Bayesian Analysis

Sensitivity analysis in Bayesian generalized linear mixed models for binary data

Małgorzata Roos and Leonhard Held

Full-text: Open access

Abstract

Generalized linear mixed models (GLMMs) enjoy increasing popularity because of their ability to model correlated observations. Integrated nested Laplace approximations (INLAs) provide a fast implementation of the Bayesian approach to GLMMs. However, sensitivity to prior assumptions on the random effects precision parameters is a potential problem. To quantify the sensitivity to prior assumptions, we develop a general sensitivity measure based on the Hellinger distance to assess sensitivity of the posterior distributions with respect to changes in the prior distributions for the precision parameters. In addition, for model selection we suggest several cross-validatory techniques for Bayesian GLMMs with a dichotomous outcome. Although the proposed methodology holds in greater generality, we make use of the developed methods in the particular context of the well-known salamander mating data. We arrive at various new findings with respect to the best fitting model and the sensitivity of the estimates of the model components.

Article information

Source
Bayesian Anal., Volume 6, Number 2 (2011), 259-278.

Dates
First available in Project Euclid: 13 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.ba/1339612046

Digital Object Identifier
doi:10.1214/11-BA609

Mathematical Reviews number (MathSciNet)
MR2806244

Zentralblatt MATH identifier
1330.62150

Subjects
Primary: 62F15: Bayesian inference
Secondary: 62J12: Generalized linear models 62P10: Applications to biology and medical sciences

Keywords
Bayesian Analysis Binary data Generalized Linear Mixed Models Hellinger distance Integrated nested Laplace approximations Model choice Sensitivity analysis

Citation

Roos, Małgorzata; Held, Leonhard. Sensitivity analysis in Bayesian generalized linear mixed models for binary data. Bayesian Anal. 6 (2011), no. 2, 259--278. doi:10.1214/11-BA609. https://projecteuclid.org/euclid.ba/1339612046


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References

  • Besag, J., Green, P., Higdon, D., and Mengersen, K. (1995). "Bayesian computation and stochastic systems." Statistical Science, 10(1): 3–66.
  • Bhattacharyya, A. (1943). "On a measure of divergence between two statistical populations defined by their probability distributions." Bulletin of the Calcutta Mathematical Society, 35: 99–109.
  • Box, G. and Tiao, G. (1973). Bayesian Inference in Statistical Analysis. Addison–Wesley Publishing Company.
  • Breslow, N. E. and Clayton, D. G. (1993). "Approximate inference in generalized linear mixed models." Journal of the American Statistical Association, 88(421): 9–25.
  • Browne, W. and Draper, D. (2006). "A comparison of Bayesian and likelihood-based methods for fitting multilevel models." Bayesian Analysis, 1(3): 473–514.
  • Chan, J. S. K. and Kuk, A. Y. C. (1997). "Maximum likelihood estimation for probit-linear mixed models with correlated random effects." Biometrics, 53(1): 86–97.
  • Fong, Y., Rue, H., and Wakefield, J. (2010). "Bayesian inference for generalized linear mixed models." Biostatistics, 11(3): 397–412.
  • Geisser, S. (1993). Predictive Inference: An Introduction. Chapman & Hall, Inc.
  • Gelfand, A. and Smith, A. (1990). "Sampling-based approaches to calculating marginal densities." Journal of the American Statistical Association, 85(410): 398–409.
  • Gelman, A. (2006). "Prior distributions for variance parameters in hierarchical models (Comment on Article by Browne and Draper)." Bayesian Analysis, 1(3): 515–534.
  • Gneiting, T. and Raftery, A. E. (2007). "Strictly proper scoring rules, prediction, and estimation"." Journal of the American Statistical Association, 102(477): 359–378.
  • Held, L., Schrödle, B., and Rue, H. (2010). "Posterior and cross-validatory predictive checks: A comparison of MCMC" and INLA. In Kneib, T. and Tutz, G. (eds.), Statistical Modelling and Regression Structures – Festschrift in Honour of Ludwig Fahrmeir, 91–110. Springer.
  • Jeffreys, H. (1961). Theory of Probability. Oxford University Press.
  • Karim, M. R. and Zeger, S. L. (1992). "Generalized linear models with random effects; salamander mating revisited." Biometrics, 48(2): 631–644.
  • Kass, R. and Natarajan, R. (2006). "A default conjugate prior for variance components in Generalized Linear Mixed Models (Comment on Article by Browne and Draper)." Bayesian Analysis, 1(3): 535–542.
  • Lambert, P. (2006). "(Comment on Article by Browne and Draper)." Bayesian Analysis, 1(3): 543–546.
  • Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer-Verlag.
  • Lee, Y. and Nelder, J. A. (1996). "Hierarchical generalized linear models." Journal of the Royal Statistical Society, Series B., 58(4): 619–678.
  • Lunn, D., Spiegelhalter, D., Thomas, A., and Best, N. (2009). "The BUGS" project: Evolution, critique and future directions. Statistics in Medicine, 28(25): 3049–3067.
  • –- (2009). "Rejoinder to commentaries on `The BUGS" project: Evolution, critique and future directions'. Statistics in Medicine, 28(25): 3081–3082.
  • McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models. Second Edition. Chapman & Hall/CRC.
  • McCulloch, R. (1989). "Local model influence." Journal of the American Statistical Association, 84(406): 473–478.
  • Narasimhan, B. (2005). "Lisp-Stat to Java to R." Journal of Statistical Software, 13(4): 1–10.
  • Pepe, M. (2003). The Statistical Evaluation of Medical Tests for Classifition and Prediction. Oxford University Press.
  • Plummer, M. (2008). "Penalized loss function for Bayesian model comparison." Biostatistics, 9(3): 523–539.
  • Rue, H. and Held, L. (2005). Gaussian Markov Random Fields. Theory and Applications. Chapman & Hall/CRC.
  • Rue, H., Martino, S., and Chopin, N. (2009). "Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations." Journal of the Royal Statistical Society, Series B., 71(2): 319–392.
  • Schmid, C. H. and Griffith, J. L. (2005). "Multivariate Classification Rules: Calibration and Discrimination." In Armitage, P. and Colton, T. (eds.), Encyclopedia of Biostatistics, Second Edition., 3491–3497. John Wiley & Sons.
  • Spiegelhalter, D., Best, N., Carlin, B., and van der Linde, A. (2002). "Bayesian measures of model complexity and fit"." Journal of the Royal Statistical Society, Series B., 64(4): 583–616.
  • Stone, M. (1977). "Asymptotic equivalence of choice of model by cross-validation and Akaike's criterion"." Journal of the Royal Statistical Society, Series B., 39(1): 44–47.
  • Verrell, P. A. and Arnold, S. J. (1989). "Behavioral observations of sexual isolation among allopatric populations of the mountain dusky salamander, Desmognathus Ochrophaeus". Evolution, 43(4): 745–755.
  • Wakefield, J. (2009). "Comments on `The BUGS" project: Evolution, critique and future directions'. Statistics in Medicine, 28(25): 3079–3080.
  • Zeger, S. and Karim, M. (1991). "Generalized linear models with random effects: A Gibbs sampling approach." Journal of the American Statistical Association, 86: 79–86.