×

Using hierarchical centering to facilitate a reversible jump MCMC algorithm for random effects models. (English) Zbl 1468.62150

Summary: Hierarchical centering has been described as a reparameterization method applicable to random effects models. It has been shown to improve mixing of models in the context of Markov chain Monte Carlo (MCMC) methods. A hierarchical centering approach is proposed for reversible jump MCMC (RJMCMC) chains which builds upon the hierarchical centering methods for MCMC chains and uses them to reparameterize models in an RJMCMC algorithm. Although these methods may be applicable to models with other error distributions, the case is described for a log-linear Poisson model where the expected value \(\lambda\) includes fixed effect covariates and a random effect for which normality is assumed with a zero-mean and unknown standard deviation. For the proposed RJMCMC algorithm including hierarchical centering, the models are reparameterized by modeling the mean of the random effect coefficients as a function of the intercept of the \(\lambda\) model and one or more of the available fixed effect covariates depending on the model. The method is appropriate when fixed-effect covariates are constant within random effect groups. This has an effect on the dynamics of the RJMCMC algorithm and improves model mixing. The methods are applied to a case study of point transects of indigo buntings where, without hierarchical centering, the RJMCMC algorithm had poor mixing and the estimated posterior distribution depended on the starting model. With hierarchical centering on the other hand, the chain moved freely over model and parameter space. These results are confirmed with a simulation study. Hence, the proposed methods should be considered as a regular strategy for implementing models with random effects in RJMCMC algorithms; they facilitate convergence of these algorithms and help avoid false inference on model parameters.

MSC:

62-08 Computational methods for problems pertaining to statistics
62J12 Generalized linear models (logistic models)
62F15 Bayesian inference
65C05 Monte Carlo methods

Software:

glmmAK
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Al-Awadhi, F.; Hurn, M.; Jennison, C., Improving the acceptance rate of reversible jump MCMC proposals, Statist. Probab. Lett., 69, 189-198, (2004) · Zbl 1116.65308
[2] Bates, D., Computational methods for mixed models. R package version 0.999375-31. tech. rep., (2009)
[3] Brooks, S. P.; Giudici, P.; Roberts, G. O., Efficient construction of reversible jump Markov chain Monte Carlo proposal distributions, J. R. Stat. Soc. Ser. B, 65, 1, 3-55, (2003) · Zbl 1063.62120
[4] Browne, W. J., An illustration of the use of reparameterisation methods for improving MCMC efficiency in crossed random effect models, Multilevel Modell. Newsl., 16, 13-25, (2004)
[5] Browne, W. J.; Steele, F.; Golalizadeh, M.; Green, M. J., The use of simple reparameterizations to improve the efficiency of Markov chain Monte Carlo estimation for multilevel models with applications to discrete time survival models, J. Roy. Statist. Soc. Ser. A, 172, 3, 579-598, (2009)
[6] Buckland, S. T.; Anderson, D. R.; Burnham, K. P.; Laake, J. L.; Borchers, D. L.; Thomas, L., Introduction to distance sampling, (2001), Oxford University Press · Zbl 1136.62407
[7] Davison, A. C., Statistical models, (2003), Cambridge University Press · Zbl 1044.62001
[8] Forster, J. J.; Gill, R. C.; Overstall, A. M., Reversible jump methods for generalised linear models and generalised linear mixed models, Stat. Comput., 22, 1, 107-120, (2012) · Zbl 1322.62195
[9] Gelfand, A. E.; Sahu, S. K.; Carlin, B. P., Efficient parametrisations for normal linear mixed models, Biometrika, 82, 3, 479-488, (1995) · Zbl 0832.62064
[10] Green, P. J., Reversible jump Markov chain Monte Carlo computation and Bayesian model determination, Biometrika, 82, 4, 711-732, (1995) · Zbl 0861.62023
[11] Green, P. J.; Mira, A., Delayed rejection in reversible jump metropolis-Hastings, Biometrika, 88, 4, 1035-1053, (2001) · Zbl 1099.60508
[12] Hastings, W. K., Monte Carlo sampling methods using Markov chains and their applications, Biometrika, 57, 1, 97-109, (1970) · Zbl 0219.65008
[13] King, R.; Morgan, J.; Gimenez, O.; Brooks, S., Bayesian analysis for population ecology, (2010), CRC Press
[14] Komárek, A.; Lesaffre, E., Generalized linear mixed model with a penalized Gaussian mixture as a random effects distribution, Comput. Statist. Data Anal., 52, 3441-3458, (2008) · Zbl 1452.62538
[15] McCulloch, E. C.; Searle, S. R., Generalized, linear, and mixed models, (2001), John Wiley & Sons, Inc. · Zbl 0964.62061
[16] Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E., Equations of state calculations by fast computing machines, J. Chem. Phys., 21, 1087-1091, (1953)
[17] Oedekoven, C. S.; Buckland, S. T.; Mackenzie, M. L.; Evans, K. O.; Burger, L. W., Improving distance sampling: accounting for covariates and non-independency between sampled sites, J. Appl. Ecol., 50, 3, 786-793, (2013)
[18] Oedekoven, C. S.; Buckland, S. T.; Mackenzie, M. L.; King, R.; Evans, K. O.; Burger, L. W., Bayesian methods for hierarchical distance sampling models, J. Agric. Biol. Environ. Stat., 19, 2, 219-239, (2014) · Zbl 1303.62088
[19] Papaspiliopoulos, O.; Roberts, G. O.; Sköld, M., A general framework for the parametrization of hierarchical models, Statist. Sci., 22, 1, 59-73, (2007) · Zbl 1246.62195
[20] Papathomas, M.; Dellaportas, P.; Vasdekis, V. G.S., A novel reversible jump algorithm for generalized linear models, Biometrika, 98, 1, 231-236, (2011) · Zbl 1215.62069
[21] Vines, S. K.; Gilks, W. R.; Wild, P., Fitting multiple random effects models. tech. rep., (1995), MRC Biostatistics Unit Cambridge
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.