Properties of Maximum Likelihood Estimators and the Likelihood Ratio Tests Under Nonstandard Conditions," Journal of the American Statistical Association, 82, 605-610.
37. Skouras, K., and Dawid, A. P. (1998), "On Efficient Point Prediction Systems," Journal of Royal Statistical Society Series B 60, 765-780.
38. Stram, D. O., and Lee, J. W. (1994), "Variance Components Testing in the Longitudinal Mixed Effects Model," Biometrics, 50, 1171-1177.
39. Sun, L., Hsu, J.S.J., Guttman, I., and Leonard, T. (1996), "Bayesian Methods for Variance Component Models," Journal of the American Statistical Association, 91, 743-752.
40. Tierney, L., and Kadane, J. B. (1986), "Accurate approximations for Posterior Moments and Marginal Densities," Journal of the American Statistical Association, 81, 82-86.
41. Verdinelli, I., and Wasserman, L. (1995), "Computing Bayes Factors Using a Generalization of the Savage-Dickey Density Ratio," Journal of the American Statistical Association, 90, 614-618.
42. Vounatsou, P., and Smith, A. (1996), "Simulation-Based Bayesian Inferences of the Variance Components in Mixed Linear Models," Journal of Statistical Planning and Inference, 59, 139-161.
43. Wakefield, J. C. (1996), "Bayesian Individualization via Sampling-Based Methods," Journal of Pharmacokinetics and Biopharmaceutics, 24,103-131.
44. Wakefield, J. C., Gelfand, A. E. and Smith, A. F. M. (1991), "Efficient Computation of Random Variatcs via the Ratio-of-Uniforms Method," Statistics and Computing, 1, 129-133.
45. Wolfinger, R. D., and Kass, R. E. (1996), "Bayesian Analysis of Variance Component Models via Rejection Sampling," technical report, Carnegie Mellon University, Dept, of Statistics.
46. Yang, R., and Chen, M.-H. (1995), "Bayesian Analysis for Random Coefficient Regression Models Using Noninformative Priors," Journal of Multivariate Analysis, 55, 283-311.